From: Martin Mares <mj@ucw.cz>
Date: Tue, 18 Oct 2011 18:01:42 +0000 (+0200)
Subject: Archivace starych kapitol, postupne vytvarim nove. Zatim 2-toky.
X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=2d2944b3208696169024ccae79180605c4ab649b;p=ads2.git

Archivace starych kapitol, postupne vytvarim nove. Zatim 2-toky.
---

diff --git a/0-demo/0-demo.tex b/0-demo/0-demo.tex
deleted file mode 100644
index bc5d7b7..0000000
--- a/0-demo/0-demo.tex
+++ /dev/null
@@ -1,37 +0,0 @@
-\input ../lecnotes.tex
-
-\prednaska{0}{Ukázka pou¾ití maker}{(zapsal Martin Mare¹)}
-
-% Takhle vypadá nadpis
-\h{Pøedposlední kapitola}
-
-% \> zaøídí, aby se odstavec neodsadil. Pomocí \I se sází kurzíva, \foot vyrobí poznámku pod èarou.
-\>Je svaèveèer. {\I Lysperní jezeleni} se vírnì vrtáèejí v~mokøavì.\foot{Viz Lewis Caroll: Jabberwocky.}
-
-\itemize\ibull
-\:Takhle.
-\:Vypadají.
-\:Odrá¾ky.
-\endlist
-
-\h{Druhá kapitola}
-
-\s{Vìta:} Paøez není strom.
-
-\proof
-Strom je speciálním pøípadem lesa. Les je (podle definice) graf bez kru¾nic.
-Paøez obsahuje alespoò jednu kru¾nici. Proto paøez nemù¾e být les, a~tedy ani
-strom.
-\qed
-
-\s{Algoritmus:} (tøídìní posloupnosti $a_1,\ldots,a_n$ pomocí {\sc Stupidsort}u)
-
-\algo
-\:$\pi \leftarrow \hbox{identická permutace na~mno¾inì $\{1,\ldots,n\}$}$.
-\:Opakuj:
-\::Ovìø, zda je posloupnost $a_{\pi(1)},a_{\pi(2)},\ldots,a_{\pi(n)}$ uspoøádána vzestupnì.
-   Pokud je, vra» ji jako výsledek.
-\::Nahraï $\pi$ jejím lexikografickým následníkem.
-\endalgo
-
-\bye
diff --git a/0-demo/Makefile b/0-demo/Makefile
deleted file mode 100644
index 98def69..0000000
--- a/0-demo/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=0-demo
-
-include ../Makerules
diff --git a/1-toky/1-toky.tex b/1-toky/1-toky.tex
deleted file mode 100644
index 126d81c..0000000
--- a/1-toky/1-toky.tex
+++ /dev/null
@@ -1,269 +0,0 @@
-\input lecnotes.tex
-
-\prednaska{1}{Toky v sítích}{(zapsala Markéta Popelová)}
-
-\s{První motivaèní úloha:} {\I Rozvod èajovodu do~v¹ech uèeben.}
-\smallskip
-
-Pøedstavme si, ¾e~by v~budovì fakulty na~Malé Stranì existoval èajovod, který
-by rozvádìl èaj do~ka¾dé uèebny. Znázornìme si to orientovaným grafem, kde by
-jeden významný vrchol pøedstavoval èajovar a~druhý uèebnu, ve~které sedíme.
-Hrany mezi vrcholy by pøedstavovaly vìtvící se trubky, které mají èaj rozvádìt.
-Jak rozvést co nejefektivnìji dostatek èaje do~dané uèebny?
-
-\figure{toky01.eps}{Èajovod}{2in}
-
-\s{Druhá motivaèní úloha:} {\I Pøenos dat.}
-\smallskip
-
-Jiným pøíkladem mù¾e být poèítaèová sí» na~pøenos dat, která se sestává z~pøenosových linek
-spojených pomocí routerù. Data se sice obvykle pøená¹ejí po~paketech, ale to
-mù¾eme pøi dne¹ních rychlostech pøenosu zanedbat a pova¾ovat data za spojitá.
-Jak pøená¹et data mezi dvìma poèítaèi v~síti co nejrychleji?
-
-\s{Definice:} {\I Sí»} je uspoøádaná pìtice $(V,E,z,s,c)$, pro ní¾ platí:
-\itemize\ibull
-\:$(V,E)$ je orientovaný graf.
-\:$c:E\to{\bb R}_{0}^{+}$ je {\I kapacita} hran.
-\:$z,s \in V$ jsou dva vrcholy grafu, kterým øíkáme {\I zdroj} a~{\I stok} (spotøebiè).
-\:Graf je symetrický, tedy $\forall u,v \in V: uv \in E \Leftrightarrow vu \in E$ (tuto podmínku si~mù¾eme zvolit bez~újmy na~obecnosti, nebo» v¾dy mù¾eme do~grafu pøidat hranu, která v~nìm je¹tì nebyla, a~dát jí nulovou kapacitu).
-\endlist
-
-\figure{sit.eps}{Pøíklad sítì. Èísla pøedstavují kapacity jednotlivých hran.}{2.5in}
-
-\s{Definice:} {\I Tok} je funkce $f:E \to {\bb R}_{0}^{+}$ taková, ¾e~platí:
-\numlist{\ndotted}
-\:Tok po~ka¾dé hranì je omezen její kapacitou: $\forall e \in E : f(e)\le c(e)$.
-\:Kirchhoffùv zákon: $$\forall v \in V \setminus \{z,s\}: \sum_{u: uv \in E}{f(uv)}=\sum_{u: vu \in E}{f(vu)}.$$ Neboli pro~ka¾dý vrchol kromì zdroje a~stoku platí, ¾e~to, co do~nìj pøitéká, je stejnì velké jako to, co z~nìj odtéká (\uv{sí» tìsní}).
-\endlist
-
-\s{Definice:} Pro libovolnou funkci $f:E \to {\bb R}$ se nám bude hodit následující znaèení:
-\itemize\ibull
-\:$f^+(v) =  \sum_{u: uv \in E}{f(uv)}$ (celkový pøítok do vrcholu)
-\:$f^-(v) =  \sum_{u: vu \in E}{f(vu)}$ (celkový odtok)
-\:$f^\Delta(v) = f^+(v) - f^-(v)$ (pøebytek ve~vrcholu)
-\endlist
-
-\>(Kirchhoffùv zákon pak øíká prostì to, ¾e $f^\Delta(v)=0$ pro v¹echna $v\ne z,s$.)
-
-\figure{tok.eps}{Pøíklad toku. Èísla pøedstavují toky po~hranách, v~závorkách jsou kapacity.}{4in}
-
-\s{Pozorování:} Nìjaký tok v¾dy existuje. V libovolné síti mù¾eme v¾dy zvolit
-konstantnì nulovou funkci (po~¾ádné hranì nic nepoteèe). To je korektní tok,
-ale sotva u¾iteèný. Budeme chtít najít tok, který pøepraví co nejvíce tekutiny
-ze~zdroje do~spotøebièe.
-
-\s{Definice:} {\I Velikost toku} $f$ budeme znaèit $\vert f\vert$ a polo¾íme ji
-rovnou rozdílu souètu velikostí toku na~hranách vedoucích do~$s$ a~souètu velikostí
-toku na~hranách vedoucích z~$s$. Neboli $\vert f\vert:=f^\Delta(s).$
-
-\s{Pozorování:} Jeliko¾ sí» tìsní, mìlo by být jedno, zda velikost toku mìøíme
-u~spotøebièe nebo u~zdroje. Vskutku, krátkým výpoètem ovìøíme, ¾e tomu tak je:
-$$
-f^\Delta(z) - f^\Delta(s) = \sum_v f^\Delta(v) = 0.
-$$
-První rovnost platí proto, ¾e podle Kirchhoffova zákona jsou zdroj a spotøebiè jediné
-dva vrcholy, jejich¾ pøebytek mù¾e být nenulový. Druhou rovnost získáme tak, ¾e si
-uvìdomíme, ¾e tok na ka¾dé hranì pøispìje do celkové sumy jednou s~kladným znaménkem
-a jednou se záporným. Zjistili jsme tedy, ¾e pøebytek zdroje a spotøebièe se li¹í
-pouze znaménkem.
-\qed
-
-\s{Poznámka:} Rádi bychom nalezli v~zadané síti tok, jeho¾ velikost je maximální.
-Máme ale zaruèeno, ¾e maximum bude existovat? V¹ech mo¾ných tokù je nekoneènì mnoho
-a v~nekoneèné mno¾inì se mù¾e snadno stát, ¾e aèkoliv existuje supremum, není maximem
-(pøíklad: $\{1-1/n \mid n\in{\bb N}^+\}$).
-Odpovìï nám poskytne matematická analýza: mno¾ina v¹ech tokù je kompaktní podmno¾inou
-prostoru ${\bb R}^{\vert E\vert}$, velikost toku je spojitá (dokonce lineární) funkce
-z~této mno¾iny do~$\bb R$, tak¾e musí nabývat minima i maxima.
-
-Nám ale bude staèít studovat sítì s~racionálními kapacitami, kde existence maximálního
-toku bude zjevná u¾ z~toho, ze sestrojíme algoritmus, který takový tok najde.
-
-\s{První pokus:} Hledejme cestu $P$ ze~$z$ do~$s$ takovou, ¾e~$\forall e \in
-P: f(e) < c(e)$ (po~v¹ech jejích hranách teèe ostøe ménì, ne¾ jim dovolují
-jejich kapacity). Pak zjevnì mù¾eme tok upravit tak, aby se~jeho velikost
-zvìt¹ila. Zvolme $$\varepsilon := \min_{e \in P} (c(e) - f(e)).$$ Nový tok $f'$
-pak definujme jako $f'(e):=f(e) + \varepsilon$. Kapacity nepøekroèíme ($\varepsilon$
-je nejvìt¹í mo¾ná hodnota, abychom tok zvìt¹ili, ale nepøekroèili kapacitu ani
-jedné z~hran cesty $P$) a~Kirchhoffovy zákony zùstanou neporu¹eny, nebo» zdroj
-a~stok neomezují a~ka¾dému jinému vrcholu na~cestì $P$ se~pøítok $f^+(v)$
-i~odtok $f^-(v)$ zvìt¹í pøesnì o~$\varepsilon$.
-
-Opakujme tento proces tak dlouho, dokud existují zlep¹ující cesty. A¾ se algoritmus
-zastaví (co¾ by obecnì nemusel, ale nás je¹tì chvíli trápit nemusí), získáme maximální tok?
-Pøekvapivì nemusíme. Napø. na~obrázku je vidìt, ¾e~kdy¾ najdeme nejdøíve cestu
-pøes hranu s~kapacitou 1 (na obrázku tuènì) a~u¾ hodnotu toku na~této hranì
-nesní¾íme, tak dosáhneme velikost toku nejvý¹e 19. Ale maximální tok této sítì
-má velikost 20.
-
-\figure{toky02.eps}{Èísla pøedstavují kapacity jednotlivých hran.}{1.5in}
-
-Zde by ov¹em situaci zachránilo, kdybychom poslali tok velikosti 1 proti smìru
-prostøední hrany -- to mù¾eme udìlat tøeba odeètením jednièky od toku po smìru
-hrany. Roz¹íøíme tedy ná¹ algoritmus tak, aby umìl posílat tok i proti smìru
-hran. O~kolik mù¾eme tok hranou zlep¹it (a» u¾ pøiètením po~smìru nebo odeètením
-proti smìru) nám bude øíkat její {\I rezerva:}
-
-\s{Definice:} {\I Rezerva hrany} $uv$ je $r(uv):=c(uv) - f(uv) + f(vu).$
-
-\smallskip
-Algoritmus bude vypadat následovnì. Postupnì dok¾eme, ¾e je koneèný a ¾e v~ka¾dé
-síti najde maximální tok.
-
-\s{Algoritmus (Fordùv-Fulkersonùv)}
-
-\algo
-\:$f \leftarrow$ libovolný tok, napø. v¹ude nulový.
-\:Dokud $\exists P$ cesta ze $z$ do $s$ taková, ¾e~$\forall e \in P: r(e) > 0$, opakujeme:
-\::$\varepsilon \leftarrow \min \{r(e) \mid e \in P\}$.
-\::Pro v¹echny hrany $uv \in P$:
-\:::$\delta \leftarrow \min \{f(vu),\varepsilon\}$
-\:::$f(vu) \leftarrow f(vu) - \delta$
-\:::$f(uv) \leftarrow f(uv) + \varepsilon - \delta$
-\:Prohlásíme $f$ za~maximální tok.
-\endalgo
-
-\s{Koneènost:} Zastaví se~Fordùv-Fulkersonùv algoritmus?
-
-\itemize\ibull
-
-\:Pro~celoèíselné kapacity se~v~ka¾dém kroku zvìt¹í velikost toku alespoò o~1.
-Algoritmus se~tedy zastaví po~nejvíce tolika krocích, kolik je nìjaká horní
-závora pro~velikost maximálního toku -- napø. souèet kapacit v¹ech hran
-vedoucích do~stoku (tedy $c^+(s)$).
-
-\:Pro~racionální kapacity vyu¾ijeme jednoduchý trik. Nech» $M$ je nejmen¹í
-spoleèný násobek jmenovatelù v¹ech kapacit. Spustíme-li algoritmus na sí»
-s~kapacitami $c'(e) = c(e)\cdot M$, bude se rozhodovat stejnì jako v~pùvodní
-síti, proto¾e bude stále platit $f'(e) = f(e)\cdot M$. Nová sí» je pøitom
-celoèíselná, tak¾e se algoritmus jistì zastaví.
-
-\:Na~síti s~iracionálními kapacitami se~algoritmus chová mnohdy divoce, nemusí
-se~zastavit, ba ani konvergovat ke~správnému výsledku. (Zkuste vymyslet pøíklad
-takové sítì.)
-
-\endlist
-
-\s{Maximalita:} Kdy¾ se algoritmus zastaví, je tok~$f$ maximální? K~tomu se
-bude hodit zavést øezy.
-
-\s{Definice:} {\I Øez} je uspoøádaná dvojice mno¾in vrcholù ($A,B$) taková, ¾e
-$A$ a $B$ jsou disjunktní, pokrývají v¹echny vrcholy, $A$ obsahuje zdroj a $B$
-obsahuje stok. Neboli $A \cap B = \emptyset$, $A \cup B = V$, $z \in A$, $s \in B$.
-
-\>Ka¾dému øezu pøirozenì pøiøadíme mno¾iny hran:
-\itemize\ibull
-\:$E^+(A,B) = E \cap (A\times B)$ (hrany \uv{zleva doprava})
-\:$E^-(A,B) = E \cap (B\times A)$ (hrany \uv{zprava doleva})
-\:$E^\Delta(A,B) = E^+(A,B) \cup E^-(A,B)$ (v¹echny hrany øezu)
-\endlist
-
-\>Také pro libovolnou funkci $f: E\rightarrow {\bb R}$ zavedeme:
-\itemize\ibull
-\:$f^+(A,B) = \sum_{e\in E^+(A,B)} f(e)$ (prùtok pøes øez zleva doprava)
-\:$f^-(A,B) = \sum_{e\in E^-(A,B)} f(e)$ (prùtok zprava doleva)
-\:$f^\Delta(A,B) = f^+(A,B) - f^-(A,B)$ (èistý prùtok)
-\endlist
-
-\>{\I Kapacita øezu} budeme øíkat souètu kapacit hran zleva doprava, tedy $c+(A,B)$.
-
-\s{Poznámka:} Øezy se~dají definovat více zpùsoby, jedna z~definic je, ¾e~øez
-je mno¾ina hran grafu takových, ¾e~po~jejich odebrání se~graf rozpadne na~více
-komponent. Tuto vlastnost mají i na¹e øezy, ale opaènì to nemusí platit.
-
-\s{Lemma:} Pro ka¾dý øez $(A,B)$ a ka¾dý tok~$f$ platí, ¾e $f^\Delta(A,B)
-= \vert f\vert$. (Jinými slovy velikost toku mù¾eme mìøit na libovolném øezu,
-nejen na triviálních øezech kolem zdroje nebo kolem spotøebièe.)
-
-\proof
-Opìt ¹ikovným seètením pøebytkù vrcholù:
-$$
-f^\Delta(A,B) = \sum_{v\in B} f^\Delta(v) = f^\Delta(s).
-$$
-První rovnost získáme poèítáním pøes hrany: ka¾dá hrana vedoucí z~vrcholu v~$B$
-do~jiného vrcholu v~$B$ pøispìje jednou kladnì a jednou zápornì; hrany le¾ící
-celé mimo~$B$ nepøispìjí vùbec; hrany s~jedním koncem v~$B$ a druhým mimo pøispìjí
-jednou, pøièem¾ znaménko se bude li¹it podle toho, který konec je v~$B$. Druhá
-rovnost je snadná: v¹echny vrcholy v~$B$ mimo spotøebièe mají podle Kirchhoffova
-zákona nulový pøebytek.
-\qed
-
-\s{Dùsledek:} Pro ka¾dý tok~$f$ a ka¾dý øez $(A,B)$ platí $\vert f \vert \le c^+(A,B)$.
-(Velikost ka¾dého toku je shora omezena kapacitou ka¾dého øezu.)
-
-\proof
-$f^\Delta(A,B) = f^+(A,B) - f^-(A,B) \le f^+(A,B) \le c^+(A,B)$.
-\qed
-
-\s{Dùsledek:} Pokud $\vert f\vert = c^+(A,B)$, pak je tok~$f$ maximální a øez~$(A,B)$
-minimální. Jinými slovy pokud najdeme dvojici tok a stejnì velký øez, mù¾eme øez pou¾ít
-jako certifikát maximality toku. Následující vìta nám zaruèí, ¾e je to mo¾né v¾dy:
-
-\s{Vìta:} Pokud se~Fordùv-Fulkersonùv algoritmus zastaví, tak vydá maximální tok.
-
-\proof
-Nech» se~Fordùv-Fulkersonùv algoritmus zastaví. Definujme mno¾inu vrcholù $A
-:= \{v \in V \mid \hbox{existuje cesta ze~$z$ do~$v$ jdoucí po~hranách s~$r
-> 0$}\}$ a~$B := V \setminus A$.
-
-Dvojice $(A,B)$ je øez, nebo» $z \in A$ (ze~$z$ do~$z$ existuje cesta délky 0)
-a~$s \in B$ (kdyby $s \not\in B$, tak by musela existovat cesta ze~$z$ do~$s$
-s~kladnou rezervou, tudí¾ by algoritmus neskonèil, nýbr¾ tuto cestu vzal
-a~stávající tok vylep¹il).
-
-Dále víme, ¾e~v¹echny hrany øezu mají nulovou rezervu, èili $\forall uv \in
-E^+(A,B) : r(uv) = 0$ (kdyby mìla hrana $uv$ rezervu nenulovou, tedy kladnou,
-tak by vrchol $v$ patøil do~$A$). Proto po~v¹ech hranách øezu vedoucích z~$A$
-do~$B$ teèe tolik, kolik jsou kapacity tìchto hran, a~po~hranách vedoucích
-z~$B$ do~$A$ neteèe nic, tedy $f(uv) = c(uv)$ a $f(vu) = 0$. Máme øez $(A,B)$
-takový, ¾e~$f^\Delta(A,B) = c^+(A,B)$. To znamená, ¾e~jsme na¹li maximální tok
-a~minimální øez. \qed
-
-Dokázali jsme tedy následující:
-
-\s{Vìta:} Pro~sí» s~racionálními kapacitami se~Fordùv-Fulkersonùv algoritmus
-zastaví a~vydá maximální tok a~minimální øez.
-
-\s{Vìta:} Sí» s~celoèíselnými kapacitami má aspoò jeden z~maximálních tokù
-celoèíselný a~Fordùv-Fulkersonùv algoritmus takový tok najde.
-
-\proof
-Kdy¾ dostane Fordùv-Fulkersonùv algoritmus celoèíselnou sí», tak najde maximální tok a~ten bude zase celoèíselný (algoritmus nikde nedìlí).
-\qed
-
-To, ¾e~umíme najít celoèíselné øe¹ení není úplnì samozøejmé. (U~jiných problémù takové ¹tìstí mít nebudeme.) Uka¾me si rovnou jednu aplikaci, která právì celoèíselný tok vyu¾ije.
-
-\s{Aplikace:} Hledání nejvìt¹ího párování v~bipartitních grafech.
-
-\s{Definice:} Mno¾ina hran $F \subseteq E$ se~nazývá {\I párování}, jestli¾e
-¾ádné dvì hrany této mno¾iny nemají spoleèný ani jeden vrchol. Neboli $\forall
-e,f \in F : e \cap f = \emptyset$. {\I Velikostí} párování myslíme poèet jeho
-hran.
-
-\s{Øe¹ení:}
-Mìjme bipartitní graf $G = (V,E)$. V~nìm hledáme nejvìt¹í párování. Sestrojme
-si~sí» takovou, ¾e~vezmeme vrcholy $V$ grafu $G$ a~pøidáme k~nim dva speciální
-vrcholy $z$ (zdroj) a~$s$ (stok) a~ze~zdroje pøidáme hrany do~v¹ech vrcholù
-levé partity a~ze~v¹ech vrcholù pravé partity povedeme hrany do~stoku. V¹echny
-kapacity nastavme na~1. Hrany bipartitního grafu zorientujme z levé partity do
-pravé. Nyní staèí jen na~tuto sí» spustit Fordùv-Fulkersonùv algoritmus (nebo
-libovolný jiný algoritmus, který najde maximální celoèíselný tok) a~a¾~dobìhne,
-tak prohlásit hrany s~tokem 1 za~maximální párování.
-
-\figure{toky04.eps}{Hledání maximálního párování v~bipartitním grafu.}{2in}
-
-Existuje toti¾ bijekce mezi párováním a~celoèíselnými toky pøi~zachování
-velikosti. Z ka¾dého celoèíselného toku na~vý¹e zmínìném grafu (viz obrázek) lze sestrojit
-párování o~stejné velikosti (velikost toku zde odpovídá poètu hran bipartitního
-grafu, po~kterých poteèe 1) a~naopak. Dùle¾ité je si uvìdomit, ¾e~definice toku
-(omezení toku kapacitou a~Kirchhoffovy zákony) nám zaruèují, ¾e~hrany
-s~nenulovým tokem (tedy jednièkovým) budou tvoøit párování (nestane se, ¾e~by
-dvì hrany zaèínaly nebo konèily ve~stejném vrcholu, nebo» by se~nutnì poru¹ila
-jedna ze~dvou podmínek definice toku). Potom i~maximální tok bude odpovídat
-maximálnímu párování a~naopak.
-
-V~bipartitním grafu najdeme maximální párování v~èase $\O(n \cdot (m+n))$. Fordùv-Fulkersonùv algoritmus stráví jednou iterací èas $\O(m+n)$ (za~prohledání do~¹íøky) a~pøi~jednotkových kapacitách bude iterací nejvý¹e~$n$, proto¾e ka¾dou se~tok zvìt¹í alespoò o~1 a v¹echny toky jsou omezené øezem kolem zdroje, který má kapacitu nejvý¹e~$n$. Výsledná èasová slo¾tost hledání maximálního párování bude tedy $\O(n \cdot (m+n))$.
-
-
-\bye
diff --git a/1-toky/Makefile b/1-toky/Makefile
deleted file mode 100644
index e82416b..0000000
--- a/1-toky/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=1-toky
-
-include ../Makerules
diff --git a/1-toky/sit.eps b/1-toky/sit.eps
deleted file mode 100644
index f66f5e1..0000000
--- a/1-toky/sit.eps
+++ /dev/null
@@ -1,828 +0,0 @@
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28.064596,53.350816 28.027214,53.319558 28.018657,53.314957 C 28.00927,53.309909 27.998955,53.306828 27.9891,53.302763 C 27.990819,52.766373 27.992538,52.229983 27.994258,51.693593 C 27.999738,51.69011 28.0047,51.685635 28.010694,51.683143 C 28.019716,51.679395 28.029743,51.678564 28.038861,51.675056 C 28.044054,51.673057 28.048079,51.66857 28.053333,51.666735 C 28.08629,51.655221 28.07957,51.67028 28.084858,51.652949 C 28.096033,51.621919 28.134512,51.617848 28.15048,51.593778 C 28.176388,51.575208 28.208201,51.563083 28.233606,51.544171 C 28.239243,51.539977 28.24282,51.533563 28.247425,51.528258 C 28.255027,51.523165 28.262625,51.518072 28.270227,51.512979 C 28.322621,51.491745 28.366511,51.455169 28.419404,51.434842 C 28.435975,51.422331 28.459434,51.418341 28.476404,51.407597 C 28.482735,51.403587 28.489133,51.385136 28.493174,51.378417 C 28.497783,51.373803 28.512143,51.360036 28.514422,51.354794 C 28.516276,51.350528 28.512599,51.34478 28.515094,51.340853 C 28.521053,51.331473 28.550468,51.309399 28.560507,51.301749 C 28.579649,51.295459 28.586833,51.272678 28.601845,51.26215 C 28.61708,51.251464 28.638007,51.245403 28.654339,51.235607 C 28.697235,51.201019 28.616275,51.264081 28.686724,51.220631 C 28.688889,51.219295 28.685696,51.21468 28.687653,51.213056 C 28.700877,51.202097 28.726293,51.195167 28.742185,51.191165 C 28.772263,51.179641 28.790158,51.153976 28.819081,51.140413 C 28.833733,51.124908 28.857181,51.121209 28.873905,51.110086 C 28.877811,51.10749 28.878441,51.101588 28.88209,51.098647 C 28.910415,51.075835 28.895948,51.100076 28.907832,51.075104 C 28.94653,51.052903 28.945037,51.048431 28.943648,51.007821 C 28.832363,50.477313 28.941469,50.995071 28.933882,52.606434 C 28.933874,52.60849 28.932872,52.601273 28.934695,52.600321 C 28.945275,52.594795 28.956105,52.593787 28.966881,52.587957 C 28.954762,52.577706 28.936131,52.581114 28.920907,52.579386 C 28.892141,52.576833 28.862342,52.57632 28.833514,52.579448 C 28.803344,52.575217 28.773157,52.579405 28.742964,52.57814 C 28.718518,52.579922 28.694184,52.580295 28.669847,52.582618 C 28.649331,52.585526 28.579726,52.584252 28.578866,52.57132 C 28.577412,52.570204 28.574825,52.569777 28.574499,52.567972 C 28.574184,52.56625 28.577803,52.565359 28.577465,52.563641 C 28.576598,52.55918 28.538587,52.552192 28.527324,52.54644 C 28.50654,52.536976 28.482601,52.540024 28.461387,52.532136 C 28.45702,52.530513 28.457504,52.523446 28.454222,52.52014 C 28.441781,52.511831 28.447476,52.514991 28.420951,52.507655 C 28.406322,52.50361 28.376689,52.511966 28.376735,52.496789 C 28.377863,52.138876 27.906082,50.937353 28.489167,50.868023 C 28.518678,50.862922 28.544608,50.850455 28.575117,50.846988 C 28.569514,50.834359 28.574572,50.850382 28.586242,50.83715 C 28.587988,50.835171 28.584427,50.832193 28.583521,50.829714 C 28.58425,50.826473 28.583203,50.822176 28.585705,50.819989 C 28.606182,50.802088 28.604842,50.821981 28.603568,50.803133 C 28.618558,50.792744 28.594504,50.790108 28.596692,50.783541 C 28.5998,50.774212 28.634472,50.764652 28.634595,50.751722 C 28.63463,50.747807 28.628351,50.746992 28.625231,50.744626 C 28.597083,50.716835 28.673247,50.719287 28.643502,50.696288 C 28.658891,50.696907 28.666857,50.682335 28.679594,50.67781 C 28.695608,50.672122 28.716861,50.677168 28.729014,50.665292 C 28.731301,50.66306 28.723158,50.662308 28.721147,50.659826 C 28.719612,50.657932 28.719677,50.655177 28.718944,50.652853 C 28.724382,50.653168 28.778887,50.631896 28.775552,50.622613 C 28.773948,50.618142 28.76714,50.618198 28.762934,50.615991 C 28.750581,50.593061 28.739475,50.596918 28.728907,50.570232 C 28.712724,50.547943 28.724252,50.552352 28.742964,50.540789 C 28.75471,50.533528 28.764178,50.523073 28.775241,50.514807 C 28.766004,50.511113 28.776216,50.490414 28.771058,50.48607 C 28.768019,50.483506 28.756337,50.48468 28.759668,50.482509 C 28.768737,50.476605 28.78041,50.476325 28.790779,50.473232 C 28.792268,50.471513 28.831657,50.456166 28.82775,50.453716 C 28.821649,50.449886 28.804299,50.458161 28.806279,50.451232 C 28.808551,50.443266 28.822447,50.447488 28.830291,50.444825 C 28.834977,50.443234 28.839252,50.44062 28.843734,50.438518 C 28.834965,50.436507 28.867246,50.430262 28.861026,50.427378 C 28.853673,50.42397 28.844758,50.426611 28.836853,50.424845 C 28.831047,50.423547 28.816902,50.416515 28.822435,50.414333 C 28.834708,50.409491 28.848312,50.409168 28.861248,50.406586 C 28.844559,50.38172 28.894643,50.389724 28.877274,50.367002 C 28.874941,50.327338 28.869986,50.345901 28.911896,50.329213 C 28.915646,50.32772 28.912242,50.321039 28.913547,50.31722 C 28.920446,50.298267 28.910711,50.318691 28.94701,50.303839 C 28.948963,50.30304 28.943215,50.301272 28.942662,50.299235 C 28.937938,50.281888 28.959977,50.289992 28.930938,50.279743 C 28.906858,50.259427 28.987224,50.2614 28.982768,50.248676 C 28.980926,50.24341 28.971589,50.247094 28.966466,50.244888 C 28.963857,50.243766 28.962356,50.240967 28.960303,50.239006 C 28.961405,50.230323 29.010967,50.233568 29.015061,50.222151 C 29.016021,50.219471 29.007106,50.216961 29.009597,50.215583 C 29.018949,50.210411 29.030588,50.211298 29.040738,50.207952 C 29.04816,50.205506 29.054976,50.201505 29.062094,50.198281 C 29.105762,50.172263 29.15007,50.147952 29.198243,50.131238 C 29.270849,50.101749 29.348387,50.087438 29.421968,50.061518 C 29.513789,50.043988 29.596968,50.002194 29.686222,49.976654 C 29.701883,49.972284 29.717318,49.966999 29.733205,49.963543 C 29.807078,49.947478 29.77158,49.959967 29.849709,49.949223 C 29.913585,49.94044 29.976955,49.922908 30.040063,49.909913 L 30.074954,51.414155 C 29.959528,51.444852 29.844494,51.474291 29.733355,51.518931 C 29.646242,51.54951 29.560937,51.585069 29.473453,51.61466 C 29.404289,51.637937 29.336878,51.667204 29.274272,51.704859 C 29.233253,51.735488 29.192203,51.765749 29.151751,51.797087 C 29.139732,51.809671 29.142015,51.802182 29.145665,51.819775 C 29.134428,51.834553 29.086374,51.840968 29.136312,51.85138 C 29.131972,51.853777 29.127854,51.856637 29.123288,51.858572 C 29.113606,51.862674 29.085496,51.866069 29.106767,51.874296 C 29.112335,51.876449 29.121745,51.873696 29.123979,51.87923 C 29.125805,51.883752 29.116215,51.885131 29.112335,51.888081 C 29.109461,51.889487 29.052262,51.917147 29.052201,51.920747 C 29.052101,51.926804 29.065168,51.920775 29.069996,51.92443 C 29.072126,51.926041 29.06589,51.927843 29.063836,51.929549 C 29.057359,51.935151 29.037373,51.939903 29.043774,51.945593 C 29.04996,51.951095 29.060843,51.944683 29.068514,51.947802 C 29.069827,51.948334 29.045677,51.968947 29.044261,51.971107 C 29.038186,51.975406 29.031497,51.978944 29.02604,51.984004 C 29.01579,51.993505 29.025799,51.991186 29.021305,52.002832 C 29.01769,52.012191 28.999036,52.018132 28.993629,52.02507 C 29.07137,52.029163 28.971114,52.036804 28.965369,52.054157 C 28.966958,52.059958 28.983854,52.06088 28.979652,52.065183 C 28.965852,52.079314 28.938057,52.069253 28.961773,52.085012 C 28.969536,52.086538 28.980784,52.082934 28.985063,52.08959 C 28.988152,52.094398 28.975711,52.096174 28.971221,52.09971 C 28.9616,52.107286 28.955606,52.119131 28.944815,52.125478 C 28.966382,52.137942 28.983666,52.152763 28.96736,52.176985 C 28.95018,52.198752 28.92867,52.213945 28.903189,52.225203 C 28.89386,52.228655 28.828572,52.248638 28.843454,52.265978 C 28.837905,52.268148 28.79758,52.278334 28.803973,52.282998 C 28.807423,52.285514 28.812469,52.285524 28.815551,52.288479 C 28.817059,52.289923 28.815977,52.292632 28.816192,52.294709 C 28.819158,52.297404 28.822128,52.3001 28.825099,52.302796 C 28.777958,52.33392 28.833165,52.329276 28.789133,52.352669 C 28.744694,52.375324 28.773246,52.374028 28.761626,52.385848 C 28.751494,52.396152 28.734859,52.397957 28.72391,52.407225 C 28.716811,52.410796 28.709707,52.414366 28.702604,52.417937 C 28.701219,52.420459 28.700827,52.42389 28.698444,52.425505 C 28.673807,52.442186 28.638632,52.447397 28.610894,52.45578 C 28.592175,52.466308 28.56573,52.473245 28.550549,52.481819 C 29.091966,52.620302 28.620972,52.41836 28.614693,50.891653 C 28.614636,50.87734 28.586077,50.893034 28.571866,50.894748 C 28.565627,50.895501 28.558051,50.894695 28.553504,50.899034 C 28.551171,50.901258 28.558527,50.903066 28.561037,50.905081 C 28.567914,50.909592 28.589055,50.910836 28.582972,50.916369 C 28.577208,50.921614 28.566536,50.916503 28.560012,50.920767 C 28.557299,50.922541 28.577907,50.932073 28.589047,50.937973 C 28.593004,50.939092 28.596964,50.940211 28.600924,50.94133 C 28.598706,50.942535 28.59186,50.944207 28.594274,50.944946 C 28.615165,50.951346 28.64181,50.941921 28.661654,50.954464 C 28.698928,50.962916 28.650026,50.963163 28.642097,50.965494 C 28.633213,50.968105 28.67675,50.960442 28.669854,50.966622 C 28.694169,50.968945 28.71848,50.969321 28.74291,50.971091 C 28.77295,50.969809 28.802991,50.973962 28.832993,50.969568 C 28.862638,50.972599 28.893112,50.971521 28.922726,50.969295 C 28.95513,50.966936 28.988306,50.9693 29.020468,50.973095 C 29.60287,51.025562 29.545253,52.469404 29.102956,52.620597 C 29.074014,52.634984 29.042051,52.646682 29.018692,52.669328 C 28.981678,52.688597 28.948464,52.712367 28.913312,52.734827 C 28.884654,52.765501 28.843707,52.777662 28.807768,52.797643 C 28.794049,52.8196 28.763364,52.823617 28.74512,52.840141 C 28.738497,52.844504 28.729517,52.847036 28.725821,52.854051 C 28.724655,52.856264 28.725944,52.859489 28.72434,52.861408 C 28.713046,52.874905 28.691755,52.881114 28.685113,52.899506 C 28.679778,52.906444 28.665495,52.928938 28.659846,52.934356 C 28.642953,52.950563 28.617894,52.959303 28.603315,52.978614 C 28.565922,53.001108 28.522078,53.00786 28.48325,53.027882 C 28.443942,53.047683 28.401928,53.066624 28.371278,53.099058 C 28.337039,53.127773 28.297025,53.150855 28.271056,53.18874 C 28.264548,53.193381 28.256896,53.196736 28.251535,53.202664 C 28.248672,53.205831 28.249616,53.21112 28.247474,53.214816 C 28.237006,53.232886 28.217384,53.248036 28.200841,53.259745 C 28.195042,53.263186 28.185978,53.263821 28.183449,53.27007 C 28.181895,53.273912 28.19135,53.273475 28.193987,53.276672 C 28.195602,53.278633 28.193219,53.282522 28.195115,53.284213 C 28.20223,53.290557 28.214556,53.293432 28.223628,53.296105 C 28.224461,52.765621 28.22529,52.235137 28.226122,51.704653 C 28.216966,51.707625 28.207004,51.708782 28.198649,51.713569 C 28.177723,51.725559 28.227293,51.783528 28.242962,51.769898 C 28.248848,51.772654 28.259436,51.771772 28.260622,51.778164 C 28.261585,51.783365 28.249873,51.779237 28.245122,51.781562 C 28.234891,51.78657 28.234565,51.811836 28.216567,51.808246 C 28.223336,51.813527 28.252805,51.829703 28.255664,51.833389 C 28.257594,51.835877 28.251251,51.839976 28.253143,51.842493 C 28.255944,51.846219 28.290106,51.862724 28.291848,51.863599 C 28.293107,51.865229 28.309079,51.886443 28.311861,51.888004 C 28.315729,51.890177 28.352669,51.900799 28.358479,51.906519 C 28.360252,51.908264 28.354297,51.909209 28.352205,51.910554 C 28.356987,51.912693 28.361768,51.914831 28.36655,51.91697 C 28.41374,51.93213 28.371876,51.923056 28.365947,51.93648 C 28.364447,51.939876 28.399119,51.953985 28.401844,51.953489 C 28.38309,51.961372 28.394395,51.954712 28.411848,51.960118 C 28.419546,51.962503 28.443907,51.983006 28.411411,51.977728 C 28.435599,51.981486 28.45864,51.975785 28.482862,51.976253 C 28.50941,51.981894 28.536718,51.978328 28.563128,51.985179 C 28.605338,51.989875 28.648568,51.985027 28.690877,51.989256 C 28.74697,51.989084 28.802772,51.992024 28.858593,51.985868 C 29.429563,52.013183 28.916325,51.977482 28.969341,53.60136 C 28.969387,53.602808 28.922557,53.607445 28.921628,53.607656 C 28.915484,53.609047 28.894443,53.620186 28.890283,53.622289 C 28.864441,53.63781 28.838311,53.657295 28.810723,53.669203 C 28.802542,53.672734 28.793473,53.673982 28.785392,53.677732 C 28.781535,53.679522 28.778699,53.682986 28.775353,53.685613 C 28.749913,53.706944 28.717816,53.718239 28.692128,53.739678 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diff --git a/10-prevody/10-prevody.tex b/10-prevody/10-prevody.tex
deleted file mode 100644
index 070d102..0000000
--- a/10-prevody/10-prevody.tex
+++ /dev/null
@@ -1,282 +0,0 @@
-\input lecnotes.tex
-
-\prednaska{10}{Pøevody problémù}{\vbox{\hbox{(zapsali Martin Chytil, Vladimír Kudelas,}\hbox{ Michal Kozák, Vojta Tùma)}}}
-
-\>Na této pøedná¹ce se budeme zabývat rozhodovacími problémy a jejich obtí¾ností.
-Za jednoduché budeme trochu zjednodu¹enì pova¾ovat ty problémy, na~nì¾ známe algoritmus
-pracující v~polynomiálním èase.
-
-\s{Definice:} {\I Rozhodovací problém} je takový problém, jeho¾ výstupem je v¾dy {\sc ano}, nebo {\sc ne}.
-[Formálnì bychom se na~nìj mohli dívat jako na~mno¾inu $L$ vstupù, na~které je odpovìï {\sc ano},
-a místo $L(x)=\hbox{\sc ano}$ psát prostì $x\in L$.]
-
-Vstupy mìjme zakódované jen pomocí nul a jednièek (obecnì je jedno, jaký základ pro soustavu
-kódování zvolíme, pøevody mezi soustavami o nìjakém základu $\neq$ 1 jsou co do velikosti
-zápisu polynomiální). Rozhodovací problém je tedy
-$f:\ \{0,1\}^{\ast} \to \{0,1\}$, to jest funkce z mno¾iny v¹ech øetìzcù jednièek a nul
-do mno¾iny $\{1,0\}$, kde 1 na výstupu znamená {\sc ano}, 0 {\sc ne}.
-
-\s{Pøíklad:} Je dán bipartitní graf $G$ a $k \in {\bb N}$. Existuje v $G$
-párování, které obsahuje alespoò $k$ hran?
-
-To, co bychom ve~vìt¹inì pøípadù chtìli, je samozøejmì nejen zjistit, zda takové párování
-existuje, ale také nìjaké konkrétní najít. V¹imnìme si ale, ¾e kdy¾ umíme rozhodovat
-existenci párování v~polynomiálním èase, mù¾eme ho polynomiálnì rychle i najít:
-
-Mìjme èernou skøíòku (fungující v polynomiálním èase), která odpoví, zda daný
-graf má nebo nemá párování o~$k$ hranách. Odebereme z~grafu libovolnou hranu
-a zeptáme se, jestli i tento nový graf má párovaní velikosti~$k$. Kdy¾ má, pak tato
-hrana nebyla pro existenci párování potøebná, a~tak ji odstraníme. Kdy¾ naopak
-nemá (hrana patøí do ka¾dého párování po¾adované velikosti), tak si danou hranu
-poznamenáme a odebereme nejen ji a její vrcholy, ale také hrany, které do tìchto
-vrcholù vedly. Toto je korektní krok, proto¾e v pùvodním grafu tyto vrcholy
-byly navzájem spárované, a tedy nemohou být spárované s~¾ádnými jinými vrcholy.
-Na~nový graf aplikujeme znovu tentý¾ postup. Výsledkem je mno¾ina hran, které patøí
-do hledaného párování. Hran, a tedy i iterací na¹eho algoritmu, je polynomiálnì
-mnoho a skøíòka funguje v polynomiálním èase, tak¾e celý algoritmus je polynomiální.
-
-A~jak ná¹ rozhodovací problém øe¹it? Nejsnáze tak, ¾e ho pøevedeme
-na~{jiný,\footnote{${}^{\dag}$}{vìrni matfyzáckým vtipùm}} který
-u¾ vyøe¹it umíme. Tento postup jsme (právì u~hledání párování) u¾ pou¾ili
-v~kapitole o~Dinicovì algoritmu. Vytvoøili jsme vhodnou sí», pro kterou
-platilo, ¾e v~ní existuje tok velikosti~$k$ právì tehdy, kdy¾
-v~pùvodním grafu existuje párování velikosti~$k$.
-
-Takovéto pøevody mezi problémy mù¾eme definovat obecnì:
-
-\s{Definice:} Jsou-li $A$, $B$ rozhodovací problémy, pak øíkáme, ¾e $A$ lze {\I
-redukovat} (neboli {\it pøevést}) na $B$ (pí¹eme $A \rightarrow B$) právì tehdy,
-kdy¾ existuje funkce $f$ spoèitatelná v polynomiálním èase taková, ¾e pro $\forall
-x: A(x) = B(f(x))$. V¹imnìme si, ¾e $f$ pracující v polynomiálním èase vstup
-zvìt¹í nejvíce polynomiálnì.
-
-\s{Pozorování:} $A\rightarrow B$ také znamená, ¾e problém~$B$ je alespoò tak tì¾ký
-jako problém~$A$ (tím myslíme, ¾e pokud lze $B$ øe¹it v~polynomiálním èase,
-lze tak øe¹it i~$A$): Nech» problém~$B$ umíme øe¹it v~èase $\O(b^k)$, kde
-$b$ je délka jeho vstupu. Nech» dále funkce $f$ pøevádìjící $A$ na $B$ pracuje
-v~èase $\O(a^\ell)$ pro vstup délky~$a$. Spustíme-li tedy $B(f(x))$ na~nìjaký
-vstup~$x$ problému~$A$, bude mít $f(x)$ délku $\O(a^\ell)$, kde $a=|q|$; tak¾e
-$B(f(x))$ pobì¾í v~èase $\O(a^\ell + (a^\ell)^k) = \O(a^{k\ell})$, co¾ je
-polynomiální v~délce vstupu~$a$.
-
-
-\s{Pozorování:} Pøevoditelnost je
-\itemize\ibull
-\:reflexivní (úlohu mù¾eme pøevést na tu stejnou identickým zobrazením): $A \rightarrow A$,
-\:tranzitivní: Je-li $A \rightarrow B$ funkcí $f$, $B \rightarrow C$ funkcí $g$,
-pak $A \rightarrow C$ slo¾enou funkcí $g \circ f$
-(slo¾ení dvou polynomiálních funkcí je zase polynomiální funkce, jak u¾ jsme zpozorovali
-v~pøedchozím odstavci).
-\endlist
-\>Takovýmto relacím øíkáme kvaziuspoøádání -- nesplòují obecnì antisymetrii, tedy mù¾e nastat
-$A\rightarrow B$ a $B\rightarrow A$. Omezíme-li se v¹ak na tøídy navzájem pøevoditelných
-problémù, dostáváme ji¾ (èásteèné) uspoøádání. Existují i navzájem nepøevoditelné problémy --
-napøíklad problém v¾dy odpovídající 1 a problém v¾dy odpovídající 0.
-Nyní se ji¾ podíváme na nìjaké zajímavé problémy. Obecnì to budou problémy, na které
-polynomiální algoritmus není znám, a vzájemnými pøevody zjistíme ¾e jsou stejnì tì¾ké.
-
-\h{1. problém: SAT}
-\>Splnitelnost (satisfiability) logických formulí, tj. dosazení 1 èi
-0 za promìnné v logické formuli tak, aby formule dala výsledek 1.
-
-\>Zamìøíme se na speciální formu zadání formulí,  {\I konjunktivní normální formu} (CNF),
-	které splòují následující podmínky:
-\itemize\ibull
-\:{\I formule} je zadána pomocí {\I klauzulí}\footnote{${}^{\dag}$}{bez politických konotací} oddìlených $\land$,
-\:ka¾dá {\I klauzule} je slo¾ená z {\I literálù} oddìlených $\lor$,
-\:ka¾dý {\I literál} je buïto promìnná nebo její negace.
-\endlist
-\>Formule mají tedy tvar:
-$$\psi = (\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$
-
-\>{\I Vstup:} Formule $\psi$ v konjunktivní normální formì.
-
-\>{\I Výstup:} $\exists$ dosazení 1 a 0 za promìnné takové, ¾e hodnota formule $\psi(\ldots) = 1$.
-
-
-
-\>Pøevod nìjaké obecné formule $\psi$ na jí ekvivalentní $\chi$ v~CNF mù¾e
-zpùsobit, ¾e $\chi$ je exponenciálnì velká vùèi $\psi$.
-Pozdìji uká¾eme, ¾e lze podniknout pøevod na takovou formuli $\chi'$ v~CNF, která sice není
-ekvivalentní s $\psi$ (pøibydou nám promìnné, a ne ka¾dý roz¹íøený model
-$\psi$ je modelem $\chi'$), ale je splnitelná právì tehdy, kdy¾ je splnitelná $\psi$ -- co¾ nám
-pøesnì staèí -- a je lineárnì velká vùèi $\psi$.
-
-\h{2. problém: 3-SAT}
-\s{Definice:} 3-SAT je takový SAT, v nìm¾ ka¾dá klauzule obsahuje nejvý¹e tøi literály.
-
-\s{Pøevod 3-SAT na SAT:}
-Vstup není potøeba nijak upravovat, 3-SAT splòuje vlastnosti SATu, proto 3-SAT
-$\rightarrow$ SAT (SAT je alespoò tak tì¾ký jako 3-SAT)
-
-\s {Pøevod SAT na 3-SAT:}
-Musíme formuli pøevést tak, abychom neporu¹ili splnitelnost.
-
-\>Trik pro dlouhé klauzule: Ka¾dou \uv{¹patnou} klauzuli
-$$(\alpha \lor \beta) \hbox{, t¾. } \vert\alpha\vert + \vert\beta\vert \ge 4
-,\ \vert\alpha\vert \geq 2,\ \vert\beta\vert\geq 2$$
-pøepí¹eme na: $$(\alpha \lor x) \land (\beta \lor \lnot x),$$
-kde $x$ je nová promìnná (pøi ka¾dém dìlení klauzule {\it jiná}
-nová promìnná).
-%kterou nastavíme tak, abychom neovlivnili splnitelnost formule.
-
-\>Tento trik opakujeme tak dlouho, dokud je to tøeba -- formuli délky $k+l$
-roztrhneme na formule délky $k+1$ a $l+1$. Pokud klauzule pùlíme, dostaneme
-polynomiální èas (strom rekurze má logaritmicky pater -- formule délky alespoò 6
-se nám pøi rozdìlení zmen¹í na dvì instance velikosti maximálnì $2/3$ pùvodní, krat¹í
-formule nás netrápí; na ka¾dém patøe se vykoná tolik co na pøedchozím + $2^{hloubka}$
-za pøidané formule). Velikost výsledné formule je tím pádem polynomiální vùèi pùvodní:
-v ka¾dém kroku se pøidají jen dva literály, tedy celkem {\it èas na pøevod}$\cdot
-2$ nových.
-
-\>Platí-li:
-\itemize\ibull
-\:$\alpha \Rightarrow$ zvolíme $x = 0$ (zajistí splnìní druhé poloviny nové formule),
-\:$\beta \Rightarrow$ zvolíme $x = 1$ (zajistí splnìní první poloviny nové formule),
-\:$\alpha ,\beta / \lnot\alpha ,\lnot\beta \Rightarrow$ zvolíme $x = 0/1$ (je nám to
-	jedno, celkové øe¹ení nám to neovlivní).
-\endlist
-
-Nabízí se otázka, proè mù¾eme pøidanou promìnnou $x$ nastavovat, jak se nám zlíbí.
-Vysvìtlení je prosté -- promìnná $x$ nám pùvodní formuli nijak neovlivní, proto¾e
-se v ní nevyskytuje, proto ji mù¾eme nastavit tak, jak chceme.
-
-\s{Poznámka:} U~3-SAT lze vynutit právì tøi literály, pro krátké klauzule
-pou¾ijeme stejný trik:
-$$(\alpha) \rightarrow (\alpha \vee \alpha) \rightarrow (\alpha \lor x) \land (\alpha \lor \lnot x).$$
-
-\h{3. problém: Hledání nezávislé mno¾iny v grafu}
-
-\>Existuje nezávislá mno¾ina vrcholù z~$G$ velikosti alespoò $k$?
-
-\s{Definice:} {\I Nezávislá mno¾ina} (NzMna) budeme øíkat ka¾dé mno¾inì vrcholù grafu
-takové, ¾e mezi nimi nevede ¾ádná hrana.
-
-\figure{nezmna.eps}{Pøíklad nezávislé mno¾iny}{1in}
-
-\>{\I Vstup:} Neorientovaný graf $G$, $k \in {\bb N}$.
-
-\>{\I Výstup:} $\exists A \subseteq V(G)$, $\vert A \vert \ge k$: $\forall u,v \in A \Rightarrow uv \not\in E(G)$?
-
-\s{Poznámka:} Ka¾dý graf má minimálnì jednu nezávislou mno¾inu, a tou je prázdná mno¾ina. Proto je potøeba zadat i minimální velikost hledané mno¾iny.
-
-\>Uká¾eme, jak na~tento probém pøevést 3-SAT.
-
-\s{Pøevod 3-SAT na NzMna:} Z ka¾dé klauzule vybereme jeden literál, jeho¾ nastavením se klauzuli
-rozhodneme splnit. Samozøejmì tak, abychom v~rùzných klauzulích nevybírali
-konfliktnì, tj.~$x$ a~$\lnot x$.
-
-\s{Pøíklad:}
-$(x \lor y \lor z) \land (x \lor \lnot y \lor \lnot z) \land (\lnot x \lor \lnot y \lor p) $.
-
-\>Pro ka¾dou klauzuli sestrojíme graf (trojúhelník) a pøidáme \uv{konfliktní}
-hrany, tj. $x$ a $\lnot x$. Poèet vrcholù grafu odpovídá poètu literálù ve formuli,
-poèet hran je maximálnì kvadratický a pøevod je tedy polynomiální.
-
-Existuje-li v grafu nezávislá mno¾ina velikosti $k$, pak z~ka¾dého z~$k$ trojúhelníkù
-vybere právì jeden vrchol, a pøitom ¾ádné dva vrcholy nebudou odpovídat literálu a
-jeho negaci -- tedy dostaneme ohodnocení promìnných splòujících alespoò $k$ klauzulí.
-Na druhou stranu, existuje-li ohodnocení $k$ klauzulí, pak pøímo odpovídá nezávislé
-mno¾inì velikosti $k$ (v ka¾dém trojúhelníku zvolíme právì jednu z ohodnocených
-promìnných, nemù¾e se stát ¾e zvolíme vrcholy konfliktní hrany). Ptáme-li se tedy
-na nezávislou mno¾inu velikosti odpovídající
-poètu klauzulí, dostaneme odpovìï {\sc ano} právì tehdy, kdy¾ je formule splnitelná.
-
-Jsou-li ve formuli i klauzule krat¹í ne¾ 3, mù¾eme je buïto prodlou¾it metodou vý¹e
-popsanou; nebo si v grafu necháme dvoj- a jedno-úhelníky, které ¾ádné z na¹ich úvah
-vadit nebudou.
-
-\figure{nezmna_graf.eps}{Ukázka pøevodu 3-SAT na nezávislou mno¾inu}{3in}
-
-\s{Pøevod NzMna na SAT:}
-
-\itemize\ibull
-\:Poøídíme si promìnné $v_1, \ldots, v_n$ odpovídající vrcholùm grafu. Promìnná $v_i$ bude
-  indikovat, zda se $i$-tý vrchol vyskytuje v~nezávislé mno¾inì (tedy pøíslu¹né ohodnocení
-  promìnných bude vlastnì charakteristická funkce nezávislé mno¾iny).
-\:Pro ka¾dou hranu $ij \in E(G)$ pøidáme klauzuli $(\lnot v_i \lor \lnot v_j)$. Tyto klauzule
-  nám ohlídají, ¾e vybraná mno¾ina je vskutku nezávislá.
-\:Je¹tì potøebujeme zkontrolovat, ¾e je mno¾ina dostateènì velká, tak¾e si její prvky
-  oèíslujeme èísly od~1 do~$k$. Oèíslování popí¹eme maticí promìnných $x_{ij}$, pøièem¾
-  $x_{ij}$ bude pravdivá právì tehdy, kdy¾ v~poøadí $i$-tý prvek nezávislé mno¾iny je vrchol~$v_j$
--- pøidáme tedy klauzule, které nám øeknou, ¾e vybrané do nezávislé mno¾iny jsou právì
-  ty vrcholy, které jsou touto maticí oèíslované: $\forall i,j$, $x_{ij} \Rightarrow v_j$
-  (jen dodejme, ¾e $a\Rightarrow b$ je definované jako $\neg a\vee b$).
-\:Je¹tì potøebujeme zajistit, aby byla v~ka¾dém øádku i sloupci nejvý¹e jedna jednièka:
-  $\forall j,i,i^{'}, i\ne i^{'} : x_{ij} \Rightarrow \lnot x_{i^{'}j}$ a
-  $\forall i,j,j^{'}, j\ne j^{'} : x_{ij} \Rightarrow \lnot x_{ij^{'}}$.
-\:A~nakonec si ohlídáme, aby v~ka¾dém øádku byla alespoò jedna jednièka, klauzulí $\forall i :
-  x_{i1} \lor x_{i2} \lor \ldots \lor x_{in}$.
-\endlist
-Tímto vynutíme NzMnu $\geq k$, co¾ jsme pøesnì chtìli. Takovýto pøevod je zøejmì polynomiální.
-
-\s{Pøíklad matice:} Jako pøíklad pou¾ijeme nezávislou mno¾inu z ukázky nezávislé mno¾iny.
-Nech» jsou vrcholy grafu oèíslované zleva a zeshora. Hledáme nezávislou mno¾inu velikosti $2$.
-Matice pak bude vypadat následovnì:
-$$ \pmatrix{1&0&0&0&0 \cr 0&0&0&1&0}$$
-\s{Vysvìtlení:} Jako první vrchol mno¾iny bude vybrán vrchol $v_1$, proto v prvním
-øádku a v prvním sloupci bude $1$. Jako druhý vrchol mno¾iny bude vybrán
-vrchol $v_4$, proto na druhém øádku a ve ètvrtém sloupci bude $1$. Na ostatních místech bude $0$.
-
-\h{4. problém: Klika}
-
-\>{\I Vstup:} Graf $G, k \in N$.
-
-\>{\I Výstup:} $\exists$ úplný podgraf grafu $G$ na $k$ vrcholech?
-\figure{klika.eps}{Pøíklad kliky}{2in}
-
-\s{Pøevod:} Prohodíme v grafu $G$ hrany a nehrany $\Rightarrow$ (hledání nezávislé mno¾iny $\leftrightarrow$ hledání kliky).
-
-\s{Dùvod:} Pokud existuje úplný graf na $k$ vrcholech, tak v~komplementárním grafu tyto vrcholy nejsou spojeny hranou, tj. tvoøí nezávislou mno¾inu,
-	a naopak.
-
-\figure{doplnek_nm.eps}{Prohození hran a nehran}{2in}
-
-
-\h{5. problém: 3,3-SAT}
-\s{Definice:} 3,3-SAT je speciální pøípad 3-SATu, kde ka¾dá promìnná se vyskytuje v~maximálnì tøech literálech.
-
-\s{Pøevod 3-SAT na 3,3-SAT:}
-Pokud se promìnná $x$ vyskytuje v~$k > 3$ literálech, tak nahradíme výskyty novými promìnnými $x_1, \ldots , x_k$ a pøidáme klauzule:
-$$
-(\lnot x_1 \lor x_2),
-(\lnot x_2 \lor x_3),
-(\lnot x_3 \lor x_4),
-\ldots,
-(\lnot x_{k-1} \lor x_k),
-(\lnot x_k \lor x_1),
-$$
-
-co¾ odpovídá:
-
-$$
-(x_1 \Rightarrow x_2),
-(x_2 \Rightarrow x_3),
-(x_3 \Rightarrow x_4),
-\ldots,
-(x_{k-1} \Rightarrow x_k),
-(x_k \Rightarrow x_1).
-$$
-
-Tímto zaruèíme, ¾e v¹echny nové promìnné budou mít stejnou hodnotu.
-
-Mimochodem, mù¾eme rovnou zaøídit, ¾e ka¾dý literál se vyskytuje nejvíce dvakrát (tedy ¾e
-ka¾dá promìnná se vyskytuje alespoò jednou pozitivnì a alespoò jednou negativnì). Pokud by
-se nìjaká promìnná objevila ve~tøech stejných literálech, mù¾eme na~ni
-také pou¾ít ná¹ trik a nahradit ji tøemi promìnnými. V~nových klauzulích se pak bude
-vyskytovat jak pozitivnì, tak negativnì.
-
-\h{6. problém: 3D párování (3D matching)}
-
-\>{\I Vstup:} Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a mno¾ina kompatibilních trojic (tìch, kteøí se spolu snesou).
-
-\>{\I Výstup:} Perfektní podmno¾ina trojic -- tj. taková podmno¾ina trojic, která obsahuje v¹echna $K$, $H$ a $Z$.
-
-\>Uká¾eme, jak na tento problém pøevést 3,3-SAT (ov¹em to a¾ na dal¹í pøedná¹ce).
-
-\figure{3d_parovani.eps}{Ukázka 3D párování}{3in}
-
-\s{Závìr:} Obrázek ukazuje problémy, jimi¾ jsme se dnes zabývali, a vztahy mezi tìmito problémy.
-\figure{prevody.eps}{Pøevody mezi problémy}{3in}
-
-\bye
diff --git a/10-prevody/3d_parovani.eps b/10-prevody/3d_parovani.eps
deleted file mode 100644
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diff --git a/11-np/11-np.tex b/11-np/11-np.tex
deleted file mode 100644
index c81d913..0000000
--- a/11-np/11-np.tex
+++ /dev/null
@@ -1,334 +0,0 @@
-\input lecnotes.tex
-
-\prednaska{11}{NP-úplné problémy}{\vbox{\hbox{(zapsali F. Kaèmarik, R. Krivák, D. Remi¹}
-	\hbox{ Michal Kozák, Vojta Tùma)}}}
-
-Dosud jsme zkoumali problémy, které se nás ptaly na to, jestli nìco existuje.
-Napøíklad jsme dostali formuli a problém splnitelnosti se nás ptal, zda
-existuje ohodnocení promìnných takové, ¾e formule platí. Nebo v~pøípadì
-nezávislých mno¾in jsme dostali graf a èíslo $k$ a ptali jsme se, jestli
-v~grafu existuje nezávislá mno¾ina, která obsahuje alespoò~$k$ vrcholù.
-Tyto otázky mìly spoleèné to, ¾e kdy¾ nám nìkdo napovìdìl nìjaký objekt, umìli
-jsme efektivnì øíci, zda je to ten, který hledáme. Napøíklad pokud dostaneme
-ohodnocení promìnných logické formule, staèí jen dosadit a spoèítat, kde
-formule dá \<true> nebo \<false>. Zjistit, ¾e nìjaký objekt je ten, který
-hledáme, umíme efektivnì. Tì¾ké na tom je takový objekt najít. Co¾ vede
-k~definici obecných vyhledávacích problémù, kterým se øíká tøída problémù NP.
-Nadefinujeme si ji poøádnì, ale nejdøíve zaèneme tro¹ièku jednodu¹¹í tøídou.
-
-\s{Definice:} P je {\I tøída rozhodovacích problémù}, které jsou øe¹itelné
-v~polynomiálním èase. Jinak øeèeno, problém
-$L \in {\rm P}  \Leftrightarrow  \exists $ polynom $f$ a~$\exists$ algoritmus~$A$
-takový, ¾e $\forall x: L(x)=A(x)$ a $A(x)$ dobìhne v~èase $\O(f(x))$.
-
-Tøída P tedy odpovídá tomu, o èem jsme se shodli, ¾e umíme efektivnì øe¹it.
-Nadefinujme nyní tøídu NP:
-
-\s{Definice:} NP je {\I tøída rozhodovacích problémù} takových, ¾e $L \in {\rm NP}$ právì tehdy, kdy¾ $\exists $ problém
-$K\in{\rm P}$ a $\exists$  polynom  $g$ takový, ¾e pro
-$\forall x$ platí $L(x)=1  \Leftrightarrow  \exists $ nápovìda $ y: \vert y \vert \leq g(\vert x \vert)$ a souèasnì $K(x,y)=1$.
-
-\s{Pozorování:} Splnitelnost logických formulí je v~NP. Staèí si toti¾ nechat napovìdìt, jak
-ohodnotit jednotlivé promìnné a pak ovìøit, jestli je formule splnìna. Nápovìda je polynomiálnì
-velká (dokonce lineárnì), splnìní zkontrolujeme také v~lineárním èase. Odpovíme tedy ano právì
-tehdy, existuje-li nápovìda, která nás pøesvìdèí, èili pokud je formule splnitelná.
-
-\s{Pozorování:} Tøída P le¾í uvnitø NP.
-V~podstatì øíkáme, ¾e kdy¾ máme problém, který umíme øe¹it v~polynomiálním èase
-bez nápovìdy, tak to zvládneme v~polynomiálním èase i s~nápovìdou.
-
-Problémy z minulé pøedná¹ky jsou v¹echny v NP (napø. pro nezávislou
-mno¾inu je onou nápovìdou pøímo mno¾ina vrcholù deklarující nezávislost),
-o jejich pøíslu¹nosti do P ale nevíme nic.
-Brzy uká¾eme, ¾e to jsou v jistém smyslu nejtì¾¹í problémy v~NP.
-Nadefinujme si:
-
-\s{Definice:} Problém $L$ je NP-{\I tì¾ký} právì tehdy, kdy¾ je na~nìj pøevoditelný
-ka¾dý problém z~NP (viz definici pøevodù z minulé pøedná¹ky).
-
-Rozmyslete si, ¾e pokud umíme øe¹it nìjaký NP-tì¾ký problém v~polynomiálním èase,
-pak umíme vyøe¹it v~polynomiálním èase v¹e v~NP, a tedy ${\rm P}={\rm NP}$.
-
-My se budeme zabývat problémy, které jsou NP-tì¾ké a samotné jsou v~NP. Takovým problémùm se øíká NP-úplné.
-
-\s{Definice:} Problém $L$ je NP-{\I úplný} právì tehdy, kdy¾ $L$ je NP-tì¾ký a $L \in {\rm NP}$.
-
-NP-úplné problémy jsou tedy ve~své podstatì nejtì¾¹í problémy, které le¾í v~NP.
-Kdybychom umìli vyøe¹it nìjaký NP-úplný problém v~polynomiálním èase, pak
-v¹echno v~NP je øe¹itelné v~polynomiálním èase. Bohu¾el to, jestli nìjaký
-NP-úplný problém lze øe¹it v~polynomiálním èase, se neví. Otázka, jestli
-${\rm P}={\rm NP}$, je asi nejznámìj¹í otevøený problém v~celé teoretické
-informatice.
-
-Kde ale nìjaký NP-úplný problém vzít? K~tomu se nám bude velice hodit následující vìta:
-
-\s{Vìta (Cookova):} SAT je NP-úplný.
-
-\>Dùkaz je znaènì technický, pøibli¾nì ho naznaèíme pozdìji. Pøímým dùsledkem
-Cookovy vìty je, ¾e cokoli v~NP je pøevoditelné na SAT.
-K dokazování NP-úplnosti dal¹ích problémù pou¾ijeme následující vìtièku:
-
-\s{Vìtièka:} Pokud problém $L$ je NP-úplný a $L$ se dá pøevést na nìjaký problém $M\in{\rm NP}$, pak $M$ je také NP-úplný.
-
-\proof
-Tuto vìtièku staèí dokázat pro NP-tì¾kost, NP-úplnost plyne okam¾itì z~toho, ¾e
-problémy jsou NP-tì¾ké a le¾í v~NP (podle pøedpokladu).
-
-Víme, ¾e $L$ se dá pøevést na~$M$ nìjakou funkcí~$f$. Jeliko¾ $L$ je NP-úplný,
-pak pro ka¾dý problém $Q\in{\rm NP}$ existuje nìjaká funkce~$g$, která pøevede
-$Q$ na~$L$. Staèí tedy slo¾it funkci~$f$ s~funkcí~$g$, èím¾ získáme funkci pracující
-opìt v~polynomiálním èase, která pøevede~$Q$ na~$M$. Ka¾dý problém z~NP se tedy
-dá pøevést na problém~$M$.
-\qed
-
-\s{Dùsledek:} Cokoliv, na co jsme umìli pøevést SAT, je také NP-úplné.
-Napøíklad nezávislá mno¾ina, rùzné varianty SATu, klika v~grafu~\dots
-
-Jak taková tøída NP vypadá? Pøedstavme si v¹echny problémy tøídy NP, jakoby seøazené
-shora dolu podle obtí¾nosti problémù (tedy navzdor gravitaci), kde porovnání dvou
-problémù urèuje pøevoditelnost (viz obrázek).
-
-\figure{p-np.eps}{Struktura tøídy NP}{2.5cm}
-
-Obecnì mohou nastat dvì situace, proto¾e nevíme, jestli ${\rm P}={\rm NP}$.
-Jestli ano, pak v¹echno je jedna a ta samá tøída. To by bylo v nìkterých
-pøípadech nepraktické, napø. ka¾dá ¹ifra by byla jednodu¹e rozlu¹titelná.
-\foot{Poznámka o ¹ifrách -- libovolnou funkci vyèíslitelnou v polynomiálním
-èase bychom umìli v polynomiálním èase také invertovat.}
-Jestli ne, NP-úplné problémy urèitì nele¾í v P, tak¾e P a NP-úplné problémy
-jsou dvì disjunktní èásti NP. Také se dá dokázat (to dìlat nebudeme, ale je
-dobré to vìdìt), ¾e je¹tì nìco le¾í mezi nimi, tedy ¾e existuje problém, který
-je v~NP, není v~P a není NP-úplný (dokonce je takových problémù nekoneènì mnoho,
-v nekoneènì tøídách).
-
-\s{Katalog NP-úplných problémù}
-
-Uká¾eme si nìkolik základních NP-úplných problémù. O~nìkterých jsme to dokázali
-na~minulé pøedná¹ce, o~dal¹ích si to doká¾eme nyní, zbylým se na~zoubek podíváme
-na~cvièeních.
-
-\itemize\ibull
-\:{\I logické:}
-  \itemize\ibull
-    \:SAT (splnitelnost logických formulí v~CNF)
-    \:3-SAT (ka¾dá klauzule obsahuje max.~3 literály)
-    \:3,3-SAT (a navíc ka¾dá promìnná se vyskytuje nejvý¹e tøikrát)
-    \:SAT pro obecné formule (nejen CNF)
-    \:Obvodový SAT (není to formule, ale obvod)
-  \endlist
-\:{\I grafové:}
-  \itemize\ibull
-    \:Nezávislá mno¾ina (existuje mno¾ina alespoò~$k$ vrcholù taková, ¾e ¾ádné dva nejsou propojeny hranou?)
-    \:Klika (existuje úplný podgraf na~$k$ vrcholech?)
-    \:3D párování (tøi mno¾iny se zadanými trojicemi, existuje taková mno¾ina disjunktních trojic, ve~které jsou v¹echny prvky?)
-    \:Barvení grafu (lze obarvit vrcholy $k$~barvami tak, aby vrcholy stejné barvy nebyly nikdy spojeny hranou? NP-úplné u¾ pro~$k=3$)
-    \:Hamiltonovská cesta (cesta obsahující v¹echny vrcholy [právì jednou])
-    \:Hamiltonovská kru¾nice (kru¾nice, která nav¹tíví v¹echny vrcholy [právì jednou])
-  \endlist
-\:{\I èíselné:}
-  \itemize\ibull
-    \:Batoh (nejjednodu¹¹í verze: dána mno¾ina èísel, zjistit, zda existuje podmno¾ina se zadaným souètem)
-	\:Batoh -- optimalizace (podobnì jako u pøedchozího problému, ale místo mno¾iny èísel máme mno¾inu
-		pøedmìtù s vahami a cenami, chceme co nejdra¾¹í podmno¾inu její¾ váha nepøesáhne zadanou kapacitu
-		batohu)
-    \:Loupe¾níci (lze rozdìlit danou mno¾inu èísel na~dvì podmno¾iny se stejným souètem)
-    \:$Ax=b$ (soustava celoèíslených lineárních rovnic; $x_i$ mohou být pouze 0 nebo 1; NP-úplné i pokud $A_{ij}\in\{0,1\}$ a $b_i\in\{0,1\}$)
-    \:Celoèíselné lineární programování (existuje vektor nezáporných celoèísených $x$ takový, ¾e $Ax \leq b$ ?)
-  \endlist
-\endlist
-
-Nyní si uká¾eme, jak pøevést SAT na nìjaký problém. Kdy¾ chceme ukázat, ¾e na
-nìco se dá pøevést SAT, potøebujeme obvykle dvì vìci:
-konstrukci, která bude simulovat promìnné, tedy nìco, co nabývá dvou stavù
-\<true>/\<false>; a nìco, co bude reprezentovat klauzule a umí zaøídit, aby
-ka¾dá klauzule byla splnìna alespoò jednou promìnnou.
-Roz¹iøme tedy ná¹ katalog problémù o následující:
-\h{3D párování (3D matching)}
-
-\>{\I Vstup:} Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a
-mno¾ina $T$ kompatibilních trojic (tìch, kteøí se spolu snesou),
-	tj. $T \subseteq K\times H\times Z$.
-
-\>{\I Výstup:} Perfektní podmno¾ina trojic $P\subseteq K\times H \times Z$ --
-	tj. taková podmno¾ina trojic, ¾e $(\forall k\in K\ \exists !p\in P: k\in p)
-	\wedge(\forall h\in H\ \exists !p\in P: h\in p)
-	\wedge(\forall z\in Z\ \exists !p\in P: z\in p)$ -- tedy ka¾dý byl vybrán
-	právì jednou.
-
-
-\h{ Uká¾eme, jak na 3D-párování pøevést 3,3-SAT }
-
-Uva¾ujme takovouto konfiguraci:
-
-\fig{3d.eps}{4cm}
-
-\>4 zvíøátka, 2 kluci, 2 dívky a 4 trojice, které oznaèíme $A, B, C, D$.
-Je¹tì pøedpokládáme, ¾e zvíøátka
- se mohou úèastnit nìjakých jiných trojic, ale
-tito ètyøi lidé se vyskytují pouze v~tìchto ètyøech trojicích a~nikde jinde.
-V¹imneme si, ¾e existují právì dvì mo¾nosti, jak tento obrázek spárovat.
-Abychom spárovali kluka $k_1$, tak si musíme vybrat $A$ nebo $B$. Kdy¾ si
-vybereme $A$, $k_1$ i $d_2$ u¾ jsou spárovaní tak¾e si nesmíme vybrat $B$ ani
-$D$. Pak jediná mo¾nost, jak spárovat $d_1$ a~$k_2$ je $C$. Jedna mo¾nost je
-tedy vybrat si $A$ a $C$ a jeliko¾ je obrázek symetrický, tak kdy¾ vybereme
-místo $A$ trojici $B$, dostaneme $B$ a~$D$. V¾dy si tedy vybereme dvì protìj¹í
-trojice v~obrázku.
-
-Tyto konfigurace budeme pou¾ívat k~reprezentaci promìnných. Pro ka¾dou
-promìnnou si nakreslíme takový obrázek a~to, ¾e $A$ bude spárované s~$C$, bude
-odpovídat tomu, ¾e $x=1$, a~spárování $B$ a~$D$ bude odpovídat
- $x=0$. Pokud jsme
-pou¾ili $A$ a~$C$, zvíøata se sudými èísly, tj. $z_2$ a~$z_4$, horní a~dolní,
-jsou nespárovaná a~pokud jsme pou¾ili $B$ a~$D$, zvíøátka $z_1$ a~$z_3$ zùstala
-nespárovaná. Pøes tato nespárovaná zvíøátka mù¾eme pøedávat informaci, jestli
-promìnná $x$ má hodnotu \<true> nebo \<false> do dal¹ích èástí vstupu.
-
-Zbývá vymyslet, jak reprezentovat klauzule. Klauzule jsou trojice popø. dvojice
-literálù, napø. $\kappa = (x \lor y \lor \lnot r)$, kde
-potøebujeme zajistit, aby $x$ bylo nastavené na $1$ nebo $y$ bylo nastavené
-na~$1$ nebo $r$ na $0$.
-
-\fig{klauzule.eps}{4cm}
-
-\>Pro takovouto klauzuli si poøídíme dvojici kluk-dívka, kteøí budou figurovat
-ve tøech trojicích se tøemi rùznými zvíøátky, co¾ mají být volná zvíøátka
-z~obrázkù pro pøíslu¹né promìnné (podle toho, má-li se promìnná vyskytnout
-s negací nebo ne). A~zaøídíme to tak, aby ka¾dé zvíøátko bylo
-pou¾ité maximálnì v~jedné takové trojici, co¾ jde proto, ¾e ka¾dý literál se
-vyskytuje maximálnì dvakrát a~pro ka¾dý literál máme dvì volná zvíøátka,
-z~èeho¾ plyne, ¾e zvíøátek je dost pro v¹echny klauzule. Pro dvojice se postupuje
-obdobnì.
-
-Je¹tì nám ale urèitì zbude $2p-k$ zvíøátek, kde $p$ je poèet promìnných, $k$
-poèet klauzulí -- ka¾dá promìnná vyrobí 4 zvíøátka, klauzule zba¹tí jedno
-a samotné ohodnocení 2 zvíøátka -- tak pøidáme je¹tì $2p-k$ párù
-kluk-dìvèe, kteøí milují
-v¹echna zvíøátka, a~ti vytvoøí zbývající trojice.
-
-Pokud formule byla splnitelná, pak ze splòujícího ohodnocení mù¾eme vyrobit
-párování s~na¹í konstrukcí. Obrázek pro ka¾dou promìnnou spárujeme podle
-ohodnocení, tj. promìnná je $0$ nebo $1$ a~pro ka¾dou klauzuli si vybereme
-nìkterou z~promìnných, kterými je ta klauzule splnìna. Funguje to ale
-i~opaènì: Kdy¾ nám nìkdo dá párovaní v~na¹í konstrukci, pak z nìho doká¾eme
-vyrobit splòující ohodnocení dané formule. Podíváme se, v~jakém stavu je
-promìnná, a~to je v¹echno. Z~toho, ¾e jsou správnì spárované klauzule, u¾
-okam¾itì víme, ¾e jsou v¹echny splnìné.
-
-Zbývá ovìøit, ¾e na¹e redukce funguje v~polynomiálním èase. Pro ka¾dou klauzuli
-spotøebujeme konstantnì mnoho èasu, $2p-k$ je také polynomiálnì mnoho a~kdy¾ v¹e
-seèteme, máme polynomiální èas vzhledem k~velikosti vstupní formule. Tím je
-pøevod hotový a~mù¾eme 3D-párování zaøadit mezi NP-úplné problémy.
-
-
-%RK
-
-
-\h{Náznak dùkazu Cookovy vìty}
-
-Abychom mohli budovat teorii NP-úplnosti, potøebujeme alespoò jeden problém,
-o kterém doká¾eme, ¾e je NP-úplný, z definice. Cookova vìta øíká o NP-úplnosti
-SAT-u, ale nám se to hodí dokázat o tro¹ku jiném problému -- {\I obvodovém SAT-u}.
-
-\>{\I Obvodový SAT} je splnitelnost, která nepracuje s~formulemi, ale s~booleovskými
-obvody (hradlovými sítìmi). Ka¾dá formule se dá pøepsat do booleovského obvodu,
-který ji poèítá, tak¾e dává smysl zavést splnitelnost i pro obvody. Na¹e obvody
-budou mít nìjaké vstupy a~jenom jeden výstup. Budeme se ptát, jestli se vstupy
-tohoto obvodu dají nastavit tak, abychom na výstupu dostali \<true>.
-
-\>Nejprve doká¾eme NP-úplnost {\I obvodového SAT-u} a~pak uká¾eme, ¾e se dá
-pøevést na obyèejný SAT v~CNF. Tím bude dùkaz Cookovy vìty hotový. Zaènìme
-s pomocným lemmatem.
-
-\s{Lemma:} Nech» $L$ je problém v P. Potom existuje polynom $p$ a algoritmus,
-který pro $\forall n \ge 0$ spoète v èase $p(n)$ hradlovou sí» $Bn$ s $n$ vstupy
-a 1 výstupem takovou, ¾e $\forall x \in \{ 0, 1 \}^n : Bn(x) = L(x)$.
-
-\proof
-Náznakem. Na základì zku¹eností z Principù poèítaèù intuitivnì chápeme poèítaèe
-jako nìjaké slo¾ité booleovské obvody, jejich¾ stav se mìní v~èase. Uva¾me tedy
-nìjaký problém $L \in {\rm P}$ a polynomiální algoritmus, který ho øe¹í. Pro vstup
-velikosti~$n$ dobìhne v~èase~$T$ polynomiálním v~$n$ a spotøebuje $\O(T)$ bunìk pamìti.
-Staèí nám tedy \uv{poèítaè s~pamìtí velkou $\O(T)$}, co¾ je nìjaký booleovský obvod
-velikosti polynomiální v~$T$, a~tedy i v~$n$. Vývoj v~èase o¹etøíme tak, ¾e sestrojíme~$T$
-kopií tohoto obvodu, ka¾dá z~nich bude odpovídat jednomu kroku výpoètu a bude
-propojena s~\uv{minulou} a \uv{budoucí} kopií. Tím sestrojíme booleovský obvod,
-který bude øe¹it problém~$L$ pro vstupy velikosti~$n$ a bude polynomiálnì velký
-vzhledem k~$n$.
-
-\s{Poznámka:}
-Pro dùkaz následující vìty si dovolíme drobnou úpravu v~definici tøídy NP.
-Budeme chtít, aby nápovìda
-mìla pevnou velikost, závislou pouze na~velikosti vstupu (tedy: $\vert y \vert
-= g(\vert x \vert)$ namísto $\vert y \vert \le g(\vert x \vert)$). Proè je taková
-úprava BÚNO? Jistì si dovedete pøedstavit,
-¾e pùvodní nápovìdu doplníme na po¾adovanou délku nìjakými \uv{mezerami}, které
-program ignoruje. (Tedy upravíme program tak, aby mu nevadilo, ¾e dostane na
-konci nápovìdy nìjak kódované mezery.)
-
-\s{Vìta:} Obvodový SAT je NP-úplný.
-
-\proof
-Máme tedy nìjaký problém $L$ z~NP a~chceme dokázat, ¾e $L$ se dá pøevést
-na~obvodový SAT (tj. NP-tì¾kost). Kdy¾ nám nìkdo pøedlo¾í nìjaký vstup $x$
-(chápeme jako posloupnost $x_1, x_2, \ldots, x_n$),
-spoèítáme velikost nápovìdy $g(n)$. Víme, ¾e kontrolní
-algoritmus~$K$ (který kontroluje, zda nápovìda je správnì) je v~P. Vyu¾ijeme
-pøedchozí lemma, abychom získali obvod, který pro konkrétní velikost vstupu
-$x$ poèítá to, co kontrolní algoritmus $K$. Na vstupu tohoto obvodu bude $x$
-(vstup problému $L$) a~nápovìda~$y$. Na výstupu nám øekne, jestli je nápovìda
-správná. Velikost vstupu tohoto obvodu bude tedy $p(g(n))$, co¾ je také polynom.
-
-\fig{kobvod.eps}{2.3cm}
-
-\>V tomto obvodu zafixujeme vstup $x$ (na místa vstupu dosadíme konkrétní hodnoty z $x$). Tím získáme obvod, jeho¾ vstup je jen $y$ a~ptáme se, zda za $y$ mù¾eme dosadit nìjaké hodnoty tak, aby na výstupu bylo \<true>. Jinými slovy, ptáme se, zda je tento obvod splnitelný.
-
-\>Pro libovolný problém z~NP tak doká¾eme sestrojit funkci, která pro ka¾dý vstup~$x$ v~polynomiálním èase vytvoøí obvod, který je splnitelný pravì tehdy, kdy¾ odpovìï tohoto problému na vstup $x$ má být \<true>. Tedy libovolný problém z~NP se dá
-v~polynomiálním èase pøevést na obvodový SAT.
-
-\>Obvodový SAT je v NP triviálnì -- staèí si nechat poradit vstup, sí»
-topologicky setøídit a v~tomto poøadí poèítat hodnoty hradel.
-\qed
-
-\s{Lemma:} Obvodový SAT se dá pøevést na 3-SAT.
-
-\proof
-Budeme postupnì budovat formuli v~konjunktivní normální formì. Ka¾dý booleovský obvod se dá pøevést v~polynomiálním èase na~ekvivalentní obvod, ve~kterém se vyskytují jen hradla {\sc and} a {\sc not}, tak¾e staèí najít klauzule odpovídající tìmto hradlùm. Pro ka¾dé hradlo v~obvodu zavedeme novou promìnnou popisující jeho výstup. Pøidáme klauzule, které nám kontrolují, ¾e toto hradlo máme ohodnocené konzistentnì.
-
-\>{\I Pøevod hradla \sc not}: na vstupu hradla budeme mít nìjakou promìnnou $x$ (která pøi¹la buïto pøímo ze~vstupu toho celého obvodu nebo je to promìnná, která vznikla na výstupu nìjakého hradla) a na výstupu promìnnou $y$. Pøidáme klauzule, které nám zaruèí, ¾e jedna promìnná bude negací té druhé:
-$$\matrix{ (x \lor y), \cr
-  (\neg{x} \lor \neg{y}). \cr }
-  \hskip 0.2\hsize
-\vcenter{\hbox{\epsfxsize=0.7cm\epsfbox{not.eps}}}
-$$
-
-\>{\I Pøevod hradla \sc and}: Hradlo má vstupy $x, y$ a~výstup $z$. Potøebujeme pøidat klauzule, které nám popisují, jak se má hradlo {\sc and} chovat. Tyto vztahy pøepí¹eme do~konjunktivní normální formy:
-$$
-\left. \matrix{
-  x\ \&\ y \Rightarrow z \cr
-  \neg{x} \Rightarrow \neg{z} \cr
-  \neg{y} \Rightarrow \neg{z} \cr
-}
-\ \quad
- \right\}
-\quad
-\matrix{
- (z \lor \neg{x} \lor \neg{y}) \cr
- (\neg{z} \lor x)              \cr
- (\neg{z} \lor y)              \cr
- }
- \hskip 0.1\hsize
-\vcenter{\hbox{\epsfxsize=0.7cm\epsfbox{and.eps}}}
-$$
-
-\>Kdy¾ chceme pøevádìt obvodový SAT na 3-SAT, obvod nejdøíve pøelo¾íme na takový, ve~kterém jsou jen hradla {\sc and} a~{\sc not}, a~pak hradla tohoto obvodu pøelo¾íme na klauzule. Formule vzniklá z~takovýchto klauzulí je splnitelná pravì tehdy, kdy¾ je splnitelný daný obvod. Pøevod pracuje v polynomiálním èase.
-\qed
-
-\s{Poznámka:}
-Kdy¾ jsme zavádìli SAT, omezili jsme se jen na formule, které jsou
-v~konjunktivní normální formì (CNF). Teï u¾ víme, ¾e splnitelnost obecné
-booleovské formule doká¾eme pøevést na obvodovou splnitelnost a tu pak
-pøevést na 3-SAT. Opaèný pøevod je samozøejmì triviální, tak¾e obecný SAT
-je ve~skuteènosti ekvivalentní s~na¹ím \uv{standardním} SATem pro CNF.
-
-\bye
-
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-\input lecnotes.tex
-\prednaska{12}{Aproximaèní algoritmy}{\vbox{\hbox{(F. Ha¹ko, J. Menda, M. Mare¹,}
-	\hbox{ Michal Kozák, Vojta Tùma)}}}
-
-\>Na~minulých pøedná¹kách jsme se zabývali rùznými tì¾kými rozhodovacími
-problémy. Tato se zabývá postupy, jak se v~praxi vypoøádat s~øe¹ením tìchto
-problémù.
-
-\h{Co dìlat, kdy¾ potkáme NP-úplný problém}
-\algo
-\:Nepanikaøit.
-\:Spokojit se s~málem.
-\:Rozmyslet, jestli opravdu potøebujeme obecný algoritmus. Mnohdy potøebujeme pouze
-speciálnìj¹í pøípady, které mohou být øe¹itelné v~polynomiálním èase.
-\:Spokojit se s~pøibli¾ným øe¹ením, (pou¾ít aproximaèní algoritmus).
-\:Pou¾ít heuristiku -- napøíklad genetické algoritmy nebo randomizované algoritmy.
-Velmi pomoci mù¾e i jen výhodnìj¹í poøadí pøi~prohledávání èi oøezávání nìkterých
-napohled nesmyslných vìtví výpoètu.
-\endalgo
-
-\h{První zpùsob: Speciální pøípad}
-
-\>Èasto si vystaèíme s~vyøe¹ením speciálního pøípadu NP-úplného problému, který
-le¾í v~$P$. Napøíklad pøi øe¹ení grafové úlohy nám mù¾e staèit øe¹ení
-pro~speciální druh grafù (stromy, bipartitní grafy, \dots). Barvení grafu je lehké
-napø. pro~dvì barvy èi pro intervalové grafy. 2-SAT, jako speciální pøípad SATu,
-se dá øe¹it v~lineárním èase.
-
-\>Uká¾eme si dva takové pøípady (budeme øe¹ení hledat, nejen rozhodovat, zda existuje)
-
-\s{Problém: Maximální nezávislá mno¾ina ve~stromì (ne rozhodovací)}
-
-\>{\I Vstup:} Zakoøenìný strom~$T$.
-
-\>{\I Výstup:} Maximální (co do~poètu vrcholù) nezávislá mno¾ina vrcholù~$M$~v~$T$.
-
-\>BÚNO mù¾eme pøedpokládat, ¾e v~$M$ jsou v¹echny listy $T$. Pokud by nìkterý
-list $l$ v~$M$ nebyl, tak se podíváme na~jeho otce:
-\itemize\ibull
-\:Pokud otec není v~$M$, tak list $l$ pøidáme do~$M$, èím¾ se nezávislost
-mno¾iny zachovala a velikost stoupla o~1.
-\:Pokud tam otec je, tak ho z~$M$ vyjmeme a na~místo nìho vlo¾íme $l$.
-Nezávislost ani velikost $M$ se nezmìnily.
-\endlist
-\>Nyní listy spolu s~jejich otci z~$T$ odebereme a postup opakujeme. $T$ se
-mù¾e rozpadnout na~les, ale to nevadí $\to$ tentý¾ postup aplikujeme na~v¹echny stromy v~lese.
-
-\s{Algoritmus:}
-\>MaxNz$(T)$
-\algo
-\:Polo¾íme $L$:=$\{$listy stromu $T\}$.
-\:Polo¾íme $O$:=$\{$otcové vrcholù z~$L\}$.
-\:Vrátíme $L \cup$ MaxNz$(T\setminus(O \cup L))$.
-\endalgo
-\>{\I Poznámka:} Toto doká¾eme naprogramovat v~$\O(n)$ (udr¾ujeme si frontu listù).
-
-\s{Problém: Batoh}
-
-\>Je daná mno¾ina $n$~pøedmìtù s~hmotnostmi $h_1,\ldots,h_n$
-a cenami $c_1,\ldots,c_n$ a~batoh, který unese hmostnost~$H$. Najdìte takovou
-podmno¾inu pøedmìtù, jejich¾ celková hmotnost je maximálnì $H$ a celková cena
-je maximální mo¾ná.
-
-\>Tento problém je zobecnìním problému batohu z~minulé pøedná¹ky dvìma smìry:
-Jednak místo rozhodovacího problému øe¹íme optimalizaèní, jednak pøedmìty
-mají ceny (pøedchozí verze odpovídala tomu, ¾e ceny jsou rovny hmotnostem).
-Uká¾eme si algoritmus pro øe¹ení tohoto obecného problému, jeho¾ èasová
-slo¾itost bude polynomiální v~poètu pøedmìtù~$n$ a souètu v¹ech cen~$C=\sum_i
-c_i$.
-
-\>Pou¾ijeme dynamické programování. Pøedstavme si problém omezený na~prvních~$k$
-pøedmìtù. Oznaème si $A_k(c)$ (kde $0\le c\le C$) minimální hmotnost
-podmno¾iny, její¾ cena je právì~$c$. Tato $A_k$ spoèteme indukcí podle~$k$:
-Pro $k=0$ je urèitì $A_0(0)=0$, $A_0(c)=infty$ pro $c>0$. Pokud ji¾ známe
-$A_{k-1}$, spoèítáme $A_k$ následovnì: $A_k(c)$ odpovídá nìjaké podmno¾inì
-pøedmìtù z~$1,\ldots,k$. V~této podmno¾inì jsme buïto $k$-tý pøedmìt nepou¾ili
-(a pak je $A_k(c)=A_{k-1}(c)$), nebo pou¾ili a tehdy bude $A_k(c) =
-A_{k-1}(c-c_k) + h_k$ (to samozøejmì jen pokud $c\ge c_k$). Z~tìchto dvou
-mo¾ností si vybereme tu, která dává mno¾inu s~men¹í hmotností. Tedy:
-$$
-A_k(c) = \min (A_{k-1}(c), A_{k-1}(c-c_k) + h_k).
-$$
-Tímto zpùsobem v~èase $\O(C)$ spoèteme $A_k(c)$ pro fixní $k$ a v¹echna $c$,
-v~èase $\O(nC)$ pak v¹echny $A_k(c)$.
-
-\>Podle $A_n$ snadno nalezneme maximální cenu mno¾iny, která se vejde do~batohu.
-To bude nejvìt¹í~$c^*$, pro nì¾ je $A_n(c^*) \le H$. Jeho nalezení nás stojí
-èas $\O(C)$.
-
-\>A~jak zjistit, které pøedmìty do~nalezené mno¾iny patøí? Upravíme algoritmus,
-aby si pro ka¾dé $A_k(c)$ pamatoval $B_k(c)$, co¾ bude index posledního pøedmìtu,
-který jsme do~pøíslu¹né mno¾iny pøidali. Pro nalezené $c^*$ tedy bude $i=B_n(c^*)$
-poslední pøedmìt v~nalezené mno¾inì, $i'=B_{i-1}(c^*-c_i)$ ten pøedposlední
-a tak dále. Takto v~èase $\O(n)$ rekonstruujeme celou mno¾inu od~posledního
-prvku k~prvnímu.
-
-\>Ukázali jsme tedy algoritmus s~èasovou slo¾itostí $\O(nC)$, který vyøe¹í
-problém batohu. Jeho slo¾itost není polynomem ve~velikosti vstupu ($C$~mù¾e
-být a¾ exponenciálnì velké vzhledem k~velikosti vstupu), ale pouze ve~velikosti
-èísel na~vstupu. Takovým algoritmùm se øíká {\I pseudopolynomiální.} Ani takové
-algoritmy ale nejsou k dispozici pro v¹echny problémy (napø. u problému obchodního
-cestujícího nám vùbec nepomù¾e, ¾e váhy hran budou malá èísla).
-
-\s{Verze bez cen:} Na verzi s~cenami rovnými hmotnostem se dá pou¾ít
-i jiný algoritmus zalo¾ený na~dynamickém programování: poèítáme mno¾iny
-$Z_k$ obsahující v¹echny hmotnosti men¹í ne¾~$H$, kterých nabývá
-nìjaká podmno¾ina prvních~$k$ prvkù. Pøitom $Z_0=\{0\}$, $Z_k$
-spoèteme ze~$Z_{k-1}$ --- udr¾ujme si $Z_{k-1}$ jako setøídìný spojový seznam,
-výpoèet dal¹ího seznamu udìláme slitím dvou seznamù $Z_{k-1}$ a $Z_{k-1}$ se
-v¹emi prvky zvý¹enými o hmotnost $k$ zahazujíce duplicitní a pøíli¹ velké hodnoty ---
-a ze~$Z_n$ vyèteme výsledek. V¹echny tyto mno¾iny
-mají nejvý¹e $H$ prvkù, tak¾e celková èasová slo¾itost algoritmu je~$\O(nH)$.
-
-\h{Druhý zpùsob: Aproximace}
-
-\>V pøedcházejících problémech jsme se zamìøili na~speciální pøípady. Obèas v¹ak
-takové ¹tìstí nemáme a musíme vyøe¹it celý NP-úplný problém. Mù¾eme si v¹ak
-pomoct tím, ¾e se ho nebudeme sna¾it vyøe¹it optimálnì -- namísto optimálního
-øe¹ení najdeme nìjaké, které je nejvý¹e $c$-krát hor¹í pro nìjakou konstantu $c$.
-
-\s{Problém: Obchodní cestující}
-
-\>{\I Vstup:} Neorientovaný graf~$G$, ka¾dá hrana
-je ohodnocená funkcí $w: E(G)\rightarrow {\bb R }^+_0$.
-
-\>{\I Výstup:} Hamiltonovská kru¾nice (obsahující v¹echny vrcholy grafu), a~to ta nejkrat¹í
-(podle ohodnocení).
-
-\>Tento problém je hned na~první pohled nároèný -- u¾ sama existence
-hamiltonovské kru¾nice je NP-úplná. Najdeme aproximaèní algoritmus nejprve za pøedpokladu,
-¾e vrcholy splòují trojúhelníkovou nerovnost (tj. $\forall x,y,z \in V: w(xz)\le
-w(xy)+w(yz)$), potom uká¾eme, ¾e v úplnì obecném pøípadé by samotná existence
-aproximaèního algoritmu implikovala ${\rm P=NP }$.
-
-\>{\I a) trojúhelníková nerovnost:}
-
-Existuje pìkný algoritmus, který najde hamiltonovskou kru¾nici o délce $\leq
-2\cdot opt$, kde $opt$ je délka nejkrat¹í hamiltonovské kru¾nice.
-Vedle pøedpokladu trojúhelníkové
-nerovnosti budeme potøebovat, aby ná¹ graf byl úplný. Souhrnnì mù¾eme
-pøedpokládat, ¾e úlohu øe¹íme v nìjakém metrickém protoru, ve kterém jsou obì
-podmínky podle definice splnìny.
-
-Najdeme nejmen¹í kostru grafu a obchodnímu cestujícímu poradíme, a» jde po~ní -- kostru
-zakoøeníme a projdeme jako strom do hloubky, pøièem¾ se zastavíme a¾ v koøeni po projití
-v¹ech vrcholù. Problém v¹ak je, ¾e prùchod po kostøe obsahuje
-nìkteré vrcholy i hrany vícekrát, a proto musíme nahradit nepovolené vracení se.
-Máme-li na nìjaký vrchol vstoupit podruhé, prostì ho ignorujeme a pøesuneme se
-rovnou na dal¹í nenav¹tívený -- dovolit si to mù¾eme, graf je úplný a obsahuje
-hrany mezi v¹emi dvojicemi vrcholù
-(jinak øeèeno, poøadí vrcholù kru¾nice bude preorder výpis prùchodem do hloubky).
-Pokud platí trojúhelníková nerovnost, tak si tìmito zkratkami neu¹kodíme.
-Nech» minimální kostra má váhu~$T$. Pokud bychom pro¹li celou kostru, bude mít
-sled váhu~$2T$ (ka¾dou hranou kostry jsme ¹li tam a zpátky), a pøeskakování
-vrcholù celkovou váhu nezvìt¹uje (pøi pøeskoku
-nahradíme cestu $xyz$ jedinou hranou $xz$, pøièem¾ z trojúhelníkové nerovnosti
-máme $xz \leq xy + xz$), tak¾e váha nalezené
-hamiltonovské kru¾nice bude také nanejvý¹ $2T$.
-
-Kdy¾ máme hamiltonovskou kru¾nici $C$ a z~ní vy¹krtneme hranu, dostaneme kostru
-grafu~$G$ s~váhou men¹í ne¾ $C$ -- ale ka¾dá kostra je alespoò tak tì¾ká
-jako minimální kostra $T$. Tedy optimální hamiltonovská kru¾nice je urèitì tì¾¹í
-ne¾ minimální kostra $T$. Kdy¾ tyto dvì nerovnosti slo¾íme
-dohromady, algoritmus nám vrátí hamiltonovskou kru¾nici $T'$ s~váhou nanejvý¹
-dvojnásobnou vzhledem k optimální hamiltonovské kru¾nici ($T' \leq 2T < 2C$). Takovéto
-algoritmy se nazývají {\I 2-aproximaèní}, kdy¾ øe¹ení je maximálnì dvojnásobné
-od~optimálního.\foot{Hezkým trikem se v obecných metrických prostorech umí
-$1{,}5$-aproximace. Ve~nìkterých metrických prostorech (tøeba v euklidovské
-rovinì) se aproximaèní pomìr dá dokonce srazit na
-libovolnì blízko k 1. Zaplatíme ale na èase -- èím pøesnìj¹í výsledek
-po algoritmu chceme, tím déle to bude trvat.}
-
-\>{\I b) bez~trojúhelníkové nerovnosti:}
-
-Zde se budeme naopak sna¾it ukázat, ¾e ¾ádný polynomiální aproximaèní
-algoritmus neexistuje.
-
-\s{Vìta:} Pokud pro~libovolné~$\varepsilon>0$ existuje polynomiální
-$(1+\varepsilon)$-aproximaèní algoritmus pro~problém obchodního cestujícího bez~trojúhelníkové nerovnosti, tak ${\rm P = NP }$.
-
-\proof Uká¾eme, ¾e v~takovém pøípadì doká¾eme v~polynomiálním èase zjistit,
-zda v grafu existuje hamiltonovská kru¾nice.
-
-\>Dostali jsme graf~$G$, ve~kterém hledáme hamiltonovskou kru¾nici. Doplníme
-$G$ na~úplný graf~$G'$ a~váhy hran~$G'$ nastavíme takto:
-\itemize\ibull
-\: $w(e) = 1$, kdy¾ $e \in E(G)$
-\: $w(e) = c \gg 1$, kdy¾ $e \not\in E(G)$
-\endlist
-\>Konstantu $c$ potøebujeme zvolit tak velkou, abychom jasnì poznali, jestli
-je ka¾dá hrana z nalezené hamiltonovské kru¾nice hranou grafu $G$ (pokud by
-nebyla, bude kru¾nice obsahovat aspoò jednu hranu s váhou $c$, která vy¾ene
-souèet poznatelnì vysoko). Pokud existuje hamiltonovská kru¾nice v~$G'$ slo¾ená jen
-z~hran, které byly
-pùvodnì v~$G$, pak optimální øe¹ení bude mít váhu~$n$, jinak bude urèitì
-minimálnì $n-1+c$. Kdy¾ máme aproximaèní algoritmus s~pomìrem~$1+\varepsilon$,
-musí tedy být
-$$
-\eqalign{
-(1+\varepsilon)\cdot n &< n-1+c \cr
-\varepsilon n+1 &< c
-}
-$$
-\>Kdyby takový algoritmus existoval, máme polynomiální algoritmus
-na~hamiltonovskou kru¾nici.
-\qed
-
-\s{Poznámka:} O existenci pseudopolynomiálního algoritmu
-platí analogická vìta, a doká¾e se analogicky -- existující hrany budou
-mít váhu 1, neexistující váhu 2.
-
-\h{Aproximaèní schéma pro problém batohu}
-
-Ji¾ víme, jak optimalizaèní verzi problému batohu vyøe¹it v~èase $\O(nC)$,
-pokud jsou hmotnosti i ceny na~vstupu pøirozená èísla a $C$ je souèet v¹ech cen.
-Jak si poradit, pokud je~$C$ obrovské? Kdybychom mìli ¹tìstí a v¹echny
-ceny byly dìlitelné nìjakým èíslem~$p$, mohli bychom je tímto èíslem
-vydìlit. Tím bychom dostali zadání s~men¹ími èísly, jeho¾ øe¹ením by byla
-stejná mno¾ina pøedmìtù jako u~zadání pùvodního.
-
-Kdy¾ nám ¹tìstí pøát nebude, mù¾eme pøesto zkusit ceny vydìlit a výsledky
-nìjak zaokrouhlit. Øe¹ení nové úlohy pak sice nebude pøesnì odpovídat optimálnímu
-øe¹ení té pùvodní, ale kdy¾ nastavíme parametry správnì, bude alespoò jeho dobrou aproximací.
-
-\s{Základní my¹lenka:}
-
-Oznaèíme si $c_{max}$ maximum z~cen~$c_i$. Zvolíme si nìjaké pøirozené èíslo~$M < c_{max}$
-a zobrazíme interval cen $[0, c_{max}]$ na $[0,M]$ (tedy ka¾dou cenu znásobíme
-$M/c_{max}$).
-Jak jsme tím zkreslili výsledek? V¹imnìme si, ¾e efekt je stejný, jako kdybychom jednotlivé
-ceny zaokrouhlili na~násobky èísla $c_{max}/M$ (prvky z intervalu
-$[i\cdot c_{max}/M,(i+1)\cdot c_{max}/M)$ se zobrazí na stejný prvek). Ka¾dé $c_i$ jsme tím
-tedy zmìnili o~nejvý¹e $c_{max}/M$, celkovou cenu libovolné podmno¾iny pøedmìtù pak
-nejvý¹e o~$n\cdot c_{max}/M$. Teï si je¹tì v¹imnìme, ¾e pokud ze~zadání odstraníme
-pøedmìty, které se samy nevejdou do~batohu, má optimální øe¹ení pùvodní úlohy cenu $OPT\ge c_{max}$,
-tak¾e chyba v~souètu je nejvý¹e $n\cdot OPT/M$. Má-li tato chyba být shora omezena
-$\varepsilon\cdot OPT$, musíme zvolit $M\ge n/\varepsilon$.\foot{Pøipomìòme, ¾e toto je¹tì není dùkaz, nebo» velkoryse pøehlí¾íme chyby dané zaokrouhlováním. Dùkaz provedeme ní¾e.}
-
-\s{Algoritmus:}
-\algo
-\:Odstraníme ze~vstupu v¹echny pøedmìty tì¾¹í ne¾~$H$.
-\:Spoèítáme $c_{max}=\max_i c_i$ a zvolíme $M=\lceil n/\varepsilon\rceil$.
-\:Kvantujeme ceny: $\forall i: \hat{c}_i \leftarrow \lfloor c_i \cdot M/c_{max} \rfloor$.
-\:Vyøe¹íme dynamickým programováním problém batohu pro upravené ceny $\hat{c}_1, \ldots, \hat{c}_n$
-a pùvodní hmotnosti i kapacitu batohu.
-\:Vybereme stejné pøedmìty, jaké pou¾ilo optimální øe¹ení kvantovaného zadání.
-\endalgo
-
-\>Kroky 1--3 a 5 jistì zvládneme v~èase $\O(n)$. Krok~4 øe¹í problém batohu
-se souètem cen $\hat{C}\le nM = \O(n^2/\varepsilon)$, co¾ stihne v~èase $\O(n\hat{C})=\O(n^3/\varepsilon)$.
-Zbývá dokázat, ¾e výsledek na¹eho algoritmu má opravdu relativní chybu nejvý¹e~$\varepsilon$.
-
-Nejprve si rozmyslíme, jakou cenu budou mít pøedmìty které daly optimální øe¹ení
-v pùvodním zadání (tedy mají v pùvodním zadání dohromady cenu $OPT$),
-kdy¾ jejich ceny nakvantujeme (mno¾inu indexù tìchto pøedmìtù si oznaèíme~$Y$):
-$$
-\eqalign{
-\widehat{OPT} &= \sum_{i\in Y} \hat{c}_i =
-\sum_i \left\lfloor c_i\cdot {M\over c_{max}} \right\rfloor \ge
-\sum_i \left( c_i\cdot {M\over c_{max}} - 1 \right) \ge \cr
-&\ge
-\biggl(\sum_i c_i \cdot {M\over c_{max}}\biggr) - n =
-OPT \cdot {M\over c_{max}} - n.
-}
-$$
-Nyní spoèítejme, jak dopadne optimální øe¹ení~$Q$ nakvantovaného problému pøi pøepoètu
-na~pùvodní ceny (to je výsledek na¹eho algoritmu):
-$$
-\eqalign{
-ALG &= \sum_{i\in Q} c_i \ge
-\sum_i \hat{c}_i \cdot {c_{max}\over M} =
-\biggl(\sum_i \hat{c}_i\biggr) \cdot {c_{max}\over M} \ge^*
-\widehat{OPT} \cdot {c_{max}\over M}.
-}
-$$
-Nerovnost $\ge^*$ platí proto, ¾e $\sum_{i\in Q} \hat{c}_i$ je optimální øe¹ení
-kvantované úlohy, zatímco $\sum_{i\in Y} \hat{c}_i$ je nìjaké dal¹í øe¹ení té¾e úlohy,
-které nemù¾e být lep¹í. Teï u¾ staèí slo¾it obì nerovnosti a dosadit za~$M$:
-$$
-\eqalign{
-ALG &\ge \biggl( { OPT \cdot M\over c_{max}} - n\biggr) \cdot {c_{max}\over M} \ge
-OPT - {n\cdot c_{max}\over n / \varepsilon} \ge OPT - \varepsilon c_{max} \ge \cr
-&\ge OPT - \varepsilon OPT = (1-\varepsilon)\cdot OPT.
-}
-$$
-Algoritmus tedy v¾dy vydá øe¹ení, které je nejvý¹e $(1-\varepsilon)$-krát hor¹í ne¾ optimum,
-a~doká¾e to pro libovolné~$\varepsilon$ v~èase polynomiálním v~$n$. Takovému algoritmu øíkáme
-{\I polynomiální aproximaèní schéma} (jinak té¾ PTAS\foot{Polynomial-Time Approximation Scheme}).
-V~na¹em pøípadì je dokonce slo¾itost polynomiální i v~závislosti na~$1/\varepsilon$, tak¾e
-schéma je {\I plnì polynomiální} (øeèené té¾ FPTAS\foot{Fully Polynomial-Time Approximation
-Scheme}). U nìkterých problémù se stává, ¾e aproximaèní schéma závisí na
-$1/\varepsilon$ exponenciálnì, co¾ tak pøíjemné není. Shròme, co jsme zjistili, do následující vìty:
-
-\s{Vìta:}
-Existuje algoritmus, který pro ka¾dé $\varepsilon > 0$ nalezne
-{\I $(1 - \varepsilon)$-aproximaci} problému batohu s $n$ pøedmìty v èase
-$\O(n^3/\varepsilon)$.
-
-\bye
diff --git a/12-apx/Makefile b/12-apx/Makefile
deleted file mode 100644
index 11f9d14..0000000
--- a/12-apx/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=12-apx
-
-include ../Makerules
diff --git a/2-dinic/2-dinic.tex b/2-dinic/2-dinic.tex
deleted file mode 100644
index 244427c..0000000
--- a/2-dinic/2-dinic.tex
+++ /dev/null
@@ -1,223 +0,0 @@
-\input lecnotes.tex
-
-\prednaska{2}{Dinicùv algoritmus}{(zapsala Markéta Popelová)}
-
-
-Na~minulé pøedná¹ce jsme si~ukázali Fordùv-Fulkersonùv algoritmus. Tento algoritmus hledal maximální tok tak, ¾e zaèal s~tokem nulovým a~postupnì ho zvìt¹oval. Pro~ka¾dé zvìt¹ení potøeboval v~síti najít cestu, na~které mají v¹echny hrany kladnou rezervu (po takovéto cestì mù¾eme poslat více, ne¾ po~ní aktuálnì teèe). Ukázali jsme, ¾e pokud takováto cesta existuje, jde tok vylep¹it (zvìt¹it). Zároveò pokud tok jde vylep¹it, pak takováto cesta existuje. Dokázali jsme, ¾e pro~racionální kapacity je algoritmus koneèný a~najde maximální tok.
-
-Fordùv-Fulkersonùv algoritmus má ov¹em znaèné nevýhody. Funguje pouze pro~racionální kapacity a~je pomìrnì pomalý. Nyní si~uká¾eme jiný algoritmus, který nevylep¹uje tok pomocí cest, ale pomocí tokù\dots Budeme k~tomu potøebovat sí» rezerv.
-
-\s{Definice:} {\I Sí» rezerv} k~toku~$f$ v~síti $S=(V,E,z,s,c)$ je sí» $R=(S,f)=(V,E,z,s,r)$, kde~$r(e)$ je rezerva hrany~$e$ v~toku~$f$.
-
-\s{Konvence:} Pro~hranu~$e$ znaèí~$\overleftarrow{e}$ hranu opaènou. Napø. pokud $e=uv$, tak $\overleftarrow{e}=vu$.
-
-Je dùle¾ité si~uvìdomit, ¾e sí» rezerv je závislá jak na~pùvodní síti~$S$, tak na~nìjakém toku~$f$ v~síti~$S$. Sí» rezerv~$R$ se~pak od sítì~$S$ li¹í pouze kapacitami -- sí»~$R$ má jako kapacitu hrany rezervu hrany v pùvodní síti. Pro~pøipomenutí: rezervu hrany~$e$ v~síti $S=(V,E,z,s,c)$ s~tokem~$f$ jsme si~definovali jako $r(e)=c(e) - f(e) + f(\overleftarrow{e})$.
-
-Ne¾ si~uká¾eme samotný algoritmus, doká¾eme si~následující lemma.
-
-\s{Lemma:} Pro~ka¾dý tok~$f$ v~síti~$S$ a~pro~ka¾dý tok~$g$ v~síti $R=(S,f)$ lze v~èase $\O(m)$ nalézt tok~$f'$ v~síti~$S$ takový, ¾e $\vert f' \vert = \vert f \vert + \vert g \vert$.
-
-\proof
-
-Dùkaz rozdìlíme do~tøí krokù. V~prvním kroku si~uká¾eme, jak budeme tok~$f'$ v~síti~$S$ konstruovat. V~druhém kroku doká¾eme, ¾e takto zkonstruované~$f'$ je opravdu tok. A~nakonec uká¾eme, ¾e splòuje po¾adovanou vlastnost, tedy ¾e jeho velikost je souèet velikostí tokù~$f$ a~$g$.
-
-\>{\it 1. konstrukce~$f'$}
-
-\noindent
-Pro ka¾dou dvojici hran~$e, \overleftarrow{e}$ urèíme~$f'(e)$ a~$f'(\overleftarrow{e})$ následovnì:
-
-\itemize\ibull
-\:Pokud~$g(e) = g(\overleftarrow{e}) = 0$, pak nastavme:
-	\itemize\ibull
-	\:$f'(e) := f(e)$,
-	\:$f'(\overleftarrow{e}) := f(\overleftarrow{e})$.
-	\endlist
-	
-\:Pokud~$g(e) > 0$ a~$g(\overleftarrow{e}) = 0$, pak polo¾me:
-	\itemize\ibull
-	\:$\varepsilon := \min (g(e), f(\overleftarrow{e}))$,
-	\:$f'(e) := f(e) + g(e) - \varepsilon$,
-	\:$f'(\overleftarrow{e}) := f(\overleftarrow{e}) - \varepsilon$.
-	\endlist
-
-\:Pøípad~$g(e) = 0$ a~$g(\overleftarrow{e}) > 0$ vyøe¹íme obdobnì.
-
-\:Pokud~$g(e) > 0$ a~$g(\overleftarrow{e}) > 0$, pak odeèteme od toku~$g$ cirkulaci po cyklu tvoøeném hranami~$e$ a $\overleftarrow{e}$:
-	\itemize\ibull
-	\:$\delta := \min (g(e),g(\overleftarrow{e}))$,
-	\:$g'(e) := g(e) - \delta$,
-	\:$g'(\overleftarrow{e}) := g(\overleftarrow{e}) - \delta$. 	
-	\endlist
-
-	Tok $g'$ nyní spadá pod nìkterý z~pøedchozích pøípadù, které u¾ umíme vyøe¹it.
-	
-\endlist
-
-\>{\it 2. $f'$ je tok}
-
-\numlist{\ndotted}
-
-\:Nejdøíve ovìøme první podmínku: $\forall e \in E: 0 \leq f(e) \leq c(e)$. Vezmìme libovolnou hranu~$e \in E$. Podle toho, co teèe po~hranách~$e$ a~$\overleftarrow{e}$ v~toku~$g$, jsme rozdìlili konstrukci toku na~tøi pøípady:
-
-	\numlist{\ndotted}	
-	
-	\:Pokud po~hranách~$e$ a~$\overleftarrow{e}$ netekl ¾ádný tok~$g$, pak jsme nastavili $f'(e) := f(e)$ a~$f'(\overleftarrow{e}) := f(\overleftarrow{e})$. Tedy pokud~$f$ dodr¾oval kapacity, tak pro~$f'$ musí platit to samé.
-
-
-	\:Pokud po~hranì~$e$ tekl tok~$g$ nenulový a~po opaèné nulový, tak jsme zvolili: $f'(e) := f(e) + g(e) - \varepsilon$. Víme, ¾e jsme si~$\varepsilon$ vybrali tak, ¾e $\varepsilon \leq g(e)$. Proto $f'(e) \geq 0$.
-
-	Teï ovìøme, ¾e $f'(e) \leq c(e)$. V~pøípadì, ¾e $\varepsilon = g(e)$, tak $f'(e) = f(e) \leq c(e)$. V~opaèném pøípadì platí, ¾e $\varepsilon = f(\overleftarrow{e})$. Pak ov¹em
-	$$f'(e) = f(e) + g(e) - f(\overleftarrow{e}) \leq $$
-	$$\leq f(e) + \left[ c(e) - f(e) + f(\overleftarrow{e}) \right] - f(\overleftarrow{e}) = c(e).$$
-	Vyu¾ili jsme, ¾e~$g$ je tok v~síti rezerv, tedy $g(e) \leq  c(e) - f(e) + f(\overleftarrow{e})$.
-
-	Pro tok $f'(\overleftarrow{e})$ platí, ¾e $\varepsilon \leq f(\overleftarrow{e})$. Proto $f'(\overleftarrow{e}) = f(\overleftarrow{e}) - \varepsilon \geq 0$. Zároveò $f'(\overleftarrow{e}) \leq f(\overleftarrow{e}) \leq c(\overleftarrow{e})$.
-
-	Tím jsme dokázali, ¾e~$f'(e)$ i~$f'(\overleftarrow{e})$ dodr¾ují kapacity.
-
-	\:V posledním pøípadì tekl po~obou hranách kladný tok~$g$. Men¹í tok z~$g(e)$ a~$g(\overleftarrow{e})$ jsme vynulovali a~od vìt¹ího odeèetli ten men¹í. Tok~$g'(e)$ a~$g'(\overleftarrow{e})$ tedy zùstal korektní a~tok~$f'$ u¾ konstruujeme podle pøedchozího pøípadu.
-
-	\endlist
-
-\:Teï musíme je¹tì dokázat, ¾e nový tok neporu¹uje Kirchhoffovy zákony: $$\forall v~\in V \setminus \{z,s\}: f'^\Delta(v)=0.$$
-	% neboli $$\forall v~\in V \setminus \{z,s\}: \sum_{u: uv \in E}{f'(uv)}=\sum_{u: vu \in E}{f'(vu)}.$$
-
-	Vezmìme si~libovolnou hranu~$e = uv \in E$. Uvìdomme si, ¾e pøi~pøechodu z~$f(e)$ na~$f'(e)$ a~z~$f(\overleftarrow{e})$ na~$f'(\overleftarrow{e})$ bylo:
-	\itemize\idot
-	\:$f^\Delta(u)$ sní¾eno o~$g(e)$
-	\:$f^\Delta(v)$ zvý¹eno o~$g(e)$.
-	\endlist
-	Seèteme-li úpravy na v¹ech hranách, dostaneme: $$f'^\Delta(v) = f^\Delta(v) + \sum_{u:uv \in E} g(uv) - \sum_{u:vu \in E} g(vu) =$$ $$= f^\Delta(v) + g^+(v) - g^-(v) = f^\Delta(v) + g^\Delta(v).$$
-
-	Jeliko¾~$f$ byl tok, tak $f^\Delta(v) = 0$ a jeliko¾~$g$ byl tok, tak $g^\Delta(v) = 0$. Proto $f'^\Delta(v) = f^\Delta(v) + g^\Delta(v) = 0$.
-	
-\endlist
-
-Tím jsme dokázali, ¾e~$f'$ je tok v~síti~$S$.
-
-\>{\it 3. $\vert f' \vert = \vert f \vert + \vert g \vert$}
-
-Pou¾ijme vztah pro souèet pøebytkù z pøedchozího kroku:
-$$\vert f' \vert = f'^\Delta(s) = f^\Delta(s) + g^\Delta(s) = \vert f \vert + \vert g \vert.$$
-\qed
-
-
-Pro algoritmus budeme potøebovat vybírat kvalitní toky~$g$ v~síti rezerv. Pokud se~nám to bude daøit, bude se~tok~$f'$ rychle zvìt¹ovat, a¾ bychom mohli dojít k~maximálnímu toku. Nejlépe by se~nám hodily co nejvìt¹í toky v~síti rezerv. Kdybychom si~dali za cíl najít v¾dy maximální tok v~síti rezerv, výsledek by byl sice krásný (dostali bychom tak rovnou i~maximální tok v~pùvodní síti), ale problém hledání maximálního toku bychom pouze pøenesli na~jinou sí». Na¹e po¾adavky na~tento tok budou tedy takové, aby byl dostateènì velký, ale abychom bìhem jeho hledání nestrávili moc èasu. Podívejme se, jak se~s~tímto problémem vyrovná {\I Dinicùv algoritmus}. Nejdøíve si~ale zadefinujme nìkolik pojmù.
-
-\s{Definice:} Tok~$f$ je {\I blokující}, jestli¾e pro~ka¾dou orientovanou cestu~$P$ ze~$z$ do~$s$ existuje hrana~$e \in P$ taková, ¾e $f(e) = c(e)$.
-
-\s{Definice:} Sí» je {\I vrstevnatá (proèi¹tìná)}, kdy¾ v¹echny vrcholy a~hrany le¾í na~nejkrat¹ích cestách ze~$z$ do~$s$.
-
-Dinicùv algoritmus zaèíná s~nulovým tokem. Potom v¾dy podle toku~$f$ sestrojí sí» rezerv a~v~ní vyma¾e hrany s~nulovou rezervou. Pokud v~této promazané síti rezerv neexistuje cesta ze~zdroje do~stoku, tak skonèí a~prohlásí tok~$f$ za maximální. Jinak proèistí sí» rezerv tak, aby se~z ní stala vrstevnatá sí» (rozdìlí vrcholy do~vrstev podle vzdálenosti od zdroje a~odstraní pøebyteèné hrany). Ve~vrstevnaté síti najde blokující tok, pomocí nìho¾ zlep¹í tok~$f$. Pak opìt pokraèuje sestrojením sítì rezerv na~tomto vylep¹eném toku~$f$ atd.
-
-\figure{dinic-cistasit.eps}{Proèi¹tìná sí» rozdìlená do~vrstev}{0.4\hsize}
-
-\s{Algoritmus (Dinicùv)}
-
-\algo
-\:$f \leftarrow$ nulový tok.
-\:Sestrojíme sí» rezerv~$R$ a~sma¾eme $e: r(e) = 0$.
-\:$l \leftarrow$ délka nejkrat¹í cesty ze~$z$ do~$s$ v~$R$.
-\:Pokud $l = \infty$, zastavíme se~a vrátíme~$f$.
-\:Proèistíme sí» $R \rightarrow$ sí»~$C$.
-\:$g \leftarrow$ blokující tok v~$C$.
-\:Zlep¹íme tok~$f$ pomocí~$g$.
-\:GOTO 2.
-\endalgo
-
-\s{Pozorování:} Pokud se~algoritmus zastaví, vydá maximální tok.
-
-\proof
-Víme, ¾e~$f$ je stále korektní tok (jediné, jak ho mìníme je pøièítání toku~$g$, co¾ je, jak jsme si~dokázali, \uv{ne¹kodná operace}). Jakmile neexistuje cesta ze~$z$ do~$s$ v~$R$, tak je $f$ maximální tok, nebo» v~tuto dobu by se~zastavil (a vydal maximální tok) i~Fordùv-Fulkersonùv algoritmus, který je korektní.
-\qed
-
-A teï je¹tì musíme ujasnit, jak budeme èistit sí» rezerv a~vybírat blokující tok~$g$.
-
-\s{Algoritmus proèi¹tìní sítì rezerv}
-
-\algo
-\:Rozdìlíme vrcholy do~vrstev podle vzdálenosti od~$z$.
-\:Odstraníme vrstvy za~$s$ (tedy vrcholy, které jsou od~$z$ vzdálenìj¹í ne¾~$s$), hrany do~minulých vrstev a~hrany uvnitø vrstev.
-\:Odstraníme \uv{slepé ulièky}, tedy vrcholy s~$\deg^{out}(v) = 0$, a~to opakovanì pomocí fronty. (Nejdøíve zaøadíme do~fronty v¹echny vrcholy s~ $\deg^{out}(v) = 0$. Pak dokud není fronta prázdná, v¾dy vybereme vrchol~$v$ z~fronty, odstraníme~$v$ a~v¹echny hrany~$uv$. Pro~ka¾dý takový vrchol~$u$ zkontrolujeme, zda se~tím nesní¾il výstupní stupeò vrcholu~$u$ na~nulu ($\deg^{out}(u) = 0$). Pokud sní¾il, tak ho zaøadíme do~fronty.)
-\endalgo
-
-\figure{dinic-neprocistenasit.eps}{Neproèi¹tìná sí». Obsahuje zpìtné hrany, hrany uvnitø vrstvy a~slepé ulièky.}{0.45\hsize}
-
-Hledání blokujícího toku zaèneme s~tokem nulovým. Pak vezmeme v¾dy orientovanou cestu ze~zdroje do~stoku v~síti~$C$. V~této cestì najdeme hranu s~nejni¾¹í hodnotou výrazu $r(e) - g(e)$ (neboli $c(e) - f(e)$ v~pùvodní síti). Tuto hodnotu oznaèíme~$\varepsilon$. Pak ke~ v¹em hranám na~této cestì pøièteme~$\varepsilon$. Pokud tok~$g$ na~nìjaké hranì dosáhne kapacity hrany, co¾ je zde~$r(e)$, tak hranu vyma¾eme. Následnì sí» doèistíme, aby splòovala podmínky vrstevnaté sítì. A~pokud je¹tì existuje nìjaká orientovaná cesta ze~zdroje do~stoku, tak opìt pokraèujeme s~touto cestou.
-
-\s{Algoritmus hledání blokujícího toku}
-
-\algo
-\:$g \leftarrow$ nulový tok.
-\:Dokud existuje orientovaná cesta~$P$ ze~$z$ do~$s$ v~$C$, opakuj:
-\::$\varepsilon \leftarrow \min_{e \in P} (r(e) - g(e))$.
-\::Pro~$\forall e \in P: g(e) \leftarrow g(e) + \varepsilon$.
-\:::Pokud $g(e) = r(e)$, sma¾eme~$e$ z~$C$.
-\::Doèistíme sí» zase pomocí fronty.
-\endalgo
-
-\s{Èasová slo¾itost} Rozeberme si~jednotlivé kroky algoritmu.
-
-\numlist{\ndotted}
-\:Inicializace toku~$f$ \dots $\O(m)$.
-\:Sestrojení sítì rezerv a~smazání hran s~nulovou rezervou \dots $\O(m + n)$.
-\:Najití nejkrat¹í cesty (prohledáváním do~¹íøky) \dots $\O(m + n)$.
-\:Zkontrolování délky nejkrat¹í cesty \dots $\O(1)$.
-\:Proèi¹tìní sítì \dots $\O(m + n)$.
-	\numlist{\ndotted}
-	\:Rozdìlení vrcholù do~vrstev -- provedlo ji¾ prohledávání do~¹íøky \dots $\O(1)$.
-	\:Odstranìní nìkterých hran \dots $\O(m + n)$.
-	\:Odstranìní \uv{slepých ulièek} pomocí fronty -- ka¾dou hranu odstraníme nejvý¹e jedenkrát, ka¾dý vrchol se~dostane do~fronty nejvý¹e jedenkrát \dots $\O(m + n)$.
-	\endlist
-\:Najití blokujícího toku~$g$ \dots $\O(m \cdot n)$.
-	\numlist{\ndotted}
-	\:Inicializace toku~$g$ \dots $\O(m)$.
-	\:Najití orientované cesty v~proèi¹tìné síti rezerv (staèí vzít libovolnou cestu ze~zdroje, nebo» ka¾dá z~nich v~této síti vede do~stoku) \dots $\O(n)$.
-	\:Výbìr minima z~výrazu $r(e) - g(e)$ pøes v¹echny hrany cesty -- ta mù¾e být dlouhá nejvý¹e~$n$ \dots $\O(n)$.
-	\:Pøepoèítání v¹ech hran cesty \dots $\O(n)$.
-	\:Smazání hran cesty, jejich¾ tok~$g(e)$ se~zvý¹il na~hodnotu~$r(e)$ \dots $\O(n)$.
-	\:Doèi¹»ování vyøe¹me zvlá¹».
-	\endlist
-	
-	Vnitøní cyklus (kroky 2 a¾ 6) provedeme nejvý¹e~$m$ krát, nebo» pøi~ka¾dém prùchodu vyma¾eme alespoò jednu hranu (tak jsme si~volili~$\varepsilon$).
-	
-	Èi¹tìní bìhem celého hledání blokujícího toku~$g$ v~proèi¹tìné síti rezerv trvá dohromady $\O(m + n)$, nebo» ka¾dou hranu a~vrchol sma¾eme nejvý¹e jedenkrát.
-	
-	Najití blokujícího toku bude tedy trvat $\O(m \cdot n + (m + n)) = \O(m \cdot n)$.
-	
-\:Zlep¹ení toku~$f$ pomocí toku~$g$ \dots $\O(m)$.
-\:Skok na~2. krok \dots $\O(1)$.
-\endlist
-
-Zbývá nám jen urèit, kolikrát projdeme vnìj¹ím cyklem (fází). Doká¾eme si~lemma, ¾e hodnota~$l$ vzroste mezi prùchody vnìj¹ím cyklem (fázemi) alespoò o~1. Z~toho plyne, ¾e vnìj¹ím cyklem mù¾eme projít nejvý¹e $n$-krát, nebo» cesta v síti na~$n$ vrcholech mù¾e být dlouhá nejvý¹e $n$.
-
-Uvìdomme si, ¾e uvnitø vnìj¹ího cyklu pøevládá èlen $\O(m \cdot n)$, tak¾e celková èasová slo¾itost bude $\O(n^2 \cdot m)$.
-
-\s{Lemma:} Hodnota~$l$ (délka nejkrat¹í cesty ze~$z$ do~$s$ v~proèi¹tìné síti) vzroste mezi fázemi alespoò o~1.
-
-\proof
-
-Podíváme se~na~prùbìh jednoho prùchodu vnìj¹ím cyklem. Délku aktuálnì nejkrat¹í cesty ze~zdroje do~stoku oznaème~$l$. V¹echny pùvodní cesty délky~$l$ se~bìhem prùchodu zaruèenì nasytí, proto¾e tok~$g$ je blokující. Musíme v¹ak dokázat, ¾e nemohou vzniknout ¾ádné nové cesty délky~$l$ nebo men¹í. V~síti rezerv toti¾ mohou hrany nejen
-ubývat, ale i~pøibývat: pokud po¹leme tok po~hranì, po~které je¹tì nic neteklo, tak v~protismìru z~dosud nulové rezervy vyrobíme nenulovou. Rozmysleme si~tedy, jaké hrany mohou pøibývat.
-
-Hrany mohou pøibývat jen tehdy, kdy¾ jsme po~opaèné hranì nìco poslali. Ale my nìco posíláme po~hranách pouze z~vrstvy do~té následující. Hrany tedy pøibývají do~minulé vrstvy.
-
-Vznikem nových hran by proto mohly vzniknout nové cesty ze~zdroje do~stoku, které pou¾ívají zpìtné hrany. Jen¾e cesta ze~zdroje do~stoku, která pou¾ije zpìtnou hranu, musí alespoò jednou skoèit o~vrstvu zpìt a~nikdy nemù¾e skoèit o~více ne¾ jednu vrstvu dopøedu, a~proto je její délka alespoò $l+2$. Pokud cesta novou zpìtnou hranu nepou¾ije, má buï délku~$> l$, co¾ je v~poøádku, nebo má délku~$= l$, pak je zablokovaná.
-
-Tím je lemma dokázáno.
-\qed
-
-\figure{dinic-cestashranouzpet.eps}{Cesta u¾ívající novou zpìtnou hranu}{0.4\hsize}
-
-V¹echna dokázaná tvrzení mù¾eme shrnout do~následující vìty:
-
-\s{Vìta:} Dinicùv algoritmus najde maximální tok v~èase $\O(n^2\cdot m)$.
-
-\s{Poznámka:} Algoritmus se~chová hezky na~sítích s~malými celoèíselnými kapacitami, ale kupodivu i~na~rùzných jiných sítích. Èasto se~pou¾ívá, nebo» se~chová efektivnì. A~je mnoho zpùsobù, jak ho je¹tì vylep¹ovat, èi odhadovat ni¾¹í slo¾itost na~speciálních sítích.
-
-\s{Poznámka:} Algoritmus nevy¾aduje racionální kapacity! Dal¹í z~dùvodù, proè maximální tok existuje i~v~síti s~iracionálními kapacitami.
-
-
-%k,s,v,na,do,ke,pro,pøi,a,u,i,po,
-
-\bye
diff --git a/2-dinic/Makefile b/2-dinic/Makefile
deleted file mode 100644
index c183fb0..0000000
--- a/2-dinic/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=2-dinic
-
-include ../Makerules
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diff --git a/2-toky/2-toky.tex b/2-toky/2-toky.tex
new file mode 100644
index 0000000..126d81c
--- /dev/null
+++ b/2-toky/2-toky.tex
@@ -0,0 +1,269 @@
+\input lecnotes.tex
+
+\prednaska{1}{Toky v sítích}{(zapsala Markéta Popelová)}
+
+\s{První motivaèní úloha:} {\I Rozvod èajovodu do~v¹ech uèeben.}
+\smallskip
+
+Pøedstavme si, ¾e~by v~budovì fakulty na~Malé Stranì existoval èajovod, který
+by rozvádìl èaj do~ka¾dé uèebny. Znázornìme si to orientovaným grafem, kde by
+jeden významný vrchol pøedstavoval èajovar a~druhý uèebnu, ve~které sedíme.
+Hrany mezi vrcholy by pøedstavovaly vìtvící se trubky, které mají èaj rozvádìt.
+Jak rozvést co nejefektivnìji dostatek èaje do~dané uèebny?
+
+\figure{toky01.eps}{Èajovod}{2in}
+
+\s{Druhá motivaèní úloha:} {\I Pøenos dat.}
+\smallskip
+
+Jiným pøíkladem mù¾e být poèítaèová sí» na~pøenos dat, která se sestává z~pøenosových linek
+spojených pomocí routerù. Data se sice obvykle pøená¹ejí po~paketech, ale to
+mù¾eme pøi dne¹ních rychlostech pøenosu zanedbat a pova¾ovat data za spojitá.
+Jak pøená¹et data mezi dvìma poèítaèi v~síti co nejrychleji?
+
+\s{Definice:} {\I Sí»} je uspoøádaná pìtice $(V,E,z,s,c)$, pro ní¾ platí:
+\itemize\ibull
+\:$(V,E)$ je orientovaný graf.
+\:$c:E\to{\bb R}_{0}^{+}$ je {\I kapacita} hran.
+\:$z,s \in V$ jsou dva vrcholy grafu, kterým øíkáme {\I zdroj} a~{\I stok} (spotøebiè).
+\:Graf je symetrický, tedy $\forall u,v \in V: uv \in E \Leftrightarrow vu \in E$ (tuto podmínku si~mù¾eme zvolit bez~újmy na~obecnosti, nebo» v¾dy mù¾eme do~grafu pøidat hranu, která v~nìm je¹tì nebyla, a~dát jí nulovou kapacitu).
+\endlist
+
+\figure{sit.eps}{Pøíklad sítì. Èísla pøedstavují kapacity jednotlivých hran.}{2.5in}
+
+\s{Definice:} {\I Tok} je funkce $f:E \to {\bb R}_{0}^{+}$ taková, ¾e~platí:
+\numlist{\ndotted}
+\:Tok po~ka¾dé hranì je omezen její kapacitou: $\forall e \in E : f(e)\le c(e)$.
+\:Kirchhoffùv zákon: $$\forall v \in V \setminus \{z,s\}: \sum_{u: uv \in E}{f(uv)}=\sum_{u: vu \in E}{f(vu)}.$$ Neboli pro~ka¾dý vrchol kromì zdroje a~stoku platí, ¾e~to, co do~nìj pøitéká, je stejnì velké jako to, co z~nìj odtéká (\uv{sí» tìsní}).
+\endlist
+
+\s{Definice:} Pro libovolnou funkci $f:E \to {\bb R}$ se nám bude hodit následující znaèení:
+\itemize\ibull
+\:$f^+(v) =  \sum_{u: uv \in E}{f(uv)}$ (celkový pøítok do vrcholu)
+\:$f^-(v) =  \sum_{u: vu \in E}{f(vu)}$ (celkový odtok)
+\:$f^\Delta(v) = f^+(v) - f^-(v)$ (pøebytek ve~vrcholu)
+\endlist
+
+\>(Kirchhoffùv zákon pak øíká prostì to, ¾e $f^\Delta(v)=0$ pro v¹echna $v\ne z,s$.)
+
+\figure{tok.eps}{Pøíklad toku. Èísla pøedstavují toky po~hranách, v~závorkách jsou kapacity.}{4in}
+
+\s{Pozorování:} Nìjaký tok v¾dy existuje. V libovolné síti mù¾eme v¾dy zvolit
+konstantnì nulovou funkci (po~¾ádné hranì nic nepoteèe). To je korektní tok,
+ale sotva u¾iteèný. Budeme chtít najít tok, který pøepraví co nejvíce tekutiny
+ze~zdroje do~spotøebièe.
+
+\s{Definice:} {\I Velikost toku} $f$ budeme znaèit $\vert f\vert$ a polo¾íme ji
+rovnou rozdílu souètu velikostí toku na~hranách vedoucích do~$s$ a~souètu velikostí
+toku na~hranách vedoucích z~$s$. Neboli $\vert f\vert:=f^\Delta(s).$
+
+\s{Pozorování:} Jeliko¾ sí» tìsní, mìlo by být jedno, zda velikost toku mìøíme
+u~spotøebièe nebo u~zdroje. Vskutku, krátkým výpoètem ovìøíme, ¾e tomu tak je:
+$$
+f^\Delta(z) - f^\Delta(s) = \sum_v f^\Delta(v) = 0.
+$$
+První rovnost platí proto, ¾e podle Kirchhoffova zákona jsou zdroj a spotøebiè jediné
+dva vrcholy, jejich¾ pøebytek mù¾e být nenulový. Druhou rovnost získáme tak, ¾e si
+uvìdomíme, ¾e tok na ka¾dé hranì pøispìje do celkové sumy jednou s~kladným znaménkem
+a jednou se záporným. Zjistili jsme tedy, ¾e pøebytek zdroje a spotøebièe se li¹í
+pouze znaménkem.
+\qed
+
+\s{Poznámka:} Rádi bychom nalezli v~zadané síti tok, jeho¾ velikost je maximální.
+Máme ale zaruèeno, ¾e maximum bude existovat? V¹ech mo¾ných tokù je nekoneènì mnoho
+a v~nekoneèné mno¾inì se mù¾e snadno stát, ¾e aèkoliv existuje supremum, není maximem
+(pøíklad: $\{1-1/n \mid n\in{\bb N}^+\}$).
+Odpovìï nám poskytne matematická analýza: mno¾ina v¹ech tokù je kompaktní podmno¾inou
+prostoru ${\bb R}^{\vert E\vert}$, velikost toku je spojitá (dokonce lineární) funkce
+z~této mno¾iny do~$\bb R$, tak¾e musí nabývat minima i maxima.
+
+Nám ale bude staèít studovat sítì s~racionálními kapacitami, kde existence maximálního
+toku bude zjevná u¾ z~toho, ze sestrojíme algoritmus, který takový tok najde.
+
+\s{První pokus:} Hledejme cestu $P$ ze~$z$ do~$s$ takovou, ¾e~$\forall e \in
+P: f(e) < c(e)$ (po~v¹ech jejích hranách teèe ostøe ménì, ne¾ jim dovolují
+jejich kapacity). Pak zjevnì mù¾eme tok upravit tak, aby se~jeho velikost
+zvìt¹ila. Zvolme $$\varepsilon := \min_{e \in P} (c(e) - f(e)).$$ Nový tok $f'$
+pak definujme jako $f'(e):=f(e) + \varepsilon$. Kapacity nepøekroèíme ($\varepsilon$
+je nejvìt¹í mo¾ná hodnota, abychom tok zvìt¹ili, ale nepøekroèili kapacitu ani
+jedné z~hran cesty $P$) a~Kirchhoffovy zákony zùstanou neporu¹eny, nebo» zdroj
+a~stok neomezují a~ka¾dému jinému vrcholu na~cestì $P$ se~pøítok $f^+(v)$
+i~odtok $f^-(v)$ zvìt¹í pøesnì o~$\varepsilon$.
+
+Opakujme tento proces tak dlouho, dokud existují zlep¹ující cesty. A¾ se algoritmus
+zastaví (co¾ by obecnì nemusel, ale nás je¹tì chvíli trápit nemusí), získáme maximální tok?
+Pøekvapivì nemusíme. Napø. na~obrázku je vidìt, ¾e~kdy¾ najdeme nejdøíve cestu
+pøes hranu s~kapacitou 1 (na obrázku tuènì) a~u¾ hodnotu toku na~této hranì
+nesní¾íme, tak dosáhneme velikost toku nejvý¹e 19. Ale maximální tok této sítì
+má velikost 20.
+
+\figure{toky02.eps}{Èísla pøedstavují kapacity jednotlivých hran.}{1.5in}
+
+Zde by ov¹em situaci zachránilo, kdybychom poslali tok velikosti 1 proti smìru
+prostøední hrany -- to mù¾eme udìlat tøeba odeètením jednièky od toku po smìru
+hrany. Roz¹íøíme tedy ná¹ algoritmus tak, aby umìl posílat tok i proti smìru
+hran. O~kolik mù¾eme tok hranou zlep¹it (a» u¾ pøiètením po~smìru nebo odeètením
+proti smìru) nám bude øíkat její {\I rezerva:}
+
+\s{Definice:} {\I Rezerva hrany} $uv$ je $r(uv):=c(uv) - f(uv) + f(vu).$
+
+\smallskip
+Algoritmus bude vypadat následovnì. Postupnì dok¾eme, ¾e je koneèný a ¾e v~ka¾dé
+síti najde maximální tok.
+
+\s{Algoritmus (Fordùv-Fulkersonùv)}
+
+\algo
+\:$f \leftarrow$ libovolný tok, napø. v¹ude nulový.
+\:Dokud $\exists P$ cesta ze $z$ do $s$ taková, ¾e~$\forall e \in P: r(e) > 0$, opakujeme:
+\::$\varepsilon \leftarrow \min \{r(e) \mid e \in P\}$.
+\::Pro v¹echny hrany $uv \in P$:
+\:::$\delta \leftarrow \min \{f(vu),\varepsilon\}$
+\:::$f(vu) \leftarrow f(vu) - \delta$
+\:::$f(uv) \leftarrow f(uv) + \varepsilon - \delta$
+\:Prohlásíme $f$ za~maximální tok.
+\endalgo
+
+\s{Koneènost:} Zastaví se~Fordùv-Fulkersonùv algoritmus?
+
+\itemize\ibull
+
+\:Pro~celoèíselné kapacity se~v~ka¾dém kroku zvìt¹í velikost toku alespoò o~1.
+Algoritmus se~tedy zastaví po~nejvíce tolika krocích, kolik je nìjaká horní
+závora pro~velikost maximálního toku -- napø. souèet kapacit v¹ech hran
+vedoucích do~stoku (tedy $c^+(s)$).
+
+\:Pro~racionální kapacity vyu¾ijeme jednoduchý trik. Nech» $M$ je nejmen¹í
+spoleèný násobek jmenovatelù v¹ech kapacit. Spustíme-li algoritmus na sí»
+s~kapacitami $c'(e) = c(e)\cdot M$, bude se rozhodovat stejnì jako v~pùvodní
+síti, proto¾e bude stále platit $f'(e) = f(e)\cdot M$. Nová sí» je pøitom
+celoèíselná, tak¾e se algoritmus jistì zastaví.
+
+\:Na~síti s~iracionálními kapacitami se~algoritmus chová mnohdy divoce, nemusí
+se~zastavit, ba ani konvergovat ke~správnému výsledku. (Zkuste vymyslet pøíklad
+takové sítì.)
+
+\endlist
+
+\s{Maximalita:} Kdy¾ se algoritmus zastaví, je tok~$f$ maximální? K~tomu se
+bude hodit zavést øezy.
+
+\s{Definice:} {\I Øez} je uspoøádaná dvojice mno¾in vrcholù ($A,B$) taková, ¾e
+$A$ a $B$ jsou disjunktní, pokrývají v¹echny vrcholy, $A$ obsahuje zdroj a $B$
+obsahuje stok. Neboli $A \cap B = \emptyset$, $A \cup B = V$, $z \in A$, $s \in B$.
+
+\>Ka¾dému øezu pøirozenì pøiøadíme mno¾iny hran:
+\itemize\ibull
+\:$E^+(A,B) = E \cap (A\times B)$ (hrany \uv{zleva doprava})
+\:$E^-(A,B) = E \cap (B\times A)$ (hrany \uv{zprava doleva})
+\:$E^\Delta(A,B) = E^+(A,B) \cup E^-(A,B)$ (v¹echny hrany øezu)
+\endlist
+
+\>Také pro libovolnou funkci $f: E\rightarrow {\bb R}$ zavedeme:
+\itemize\ibull
+\:$f^+(A,B) = \sum_{e\in E^+(A,B)} f(e)$ (prùtok pøes øez zleva doprava)
+\:$f^-(A,B) = \sum_{e\in E^-(A,B)} f(e)$ (prùtok zprava doleva)
+\:$f^\Delta(A,B) = f^+(A,B) - f^-(A,B)$ (èistý prùtok)
+\endlist
+
+\>{\I Kapacita øezu} budeme øíkat souètu kapacit hran zleva doprava, tedy $c+(A,B)$.
+
+\s{Poznámka:} Øezy se~dají definovat více zpùsoby, jedna z~definic je, ¾e~øez
+je mno¾ina hran grafu takových, ¾e~po~jejich odebrání se~graf rozpadne na~více
+komponent. Tuto vlastnost mají i na¹e øezy, ale opaènì to nemusí platit.
+
+\s{Lemma:} Pro ka¾dý øez $(A,B)$ a ka¾dý tok~$f$ platí, ¾e $f^\Delta(A,B)
+= \vert f\vert$. (Jinými slovy velikost toku mù¾eme mìøit na libovolném øezu,
+nejen na triviálních øezech kolem zdroje nebo kolem spotøebièe.)
+
+\proof
+Opìt ¹ikovným seètením pøebytkù vrcholù:
+$$
+f^\Delta(A,B) = \sum_{v\in B} f^\Delta(v) = f^\Delta(s).
+$$
+První rovnost získáme poèítáním pøes hrany: ka¾dá hrana vedoucí z~vrcholu v~$B$
+do~jiného vrcholu v~$B$ pøispìje jednou kladnì a jednou zápornì; hrany le¾ící
+celé mimo~$B$ nepøispìjí vùbec; hrany s~jedním koncem v~$B$ a druhým mimo pøispìjí
+jednou, pøièem¾ znaménko se bude li¹it podle toho, který konec je v~$B$. Druhá
+rovnost je snadná: v¹echny vrcholy v~$B$ mimo spotøebièe mají podle Kirchhoffova
+zákona nulový pøebytek.
+\qed
+
+\s{Dùsledek:} Pro ka¾dý tok~$f$ a ka¾dý øez $(A,B)$ platí $\vert f \vert \le c^+(A,B)$.
+(Velikost ka¾dého toku je shora omezena kapacitou ka¾dého øezu.)
+
+\proof
+$f^\Delta(A,B) = f^+(A,B) - f^-(A,B) \le f^+(A,B) \le c^+(A,B)$.
+\qed
+
+\s{Dùsledek:} Pokud $\vert f\vert = c^+(A,B)$, pak je tok~$f$ maximální a øez~$(A,B)$
+minimální. Jinými slovy pokud najdeme dvojici tok a stejnì velký øez, mù¾eme øez pou¾ít
+jako certifikát maximality toku. Následující vìta nám zaruèí, ¾e je to mo¾né v¾dy:
+
+\s{Vìta:} Pokud se~Fordùv-Fulkersonùv algoritmus zastaví, tak vydá maximální tok.
+
+\proof
+Nech» se~Fordùv-Fulkersonùv algoritmus zastaví. Definujme mno¾inu vrcholù $A
+:= \{v \in V \mid \hbox{existuje cesta ze~$z$ do~$v$ jdoucí po~hranách s~$r
+> 0$}\}$ a~$B := V \setminus A$.
+
+Dvojice $(A,B)$ je øez, nebo» $z \in A$ (ze~$z$ do~$z$ existuje cesta délky 0)
+a~$s \in B$ (kdyby $s \not\in B$, tak by musela existovat cesta ze~$z$ do~$s$
+s~kladnou rezervou, tudí¾ by algoritmus neskonèil, nýbr¾ tuto cestu vzal
+a~stávající tok vylep¹il).
+
+Dále víme, ¾e~v¹echny hrany øezu mají nulovou rezervu, èili $\forall uv \in
+E^+(A,B) : r(uv) = 0$ (kdyby mìla hrana $uv$ rezervu nenulovou, tedy kladnou,
+tak by vrchol $v$ patøil do~$A$). Proto po~v¹ech hranách øezu vedoucích z~$A$
+do~$B$ teèe tolik, kolik jsou kapacity tìchto hran, a~po~hranách vedoucích
+z~$B$ do~$A$ neteèe nic, tedy $f(uv) = c(uv)$ a $f(vu) = 0$. Máme øez $(A,B)$
+takový, ¾e~$f^\Delta(A,B) = c^+(A,B)$. To znamená, ¾e~jsme na¹li maximální tok
+a~minimální øez. \qed
+
+Dokázali jsme tedy následující:
+
+\s{Vìta:} Pro~sí» s~racionálními kapacitami se~Fordùv-Fulkersonùv algoritmus
+zastaví a~vydá maximální tok a~minimální øez.
+
+\s{Vìta:} Sí» s~celoèíselnými kapacitami má aspoò jeden z~maximálních tokù
+celoèíselný a~Fordùv-Fulkersonùv algoritmus takový tok najde.
+
+\proof
+Kdy¾ dostane Fordùv-Fulkersonùv algoritmus celoèíselnou sí», tak najde maximální tok a~ten bude zase celoèíselný (algoritmus nikde nedìlí).
+\qed
+
+To, ¾e~umíme najít celoèíselné øe¹ení není úplnì samozøejmé. (U~jiných problémù takové ¹tìstí mít nebudeme.) Uka¾me si rovnou jednu aplikaci, která právì celoèíselný tok vyu¾ije.
+
+\s{Aplikace:} Hledání nejvìt¹ího párování v~bipartitních grafech.
+
+\s{Definice:} Mno¾ina hran $F \subseteq E$ se~nazývá {\I párování}, jestli¾e
+¾ádné dvì hrany této mno¾iny nemají spoleèný ani jeden vrchol. Neboli $\forall
+e,f \in F : e \cap f = \emptyset$. {\I Velikostí} párování myslíme poèet jeho
+hran.
+
+\s{Øe¹ení:}
+Mìjme bipartitní graf $G = (V,E)$. V~nìm hledáme nejvìt¹í párování. Sestrojme
+si~sí» takovou, ¾e~vezmeme vrcholy $V$ grafu $G$ a~pøidáme k~nim dva speciální
+vrcholy $z$ (zdroj) a~$s$ (stok) a~ze~zdroje pøidáme hrany do~v¹ech vrcholù
+levé partity a~ze~v¹ech vrcholù pravé partity povedeme hrany do~stoku. V¹echny
+kapacity nastavme na~1. Hrany bipartitního grafu zorientujme z levé partity do
+pravé. Nyní staèí jen na~tuto sí» spustit Fordùv-Fulkersonùv algoritmus (nebo
+libovolný jiný algoritmus, který najde maximální celoèíselný tok) a~a¾~dobìhne,
+tak prohlásit hrany s~tokem 1 za~maximální párování.
+
+\figure{toky04.eps}{Hledání maximálního párování v~bipartitním grafu.}{2in}
+
+Existuje toti¾ bijekce mezi párováním a~celoèíselnými toky pøi~zachování
+velikosti. Z ka¾dého celoèíselného toku na~vý¹e zmínìném grafu (viz obrázek) lze sestrojit
+párování o~stejné velikosti (velikost toku zde odpovídá poètu hran bipartitního
+grafu, po~kterých poteèe 1) a~naopak. Dùle¾ité je si uvìdomit, ¾e~definice toku
+(omezení toku kapacitou a~Kirchhoffovy zákony) nám zaruèují, ¾e~hrany
+s~nenulovým tokem (tedy jednièkovým) budou tvoøit párování (nestane se, ¾e~by
+dvì hrany zaèínaly nebo konèily ve~stejném vrcholu, nebo» by se~nutnì poru¹ila
+jedna ze~dvou podmínek definice toku). Potom i~maximální tok bude odpovídat
+maximálnímu párování a~naopak.
+
+V~bipartitním grafu najdeme maximální párování v~èase $\O(n \cdot (m+n))$. Fordùv-Fulkersonùv algoritmus stráví jednou iterací èas $\O(m+n)$ (za~prohledání do~¹íøky) a~pøi~jednotkových kapacitách bude iterací nejvý¹e~$n$, proto¾e ka¾dou se~tok zvìt¹í alespoò o~1 a v¹echny toky jsou omezené øezem kolem zdroje, který má kapacitu nejvý¹e~$n$. Výsledná èasová slo¾tost hledání maximálního párování bude tedy $\O(n \cdot (m+n))$.
+
+
+\bye
diff --git a/2-toky/Makefile b/2-toky/Makefile
new file mode 100644
index 0000000..e4e9173
--- /dev/null
+++ b/2-toky/Makefile
@@ -0,0 +1,3 @@
+P=2-toky
+
+include ../Makerules
diff --git a/2-toky/sit.eps b/2-toky/sit.eps
new file mode 100644
index 0000000..f66f5e1
--- /dev/null
+++ b/2-toky/sit.eps
@@ -0,0 +1,828 @@
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+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
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+%%CreationDate: 1991 Aug 20 16:38:05
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
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+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ]
+ipeMakeFont
+%%BeginResource: font OKRINM+CMMI12
+%!PS-AdobeFont-1.1: CMMI12 1.100
+%%CreationDate: 1996 Jul 27 08:57:55
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
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+end readonly def
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+/Encoding 256 array
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+dup 25 /pi put
+readonly def
+/FontBBox{-30 -250 1026 750}readonly def
+currentdict end
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+/G { setgray } def
+/rg { setrgbcolor } def
+/RG { setrgbcolor } def
+/S { stroke } def
+/f* { eofill } def
+/f { fill } def
+/ipeMakeFont {
+  exch findfont
+  dup length dict begin
+    { 1 index /FID ne { def } { pop pop } ifelse } forall
+    /Encoding exch def
+    currentdict
+  end
+  definefont pop
+} def
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+/Tf { dup /ipeFontSize exch store selectfont } def
+/Td { translate } def
+/BT { gsave } def
+/ET { grestore } def
+/TJ { 0 0 moveto { dup type /stringtype eq
+ { show } { ipeFontSize mul -0.001 mul 0 rmoveto } ifelse
+} forall } def
+end
+%%EndResource
+%%EndProlog
+%%BeginSetup
+ipe begin
+%%BeginResource: font IEYHEQ+CMR10
+%!PS-AdobeFont-1.1: CMR10 1.00B
+%%CreationDate: 1992 Feb 19 19:54:52
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.00B) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR10) readonly def
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
+/ItalicAngle 0 def
+/isFixedPitch false def
+end readonly def
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+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/Encoding 256 array
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+dup 50 /two put
+dup 55 /seven put
+dup 57 /nine put
+readonly def
+/FontBBox{-251 -250 1009 969}readonly def
+currentdict end
+currentfile eexec
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+%%EndResource
+/F8 /IEYHEQ+CMR10
+[ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
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+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
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+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ]
+ipeMakeFont
+%%BeginResource: font OLBANC+CMR12
+%!PS-AdobeFont-1.1: CMR12 1.0
+%%CreationDate: 1991 Aug 20 16:38:05
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
+/FontInfo 7 dict dup begin
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+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
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+end readonly def
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+dup 40 /parenleft put
+dup 41 /parenright put
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+dup 52 /four put
+dup 53 /five put
+dup 54 /six put
+dup 56 /eight put
+dup 57 /nine put
+readonly def
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+currentdict end
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+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ]
+ipeMakeFont
+%%BeginResource: font BYMWUK+CMMI12
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+%%CreationDate: 1996 Jul 27 08:57:55
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
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+ipeMakeFont
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+%%CreationDate: 1991 Aug 15 07:20:57
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
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+ipeMakeFont
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+%%CreationDate: 1991 Aug 20 16:38:24
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+11 dict begin
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diff --git a/3-goldberg/3-goldberg.tex b/3-goldberg/3-goldberg.tex
deleted file mode 100644
index 70544d4..0000000
--- a/3-goldberg/3-goldberg.tex
+++ /dev/null
@@ -1,283 +0,0 @@
-\input lecnotes.tex
-
-\prednaska{3}{Goldbergùv algoritmus}{(zapsala Markéta Popelová)}
-
-Pøedstavíme si~nový algoritmus pro~hledání maximálního toku v~síti, který se~uká¾e být stejnì dobrý jako {\I Dinicùv algoritmus} ($\O(MN^{2})$) a~po~nìkolika vylep¹eních bude i~lep¹í. Nejdøíve si~pøipomeòme definice, které budeme potøebovat:
-
-\s{Definice:} Mìjme sí» $S=(V,E,z,s,c)$, tok~$f$ a~libovolný vrchol~$v$. Pak $f^{\Delta}(v)$ nazýváme {\I pøebytek} ve~vrcholu~$v$ a~definujeme ho takto: $$f^{\Delta}(v):=\sum_{uv \in E}{f(uv)} - \sum_{vu \in E}{f(vu)}.$$ Pøebytek ve~vrcholu~$v$ je tedy souèet v¹eho, co do~vrcholu~$v$ pøiteèe, minus souèet v¹eho, co z~$v$ odteèe.
-
-\s{Definice:} Dále pro~libovolnou hranu~$uv \in E$ definujeme její {\I rezervu} následovnì:
-$$r(uv) = c(uv) - f(uv) + f(vu).$$ Rezerva hrany znaèí, co je¹tì je mo¾no po~této hranì poslat.
-
-\s{Poznámka:} Dále budeme oznaèovat písmenem~$N$ poèet vrcholù a~$M$ poèet hran, tedy~$N = \vert V \vert$ a~$M = \vert E \vert$.
-
-Goldbergùv algoritmus na~rozdíl od~Dinicova algoritmu zaèíná s~ohodnocením hran, které pravdìpodobnì není tokem (budeme ho nazývat {\I vlna}), a~postupnì ho zmen¹uje a¾ na~korektní tok.
-
-\s{Definice:} Funkce $f:E \rightarrow {\bb R}_{0}^{+}$ je {\I vlna} v~síti~$(V, E, z, s, c)$ tehdy, kdy¾ jsou splnìny následující dvì podmínky:
-	\numlist\ndotted
-	\:$\forall e \in E : f(e) \leq c(e)$ (vlna na hranì nepøekroèí kapacitu hrany)
-	\:$ \forall v \in V \setminus \{z, s\} : f^{\Delta}(v) \geq 0$ (pøebytek ve vrcholu je nezáporný).
-	\endlist
-
-\s{Pozorování:} Ka¾dý tok~$f$ je také vlna, ale opaènì to obvykle platit nemusí.
-
-\s{Operace:} {\I Pøevedení pøebytku}
-
-Algoritmus bude potøebovat pøevádìt pøebytky z~vrcholu~$u$ na~sousední vrchol~$v$. Mìjme hranu~$uv$ s~kladnou rezervou $r(uv) > 0$ a~kladným pøebytkem ve~vrcholu~$u$: $f^\Delta(u) > 0$. Èást pøebytku budeme chtít poslat z~vrcholu~$u$ do~vrcholu~$v$. Vezmeme $\delta := \min (f^\Delta(u), r(uv))$ a~po~hranì~$uv$ po¹leme tok o velikosti~$\delta$. Výsledná situace bude vypadat následovnì:
-	\itemize\ibull
-	\:$f'^\Delta(u) = f^\Delta(u) - \delta$.
-	\:$f'^\Delta(v) = f^\Delta(v) + \delta$.
-	\:$r'(uv) = r(uv) - \delta$.
-	\:$r'(vu) = r(vu) + \delta$.
-	\endlist
-	
-Kdybychom ov¹em nepøidali ¾ádnou jinou podmínku, ná¹ algoritmus by se~mohl krásnì zacyklit (napø. posílat pøebytek z~$u$ do~$v$ a~zase zpátky). Abychom se~tomu vyhnuli, zavedeme {\I vý¹ku vrcholu} $h: V \to {\bb N}$ a~dovolíme pøevádìt pøebytek pouze z~vy¹¹ího vrcholu~$u$ na~ni¾¹í $v$: $h(u) > h(v)$.
-
-\s{Shrnutí:} Podmínky pro~pøevedení pøebytku po~hranì $uv \in E$:
-	\numlist\ndotted
-	\:Ve~vrcholu~$u$ je nenulový pøebytek: $f^{\Delta}(u) > 0$.
-	\:Vrchol~$u$ je vý¹ ne¾ vrchol~$v$: $h(u) > h(v)$.
-	\:Hrana~$uv$ má nenulovou rezervu: $r(uv)>0$.
-	\endlist
-
-
-\s{Operace:} Pro~vrchol~$u \in V$ definujme {\I zvednutí vrcholu}:
-Pokud bìhem výpoètu narazíme ve~vrcholu~$u$ na~pøebytek, který nelze nikam pøevést, zvìt¹íme vý¹ku vrcholu~$u$ o~jednièku, tj. $h(u) \leftarrow h(u)+1$.
-
-
-\s{Algoritmus (Goldbergùv)}
-
-\algo
-\:$\forall v \in V: h(v)\leftarrow 0$ (v¹em vrcholùm nastavíme poèáteèní vý¹ku nula) a~$h(z)\leftarrow N$ (zdroj zvedneme do~vý¹ky~$N$).
-\:$\forall e \in E: f(e)\leftarrow 0$ (po~hranách nejdøíve nenecháme protékat nic) a~$\forall zu \in E : f(zu)\leftarrow c(zu)$ (ze~zdroje pustíme maximální mo¾nou vlnu).
-\:Dokud $\exists u \in V \setminus \{z,s\}: f^{\Delta}(u)>0$:
-\::Pokud $\exists v \in V: uv \in E,~r(uv)>0$ a~$h(u)>h(v)$, pak pøevedeme pøebytek po~hranì z~$u$ do~$v$.
-\::V~opaèném pøípadì zvedneme $u$:~$h(u) \leftarrow h(u) + 1$.
-\:Vrátíme tok~$f$ jako výsledek.
-\endalgo
-
-\noindent
-Nyní bude následovat nìkolik lemmat a~invariantù, jimi¾ doká¾eme správnost a~èasovou slo¾itost Goldbergova algoritmu.
-
-\s{Invariant A (základní):}
-	\numlist \ndotted
-	\:Funkce~$f$ je v~ka¾dém kroku algoritmu vlna.
-	\:$h(v)$ nikdy neklesá pro~¾ádné~$v$.
-	\:$h(z)=N$ a~$h(s)=0$ po~celou dobu bìhu algoritmu.
-	\endlist
-
-\proof Indukcí dle poètu prùchodù cyklem (3. -- 5. krok algoritmu).
-
-Na zaèátku je v¹e v~poøádku ($f$ je nulová funkce, pøebytky v¹ech vrcholù jsou nezáporné, tedy~$f$ je vlna, $h(z)=N$ a~$h(s)=0$). V~prùbìhu se~tyto hodnoty mìní pouze pøi:
-	\itemize\ibull
-	\:Pøevedení po~hranì~$uv$: Po hranì~$uv$ se~nepo¹le více ne¾ její rezerva. Pøebytek~$u$ se~sní¾í, ale nejménì na~nulu. Pøebytek~$v$ se~zvý¹í. Tedy~$f$ zùstává vlnou. Vý¹ky se~nemìní.
-	\:Zvednutí vrcholu~$u$: Mìní pouze vý¹ky -- a~to vrcholù rùzných od zdroje èi stoku -- a~pouze se zvy¹ují.
-	\qeditem
-	\endlist
-
-\s{Invariant S (o~Spádu):} Neexistuje hrana $uv \in E: r(uv)>0$ \& $h(u) > h(v)+1$ (s~kladnou rezervou a~spádem vìt¹ím ne¾ jedna).
-
-\proof Indukcí dle bìhu algoritmu.
-
-Na zaèátku mají v¹echny hrany ze~zdroje rezervu nulovou a~v¹echny ostatní vedou mezi vrcholy s~vý¹kou 0. V~prùbìhu by se~tento invariant mohl pokazit pouze dvìma zpùsoby:
-	\itemize\ibull
-	\:Zvednutím vrcholu~$u$, ze~kterého vede hrana~$uv$ s~kladnou rezervou a~spádem 1. Tento pøípad nemù¾e nastat, nebo» hranu zvedáme pouze tehdy, kdy¾ neexistuje vrchol~$v$ takový, ¾e hrana~$uv$ má kladnou rezervu a~spád alespoò 1. Takový vrchol v~na¹em pøípadì existuje, proto se~místo zvednutí vrcholu~$u$ po¹le pøebytek po~hranì~$uv$.
-	\:Zvìt¹ením rezervy hrany se~spádem vìt¹ím ne¾ 1. Toto také nemù¾e nastat, nebo» rezervu bychom mohli zvìt¹it jedinì tak, ¾e bychom poslali nìco v~protismìru -- a~to nesmíme, jeliko¾ bychom poslali pøebytek z~ni¾¹ího vrcholu na~vy¹¹í.
-	\qeditem
-	\endlist
-
-\s{Definice:} Cestu~$P$ nazveme {\I nenasycenou}, pokud v¹echny její hrany mají kladnou rezervu. Neboli $\forall e \in P: r(e) > 0$.
-
-\s{Lemma K (o~Korektnosti):} Kdy¾ se~algoritmus zastaví, je~$f$ maximální tok.
-
-\proof Dùkaz rozlo¾me do~dvou krokù. Nejdøíve uká¾eme, ¾e~$f$ je tok, a~pak jeho maximalitu.
-
-	\numlist\ndotted
-	\:Nech» se~algoritmus zastavil. Pak nemohl existovat ¾ádný vrchol~$v$ (kromì zdroje a~stoku) s~kladným pøebytkem. Tedy $\forall v \in V~\setminus \{z,s\}: f^\Delta(v) = 0$. (Víme ji¾, ¾e~$f$ je po~celou dobu vlna, tak¾e pøebytek nemù¾e být nikdy záporný.) V~tom pøípadì splòuje~$f$ podmínky toku.
-	\:Pro spor pøedpokládejme, ¾e tok~$f$ není maximální. Pak existuje nenasycená cesta ze~zdroje do~stoku. Vezmìme si~libovolnou takovou cestu. Zdroj je stále ve~vý¹ce~$N$ a~spotøebiè ve~vý¹ce 0 (viz invariant A). Tato cesta tedy pøekonává vý¹ku~$N$, ale mù¾e mít nejvý¹e~$N-1$ hran. Proto existuje alespoò jedna hrana se~spádem alespoò 2. Tato hrana tedy nemù¾e mít kladnou rezervu (viz invariant S). Tato cesta proto nemù¾e být zlep¹ující, co¾ je spor. Tím jsme dokázali, ¾e~$f$ je nutnì maximální tok.
-	\qeditem
-	\endlist
-	
-\s{Lemma C (Cesta):} Mìjme vrchol $v \in V$. Pokud $f^{\Delta}(v) > 0$, pak existuje nenasycená cesta z~vrcholu~$v$ do~zdroje.
-
-\proof
-Pro vrchol~$v \in V$ s $f^{\Delta}(v) > 0$ definujme mno¾inu $A := \{ u \in V : \exists$ nenasycená cesta z~$v$ do~$u \}$.
-
-Seètìme pøebytky ve~v¹ech vrcholech mno¾iny~$A$. Pøebytek ka¾dého vrcholu se~spoèítá jako souèet tokù do~nìj vstupujících minus souèet tokù z~nìj vystupujících. V¹echny hrany, jejich¾ oba vrcholy le¾í v~$A$, se~jednou pøiètou a~jednou odeètou. Proto nás budou zajímat pouze hrany mezi~$A$ a~$V \setminus A$.
-
- $$\sum_{u \in A}f^{\Delta}(u) = \underbrace{ \sum_{ab \in E \cap ( (V \setminus A) \times A )} f(ab) }\limits_{=0} -  \underbrace{ \sum_{ab \in E \cap (  A \times (V \setminus A) )} f(ab) }\limits_{\geq 0}~\leq~0.$$
-
-Uka¾me si, proè je první svorka rovna nule. Mìjme vrcholy $a \in V \setminus A$ a~$b \in A$ takové, ¾e $ab\in E$. O~nich víme, ¾e $r(ba) = 0$ (jinak by~$a$ patøilo do~$A$) $\Rightarrow f(ba) = c(ba) \Rightarrow f(ab)=0$. Proto do~$A$ nic nepøitéká.
-
-\figure{Goldberg01.eps}{Obrázek k dùkazu lemmatu C}{0.2\hsize}
-
-Proè je druhá svorka nezáporná, je zøejmé, nebo» tok na~hranì je v¾dy nezáporný a~souèet nezáporných èísel je nezáporné èíslo.
-
-Proto $\sum_{u \in A}{f^\Delta(u) \le 0}$. Zároveò v¹ak v~$A$ je aspoò jeden vrchol s~kladným pøebytkem, toti¾~$v$, proto v~$A$ musí být také vrchol se~záporným pøebytkem -- a~jediný takový je zdroj. Tím je dokázáno, ¾e $z \in A$, tedy ¾e vede nenasycená cesta z~vrcholu~$v$ do~zdroje.
-\qed
-
-\s{Invariant V (Vý¹ka):} $\forall v \in V$ platí $h(v)\leq 2N$.
-
-\proof
-Kdyby existoval vrchol~$v$ s~vý¹kou $h(v) > 2N$, tak by musel být nìkdy zvednut z~vý¹ky~$2N$. Tehdy musel mít kladný pøebytek $f^\Delta(v)>0$ (jinak by nemohl být zvednut). Dle lemmatu C musela existovat nenasycená cesta z~$v$ do~zdroje. Tato cesta mìla spád alespoò~$N$, ale mohla mít nejvý¹e~$N-1$ hran (jinak by to nebyla cesta v~síti na~$N$ vrcholech). Tudí¾ musela na~této cestì existovat hrana se~spádem alespoò 2, co¾ je spor s~invariantem S (nebo» v¹echny hrany této cesty mají z~definice nenasycené cesty kladné rezervy).
-\qed
-
-\s{Lemma Z (poèet Zvednutí):} Poèet v¹ech zvednutí je maximálnì~$2N^{2}$.
-
-\proof
-Staèí si~uvìdomit, ¾e ka¾dý vrchol mù¾eme zvednout maximálnì~$2N$-krát a~vrcholù je~$N$.
-\qed
-
-Teï nám je¹tì zbývá urèit poèet provedených pøevedení. Bude se~nám hodit, kdy¾ pøevedení rozdìlíme na~dva druhy:
-
-\s{Definice:} Øekneme, ¾e pøevedení je {\I nasycené}, pokud po~pøevodu rezerva na~hranì~$uv$ klesla na~nulu, tedy $r(uv)=0$. V~opaèném pøípadì je {\I nenasycené}, a~tehdy urèitì klesne pøebytek ve~vrcholu~$u$ na~nulu, tedy $f^{\Delta}(u) = 0$ (pøi~nasyceném pøevedení se~to~ale mù¾e stát také).
-
-\s{Lemma S (naSycená pøevedení):} Poèet v¹ech nasycených pøevedení je nejvý¹~$NM$.
-
-\proof
-Pro ka¾dou hranu~$uv$ spoèítejme poèet nasycených pøevedení (tedy takových pøevedení, ¾e po~nich klesne rezerva hrany na~nulu). Abychom dvakrát nasycenì pøevedli pøebytek (nebo jeho èást) z~vrcholu~$u$ do~vrcholu~$v$, tak jsme museli~$u$ mezitím alespoò dvakrát zvednout:
-
-Po~prvním nasyceném pøevedení z~vrcholu~$u$ do~vrcholu~$v$ se~vynulovala rezerva hrany~$uv$. Uvìdomme si, ¾e pøi~této operaci muselo být~$u$ vý¹e ne¾~$v$, a~dokonce víme, ¾e bylo vý¹e pøesnì o~1 (viz lemma~S). Po~této hranì tedy nemù¾eme u¾~nic více pøevést. Aby do¹lo k~druhému nasycenému pøevedení z~$u$ do~$v$, musíme nejprve opìt zvý¹it rezervu hrany~$uv$. Jediný zpùsob, jak toho lze dosáhnout, je pøevést èást pøebytku z~$v$ zpátky do~$u$. K~tomu se~musí~$v$ dostat (alespoò o~1) vý¹e ne¾~$u$. Po~pøelití bude rezerva~$uv$ opìt kladná. A~abychom provedli nasycené pøevedení znovu ve~smìru z~$u$ do~$v$, musíme zase~$u$ dostat (alespoò o~1) vý¹e ne¾~$v$. Proto musíme~$u$ alespoò o~2 zvednout -- nejprve na~úroveò~$v$ a~pak je¹tì o~1 vý¹e.
-
-
-Ukázali jsme si~tedy, ¾e mezi ka¾dými dvìma nasycenými pøevedeními jsme vrchol~$u$ zvedli alespoò dvakrát. Nicménì libovolnou hranu mù¾eme zvednout nejvý¹e~$2N$-krát (viz invariant V). V¹ech hran je~$M$, tudí¾ poèet v¹ech nasycených pøevedení je nejvý¹e~$NM$.
-\qed
-
-\s{Lemma N (Nenasycená pøevedení):} Poèet v¹ech nenasycených pøevedení je~$\O(N^2M)$.
-
-\proof
-Dùkaz provedeme pomocí potenciálové metody -- nadefinujme si~následující funkci jako potenciál:
- $$ \Phi := \sum_{\scriptstyle{v: f^{\Delta}(v) > 0} \atop \scriptstyle{v \ne z,s}} h(v). $$
-Nyní se~podívejme, jak se~ná¹ potenciál bìhem algoritmu vyvíjí a~jaké má vlastnosti:
-
-	\itemize\ibull
-	\:Na poèátku je $ \Phi = 0 $.
-	\:Bìhem celého algoritmu je $ \Phi \ge 0 $, nebo» je souètem nezáporných èlenù.
-	\:Zvednutí vrcholu zvý¹í $\Phi$ o~jednièku (Aby byl vrchol zvednut, musel mít kladný pøebytek $\Rightarrow$ vrchol do~sumy ji¾ pøispíval, teï jen pøispìje èíslem o 1 vy¹¹ím.). Ji¾ víme, ¾e za~celý prùbìh algoritmu je v¹ech zvednutí maximálnì~$2N^2$, proto zvedáním vrcholù zvý¹íme potenciál dohromady nejvý¹e o~$2N^2$.
-	\:Nasycené pøevedení zvý¹í~$\Phi$ nejvý¹e o~$2N$, proto¾e buï po~pøevodu hranou~$uv$ v~$u$ zùstal nìjaký pøebytek, tak¾e se~mohl potenciál zvý¹it nejvý¹e o~$h(v) \leq 2N$, nebo je pøebytek v~$u$ po~pøevodu nulový a~potenciál se~dokonce o~jedna sní¾il. Za~celý prùbìh tak dojde k~maximálnì~$NM$ takovýmto pøevedením, díky nim¾ se~potenciál zvý¹í maximálnì o~$2N^2M$.
-	\:Koneènì kdy¾ pøevádíme po~hranì~$uv$ nenasycenì, tak od~potenciálu urèitì odeèteme vý¹ku vrcholu~$u$ (nebo» se~vynuluje pøebytek ve~vrcholu~$u$) a~mo¾ná pøièteme vý¹ku vrcholu~$v$. Jen¾e $h(v) = h(u) - 1$, a~proto nenasycené pøevedení potenciál v¾dy sní¾í alespoò o~jedna.
-	\endlist
-
-\>Z~tohoto rozboru chování potenciálu~$\Phi$ v~prùbìhu algoritmu získáváme, ¾e poèet v¹ech nenasycených pøevedení mù¾e být nejvý¹e $2N^2 + 2N^2M$, co¾ je $\O(N^2M)$.
-\qed
-
-\s{Implementace:}
-
-Budeme si~pamatovat seznam~$P$ v¹ech vrcholù~$v \ne z,s$ s~kladným pøebytkem. Neboli
-$$P = \{ v \in V \setminus \{z,s\} ~\vert~ f^{\Delta}(v) > 0 \}.$$
-Kdy¾ mìníme pøebytek nìjakého vrcholu, mù¾eme ná¹ seznam v~konstantním èase aktualizovat (napø. tak, ¾e si~ka¾dý vrchol pamatuje pozici, na~které v~seznamu~$P$ je). V~konstantním èase také umíme odpovìdìt, zda existuje nìjaký vrchol s~pøebytkem.
-
-Dále si~pro ka¾dý vrchol~$u \in V$ budeme pamatovat~$L(u)$-seznam hran~$uv \in E$ takových, které vedou dolù (mají spád alespoò 1) a~kladnou rezervu. Neboli
-$$L(u) = \{ uv \in E ~\vert~ v \in V,~ r(uv) > 0,~ h(v) < h(u)\}.$$
-Díky tomu mù¾eme pøistupovat k~patøièným sousedùm~$u$ v~èase $\O(1)$, stejnì jako pøidávat hrany do~$L(u)$, resp. je mazat. Opìt ka¾dá hrana si~bude pamatovat pozici, na~které se~nachází v~seznamu~$L$.
-
-\s{Rozbor èasové slo¾itosti algoritmu:}
-
-\numlist\ndotted
-\:Inicializace vý¹ek \dots\ $\O(N)$.
-\:Inicializace vlny~$f$ \dots\ $\O(M)$.
-\:Výbìr vrcholu~$u$ s~kladným pøebytkem -- vezmeme první vrchol v~$P$ \dots\ $\O(1)$.
-\:Výbìr vrcholu~$v$, do~kterého vede z~$u$ hrana s~kladnou rezervou a~který je ní¾e ne¾~$u$ -- vezmeme první hranu z~$L(u)$ \dots\ $\O(1)$.
-	
-	Pøevedení pøebytku: \dots\ $\O(1)$.
-		\itemize\idot
-		\:Nasycené pøevedení \dots\ $\O(1)$.
-			\itemize\idot
-			\:Rezerva hrany~$uv$ klesne na~nulu $\Rightarrow$ hrana~$uv$ vypadne z~$L(u)$ \dots\ $\O(1)$.
-			\:Pøebytek vrcholu~$v$ se~zvý¹í $\Rightarrow$ pokud je¹tì nebyl v~seznamu~$P$, tak se~tam pøidá \dots\ $\O(1)$.
-			\:Pøebytek vrcholu~$u$ mo¾ná také klesne na~nulu $\Rightarrow$ pak by vrchol~$u$ vypadnul z~$P$ \dots\ $\O(1)$.
-			\endlist
-		\:Nenasycené pøevedení \dots\ $\O(1)$.
-			\itemize\idot
-			\:Rezerva hrany~$uv$ zùstane nezáporná $\Rightarrow$ hrana~$uv$ zùstane v~$L(u)$ \dots\ $\O(1)$.
-			\:Vynuluje se~pøebytek vrcholu~$u$~$\Rightarrow$ vrchol $u$ vypadne z~$P$ \dots~$\O(1)$.
-			\:Pøebytek vrcholu~$v$ se~zvý¹í~$\Rightarrow$ pokud je¹tì nebyl v~seznamu~$P$, tak se~tam pøidá \dots\ $\O(1)$.
-			\endlist
-		\endlist
-\:Zvednutí vrcholu~$u$ \dots $\O(N)$.
-
-Musíme obejít v¹echny hrany do~$u$ a~z~$u$, kterých je nejvý¹e~$2N-2$, porovnat
-vý¹ky a~pøípadnì tyto hrany~$uv$ odebrat ze~seznamu~$L(v)$ resp. pøidat
-do~$L(u)$. Abychom pro~odebrání hrany~$uv$ ze~seznamu~$L(v)$ nemuseli procházet
-celý seznam, budeme si~$\forall u \in V$ pamatovat je¹tì $L^{-1}(u) := $ seznam
-ukazatelù na~hrany~$uv$ v~seznamech~$L(v)$.
-
-\endlist
-
-Vidíme, ¾e ka¾dé zvednutí je sice drahé, ale je jich zase pomìrnì málo. Naopak pøevádìní pøebytkù je èastá operace, tak¾e je výhodné, ¾e trvá konstantní èas.
-
-\s{Shrnutí:}
-
-\itemize\ibull
-\:V¹ech zvednutí je $\O(N^2)$ (viz lemma Z), ka¾dé trvá $\O(N)$ \dots\ $\O(N^3).$
-\:V¹ech nasycených pøevedení je $\O(NM)$ (viz lemma S), ka¾dé trvá $\O(1)$ \dots\ $\O(NM).$
-\:V¹ech nenasycených pøevedení je $\O(N^2M)$ (viz lemma N), ka¾dé trvá $\O(1)$ \dots\ $\O(N^2M).$
-\endlist
-
-Dohromady má tedy Goldbergùv algoritmus èasovou slo¾itost $\O(N^2M)$. Vidíme, ¾e u¾ v~tomto obecném pøípadì to není hor¹í ne¾ Dinicùv algoritmus. Pøí¹tì si~uká¾eme, ¾e mù¾e mít i~mnohem lep¹í. Nejdøíve ale zformulujme v¹echna dokázaná tvrzení do~následující vìty:
-
-\s{Vìta:} Goldbergùv algoritmus najde maximální tok v~èase $\O(N^2M)$.
-
-\s{Pozorování:} Pokud bychom volili v¾dy nejvy¹¹í z~vrcholù s~pøebytkem, tak by se~mohl algoritmus chovat lépe. Podívejme se~na~to pozornìji a~vylep¹ený Goldebrgùv algoritmus oznaème G'.
-
-\s{Algoritmus (Vylep¹ený Goldbergùv algoritmus)}
-
-\algo
-\:$\forall v \in V: h(v)\leftarrow 0$ (v¹em vrcholùm nastavíme poèáteèní vý¹ku nula) a~$h(z)\leftarrow N$ (zdroj zvedneme do~vý¹ky~$N$).
-\:$\forall e \in E: f(e)\leftarrow 0$ (po~hranách nejdøíve nenecháme protékat nic) a~$\forall zu \in E : f(zu)\leftarrow c(zu)$ (ze~zdroje pustíme maximální mo¾nou vlnu).
-\:Dokud $\exists u \in V \setminus \{z,s\}: f^{\Delta}(u)>0$:
-\::Vybereme z~vrcholù s~pøebytkem ten s~nejvy¹¹í vý¹kou, oznaèíme ho~$u$.
-\:::Pokud $\exists v \in V: uv \in E,~r(uv)>0$ a~$h(u)>h(v)$, pak pøevedeme pøebytek po~hranì z~$u$ do~$v$.
-\:::V~opaèném pøípadì zvedneme $u$:~$h(u) \leftarrow h(u) + 1$.
-\:Vrátíme tok~$f$ jako výsledek.
-\endalgo
-
-Rozmysleme si, o~kolik bude vylep¹ený algoritmus G' lep¹í ne¾ ten pùvodní. Ten pùvodní mìl èasovou slo¾itost $\O(N^2M)$ a~pøevládal èlen, který odpovídal nenasyceným pøevedením. Zkusme tedy právì poèet nenasycených pøevedení odhadnout ve~vylep¹eném algoritmu o~nìco tìsnìji.
-
-\s{Lemma N' (Nenasycená pøevedení):} Algoritmus G' provede~$\O(N^3)$ nenasycených pøevedení.
-
-\proof
-Dokazovat budeme opìt pomocí potenciálové metody. Zadefinujme si~potenciál {\I nejvy¹¹í hladinu s~pøebytkem}:
-$$H := \max \{ h(v) \mid v \neq z,s ~\&~ f^\Delta(v) > 0\}.$$
-Rozdìlíme bìh algoritmu na~{\I fáze}. Ka¾dá fáze konèí tím, ¾e~se~$H$ zmìní. Jak se~mù¾e zmìnit? Buï se~$H$ zvý¹í, co¾ znamená, ¾e~nìjaký vrchol s~pøebytkem v~nejvy¹¹í hladinì byl o~1 zvednut, nebo se~$H$ sní¾í. My víme, ¾e zvednutí je v~celém algoritmu $\O(N^2)$. Zároveò si~mù¾eme uvìdomit, ¾e~$H$ je nezáporný potenciál, kdy sní¾ení i~zvý¹ení ho zmìní o~1, tedy poèet sní¾ení bude stejný jako poèet zvý¹ení, a~proto obojího je~$\O(N^2)$. Tudí¾ poèet fází je také~$\O(N^2)$.
-
-Je dùle¾ité, ¾e~bìhem jedné fáze provedeme nejvý¹e jedno nenasycené pøevedení z~ka¾dého vrcholu. Po~ka¾dém nenasyceném pøevedení po~hranì $uv$ se~toti¾ vynuluje pøebytek v~$u$ a~aby se~provedlo dal¹í nenasycené pøevedení z~vrcholu~$u$, muselo by nejdøíve být co~pøevádìt. Muselo by tedy do~$u$ nìco pøitéci. My ale víme, ¾e pøevádíme pouze shora dolù a~$u$ je v~nejvy¹¹í hladinì (to zajistí právì to vylep¹ení algoritmu), tedy nejdøíve by musel být nìjaký jiný vrchol zvednut. Tím by se~ale zmìnilo~$H$ a~skonèila by tato fáze.
-
-Proto poèet v¹ech nenasycených pøevedení bìhem jedné fáze je nejvý¹e~$N$. A ji¾ jsme dokázali, ¾e~fází je~$\O(N^2)$. Tedy poèet v¹ech nenasycených pøevedení je~$\O(N^3)$.
-\qed
-
-Tento odhad je hezký, ale stále není tìsný a~algoritmus se~chová lépe. Doka¾me si~je¹tì jeden tìsnìj¹í odhad na~poèet nenasycených pøevedení.
-
-\s{Lemma N'' (Nenasycená pøevedení):} Poèet nenasycených pøevedení je~$\O(N^2 \sqrt{M})$.
-
-\s{Poznámka:} Tato èasová slo¾itost je výhodná napøíklad pro~øídké grafy. Ty mají toti¾ pomìrnì malý poèet hran.
-
-\proof
-Rozdìlme si~fáze na~dva druhy: laciné a~drahé podle toho, kolik se~v~nich provede nenasycených pøevedení. Zvolme si~nìjaké nezáporné~$K$. Zatím nebudeme urèovat jeho hodnotu. Uvidíme, ¾e~èasová slo¾itost algoritmu bude závislá na~tomto parametru~$K$. Proto jeho hodnotu zvolíme a¾ pozdìji a~to tak, aby byla slo¾itost co nejni¾¹í.
-
-{\I Laciné fáze} budou ty, bìhem nich¾ se~provede nejvý¹e~$K$ nenasycených pøevedení. {\I Drahé fáze} budou ty ostatní, tedy takové, bìhem nich¾ se~provede více jak~$K$ nenasycených pøevedení.
-
-Teï potøebujeme odhadnout, kolik nás budou stát oba typy fází. Zaènìme s~tìmi jednodu¹¹ími -- s~lacinými. Víme, ¾e~v¹ech fází je~$\O(N^2)$. Tìch laciných bude tedy urèitì také~$O(N^2)$. Nenasycených pøevedení se~bìhem jedné laciné fáze provede nejvíce~$K$. Tedy celkem se~bìhem laciných fází provede~$\O(N^2K)$ nenasycených pøevedení.
-
-Pro~poèet nenasycených pøevedení v~drahých fázích si~zaveïme nový potenciál definovaný následovnì:
-$$\Phi := \sum_{\scriptstyle{v \ne z,s} \atop \scriptstyle{f^{\Delta}(v) \ne 0}} {p(v) \over K},$$
-kde~$p(v)$ je poèet takových vrcholù~$u$, které nejsou vý¹e ne¾~$v$. Neboli
-$$p(v) = \vert \{ u \in V \mid h(u) \leq h(v) \} \vert.$$
-Tedy platí, ¾e~$p(v)$ je v¾dy nezáporné a~nejvý¹e má hodnotu~$N$. Dále víme, ¾e~$\Phi$ bude v¾dy nezáporné (nebo» je to souèet nezáporných èlenù) a~nejvý¹e bude nabývat hodnoty~$N^2 \over K$. Rozmysleme si, jak nám ovlivní tento potenciál na¹e tøi operace:
-\itemize\ibull
-\:{\bf Zvednutí}: Za~ka¾dý zvednutý vrchol pøibude nejvý¹e~$N \over K$ (tento vrchol mù¾e být nadzvednut nejvý¹e nad~v¹echny ostatní vrcholy) a~mo¾ná nìco ubude (napø. kdy¾ vrchol vyzvedneme na~úroveò k~ostatním).
-\:{\bf Nasycené pøevedení} po~hranì $uv$: Mù¾e vynulovat pøebytek ve~vrcholu~$u$, pak se~$\Phi$ sní¾í. Mù¾e zvý¹it pøebytek ve~$v$ z~nuly, pak se~$\Phi$ zvý¹í. Ale nejvý¹e se~zvý¹í o~$N \over K$, nebo» do~$\Phi$ pøibude jen jeden sèítanec za~vrchol $v$ a~ten pøispìje nejvý¹e hodnotou~$N \over K$ (pod ním mù¾e být nejvíce~$N$ vrcholù).
-\:{\bf Nenasycená pøevedení} po~hranì $uv$ v~drahých fázích: Tato operace vynuluje pøebytek v~$u$, tedy~$\Phi$ klesne alespoò o~$p(u) \over K$. Zároveò mù¾e zvý¹it pøebytek ve~$v$ z~nuly, ale~$\Phi$ stoupne nejvý¹e o~$p(v) \over K$. Celkem tedy~$\Phi$ klesne alespoò o~$p(u) - p(v) \over K$.
-\endlist
-Uvìdomme si, ¾e~pokud pøevádíme po~hranì~$uv$, tak platí, ¾e~$h(u) = h(v) + 1$. Pak~$p(u) - p(v)$ je pøesnì poèet vrcholù na~hladinì~$H$. Tìch je alespoò tolik, kolik je nenasycených pøevedení bìhem jedné fáze (to jsme dokázali ji¾ v~lemmatu N'), a~my jsme si~zadefinovali, ¾e v~drahé fázi je poèet nenasycených pøevedení alespoò~$K$. Tedy~$p(u) - p(v) > K$. Proto bìhem jednoho nenasyceného pøevedení~$\Phi$ klesne alespoò o~${K \over K} = 1$. Nenasycená pøevedení potenciál nezvy¹ují.
-
-Potenciál~$\Phi$ se~mù¾e zvìt¹it pouze pøi~operacích zvednutí a~nasycené pøevedení. Zvednutí se~provede celkem~$\O(N^2)$ a~ka¾dé zvý¹í potenciál nejvý¹e o~$N \over K$. Nasycených pøevedení se provede celkem~$\O(NM)$ a~ka¾dé zvý¹í potenciál takté¾ nejvý¹e o~$N \over K$. Celkem se~tedy~$\Phi$ zvý¹í nejvý¹e o
-$${N \over K} \O(N^2) + {N \over K} \O(NM) = \O \left({N^3 \over K} + {N^2M \over K}\right).$$
-
-Teï vyu¾ijeme toho, ¾e~$\Phi$ je nezáporný potenciál, tedy kdy¾ ka¾dé nenasycené pøevdení v~drahé fázi sní¾í~$\Phi$ alespoò o~1, tak v¹ech nenasycených pøevdení v~drahých fázích je~$\O({N^3 \over K} + {N^2M \over K})$. U¾ jsme ukázali, ¾e~nenasycených pøevední v~laciných fázích je~$\O(N^2K)$. Proto celkem v¹ech nenasycených pøevedení je
-$$\O \left(N^2K + {N^3 \over K} + {N^2M \over K} \right) = \O \left(N^2K + {N^2M \over K} \right)$$
-(nebo» pro~souvislé grafy platí, ¾e~$M \geq N \Rightarrow N^2M \geq N^3$). A~my chceme, aby jich bylo co nejménì. Tato funkce má minimum tehdy, kdy¾ $N^2K = {N^2M \over K}$, èili $K = \sqrt{M}$.
-
-Proto v¹ech nenasycených pøevedení je	$\O(N^2\sqrt{M})$.
-\qed
-\bye
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diff --git a/3-goldberg/Makefile b/3-goldberg/Makefile
deleted file mode 100644
index 6330e4c..0000000
--- a/3-goldberg/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=3-goldberg
-
-include ../Makerules
diff --git a/4-hradla/4-hradla.tex b/4-hradla/4-hradla.tex
deleted file mode 100644
index ccb0c50..0000000
--- a/4-hradla/4-hradla.tex
+++ /dev/null
@@ -1,139 +0,0 @@
-\input ../lecnotes.tex
-
-\prednaska{4}{Hradlové sítì}{(zapsal: Petr Jankovský)}
-
-\def\land{\mathbin{\&}}
-
-Výkon poèítaèù nelze zvy¹ovat donekoneèna a pøesto¾e ji¾ pìkných pár let platí,
-¾e se jejich rychlost s~èasem exponenciálnì zvìt¹uje, jednou urèitì narazíme
-pøinejmen¹ím na~fyzikální limity.
-
-Kdy¾ tedy nemù¾eme zvy¹ovat rychlost jednoho procesoru, jak poèítat rychleji?
-Øe¹ením by mohlo být poøídit si procesorù víc. U¾~dnes na~bì¾ném PéCéèku máme
-k~dispozici vícejádrové procesory, díky nim¾ mù¾eme vyu¾ít pararelní poèítání
-a úlohu øe¹it tak, ¾e práci ¹ikovnì rozdìlíme mezi procesory (èi jádra) a
-zamìstnáme je pøi~výpoètu v¹echny.
-
-My se podíváme na~abstraktní výpoèetní model, který je je¹tì paralelnìj¹í.
-Techniky, které si uká¾eme na~tomto modelu, se v¹ak dají pøekvapivì vyu¾ít i~pøi~reálném
-paralelizování na~nìkolika málo procesorech. Konec koncù i proto, ¾e vnitøní
-architektura procesoru se na¹emu modelu velmi podobá. Budeme se zabývat 
-jednoduchým modelem paralelního poèítaèe, toti¾ hradlovou sítí.
-
-\h{Hradlové sítì}
-
-\s{Definice:} {\I Hradlo} je prvek, který umí vyhodnocovat nìjakou funkci
-nad~koneènou abecedou $\Sigma$.
-
-Obecnì se na~hradlo díváme jako na~funkci $f: {\Sigma}^{k} \rightarrow \Sigma$, která dostane $k$ vstupù
-a~vrátí jeden výstup, pøièem¾ hodnoty, nad~kterými pracuje, budou z~nìjaké koneèné
-abecedy -- tedy z~nìjaké koneèné mno¾iny symbolù $\Sigma$. Písmenku $k$ zde øíkáme {\I arita
-hradla}.
-
-\s{Pøíklad:} Èasto studujeme hradla booleovská (pracující nad abecedou $\{0,1\}$), která poèítají logické funkce.
-
-Z~nich nejèastìji potkáme:
-
-\itemize\ibull
-\:nulární: to jsou konstanty (FALSE=0, TRUE=1),
-\:unární: napø. negace (znaèíme~$\lnot$),
-\:binární: logický souèin ({\sc and},~$\land$), souèet ({\sc or},~$\lor$), ...
-\endlist
-
-\>Hradla kreslíme tøeba následovnì:
-
-\figure{hradlo_and.eps}{Binární hradlo provádící logickou operaci {\sc and}.}{1in}
-
-Jednotlivá hradla mù¾eme navzájem urèitým zpùsobem propojovat a vytváøet
-z nich {\I hradlové sítì}. Pokud pou¾íváme pouze booleovská hradla, øíkáme takto vytvoøeným
-sítím {\I booleovské obvody}. Pokud pracujeme s~operacemi nad nìjakou obecnìj¹í (ale koneènou) 
-mno¾inou symbolù (abecedou), nazývají se {\I kombinaèní obvody.}
-
-Ka¾dá hradlová sí» má nìjaké vstupy, nìjaké výstupy a uvnitø jsou propojovaná
-hradla. Ka¾dý vstup hradla je pøipojen buïto na~nìkterý ze~vstupù sítì nebo
-na~výstup jiného hradla. Výstupy hradel mohou být propojeny na~vstupy dal¹ích
-hradel (mohou se vìtvit), nebo na výstupy sítì. Pøitom máme zakázáno vytváøet
-cykly. 
-
-Ne¾ si øekneme formální definici, podívejme se na obrázek.
-
-\figure{hradlova_sit.eps}{Hradlová sí» -- tøívstupová verze funkce {\I majorita}.}{3in}
-
-Obrazek znázoròuje hradlovou sí», která poèítá, zda je alespoò na~dvou ze~vstupù
-jednièka. Pojïme si ale {\I hradlovou sí»} definovat formálnì.
-
-\s{Definice:} {\I Hradlová sí»} je urèena:
-\itemize\ibull
-\:{\I abecedou} $\Sigma$ (to je nìjaká koneèná mno¾ina symbolù, obvykle $\Sigma=\{0,1\}$);
-\:po dvou disjunktními koneènými mno¾inami $I$~({\I vstupy}), $O$~({\I výstupy}) \hfil\break a~$H$~({\I hradla});
-\:acyklickým orientovaným multigrafem~$(V,E)$, kde~$V = I \cup O \cup H$;
-\:zobrazením~$F$, které ka¾dému hradlu $h\in H$ pøiøadí nìjakou funkci~$F(h):
-  \Sigma^{a(h)} \rightarrow \Sigma$, co¾ je funkce, kterou toto hradlo vykonává.
-  Èíslu $a(h)$ øíkáme {\I arita} hradla~$h$;
-\:zobrazením~$z: E \rightarrow {\bb N}$, které ka¾dé hranì vedoucí do~nìjakého
-  hradla pøiøazuje nìkterý ze vstupù tohoto hradla.
-\endlist
-
-\>Pøitom jsou splnìny následující podmínky:
-
-\itemize\ibull
-\:$\forall i \in I: \deg^{in}(i)=0$ (do~vstupù nic nevede);
-\:$\forall o \in O: \deg^{in}(o)=1~\land~\deg^{out}(o)=0$ (z~výstupù nic nevede a do~ka¾dého vede právì jedna hrana);
-\:$\forall h \in H: \deg^{in}(v)=a(v)$ (do~ka¾dého hradla vede tolik hran, kolik je jeho arita);
-\:$\forall h \in H~\forall j: 1\le j\le a(h)$ existuje právì jedena hrana~$e$ taková, ¾e~$e$ konèí v~$h$ a~$z(e)=j$, 
-  (v¹echny vstupy hradel jsou zapojeny).
-\endlist
-
-\s{Pozorování:} Kdybychom pøipustili hradla s~libovolnì vysokým poètem vstupù, mohli
-bychom libovolný problém se vstupem délky~$n$ vyøe¹it jedním hradlem o~$n$~vstupech,
-kterému bychom pøiøadili funkci, která by na¹i úlohu rovnou vyøe¹ila. Tento model
-v¹ak není ani realistický, ani pìkný. Proto pøijmìme omezení, ¾e~arity v¹ech hradel
-budou omezeny nìjakou pevnou konstantou $k$ (uká¾e se, ¾e nám bude staèit dvojka
-a~vystaèíme si tedy pouze s nulárními, unárními a binárními hradly). 
-Následující obrázky ukazují, jak hradla o~více vstupech nahradit dvouvstupovými:
-
-\twofigures{hradlo_ternor.eps}{Trojvstupové hradlo \sc or.}{0.5in}{hradlo_ternbior.eps}{Jeho nahrazení 2-vstupovými hradly.}{0.6in}
-
-
-Nyní bychom je¹tì mìli definovat, co taková hradlová sí» vlastnì poèítá a~jak
-její výpoèet probíhá.
-
-\s{Definice:} {\I Výpoèet sítì} probíhá po~{\I taktech.} V nultém taktu jsou definovány pouze 
-hodnoty na~vstupech sítì a na~výstupech hradel arity 0. Mù¾eme si to pøedstavit
-tak, ¾e na~zaèátku nemá ¾ádné hradlo definovánu výstupní hodnotu (a¾ na ji¾ 
-zmínìná hradla nulární). V~ka¾dém dal¹ím taktu pak vydají výstup hradla, která
-na~konci minulého taktu mìla definovány v¹echny hodnoty na vstupech. Jakmile
-budou po~nìjakém koneèném poètu taktù definované i hodnoty v¹ech výstupù, sí»
-se zastaví a~vydá výsledek.
-
-\s{Pozorování:} Proto¾e je sí» acyklická, je jasné, ¾e jakmile jednou nìjaké hradlo vydá
-výstup, tak se tento výstup bìhem dal¹ího výpoètu sítì ji¾ nezmìní.
-
-\figure{vypocet_site.eps}{Výpoèet hradlové sítì.}{6cm}
-
-\>Podle toho, jak sí» poèítá, si ji mù¾eme rozdìlit na~vrstvy:
-
-\s{Definice:} {\I $i$-tá vrstva $S_i$} obsahuje právì takové vrcholy~$v$, pro~které nejdel¹í cesta ze~vstupù
-sítì do $v$ má délku rovnou~$i$.
-
-\s{Pozorování:} V¹imnìme si, ¾e v $i$-tém taktu vydají hodnoty právì hradla z $S_i$.
- 
-Dává tedy smysl prohlásit za~{\I èasovou slo¾itost} sítì poèet
-jejích vrstev. Podobnì {\I prostorovou slo¾itost} definujeme jako poèet hradel v~síti.
-
-\s{Pøíklad:} Sestrojme sí», která zjistí, zda se mezi jejími~$n$ vstupy
-vyskytuje alespoò jedna jednièka.
-
-\>{\I První øe¹ení:} zapojíme hradla za~sebe (sériovì). Èasová i prostorová
-slo¾itost odpovídají~$\Theta(n)$. Zde ov¹em vùbec nevyu¾íváme toho, ¾e by mohlo poèítat více
-hradel souèasnì.
-
-\figure{hloupy_or.eps}{Hradlová sí», která zjistí, zdali je na vstupu alespoò jedna jednièka.}{0.7in}
-
-\>{\I Druhé øe¹ení:} Hradla budeme spojovat do~dvojic, pak výsledky z~tìchto dvojic opìt
-do~dvojic a tak dále. Díky paralelnímu zapojení dosáhneme èasové slo¾itosti $\Theta(\log n)$,
-prostorová slo¾itost zùstane lineární.
-
-\figure{chytry_or.eps}{Chytøej¹í øe¹ení stejného problému pro vstup velikosti 16.}{3in}
-
-\bye
diff --git a/4-hradla/Makefile b/4-hradla/Makefile
deleted file mode 100644
index 65a25ae..0000000
--- a/4-hradla/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=4-hradla
-
-include ../Makerules
diff --git a/4-hradla/chytry_or.eps b/4-hradla/chytry_or.eps
deleted file mode 100644
index 3d88cff..0000000
--- a/4-hradla/chytry_or.eps
+++ /dev/null
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diff --git a/5-addsort/1_9_deleni_bloku.eps b/5-addsort/1_9_deleni_bloku.eps
deleted file mode 100644
index 2c96b1c..0000000
--- a/5-addsort/1_9_deleni_bloku.eps
+++ /dev/null
@@ -1,3225 +0,0 @@
-%!PS-Adobe-3.0 EPSF-3.0
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-%%BeginResource: procset MayuraDraw_ops
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-% width height matrix proc key cache
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-\input ../lecnotes.tex
-
-\prednaska{5}{Paralelní sèítání, bitonické tøídìní}{(zapsal: Petr Jankovský)}
-
-Minule jsme si zavedli paralelní výpoèetní model, ve kterém si nyní nìco naprogramujeme \dots
-
-\h{Sèítání binárních èísel}
-
-Mìjme dvì èísla $x$ a $y$ zapsané ve~dvojkové soustavì. Jejich èíslice oznaème
-$x_{n-1}\ldots x_0$ a $y_{n-1}\ldots y_0$, kde $i$-tý øád má váhu $2^i$. Nyní budeme chtít tato èísla
-seèíst.
-
-K tomuto úèelu se~ihned nabízí pou¾ít starý dobrý \uv{¹kolní algoritmus}, který
-funguje ve~dvojkové soustavì stejnì dobøe jako v~desítkové. Zkrátka sèítáme èísla
-zprava doleva. V¾dy seèteme pøíslu¹né èíslice pod~sebou a~pøièteme pøenos z~ni¾¹ího
-øádu. Tím dostaneme jednu èíslici výsledku a~pøenos do~vy¹¹ího øádu. Formálnì
-bychom to mohli zapsat tøeba takto:
-$$
-z_i=x_i \oplus y_i \oplus c_i,
-$$
-kde $z_i$ je $i$-tá èíslice souètu, $\oplus$ znaèí operaci {\sc xor} (souèet modulo~2) a~$c_i$ je {\I pøenos}
-z~$(i-1)$-ního øádu do~$i$-tého. Pøenos pøitom nastane tehdy, pokud se~nám potkají
-dvì jednièky pod~sebou, nebo kdy¾ se~vyskytne alespoò jedna jednièka a~k~tomu
-pøenos z~ni¾¹ího øádu. Jinými slovy tehdy, kdy¾ ze~tøí xorovaných èíslic jsou alespoò
-dvì jednièky (pomocí obvodu pro majoritu z minulé pøedná¹ky lehce zkonstruujeme):
-$$
-\eqalign{
-c_{0} &= 0, \cr
-c_{i+1} &= (x_i~\&~y_i)\lor(x_i~\&~c_i)\lor(y_i~\&~c_i).\cr
-}
-$$
-
-Takovéto sèítání sice perfektnì funguje, nicménì je bohu¾el pomìrnì pomalé.
-Pokud bychom stavìli hradlovou sí» podle tohoto pøedpisu, byla by slo¾ená z~nìjakých
-podsítí (\uv{krabièek}), které budou mít na~vstupu $x_i$, $y_i$ a~$c_i$ a jejich výstupem
-bude $z_i$ a~$c_{i+1}$. 
-
-\figure{hloupe_scitani.eps}{Sèítání ¹kolním algoritmem.}{1.5in}
-
-V¹imìme si, ¾e~ka¾dá krabièka závisí na~výstupu té pøedcházející. Jednotlivé
-krabièky tedy musí urèitì le¾et na~rùzných hladinách. Celkovì bychom museli pou¾ít
-$\Theta{(n)}$ hladin a~jeliko¾ je ka¾dá krabièka konstantnì velká, také $\Theta{(n)}$ hradel. To dává
-lineární èasovou i~prostorovou slo¾itost, èili oproti sekvenènímu algoritmu jsme si nepomohli.
-
-Zamysleme se nad tím, jak by se proces sèítání mohl zrychlit.
-
-\h{Pøenosy v~blocích}
-Jediné, co nás pøi sèítání brzdí, jsou pøenosy mezi jednotlivými øády. Ka¾dý øád,
-aby~vydal souèet, musí poèkat na~to, a¾~dopoèítají v¹echny pøedcházející øády.
-Teprve pak se~toti¾ dozví pøenos. Kdybychom ov¹em pøenosy dokázali spoèítat
-nìjakým zpùsobem paralelnì, máme vyhráno. Jakmile známe v¹echny pøenosy, souèet
-u¾~zvládneme dopoèítat na~konstantnì mnoho hladin -- tedy v~konstantním èase.
-
-Podívejme se na~libovolný {\I blok} v~na¹em souètu. Tak budeme øíkat èíslùm
-$x_j\ldots x_i$ a $y_j\ldots y_i$ v~nìjakém intervalu indexù $\left<i,j\right>$. Pøenos $c_{j+1}$ vystupující z~tohoto bloku závisí mimo hodnot sèítancù u¾ pouze na~pøenosu $c_{i}$, který do bloku vstupuje.
-
-\figure{blok_scitani.eps}{Blok souètu.}{3in}
-
-Z tohoto pohledu se mù¾eme na blok také dívat jako na nìjakou funkci, která
-dostane jednobitový vstup a vydá jednobitový výstup. To je nám pøíjemné, nebo» 
-takových funkcí existují jenom ètyøi typy:
-\numlist\ndotted
-\:$f(x) = 0$,  ~~~~(0)
-\:$f(x) = 1$,  ~~~~(1)
-\:$f(x) = x$,  ~~~~($<$ -- kopírování)
-\:$f(x) = \neg{x}$.
-\endlist
-Jak se za~chvíli uká¾e, poslední pøípad, kdy by~nìjaký blok pøedával opaèný
-pøenos, ne¾ do~nìj vstupuje, navíc nikdy nemù¾e nastat. Pojïme si to rozmyslet.
-Jednobitové bloky se chovají velice jednodu¹e:
-
-\figure{bloky_1bit.eps}{Tabulka triviálních blokù.}{1.1in}
-
-Z prvního bloku evidentnì v¾dy vyleze 0, a»~do~nìj vstoupí jakýkoli pøenos.
-Poslední blok naopak sám o~sobì pøenos vytváøí, a»~ji¾ do~nìj vleze jakýkoliv.
-Bloky prostøední se chovají stejnì, a~to tak, ¾e~samy o~sobì ¾ádný pøenos nevytvoøí,
-ale~pokud do~nich nìjaký pøijde, tak~také odejde.
-
-Mìjme nyní nìjaký vìt¹í blok~$C$ slo¾ený ze~dvou men¹ích podblokù $A$ a~$B$, jejich¾
-chování u¾ známe. Z~toho mù¾eme odvodit, jak se chová celý blok:
-
-\figure{tabulka_skladani_bloku.eps}{Skládání chování blokù.}{1.3in}
-
-Pokud vy¹¹í blok pøenos pohlcuje, pak a»~se u¾~ni¾¹í blok chová jakkoli, slo¾ení
-tìchto blokù musí v¾dy pohlcovat. V~prvním øádku tabulky jsou tudí¾ nuly. Analogicky,
-pokud vy¹¹í blok generuje pøenos, tak~ten ni¾¹í na~tom nic nezmìní. V~druhém
-øádku tabulky jsou tedy samé jednièky. Zajímavìj¹í pøípad nastává, pokud vy¹¹í blok
-kopíruje -- tehdy zále¾í èistì na~chování ni¾¹ího bloku.
-
-V¹imnìme si, ¾e~skládání chování blokù je vlastnì úplnì obyèejné skládání
-funkcí. Nyní bychom mohli prohlásit, ¾e~budeme poèítat nad~tøíprvkovou abecedou,
-a~¾e~celou tabulku doká¾eme spoèítat jedním jediným hradlem. Pojïmì si v¹ak
-rozmyslet, jak~bychom takovouto vìc popsali èistì binárnì. Jak tedy tyto tøi stavy
-popisovat pouze nìkolika bity?
-
-Evidentnì nám k tomuto binárnímu zakódování tøí stavù budou staèit bity dva.
-Oznaème si je jako $p$ a $q$. Tato dvojice mù¾e nabývat hned ètyø mo¾ných hodnot,
-kterým pøiøadíme tøi mo¾ná chování bloku. Toto kódování mù¾eme zvolit zcela
-libovolnì, ale pokud si ho zvolíme ¹ikovnì, u¹etøíme si dále práci pøi kompozici.
-Zvolme si tedy kódování takto:
-
-\itemize\ibull
-\:$(1,*) = <$,
-\:$(0,0) = 0$,
-\:$(0,1) = 1$
-\endlist
-
-Tomu, ¾e blok kopíruje, odpovídá dvojice $p = 1$; $q =$ \<cokoliv>. V~ostatních
-pøípadech bude~$p$ nulové a~$q$ nám bude øíkat, co je na~výstupu pøíslu¹ného bloku.
-Jinými slovy $p = 0$ znamená, ¾e funkce je konstanta, pøièem¾ $q$ øíká jaká; naproti
-tomu $p = 1$~znamená, ¾e funkce je identita, a»~u¾~je $q$ cokoli.
-
-Pojïme si nyní ukázat, jak bude celé skládání blokù vypadat. Rozmysleme si,
-kdy je~$p$ celého dvojbloku jednièkové, tedy kdy celý dvojblok kopíruje. To nastane
-tehdy, pokud kopírují obì jeho èásti, a tedy  $p = p_a~\&~p_b$. Dále $q$ bude rovno jednièce,
-pokud $q = (\neg{p_a}~\&~q_a) \lor (p_a~\&~q_b)$.
-
-Skládání chování blokù lze tedy popsat buï ternárnì -- tabulkou, ale lze to
-i~binárnì vý¹e uvedeným pøedpisem.
-
-Nyní si tedy mù¾eme dopøedu vypoèítat chování bloku velikosti jedna, poté
- z~nich skládáním blokù velikosti dva, dál velikosti ètyøi, osm, atd \dots
-
-\h{Paralelní sèítání}
-
-Paralelní algoritmus na~sèítání u¾~zkonstruujeme pomìrnì snadno. Bez újmy
-na~obecnosti budeme pøedpokládat, ¾e~poèet bitù vstupních èísel je mocnina dvojky,
-jinak si vstup doplníme nulami, co¾ výsedný èas bìhu algoritmu zhor¹í maximálnì
-konstanta-krát.
-
-\algo
-\:Spoèteme chování blokù velikosti~1. ($\O(1)$ hladin)
-\:Postupnì poèítáme chování blokù\foot{myslíme \uv{pøirozenì zarovnané} bloky, tedy takové, jejich¾ poloha je násobkem velikosti} velikosti 2, 4, 8, ..., $2^k$. 
-  ($\O(\log n)$ hladin, na~nich¾ se skládají bloky)
-\:$c_0 \leftarrow 0$ (pøenos do nejni¾¹ího øádu je v¾dy 0)
-\:Urèíme $c_n$ podle $c_0$ a chování (jediného) bloku velikosti~$n$.
-\:Postupnì dopoèítáme pøenosy na~hranicích dìlitelných $2^k$ \uv{zahu¹»ováním}:
-  jakmile víme $c_{2^k}$, mù¾eme dopoèítat $c_{2^k+2^{k-1}}$ podle
-  chování bloku $\left< 2^k, 2^k+2^{k-1}\right>$. ($\O(\log n)$ hladin,
-  na~nich¾ se dosazuje)
-\:Seèteme: $\forall i: z_i = x_i \oplus y_i \oplus c_i$.
-\endalgo
-
-\figure{1_9_deleni_bloku.eps}{Výpoèet pøenosu.}{2.5in}
-
-Algoritmus pracuje v~èase $\O(\log n)$. Hradel je pou¾ito lineárnì: na~jednotlivých
-hladinách kroku~2 poèet hradel exponenciálnì klesá od~$n$ k~1, na~hladinách kroku~5
-exponenciálnì stoupá od~1 k~$n$, tak¾e dohromady se seète na~$\Theta(n)$.
-
-\h{Tøídìní}
-
-Nyní se podíváme na~druhý problém, a~to na~problém tøídìní. Ji¾ známe pomìrnì efektivní sekvenèní algoritmy, které dovedou tøídit v~èase $\O(n\log n)$. Byli bychom jistì rádi, kdybychom to zvládli je¹tì rychleji. Pojïme se podívat, zda by nám v tom nepomohlo problém paralelizovat.
-
-Budeme pøi tom pracovat ve~výpoèetním modelu, kterému se øíká komparátorová sí». Ta je postavená z~hradel, kterým se øíká komparátory.
-
-\s{Definice:} {\I Komparátorová sí»} je kombinaèní obvod, jeho¾ hradla jsou
-komparátory.
-
-\figure{sortnet.0}{Komparátor}{0.7in}
-
-Komparátor umí porovnat dvì hodnoty a~rozhodnout, která z~nich je vìt¹í
-a~která men¹í. Nevrací v¹ak booleovský výsledek jako bì¾né hradlo, ale má dva
-výstupy, pøièem¾ na~jednom vrátí men¹í ze~vstupních hodnot a~na~druhém vìt¹í.
-
-Výstupy komparátorù se nevìtví. Nemù¾eme tedy jeden výstup \uv{rozdvojit} a~pøipojit ho na~dva vstupy. (Vìtvení by dokonce ani nemìlo smysl, proto¾e zatímco rozdvojit bychom mohli, slouèit u¾~ne. Pokud tedy chceme, aby sí» mìla $n$~vstupù i~$n$~výstupù, rozdvojení stejnì nesmíme provést, i kdybychom jej mìli povolené.)
-
-\s{Pøíklad:} {\sl Bubble sort}
-
-Obrázek Bubble 1 ilustruje pou¾ití komparátorù pro tøídìní Bubble sortem.
-©ipky pøedstavují jednotlivé komparátory. Výpoèet v¹ak je¹tì mù¾eme vylep¹it.
-
-\twofigures{sortnet.1}{Bubble 1}{143pt}{sortnet.2}{Bubble 2}{143pt}
-
-Sna¾íme se výpoèet co nejvíce paralelizovat (viz obrázek Bubble 2).
-Jak je vidìt, komparátory na sebe nemusejí èekat. Tím mù¾eme výpoèet urychlit a místo èasu $\Theta{(n^2)}$ docílit èasové slo¾itosti $\Theta{(n)}$. V obou pøípadech je zachován kvadratický prostor.
-
-Nyní si uká¾eme je¹tì rychlej¹í tøídící algoritmus. Pùjdeme na nìj v¹ak trochu \uv{od lesa}. Nejdøíve vymyslíme sí», která bude umìt tøídit jenom nìco -- toti¾ bitonické posloupnosti. Bez újmy na obecnosti budeme pøedpokládat, ¾e ka¾dé dva prvky na vstupu jsou navzájem rùzné.
-
-\medskip
-\s{Definice:} Øekneme, ¾e posloupnost $x_0,\dots,x_{n-1} $ je {\I èistì bitonická} právì tehdy, kdy¾
-pro nìjaké $x_k, k\in\{0, \dots, n-1\} $ platí, ¾e~v¹echny prvky pøed ním (vèetnì jeho samotného)
-tvoøí rostoucí poslopnost, kde¾to prvky stojící za~ním tvoøí poslopnost klesající.
-Formálnì zapsáno musí platit, ¾e:
-$$x_0\leq x_1\leq \dots \leq x_k \geq x_{k+1}\geq\dots \geq x_{n-1}.$$
-
-\s{Definice:} Posloupnost $x_0 \dots x_{n-1}$ je {\I bitonická}, právì kdy¾ $\exists~j\in \{0,\dots ,n-1\}$, pro
-které je rotace pùvodní posloupnosti o $j$ prvkù, tedy posloupnost
-$$x_j,x_{(j+1) \bmod n},\dots, x_{(j+n-1) \bmod n},$$ èistì bitonická.
-
-\s{Definice:} {\I Separátor $S_n$} je sí», ve které jsou v¾dy~$i$-tý a~$(i+{n/2})$-tý prvek vstupu
-(pro $i=0,\dots, {n/2}-1$) propojeny komparátorem. Minimum se pak stane~$i$-tým,
-maximum  $(i+{n/2})$-tým prvkem výstupu.
-\figure{sortnet.3}{$(y_i, y_{i+{n/2}}) = \<CMP>(x_i, x_{i+{n/2}})$} {300pt}
-
-\s{Lemma:} Pokud separátor dostane na~vstupu bitonickou posloupnost, pak jeho výstup $y_0, \dots, y_{n-1}$
-splòuje:
-
-(i) $y_0,\dots, y_{n/2 -1}$ a~$y_{n/2},\dots, y_{n-1}$ jsou 
-bitonické posloupnosti,
-
-(ii) Pro v¹echna $i,j< {n/2}$ platí $y_i < y_{j + {n/2}}$.
-
-Separátor nám tedy jednu bitonickou posloupnost na~vstupu rozdìlí na~dvì
-bitonické posloupnosti, pøièem¾ navíc ka¾dý prvek první posloupnosti ($y_0,\dots, y_{n/2 -1}$)
-je men¹í nebo roven prvkùm druhé posloupnosti ($y_{n/2}, \dots, y_{n-1}$).
-
-\>Ne¾ pøistoupíme k dùkazu lemmatu, uka¾me si, k èemu se nám bude hodit.
-
-
-\s{Definice:} {\I Bitonická tøídièka $B_n$} je obvod sestavený ze separátorù, který dostane-li na vstupu bitonickou posloupnost délky $n$ (BÚNO konstruujeme tøídièku pro $n=2^k$), vydá setøídìnou zadanou posloupnost délky $n$. 
-
-Tøídièka dostane na~vstupu bitonickou posloupnost. Separátor~$S_n$ ji pak dle lemmatu rozdìlí na~dvì bitonické posloupnosti, kdy je ka¾dý prvek z~první posloupnosti men¹í ne¾ libovolný prvek z~druhé. Tyto poloviny pak dal¹í separátory rozdìlí na~ètvrtiny, ..., a¾~na~konci zbudou pouze jednoduché posloupnosti délky jedna (zjevnì setøídìné), které mezi sebou mají po¾adovanou nerovnost -- tedy ka¾dá posloupnost (nebo spí¹e prvek) nalevo je $\leq$ ne¾ prvek napravo od~nìj.
-
-\centerline{\epsfbox{sortnet.5}}
-
-Jak je vidìt, bitonická tøídièka nám libovolnou bitonickou posloupnost délky~$n$
-setøídí na~$\Theta(\log n)$ hladin.
-
-Nyní se dá odvodit, ¾e pokud umíme tøídit bitonické posloupnosti, umíme setøídit v¹echno.
-Vzpomeòme si na~tøídìní sléváním -- Merge sort. To funguje tak, ¾e zaène s~jednoprvkovými posloupnostmi, které jsou evidentnì setøídìné, a~poté v¾dy dvì setøídìné posloupnosti slévá do~jedné. Kdybychom nyní umìli paralelnì slévat, mohli bychom vytvoøit i~paralelní Merge sort. Jinými slovy, potøebujeme dvì rostoucí posloupnosti nìjak efektivnì slít do~jedné setøídìné. Uvìdomme si, ¾e to zvládneme jednodu¹e -- staèí druhou z~posloupností obrátit a~\uv{pøilepit} za~první, èím¾ vznikne bitonická posloupnost, kterou poté mù¾eme setøídit na¹í bitonickou tøídièkou.
-
-\s{Pøíklad:} {\sl Merge sort}
-
-Bitonická tøídièka se tedy dá pou¾ít ke~slévání setøídìných posloupností. 
-Uka¾me si, jak s~její pomocí sestavíme souèástky {\I slévaèky} $M_n$:
-
-Setøídìné posloupnosti $x_0,\dots, x_{n-1}$ a~$y_0,\dots, y_{n-1}$ spojíme do jedné bitonické
-posloupnosti $x_0,\dots, x_{n-1},y_{n-1},\dots, y_0$. Z~této posloupnosti vytvoøíme pomocí bitonické tøídièky
-$B_{2n}$ setøídìnou posloupnost. Vytvoøíme tedy blok $M_{2n}$, který se ov¹em sestává de~facto pouze z~bloku $B_{2n}$, jeho¾ druhá polovina vstupù je zapojena v~obráceném poøadí.
-\figure{sortnet.6}{Paralelní MergeSort.}{3in}
-
-Nyní se pokusme odhadnout èasovou slo¾itost. Ná¹ MergeSort bude mít øádovì hloubku blokù $\log n$.
-V ka¾dém bloku $M_n$ je navíc ukryta bitonická tøídièka s takté¾ logaritmickou hloubkou.
-Celková hloubka tedy bude $\log 2 + \log4 + \dots + \log 2^k + \dots + \log n$.
-Po seètení nakonec dostáváme výslednou èasovou slo¾itost $\Theta (\log^2 n)$.
-
-Dodejme je¹tì, ¾e existuje i~tøídicí algoritmus, kterému staèí jen ${\O}(\log n)$ hladin.
-Jeho multiplikativní konstanta je v¹ak pøíli¹ veliká, tak¾e je v~praxi nepou¾itelný.
-
-Vra»me se nyní k dùkazu lemmatu, který jsme na pøedminulé stránce vynechali.
-
-\h{Dùkaz Lemmatu:}
-(i) Nejprve nahlédneme, ¾e lemma platí, je-li vstupem èistì bitonická
-posloupnost. Dále BÚNO pøedpokládejme, ¾e vrchol posloupnosti je v~první polovinì (kdyby
-byl vrchol za~polovinou, staèilo by zrcadlovì obrátit posloupnost i~komparátory a~øe¹ili
-bychom stejný problém). Nyní si definujme $k := \min j: x_j > x_{j+n/2}$. (Pokud
-by~takové~$k$ neexistovalo, znamenalo by~to, ¾e vstupní posloupnost je monotónní.
-Separátor by tedy nic nedìlal a~pouze zkopíroval vstup na~výstup, co¾ jistì lemma splòuje.) Nyní si v¹imìme,
-¾e jakmile jednou zaène platit, ¾e prvek na~levé stranì je men¹í ne¾ na~pravé, bude
-nám tato relace platit a¾~do~konce. Oznaème vrchol vstupní posloupnosti jako~$x_m$.
-Pak~$k$ bude jistì men¹í ne¾~$m$ a~$k+{n/2}$ bude vìt¹í ne¾~$m$. Mezi~$k$ a~$m$ je tedy
-vstupní posloupnost neklesající, mezi $k+{n/2}$ a~$n-1$ nerostoucí.
-
-Do pozice~$k$ tedy separátor bude pouze kopírovat vstup na~výstup, od~pozice~$k$
-dál u¾~jen prohazuje. Pro ka¾dé~$i$, $(k\leq i\leq {n/2}-1)$ se prvky
-$x_i$ a~$x_{i+{n/2}}$ prohodí. Úsek mezi~$k$ a~${n/2}-1$ tedy
-nahradíme nerostoucí posloupností, první polovina výstupu tedy bude
-(dokonce èistì) bitonická. Podobnì úsek $k+{n/2}$ a¾~$n-1$ nahradíme èistì
-bitonickou posloupností. Obì poloviny tedy budou bitonické.
-
-Dostaneme-li na vstupu obecnou bitonickou posloupnost, pøedstavíme si,
-¾e je to èistì bitonická posloupnost zrotovaná o~$r$ prvkù (BÚNO
-doprava). Zjistíme, ¾e v~komparátorech se porovnávají tyté¾ prvky,
-jako kdyby zrotovaná nebyla. Výstup se od výstupu èistì bitonické
-posloupnosti zrotovaného o~$r$ bude li¹it prohozením úsekù $x_0$ a¾
-$x_{r-1}$ a~$x_{n/2}$ a¾ $x_{{n/2}+r-1}$. Obì výstupní
-posloupnosti tedy budou zrotované o~$r$ prvkù, ale na jejich
-bitoniènosti se nic nezmìní.
-
-(ii) Z dùkazu (i) pro èistì bitonickou posloupnost víme, ¾e $y_0\dots y_{n/2-1}$ je èistì bitonická a bude rovna $x_0\dots x_{k-1},x_{k+n/2}\dots x_{n-1}$ pro vhodné $k$ a navíc bude mít maximum v $x_{k-1}$ nebo $x_k+{n/2}$. Mezi tìmito body ov¹em ve vstupní posloupnosti urèitì nele¾el ¾ádný $x_i$ men¹í ne¾ $x_k-1$ nebo $x_k+{n/2}$ (jak je vidìt z obrázku) a posloupnost  $x_k \dots x_{k-1+{n/2}}$ je rovna $y_{n/2}\dots y_{n-1}$. Pro obecné bitonické posloupnosti uká¾eme stejnì jako v (i).
-\qed
-
-\medskip
-
-\centerline{\epsfbox{sortnet.7}}
-
-
-\h{Paralelní násobení}
-
-Podobnì jako u~sèítání si vzpomeneme na~¹kolní algoritmus -- tentokráte v¹ak pro násobení.
-To fungovalo tak, ¾e~jsme si jedno ze~dvou binárních èísel na~vstupu (øíkejme mu tøeba $x$) $n$-krát posouvali. Tam kde pak byly v~èísle~$y$ jednièky, pøíslu¹né kopie $x$ jsme seèetli. Jinými slovy tedy násobení umíme pøevést na~nìjaké posuny (ty lze realizovat pouze \uv{pøedrátováním} -- nic nás nestojí), násobení jedním bitem (co¾ je 
-{\sc and} ) a~nakonec potøebujeme výsledných~$n$ èísel seèíst.
-\figure{skolni_scitani.eps}{©kolní sèítání.}{2in}
-
-Jak nyní seèíst $n$ $n$-bitových èísel..? Nabízí se vyu¾ít osvìdèený \uv{stromeèek} -- sèítat dvojice èísel, výsledky pak opìt po~dvojicích seèíst, a¾ na~konci vyjde jediný výsledek.
-\figure{stromecek.eps}{Stromeèek}{1.4in}
-
-Toto øe¹ení by v¹ak vedlo na~èasovou slo¾itost $\Theta (\log^2 n)$, nebo» sèítat umíme v~èase $\Theta (\log n)$. To je sice dle na¹ich mìøítek
-docela efektivní, ale pøekvapivì to jde je¹tì lépe -- toti¾ na~$\Theta (\log n)$ hladin. Této 
-slo¾itosti dosáhneme malým trikem.
-
-Vymyslíme obvod konstantní hloubky, který na~vstupu dostane tøi èísla. Odpoví pak dvìma èísly
-takovými, ¾e budou mít stejný souèet jako pùvodní tøi èísla. Jinými slovy pomocí
-tohoto obvodu budeme umìt seètení tøí èísel pøevést na seètení dvou èísel.
-
-\figure{obvod.eps}{$x+y+z = p+q$}{0.3in}
-
-V¹imnìme si, ¾e~kdy¾ sèítáme tøi bity, mù¾e být pøenos do~vy¹¹ího øádu nula èi~jednièka. Vezmeme si tedy bity $x_i$, $y_i$, $z_i$ a~ty seèteme. To nám dá dvoubitový výsledek, pøièem¾ ni¾¹í bit z~tohoto výsledku po¹leme do~èísla~$p$, vy¹¹í do~èísla~$q$.
-
-\figure{obvod_real.eps}{Redukování sèítání}{1.2in}
-
-Toto zredukování sèítání nám nyní umo¾ní opìt stavit strom, by» o malièko slo¾itìj¹í.
-
-\figure{sl_stromecek.eps}{Slo¾itìj¹í stromeèek}{0.9in}
-
-Pokud jsme mìli na~zaèátku $n$ èísel, po~první redukci nám jich zbývá jen $2/3 \cdot n$ a~obecnì v~$k$-té hladiné $(2/3)^k \cdot n$. Znamená to, ¾e èísel nám ubývá exponenciálnì, tak¾e poèet hladin bude logaritmický. Redukující obvod je pøi tom jen konstantnì hluboký, tak¾e celé redukování zvládneme v~èase $\Theta (\log n)$. Na~konci Slo¾itìj¹ího stromeèku pak máme umístìnou jednu obyèejnou sèítaèku, která zbývající dvì èísla seète v~logaritmickém èase.
-
-Seètení v¹ech $n$ èísel tedy zabere $\Theta (\log n)$ hladin.
-
-Kdy¾ se nyní vrátíme k~násobení, zbývá nám vyøe¹it posouvání a~{\sc and}ování. Uvìdomme si, ¾e to je plnì paralelení a~zvládneme ho za~konstantnì mnoho hladin. Celé násobení tedy zvládneme v~logaritmickém èase.
-
-\bye
diff --git a/5-addsort/Makefile b/5-addsort/Makefile
deleted file mode 100644
index 4f7be80..0000000
--- a/5-addsort/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=5-addsort
-
-include ../Makerules
diff --git a/5-addsort/blok_scitani.eps b/5-addsort/blok_scitani.eps
deleted file mode 100644
index 225a323..0000000
--- a/5-addsort/blok_scitani.eps
+++ /dev/null
@@ -1,1673 +0,0 @@
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-z110=(0v,10v);
-
-z21=(2v,1v);
-z22=(2v,2v);
-z23=(2v,3v);
-z24=(2v,4v);
-z25=(2v,5v);
-z26=(2v,6v);
-z27=(2v,7v);
-z28=(2v,8v);
-z29=(2v,9v);
-z210=(2v,10v);
-
-z31=(4v,1v);
-z32=(4v,2v);
-z33=(4v,3v);
-z34=(4v,4v);
-z35=(4v,5v);
-z36=(4v,6v);
-z37=(4v,7v);
-z38=(4v,8v);
-z39=(4v,9v);
-z310=(4v,10v);
-
-z41=(6v,1v);
-z42=(6v,2v);
-z43=(6v,3v);
-z44=(6v,4v);
-z45=(6v,5v);
-z46=(6v,6v);
-z47=(6v,7v);
-z48=(6v,8v);
-z49=(6v,9v);
-z410=(6v,10v);
-
-z51=(8v,1v);
-z52=(8v,2v);
-z53=(8v,3v);
-z54=(8v,4v);
-z55=(8v,5v);
-z56=(8v,6v);
-z57=(8v,7v);
-z58=(8v,8v);
-z59=(8v,9v);
-z510=(8v,10v);
-
-z111=(0v,11v);
-z211=(2v,11v);
-z311=(4v,11v);
-z411=(6v,11v);
-z511=(8v,11v);
-
-z1015=(-1v,1.5v);
-z1615=(9v,1.5v);
-z1035=(-1v,3.5v);
-z1635=(9v,3.5v);
-z1065=(-1v,6.5v);
-z1665=(9v,6.5v);
-
-pickup pencircle scaled 0.4pt;
-draw(z10--z111);
-draw(z20--z211);
-draw(z30--z311);
-draw(z40--z411);
-draw(z50--z511);
-drawarrow(z11--z21);
-drawarrow(z13--z23);
-drawarrow(z16--z26);
-drawarrow(z110--z210);
-drawarrow(z22--z32);
-drawarrow(z25--z35);
-drawarrow(z29--z39);
-drawarrow(z34--z44);
-drawarrow(z38--z48);
-drawarrow(z47--z57);
-
-pickup pencircle scaled 0.7pt;
-draw(z1015--z1615) dashed withdots scaled 0.7;
-draw(z1035--z1635) dashed withdots scaled 0.7;
-draw(z1065--z1665) dashed withdots scaled 0.7;
-
-label.top(btex x1 etex,z111);
-label.top(btex x2 etex,z211);
-label.top(btex x3 etex,z311);
-label.top(btex x4 etex,z411);
-label.top(btex x5 etex,z511);
-
-endfig;
-
-beginfig(2);
-v:=u*5mm;
-z13=(0v,3v);
-z23=(2v,3v);
-z33=(4v,3v);
-z43=(6v,3v);
-z53=(8v,3v);
-
-z14=(0v,4v);
-z15=(0v,5v);
-z16=(0v,6v);
-z17=(0v,7v);
-z18=(0v,8v);
-z19=(0v,9v);
-z110=(0v,10v);
-
-z24=(2v,4v);
-z25=(2v,5v);
-z26=(2v,6v);
-z27=(2v,7v);
-z28=(2v,8v);
-z29=(2v,9v);
-z210=(2v,10v);
-
-z34=(4v,4v);
-z35=(4v,5v);
-z36=(4v,6v);
-z37=(4v,7v);
-z38=(4v,8v);
-z39=(4v,9v);
-z310=(4v,10v);
-
-z44=(6v,4v);
-z45=(6v,5v);
-z46=(6v,6v);
-z47=(6v,7v);
-z48=(6v,8v);
-z49=(6v,9v);
-z410=(6v,10v);
-
-z54=(8v,4v);
-z55=(8v,5v);
-z56=(8v,6v);
-z57=(8v,7v);
-z58=(8v,8v);
-z59=(8v,9v);
-z510=(8v,10v);
-
-z111=(0v,11v);
-z211=(2v,11v);
-z311=(4v,11v);
-z411=(6v,11v);
-z511=(8v,11v);
-
-pickup pencircle scaled 0.4pt;
-draw(z13--z111);
-draw(z23--z211);
-draw(z33--z311);
-draw(z43--z411);
-draw(z53--z511);
-drawarrow(z14--z24);
-drawarrow(z16--z26);
-drawarrow(z18--z28);
-drawarrow(z110--z210);
-drawarrow(z25--z35);
-drawarrow(z27--z37);
-drawarrow(z29--z39);
-drawarrow(z36--z46);
-drawarrow(z38--z48);
-drawarrow(z47--z57);
-
-label.top(btex x1 etex,z111);
-label.top(btex x2 etex,z211);
-label.top(btex x3 etex,z311);
-label.top(btex x4 etex,z411);
-label.top(btex x5 etex,z511);
-
-endfig;
-
-
-beginfig(3);
-v:=u*5mm;
-
-z10=(0v,0v);
-z15=(0v,5v);
-z16=(0v,6v);
-
-z20=(2v,0v);
-z24=(2v,4v);
-z26=(2v,6v);
-
-z30=(4v,0v);
-z33=(4v,3v);
-z36=(4v,6v);
-
-z40=(6v,0v);
-z42=(6v,2v);
-z46=(6v,6v);
-
-z50=(8v,0v);
-z51=(8v,1v);
-z56=(8v,6v);
-
-z60=(10v,0v);
-z65=(10v,5v);
-z66=(10v,6v);
-
-z70=(12v,0v);
-z74=(12v,4v);
-z76=(12v,6v);
-
-z80=(14v,0v);
-z83=(14v,3v);
-z86=(14v,6v);
-
-z90=(16v,0v);
-z92=(16v,2v);
-z96=(16v,6v);
-
-z100=(18v,0v);
-z101=(18v,1v);
-z106=(18v,6v);
-
-z110=(20v,0v);
-z116=(20v,6v);
-
-
-pickup pencircle scaled 0.4pt;
-draw(z10--z16);
-draw(z20--z26);
-draw(z30--z36);
-draw(z40--z46);
-draw(z50--z56);
-draw(z60--z66);
-draw(z70--z76);
-draw(z80--z86);
-draw(z90--z96);
-draw(z100--z106);
-drawarrow(z15--z65);
-drawarrow(z24--z74);
-drawarrow(z33--z83);
-drawarrow(z42--z92);
-drawarrow(z51--z101);
-
-label.top(btex $x_0$ etex,z16);
-label.top(btex $x_1$ etex,z26);
-label.top(btex $x_2$ etex,z36);
-label.top(btex \dots etex,z66);
-label.top(btex $x_{n-2}$ etex,z96);
-label.top(btex $x_{n-1}$ etex,z106);
-
-endfig;
-
-
-
-beginfig(4);
-v:=u*5mm;
-
-z1075=(7.5v,0v);
-
-z10=(0v,1v);
-z175=(7.5v,1v);
-z115=(15v,1v);
-
-z20=(0v,2v);
-z235=(3.5v,2v);
-z211=(11.5v,2v);
-z215=(15v,2v);
-
-z30=(0v,3v);
-z335=(3.5v,3v);
-z37=(7v,3v);
-z38=(8v,3v);
-z311=(11.5v,3v);
-z315=(15v,3v);
-
-z40=(0v,4v);
-z415=(1.5v,4v);
-z455=(5.5v,4v);
-z47=(7v,4v);
-z48=(8v,4v);
-z495=(9.5v,4v);
-z413=(13.5v,4v);
-z416=(15v,4v);
-
-z50=(0v,5v);
-z515=(1.5v,5v);
-z53=(3v,5v);
-z54=(4v,5v);
-z555=(5.5v,5v);
-z57=(7v,5v);
-z58=(8v,5v);
-z595=(9.5v,5v);
-z511=(11v,5v);
-z512=(12v,5v);
-z513=(13.5v,5v);
-z516=(15v,5v);
-
-z60=(0v,6v);
-z61=(1v,6v);
-z62=(2v,6v);
-z63=(3v,6v);
-z64=(4v,6v);
-z65=(5v,6v);
-z66=(6v,6v);
-z67=(7v,6v);
-z68=(8v,6v);
-z69=(9v,6v);
-z610=(10v,6v);
-z611=(11v,6v);
-z612=(12v,6v);
-z613=(13v,6v);
-z614=(14v,6v);
-z615=(15v,6v);
-
-z71=(1v,7v);
-z72=(2v,7v);
-z75=(5v,7v);
-z76=(6v,7v);
-z79=(9v,7v);
-z710=(10v,7v);
-z713=(13v,7v);
-z714=(14v,7v);
-
-
-
-
-pickup pencircle scaled 0.4pt;
-draw(z10--z115--z215--z20--cycle);
-draw(z30--z37--z47--z40--cycle);
-draw(z38--z315--z416--z48--cycle);
-draw(z50--z53--z63--z60--cycle);
-draw(z54--z57--z67--z64--cycle);
-draw(z58--z511--z611--z68--cycle);
-draw(z512--z516--z615--z612--cycle);
-drawarrow(z71--z61);
-drawarrow(z72--z62);
-drawarrow(z75--z65);
-drawarrow(z76--z66);
-drawarrow(z79--z69);
-drawarrow(z710--z610);
-drawarrow(z713--z613);
-drawarrow(z714--z614);
-drawarrow(z515--z415);
-drawarrow(z555--z455);
-drawarrow(z595--z495);
-drawarrow(z513--z413);
-drawarrow(z335--z235);
-drawarrow(z311--z211);
-drawarrow(z175--z1075);
-
-endfig;
-
-
-beginfig(5);
-v:=u*5mm;
-
-z075=(7.5v,8v);
-
-z10=(0v,7v);
-z175=(7.5v,7v);
-z115=(15v,7v);
-
-z20=(0v,6v);
-z235=(3.5v,6v);
-z211=(11.5v,6v);
-z215=(15v,6v);
-
-z30=(0v,5v);
-z335=(3.5v,5v);
-z37=(7v,5v);
-z38=(8v,5v);
-z311=(11.5v,5v);
-z315=(15v,5v);
-
-z40=(0v,4v);
-z415=(1.5v,4v);
-z455=(5.5v,4v);
-z47=(7v,4v);
-z48=(8v,4v);
-z495=(9.5v,4v);
-z413=(13.5v,4v);
-z416=(15v,4v);
-
-z50=(0v,3v);
-z515=(1.5v,3v);
-z53=(3v,3v);
-z54=(4v,3v);
-z555=(5.5v,3v);
-z57=(7v,3v);
-z58=(8v,3v);
-z595=(9.5v,3v);
-z511=(11v,3v);
-z512=(12v,3v);
-z513=(13.5v,3v);
-z516=(15v,3v);
-
-z60=(0v,2v);
-z61=(1v,2v);
-z62=(2v,2v);
-z63=(3v,2v);
-z64=(4v,2v);
-z65=(5v,2v);
-z66=(6v,2v);
-z67=(7v,2v);
-z68=(8v,2v);
-z69=(9v,2v);
-z610=(10v,2v);
-z611=(11v,2v);
-z612=(12v,2v);
-z613=(13v,2v);
-z614=(14v,2v);
-z615=(15v,2v);
-
-% ve skutecnosti dle znaceni 
-% by melo byt z6* ale uz 
-% obsazeno
-z815=(1.5v,2v);
-z855=(5.5v,2v);
-z895=(9.5v,2v);
-z813=(13.5v,2v);
-
-z715=(1.5v,1v);
-z755=(5.5v,1v);
-z795=(9.5v,1v);
-z713=(13.5v,1v);
-
-z9=(4v,0v);
-
-pickup pencircle scaled 0.4pt;
-draw(z10--z115--z215--z20--cycle);
-draw(z30--z37--z47--z40--cycle);
-draw(z38--z315--z416--z48--cycle);
-draw(z50--z53--z63--z60--cycle);
-draw(z54--z57--z67--z64--cycle);
-draw(z58--z511--z611--z68--cycle);
-draw(z512--z516--z615--z612--cycle);
-drawarrow(z075--z175);
-drawarrow(z415--z515);
-drawarrow(z455--z555);
-drawarrow(z495--z595);
-drawarrow(z413--z513);
-drawarrow(z235--z335);
-drawarrow(z211--z311);
-drawarrow(z815--z715);
-drawarrow(z855--z755);
-drawarrow(z895--z795);
-drawarrow(z813--z713);
-
-label.llft(btex $n$ etex,z075);
-label.bot(btex $S_n$ etex,z175);
-label.bot(btex $S_{n\over 2}$ etex,z335);
-label.bot(btex $S_{n\over 2}$ etex,z311);
-label.bot(btex $S_{n\over 4}$ etex,z515);
-label.bot(btex $S_{n\over 4}$ etex,z555);
-label.bot(btex $S_{n\over 4}$ etex,z595);
-label.bot(btex $S_{n\over 4}$ etex,z513);
-label.rt(btex Bitonick\'a t\v r\'\i di\v cka $B_{n}$ etex,z9);
-
-endfig;
-
-
-
-beginfig(6);
-v:=u*5mm;
-
-z1075=(7.5v,0v);
-
-z10=(0v,1v);
-z175=(7.5v,1v);
-z115=(15v,1v);
-
-z20=(0v,2v);
-z235=(3.5v,2v);
-z211=(11.5v,2v);
-z215=(15v,2v);
-
-z30=(0v,3v);
-z335=(3.5v,3v);
-z37=(7v,3v);
-z38=(8v,3v);
-z311=(11.5v,3v);
-z315=(15v,3v);
-
-z40=(0v,4v);
-z415=(1.5v,4v);
-z455=(5.5v,4v);
-z47=(7v,4v);
-z48=(8v,4v);
-z495=(9.5v,4v);
-z413=(13.5v,4v);
-z416=(15v,4v);
-
-z50=(0v,5v);
-z515=(1.5v,5v);
-z53=(3v,5v);
-z54=(4v,5v);
-z555=(5.5v,5v);
-z57=(7v,5v);
-z58=(8v,5v);
-z595=(9.5v,5v);
-z511=(11v,5v);
-z512=(12v,5v);
-z513=(13.5v,5v);
-z516=(15v,5v);
-
-z60=(0v,6v);
-z61=(1v,6v);
-z62=(2v,6v);
-z63=(3v,6v);
-z64=(4v,6v);
-z65=(5v,6v);
-z66=(6v,6v);
-z67=(7v,6v);
-z68=(8v,6v);
-z69=(9v,6v);
-z610=(10v,6v);
-z611=(11v,6v);
-z612=(12v,6v);
-z613=(13v,6v);
-z614=(14v,6v);
-z615=(15v,6v);
-
-z71=(1v,7v);
-z72=(2v,7v);
-z75=(5v,7v);
-z76=(6v,7v);
-z79=(9v,7v);
-z710=(10v,7v);
-z713=(13v,7v);
-z714=(14v,7v);
-
-
-
-
-pickup pencircle scaled 0.4pt;
-draw(z10--z115--z215--z20--cycle);
-draw(z30--z37--z47--z40--cycle);
-draw(z38--z315--z416--z48--cycle);
-draw(z50--z53--z63--z60--cycle);
-draw(z54--z57--z67--z64--cycle);
-draw(z58--z511--z611--z68--cycle);
-draw(z512--z516--z615--z612--cycle);
-drawarrow(z71--z61);
-drawarrow(z72--z62);
-drawarrow(z75--z65);
-drawarrow(z76--z66);
-drawarrow(z79--z69);
-drawarrow(z710--z610);
-drawarrow(z713--z613);
-drawarrow(z714--z614);
-drawarrow(z515--z415);
-drawarrow(z555--z455);
-drawarrow(z595--z495);
-drawarrow(z513--z413);
-drawarrow(z335--z235);
-drawarrow(z311--z211);
-drawarrow(z175--z1075);
-
-label.top(btex $M_8$ etex,z175);
-label.top(btex $M_4$ etex,z335);
-label.top(btex $M_4$ etex,z311);
-label.top(btex $M_2$ etex,z515);
-label.top(btex $M_2$ etex,z555);
-label.top(btex $M_2$ etex,z595);
-label.top(btex $M_2$ etex,z513);
-
-endfig;
-
-
-beginfig(7);
-v:=u*7mm;
-
-z12=(1v,2v);
-z13=(1v,3v);
-z16=(1v,6v);
-z356=(3.5v,6v);
-z40=(5v,1v);
-z42=(5v,2v);
-z47=(5v,6.5v);
-z72=(9v,2v);
-z74=(9v,4v);
-z76=(9v,6v);
-
-z100=(0.5v,0.5v);
-z101=(1.5v,0.5v);
-
-z0=whatever[z13,z356];
-z1=whatever[z356,z74];
-z1=z0+4v*right;
-z2=whatever[z12,z72];
-z2=z0+whatever*down;
-z3=whatever[z12,z72];
-z3=z1+whatever*down;
-z4=z356+4v*right;
-z5=whatever[z72,z76];
-z5=whatever[z4,z1+4v*right];
-z6=whatever[z40,z47];
-z6=z74+4v*left;
-
-
-pickup pencircle scaled 0.4pt;
-draw(z16--z12--z72--z76);
-draw(z13--z356--z74);
-draw(z40--z47) dashed evenly;
-draw(z1--z4) dashed withdots scaled 0.7;
-draw(z4--z5) dashed withdots scaled 0.7;
-draw(z0--z6) dashed withdots scaled 0.7;
-draw(z0--z2) dashed evenly;
-draw(z1--z3) dashed evenly;
-
-draw(z100--z101) dashed withdots scaled 0.7;
-
-pickup pencircle scaled 3pt;
-drawdot(z0);
-drawdot(z1);
-
-label.bot(btex \strut 0 etex,z12);
-label.bot(btex $k$ etex,z2);
-label.llft(btex \strut ${n\over 2} - 1$ etex,z42);
-label.bot(btex \strut $k+{n\over 2}$ etex,z3);
-label.bot(btex \strut $n-1$ etex,z72);
-label.rt(btex posloupnost prohozen\'a separ\'atorem etex,z101);
-
-endfig;
-
-
-
-
-
-
-
-end;
diff --git a/5-addsort/sortnet.mpx b/5-addsort/sortnet.mpx
deleted file mode 100644
index 9814d6d..0000000
--- a/5-addsort/sortnet.mpx
+++ /dev/null
@@ -1,548 +0,0 @@
-% Written by DVItoMP, Version 0.64/color (Web2C 7.5.4)
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("a",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,-3.4869)--(4.9813,-3.4869)--
- (4.9813,8.4682)--(0,8.4682)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("b",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,-3.4869)--(5.5348,-3.4869)--
- (5.5348,8.4682)--(0,8.4682)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("min",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(16.6045,0.0000)--
- (16.6045,6.6536)--(0,6.6536)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("max",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(18.5416,0.0000)--
- (18.5416,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x1",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x2",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x3",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x4",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x5",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x1",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x2",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x3",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x4",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("x5",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(10.2394,0.0000)--
- (10.2394,6.4204)--(0,6.4204)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("0",_n2,1.00000,5.6939,-1.4944,);
-setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
- (10.1633,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("1",_n2,1.00000,5.6939,-1.4944,);
-setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
- (10.1633,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("2",_n2,1.00000,5.6939,-1.4944,);
-setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
- (10.1633,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s(":",_n1,1.00000,0.0000,0.0000,);
-_s(":",_n1,1.00000,4.4278,0.0000,);
-_s(":",_n1,1.00000,8.8556,0.0000,);
-setbounds _p to (0,0.0000)--(13.2834,0.0000)--
- (13.2834,1.0516)--(0,1.0516)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n3="cmmi7";
-_s("n",_n3,1.00000,5.6939,-1.4944,);
-_n4="cmsy7";
-_s(char0,_n4,1.00000,10.6188,-1.4944,);
-_n2="cmr7";
-_s("2",_n2,1.00000,16.8454,-1.4944,);
-setbounds _p to (0,-2.3246)--(21.3148,-2.3246)--
- (21.3148,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("x",_n1,1.00000,0.0000,0.0000,);
-_n3="cmmi7";
-_s("n",_n3,1.00000,5.6939,-1.4944,);
-_n4="cmsy7";
-_s(char0,_n4,1.00000,10.6188,-1.4944,);
-_n2="cmr7";
-_s("1",_n2,1.00000,16.8454,-1.4944,);
-setbounds _p to (0,-2.3246)--(21.3148,-2.3246)--
- (21.3148,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("n",_n1,1.00000,0.0000,0.0000,);
-setbounds _p to (0,0.0000)--(5.9799,0.0000)--
- (5.9799,4.2895)--(0,4.2895)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n3="cmmi7";
-_s("n",_n3,1.00000,6.1090,-1.4944,);
-setbounds _p to (0,-1.4944)--(11.5320,-1.4944)--
- (11.5320,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("2",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("2",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("4",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("4",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("4",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("S",_n1,1.00000,0.0000,0.0000,);
-_n5="cmmi5";
-_s("n",_n5,1.00000,7.3045,1.1829,);
-interim linecap:=0;
-vardef _r(expr _a,_w)(text _t) =
-  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
-_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
-_n6="cmr5";
-_s("4",_n6,1.00000,7.8033,-3.8949,);
-setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
- (13.3857,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("Bitonic",_n0,1.00000,0.0000,0.0000,);
-_s("k"&char19,_n0,1.00000,31.1333,0.0000,);
-_s("a",_n0,1.00000,36.3914,0.0000,);
-_s("t",_n0,1.00000,44.6936,0.0000,);
-_s(char20,_n0,1.00000,48.0283,0.0000,);
-_s("r",_n0,1.00000,48.5680,0.0000,);
-_s(char19,_n0,1.00000,51.3631,0.0000,);
-_s(char16&"di",_n0,1.00000,52.4700,0.0000,);
-_s(char20,_n0,1.00000,63.2629,0.0000,);
-_s("ck",_n0,1.00000,63.5397,0.0000,);
-_s("a",_n0,1.00000,72.6721,0.0000,);
-_n1="cmmi10";
-_s("B",_n1,1.00000,80.9743,0.0000,);
-_n3="cmmi7";
-_s("n",_n3,1.00000,88.5311,-1.4944,);
-setbounds _p to (0,-1.4944)--(93.9541,-1.4944)--
- (93.9541,6.9185)--(0,6.9185)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("M",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("8",_n2,1.00000,9.6652,-1.4944,);
-setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
- (14.1345,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("M",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("4",_n2,1.00000,9.6652,-1.4944,);
-setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
- (14.1345,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("M",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("4",_n2,1.00000,9.6652,-1.4944,);
-setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
- (14.1345,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("M",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("2",_n2,1.00000,9.6652,-1.4944,);
-setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
- (14.1345,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("M",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("2",_n2,1.00000,9.6652,-1.4944,);
-setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
- (14.1345,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("M",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("2",_n2,1.00000,9.6652,-1.4944,);
-setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
- (14.1345,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("M",_n1,1.00000,0.0000,0.0000,);
-_n2="cmr7";
-_s("2",_n2,1.00000,9.6652,-1.4944,);
-setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
- (14.1345,6.8078)--(0,6.8078)--cycle;
-_p endgroup
-mpxbreak
-begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
-string _n[];
-vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
-  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n0="cmr10";
-_s("0",_n0,1.00000,0.0000,0.0000,);
-setbounds _p to (0,-3.4869)--(4.9813,-3.4869)--
- (4.9813,8.4682)--(0,8.4682)--cycle;
-_p endgroup
-mpxbreak
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-\input lecnotes.tex
-
-\prednaska{6}{Vyhledávání v~textu}{(zapsal: Petr Jankovský)}
-
-Nyní se budeme vìnovat následujícímu problému: v~textu délky $S$ (senì) budeme chtít najít v¹echny výskyty hledaného slova délky $J$ (jehly). Nejprve se podívejme na~jeden primitivní algoritmus, který nefunguje. Je ale zajímavé rozmyslet si, proè.
-
-\h{Hloupý algoritmus} 
-Zaèneme prvním písmenkem hledaného slova a~budeme postupnì procházet text, a¾ najdeme první výskyt poèáteèního písmenka. Poté budeme testovat, zda souhlasí i~písmenka dal¹í. Pokud nastane neshoda, v~hledaném slovì se vrátíme na~zaèátek a~v~textu pokraèujeme znakem, ve~kterém neshoda nastala. Podívejme se na~pøíklad.
-
-\s{Pøíklad:} Budeme hledat slovo |jehla| v~textu |jevkupcejejehla|. Vezmeme si tedy první písmenko |j| v~hledaném slovì a~zjistíme, ¾e v~textu se nachází hned na~zaèátku. Vezmeme tedy dal¹í písmenko |e|, které se vyskytuje jako druhé i~v~textu. Pøi tøetím písmenku ale narazíme na~neshodu. V~tuto chvíli tedy zresetujeme a~opìt hledáme výskyt písmenka |j|, tentokrát v¹ak a¾ od~tøetího písmene v~textu. Takto postupujeme postupnì dál, a¾ narazíme na~dal¹í |je|, které ov¹em není následováno písmenem~|h|, tudí¾ opìt zresetujeme a~nakonec najdeme shodu s~celým hledaným øetìzcem. V~tomto pøípadì tedy algoritmus na¹el hledané slovo.
-
-Tento algoritmus v¹ak zjevnì mù¾e hanebnì selhat. Mù¾e se stát, ¾e zaèneme porovnávat, a¾ v~jednu chvíli narazíme na~neshodu. Celý tento kus tedy pøeskoèíme. Pøi tom se ale v~tomto kusu textu mohl vyskytovat nìjaký pøekrývající se výskyt hledané \uv{jehly}. Hledejme napøíklad øetìzec |kokos| v~textu |clanekokokosu|. Algoritmus tedy zaène porovnávat. Ve~chvíli kdy najde prefix |koko| a~na~vstupu dostane~|k|, dochází k~neshodì. Proto algoritmus zresetuje a~pokraèuje v~hledání od~tohoto znaku. Najde sice je¹tì výskyt |ko|, ov¹em s~dal¹ím písmenkem |s| ji¾ dochází k~neshodì a~algotimus sel¾e. Nesprávnì se toti¾ \uv{upnul} na~první nalezené |koko| a~s~dal¹ím~|k| pak \uv{zahodil} i~správný zaèátek.
-
-Máme tedy algoritmus, který i~kdy¾ je ¹patnì, tak funguje urèitì kdykoli se první písmenko hledaného slova v~tomto slovì u¾ nikde jinde nevyskytuje -- co¾ |jehla| splòovala, ale |kokos| u¾ ne.
-
-{\I Hloupý algoritmus} se na~ka¾dé písmenko textu podívá jednou, tudí¾ èasová slo¾itost bude lineární s~délkou textu ve~kterém hledáme -- tedy $\O(S)$.
-
-\h{Pomalý algoritmus}
-Zkusíme algoritmus vylep¹it tak, aby fungoval správnì: pokud nastane nìjaká neshoda, vrátíme se zpátky tìsnì za~zaèátek toho, kdy se nám to zaèalo shodovat. To je ov¹em vlastnì skoro toté¾, jako brát postupnì v¹echny mo¾né zaèátky v~\uv{senì} a~pro ka¾dý z~nìj ovìøit, jestli se tam \uv{jehla} nachází èi nikoliv.
-
-Tento algoritmus evidentnì funguje. Bì¾í v¹ak v~èase: $S$ mo¾ných zaèátkù, krát èas potøebný na~jedno porovnání (zda se na~dané pozici nenachází \uv{jehla}), co¾ nám mù¾e trvat a¾ $J$. Proto je èasová slo¾itost $\O(SJ)$. V praxi bude algoritmus èasto rychlej¹í, proto¾e typicky velmi brzo zjistíme, ¾e se øetìzce neshodují, ale je mo¾né vymyslet vstup, kde bude potøeba porovnání opravdu tolik.
-
-Nyní se pokusme najít takový algoritmus, který by byl tak rychlý, jako {\I Hloupý algoritmus}, ale chytrý, jako ten {\I Pomalý}.
-
-\h{Chytrý algoritmus}
-Ne¾ vlastní algoritmus vybudujeme, zkusíme se cestou nauèit pøemý¹let o~øetìzcích obèas trochu pøekrouceným zpùsobem. Podívejme se na~je¹tì jeden pøíklad.
-
-\s{Pøíklad:} Vezmìme si napøíklad staré italské pøízvisko |barbarossa|, které znamená Rudovous. Pøedstavme si, ¾e takovéto slovo hledáme v~nìjakém textu, který zaèíná |barbar|. Víme, ¾e a¾ sem se nám hledaný øetìzec shodoval. Øeknìme, ¾e dal¹í písmenko textu se shodovat pøestane -- místo |o| naèteme napøíklad opìt |b|. {\I Hloupý algoritmus} by velil vrátit se k~|a| a~od~nìj hledat dál. Uvìdomme si ale, ¾e kdy¾ se vracíme z~|barbar| do~|arbar| (tedy øetìzce, který ji¾ známe), mù¾eme si pøedpoèítat, jak dopadne hledání, kdy¾ ho pustíme na~nìj. V~pøedpoèítaném bychom tedy chtìli ukládat, ¾e kdy¾ máme øetìzec |arbar|, tak |ar| a~|r| nám do~hledaného nepasuje a~a¾~|bar| se bude shodovat. Tedy místo toho, abychom spustili nové hledání od~|a|, mù¾eme ho spustit a¾~od~|b|. Co víc, my dokonce víme, jak dopadne to -- pokud toti¾ nastane neshoda po~pøeètení |barbar|, je to stejné, jako kdybychom pøeèetli pouze |bar|, na~které se (pùvodne neshodující se) |b| u¾ navázat dá. Kdyby se nedalo navázat ani tam, tak bychom opìt zkracovali... Nejen, ¾e tedy víme, kam se máme vrátit, ale víme dokonce i~to, co tam najdeme. 
-
-My¹lenka, ke které míøíme, je pøedpoèítat si nìjakou tabulku, která nám bude øíkat, jak se máme pøi hledání vracet a~jak to dopadne, a~pak u¾ jenom prohlédávat s~pou¾itím této tabulky. 
-
-Aby se nám o~tìchto algoritmech lépe mluvilo a~pøedev¹ím psalo, pojïme si povìdìt nìkolik definic.
- 
-\s{Definice:}
-\itemize\ibull
-\:{\I Abeceda $\Sigma$} je koneèná mno¾ina znakù\foot{Mù¾eme pøi tom jít a¾~do~extrémù. Pøíkladem extrémních abeced je binární abeceda slo¾ená pouze z~nul a~jednièek. Pøíklad z~druhého konce (který rádi dìlají lingvisti) je abeceda, která má jako abecedu v¹echna èeská slova. V¹echny èeské vìty, pak nejsou nic jiného, ne¾ slova nad touto abecedou. Pou¾itá abeceda tedy mù¾e být i~relativnì obrovská. Dal¹ím takovým pøíkladem mù¾e být Unicode. Pro na¹e potøeby ale zatím budeme pøedpokládat, ¾e abeceda je nejen konstantnì velká, ale i~rozumnì malá. Budeme si moci tedy dovolit napøíklad indexovat pole znakem abecedy (kdybychom nemohli, tak bychom místo pole pou¾ili napøíklad hashovací tabulku, èi nìco podobného\dots).}, ze~kterých tvoøíme text, øetìzce, slova. 
-
-
-\:{\I $\Sigma^*$} je mno¾ina v¹ech slov nad abecedou $\Sigma$. Èili mno¾ina v¹ech neprázdných koneèných posloupností znakù ze $\Sigma$.
-\endlist
-\s{Znaèení:}
-Aby se nám nepletlo znaèení, budeme rozli¹ovat promìnné pro slova, promìnné pro písmenka a~promìnné pro èísla.
-
-\itemize\ibull
-\:{\I Slova} budeme znaèit malými písmenky øecké abecedy $\alpha$, $\beta$, ... .
-\:$\iota$ bude oznaèovat \uv{jehlu}
-\:$\sigma$ bude oznaèovat \uv{seno}
-\:{\I Znaky} oznaèíme malými písmeny latinky $a$, $b$, \dots .
-\:{\I Èísla} budeme znaèit velkými písmeny $A$, $B$, \dots .
-\:{\I Délka slova} $\vert \alpha  \vert$ pro $\alpha \in \Sigma^*$ je poèet jeho znakù.
-\:{\I Prázdné slovo} znaèíme písmenem $\varepsilon$, $\vert \varepsilon \vert = 0$.
-\:{\I Zøetìzení} $\alpha\beta$ vznikne zapsáním slov $\alpha$ a~$\beta$ za sebe. Platí $\vert \alpha\beta  \vert=\vert \alpha \vert+\vert \beta \vert$, $\alpha\varepsilon=\varepsilon\alpha=\alpha$.
-\:$\alpha[k]$ je $k$-tý znak slova $\alpha$, indexujeme od~$0$.
-\:$\alpha[k:l]$ je podslovo zaèínající $k$-tým znakem a~$l$-tý znak je první, který v~nìm není. Jedná se tedy o~podslovo skládající se z~$\alpha[k]$,$\alpha[k+1]$,\dots,$\alpha[l-1]$. Platí tedy: $\alpha[k:k]=\varepsilon$, $\alpha[k:k+1]=\alpha[k]$. Jednu (èi obì) meze mù¾eme i~vynechat, tento zápis pak bude znamenat buï \uv{od zaèátku slova a¾ nìkam}, nebo \uv{odnìkud a¾ do~konce}:
-%TODO - zaøadit pod pøedchozí bod
-\:$\alpha[:k]$ je {\I prefix} obsahující prvních $k$ znakù slova $\alpha$ ($\alpha[0]$,\dots,$\alpha[k-1]$).
-\:$\alpha[k:]$ je {\I suffix} obsahující znaky slova $\alpha$ poèínaje $k$-tým znakem a¾ do~konce.
-\:$\alpha[:] = \alpha$
-\endlist
-
-
-V¹imnìme si, ¾e prázdné slovo je prefixem, suffixem i~podslovem jakéhokoliv slova vèetnì sebe sama.
-Ka¾dé slovo je také prefixem, suffixem i~podslovem sebe sama. To se ne v¾dy hodí. Nìkdy budeme chtít øíct, ¾e nìjaké slovo je {\I vlastním} prefixem nebo suffixem. To bude znamenat, ¾e to nebude celé slovo.
-
-\> $\alpha$ je {\I vlastní prefix} slova $\beta \equiv \alpha$ je prefix $\beta~\&~\alpha \neq \beta$.
-
-\h{Vyhledávací automat (Knuth, Morris, Pratt)}
-{\I Vyhledávací automat} bude graf, jeho¾ vrcholùm budeme øíkat {\I stavy}. Jejich jména budou prefixy hledaného slova a~hrany budou odpovídat tomu, jak jeden prefix mù¾eme získat z~pøedchozího prefixu pøidáním jednoho písmene. Poèáteèní stav je prázdné slovo $\varepsilon$ a~koncový je celá $\iota$. Dopøedné hrany grafu budou popisovat pøechod mezi stavy ve~smyslu zvìt¹ení délky jména stavu (dopøedná funkce $h(\alpha)$, urèující znak na~dopøedné hranì z~$\alpha$). Zpìtné hrany grafu budou popisovat pøechod (zpìtná funkce $z(\alpha)$) mezi stavem $\alpha$ a~nejdel¹ím vlastním suffixem $\alpha$, který je prefixem $\iota$, kdy¾ nastane neshoda.
-
-\figure{barb.eps}{Vyhledávací automat.}{4.1in}
-
-\s{Hledej($\sigma$):}
-\algo
-\:$\alpha \leftarrow \varepsilon$.
-\:Pro $x\in\sigma$ postupnì:
-\:$\indent$Dokud $h(\alpha) \neq x~\&~\alpha \neq \varepsilon : \alpha \leftarrow z(\alpha)$. 
-\:$\indent$Pokud $h(\alpha) = x: \alpha \leftarrow \alpha x$.
-\:$\indent$Pokud $\alpha = \iota$, ohlásíme výskyt.
-\endalgo
-
-\>Vstupem je $\iota$, hledané slovo (jehla) délky $J=\vert \iota \vert$ a~$\sigma$, text (seno) délky $S=\vert \sigma \vert$.
-\>Výstupem jsou v¹echny výskyty hledaného slova $\iota$ v~textu $\sigma$, tedy mno¾ina $\left\{ k \mid \sigma[k:k+J]=\iota \right\}$
-
-Pojïme nyní dokázat, ¾e tento algoritmus správnì ohlásí v¹echny výskyty.
-
-\s{Definice}: $\alpha(\tau) := $ stav automatu po~pøeètení $\tau$
-
-\s{Invariant:} Pokud algoritmus pøeète nìjaký vstup, nachází se ve~stavu, který je nejdel¹ím suffixem pøeèteného vstupu, který je nìjakým stavem.
-$\alpha(\tau) =$ nejdel¹í stav (nejdel¹í prefix jehly), který je suffixem $\tau$ (pøeèteného vstupu).
-
-Pojïme si rozmyslet, ¾e z~tohoto invariantu ihned plyne, ¾e algoritmus najde to, co má. Kdykoli toti¾ ohlásí nìjaký výskyt, tak tam tento výskyt opravdu je. Kdykoli pak má nìjaký výskyt ohlásit, tak se v~této situaci jako suffix toho právì pøeèteného textu vyskytuje hledané slovo, pøièem¾ hledané slovo je urèitì stav a~zároveò nejdel¹í ze v¹ech existujících stavù. Tak¾e invariant opravdu øíká, ¾e jsme právì v~koncovém stavu a~algoritmus nám tedy ohlásí výskyt.
-
-\proof {\I (invariantu)}
-Indukcí podle kroku algoritmu. Na~zaèátku pro prázdný naètený vstup invariant triviálnì platí, tedy prázdný suffix $\tau$ je prefixem $\iota$. V~kroku $n$ máme naètený vstup $\tau$ a~k~nìmu pøipojíme znak $x$. Invariant nám øíká, ¾e nejdel¹í stav, který je suffixem, je nejdel¹í suffix, který je stavem. Nyní se ptáme, jaký je nejdel¹í stav, který se dá \uv{napasovat} na~konec øetìzce $\tau x$. Kdykoli v¹ak takovýto suffix máme, tak z~nìj mù¾eme $x$ na~konci odebrat, èím¾ dostaneme suffix slova $\tau$.
-
-\>Tedy: pokud $\beta$ je neprázdným suffixem slova $\tau x$, pak $\beta = \gamma x$, kde $\gamma$ je suffix $\tau$.
-
-Suffix, který máme sestrojit, tedy vznikne z~nìjakého suffixu slova $\tau$ pøipsáním~$x$. Chceme najít nejdel¹í suffix slova $\tau x$, který je stavem, tak¾e chceme najít i~nejdel¹í suffix pùvodního slova $\tau$, za který se dá pøidat $x$ tak, aby vy¹lo jméno stavu. Staèí tedy u¾ jen \uv{probírat} suffixy slova $\tau$ od~nejdel¹ího po~nejkrat¹í, zkou¹et k~nim pøidávat $x$ a~a¾ to pùjde, tak jsme na¹li nejdel¹í suffix $\tau x$. Pøesnì toto ov¹em algoritmus dìlá, nebo» zpìtná funkce mu v¾dy øekne nejbli¾¹í krat¹í suffix, který je stavem. Pokud pak nemù¾eme $x$ pøidat ani do~$\varepsilon$, pak je øe¹ením prázdný suffix. Algoritmus tedy funguje. \qed
-
-Nyní pojïmì zkoumat to, jak je ve~skuteènosti ná¹ algoritmus rychlý. K tomu bychom si ale nejdøív mìli øíct, jak pøesnì budeme automat reprezentovat. V~algoritmu vystupují nìjaká porovnávání stavù, pøièem¾ není úplnì jasné, jak zaøídit, aby v¹e trvalo konstantnì dlouho. Vyjde nám to ale docela snadno. K reprezentaci automatu nám toti¾ budou staèit pouze dvì pole.
-
-\s{Reprezentace automatu:}
-Oèíslujeme si stavy délkami pøíslu¹ných prefixù, tedy èísly $0 \dots J$. Poté je¹tì potøebujeme nìjakým zpùsobem zakódovat dopøedné a~zpìtné hrany. Vzhledem k~tomu, ¾e z~ka¾dého vrcholu vede v¾dy nejvý¹e jedna dopøedná a~nejvý¹e jedna zpìtná, tak nám evidentnì staèí pamatovat si pro ka¾dý typ hran pouze jedno èíslo na~vrchol. Budeme mít tedy nìjaké pole dopøedných hran, které nám pro ka¾dý stav øekne, jakým písmenkem je nadepsaná dopøedná hrana ze stavu $I$ do~$I+1$. To jsou ale pøesnì písmenka jehly, tak¾e si staèí pamatovat jehlu samotnou. Èili z~$I$ do~$I+1$ vede hrana nadepsaná $\iota [I]$. Pro zpìtné hrany pak budeme potøebovat pole $Z$, které nám pro stav $I$ øekne èíslo stavu, do~kterého vede zpìtná hrana. Tedy $Z[I]$ je cíl zpìtné hrany ze stavu $I$.
-S~touto reprezentací ji¾ doká¾eme na¹i hledací proceduru pøímoèaøe pøepsat tak, aby sahala pouze do~tìchto dvou polí:
-\algo
-\:$I \leftarrow 0$.
-\:Pro znaky $x$ z~textu:
-\:$\indent$Dokud $\iota[I] \neq x~\&~I \neq 0: I \leftarrow Z[I]$.
-\:$\indent$Pokud $\iota[I] = x$, pak $I \leftarrow I + 1$.
-\:$\indent$Pokud $I = J$, ohlásíme výskyt.
-\endalgo
-
-Zatím se v~algoritmu je¹tì skrývá drobná chyba -- toti¾ algoritmus se obèas zeptá na~dopøednou hranu z~posledního stavu. Pokud jsme právì ohlásili výskyt (jsme tedy v~posledním stavu) a~pøijde nìjaký dal¹í znak, algoritmus se ptá, zda je roven tomu, co je na~dopøedné hranì z~posledního stavu. Ta ale ov¹em neexistuje. Jednodu¹e to ale napravíme tak, ¾e si pøidáme fiktivní hranu, na~které se vyskytuje nìjaké \uv{nepísmenko} -- nìco, co se nerovná ¾ádnému jinému písmenku. Zajistíme tak, ¾e se po~této hranì nikdy nevydáme. Dodefinujeme tedy $\iota[J]$ odli¹nì od~v¹ech znakù.\foot{V jazyce C se toto dodefinování provede vlastnì zadarmo, nebo» ka¾dý øetìzec je v~nìm ukonèen znakem s~kódem nula, který se ve~vstupu nevyskytne\dots Algoritmus bude tedy fungovat i~bez tohoto dodefinování. V jiných jazycích je ale tøeba na~nìj nezapomenout!}
-
-\s{Lemma:} Funkce {\I Hledej} bì¾í v~èase $\O(S)$.
-
-\proof
-Funkce {\I Hledej} chodí po~dopøedných a~zpìtných hranách. Dopøedných hran projdeme urèitì maximálnì tolik, kolik je délka sena. Pro ka¾dý znak pøeètený ze sena toti¾ jdeme nejvý¹e jednou po~dopøedné hranì. Se zpìtnými hranami se to má tak, ¾e na~jeden pøeètený znak z~textu se mù¾eme po~zpìtné hranì vracet maximálnì $J$-krát. Z~tohoto by nám v¹ak vy¹la slo¾itost $\O(JS)$, èím¾ bychom si nepomohli. Zachrání nás ale pøímoèarý potenciál. Uvìdomme si, ¾e chùze po~dopøedné hranì zvý¹í $I$ o~jedna a~chùze po~zpìtné hranì $I$ sní¾í alespoò o~jedna. Vzhledem k~tomu, ¾e $I$ není nikdy záporné a~na~zaèátku je nulové, zjistíme, ¾e krokù zpìt mù¾e být maximálnì tolik, kolik krokù dopøedu. Èasová slo¾itost hledání je tedy lineární vzhledem k~délce sena. \qed
-
-Nyní nám zbývá na~první pohled malièkost -- toti¾ zkonstruovat automat. Zkonstruovat dopøedné hrany zvládneme zjevnì snadno, jsou toti¾ explicitnì popsané hledaným slovem. Tì¾¹í u¾ to bude pro hrany zpìtné. Vyu¾ijeme k~tomu následující pozorování:
-
-\s{Pozorování:}
-Pøedstavme si, ¾e automat u¾ máme hotový a~tím, ¾e budeme sledovat jeho chování, chceme zjistit, jak v~nìm vedou zpìtné hrany.
-Vezmìme si nìjaký stav~$\beta$. To, kam z~nìj vede zpìtná hrana zjistíme tak, ¾e spustíme automat na~øetìzec $\beta$~bez prvního písmenka a~stav, ve~kterém se automat zastaví, je pøesnì ten, kam má vést i~zpìtná hrana z~$\beta$. Jinými slovy víme, ¾e $z(\beta) = \alpha (\beta[1:])$. 
-Proè takováto vìc funguje? V¹imìme si, ¾e definice $z$ a~to, co nám o~$\alpha$ øíká invariant, je témìø toto¾né -- $z(\beta)$ je nejdel¹í vlastní suffix $\beta$, který je stavem, $\alpha(\beta)$ je nejdel¹í suffix $\beta$, který je stavem. Jediná odli¹nost je v~tom, ¾e definice $z$ narozdíl od~definice $\alpha$ zakazuje nevlastní suffixy. Jak nyní vylouèit suffix $\beta$, který by byl roven $\beta$ samotné? Zkrátíme $\beta$ o~první znak. Tím pádem v¹echny suffixy $\beta$ bez prvního znaku jsou stejné jako v¹echny vlastní suffixy $\beta$.
-
-K èemu je toto pozorování dobré? Rozmysleme si, ¾e pomocí nìj u¾ doká¾eme zkonstruovat zpìtné hrany. Není to ale trochu divné, kdy¾ pøi simulování automatu na~øetìzec bez prvního znaku u¾ zpìtné hrany potøebujeme? Není. Za chvíli zjistíme, ¾e takto mù¾eme zji¹»ovat zpìtné hrany postupnì -- a~to tak, ¾e pou¾íváme v¾dy jenom ty, které jsme u¾ sestrojili.
- 
-Takovémuhle pøístupu, kdy pøi konstruování chtìného u¾ pou¾íváme to, co chceme sestrojit, ale pouze ten kousek, který ji¾ máme hotový, se v~angliètinì øíká {\I bootstrapping}\foot{Z~tohoto slova vzniklo i~{\I bootování} poèítaèù, kdy operaèní systém v~podstatì zavádí sám sebe. Bootstrap znamená èesky ¹truple -- tedy oèko na~konci boty, které slou¾í k~usnadnìní nazouvání. A~jak souvisí ¹truple s~algoritmem? To se zase musíme vrátit k~pøíbìhùm o~baronu Prá¹ilovi, mezi nimi¾ je i~ten, ve~kterém baron Prá¹il vypráví o~tom, jak sám sebe vytáhl z~ba¾iny za ¹truple. Stejnì tak i~my budeme algoritmus konstruovat tím, ¾e se budeme sami vytahovat za ¹truple, tedy bootstrappovat.}.
-V¹imnìme si, ¾e pøi výpoètu se vstupem $\beta$ projde automat jenom prvních $\vert \beta  \vert$ stavù. Automat se evidentnì nemù¾e dostat dál, proto¾e na~ka¾dý krok dopøedu (doprava) spotøebuje písmenko $\beta$. Tak¾e krokù doprava je maximálnì tolik, kolik je  $\vert \beta  \vert$. Jinými slovy kdybychom ji¾ mìli zkonstruované zpìtné hrany pro prvních  $\vert \beta  \vert$ stavù (tedy $0 \dots \vert \beta  \vert - 1$), tak pøi tomto výpoètu, který potøebujeme na~zkonstruování zpìtné hrany z~$\beta$, je¹tì tuto zpìtnou hranu nemù¾eme potøebovat. Vystaèíme si s~tìmi, které ji¾ máme zkonstruované.
-
-Nabízí se tedy zaèít zpìtnou hranou z~prvního znaku (která vede evidentnì do~$\varepsilon$), pak postupnì brát dal¹í stavy a~pro ka¾dý z~nich si spoèítat, kdy spustíme automat na~jméno stavu bez prvního znaku a~tím získáme dal¹í zpìtnou hranu. Toto funguje, ale je to kvadratické \dots. Máme toti¾ $J$ stavù a~pro ka¾dý z~nich nám automat bì¾í v~èase a¾ lineárním s~$J$. Jak z~toho ven?
-
-Z~prvního stavu povede zpìtná funkce do~$\varepsilon$. Pro dal¹í stavy chceme spoèítat zpìtnou funkci. Z~druhého stavu $\iota[0:2]$ tedy automat spustíme na~$\iota[1:2]$, dále pak na~$\iota[1:3]$, $\iota[1:4]$, atd. Ty øetìzce, pro které potøebujeme spo¹tìt automat, abychom dostali zpìtné hrany, jsou tedy ve~skuteènosti takové, ¾e ka¾dý dal¹í dostaneme roz¹íøením pøedchozího o~jeden znak. To jsou ale pøesnì ty stavy, kterými projde automat pøi zpracovávání øetìzce $\iota$ od~prvního znaku dál. Jedním prùchodem automatu nad jehlou bez prvního písmenka se tím pádem rovnou dozvíme v¹echny údaje, které potøebujeme.
-Z~pøedchozího pozorování plyne, ¾e nikdy nebudeme potøebovat zpìtnou hranu, kterou jsme je¹tì nezkonstruovali a~jeliko¾ víme, ¾e jedno prohledání trvá lineárnì s~délkou toho, v~èem hledáme, tak toto celé pobì¾í v~lineárním èase. Dostaneme tedy následující algoritmus:
-
-\s{Konstrukce zpìtné funkce:}
-\algo
-\:$Z[0] \leftarrow ?$, $Z[1] \leftarrow 0$.
-\:$I \leftarrow 0$.
-\:Pro $k = 2 \dots J$:
-\::$I \leftarrow \<Krok>( I , \iota [k])$.
-\::$Z[k] \leftarrow I$.
-\endalgo
-
-Zaèínáme tím, ¾e nastavíme zpìtnou hranu z~prvních dvou stavù, pøièem¾ $z[0]$ je nedefinované, proto¾e tuto zpìtnou hranu nikdy nepou¾íváme. Dále postupnì simulujeme výpoèet automatu nad slovem bez prvního znaku a~po ka¾dém kroku se dozvíme novou zpìtnou hranu. {\I Krokem} automatu pak není nic jiného ne¾ vnitøek (3. a~4. bod) na¹í hledací procedury. To, kam jsme se dostali, pak zaznamenáme jako zpìtnou funkci z~$k$.
-Èili pou¹tíme automat na~jehlu bez prvního písmenka, provedeme v¾dy jeden krok automatu (pøes dal¹í písmenko jehly) a~zapamatujeme si, jakou zpìtnou funkci jsme zrovna dostali. Díky pozorováním navíc víme, ¾e zpìtné hrany konstruujeme správnì, nikdy nepou¾ijeme zpìtnou hranu, kterou jsme je¹tì nesestrojili a~víme i~to, ¾e celou konstrukci zvládneme v~lineárním èase s~délkou jehly.
-
-\s{Vìta:} Algoritmus KMP najde v¹echny výskyty v~èase $O(J+S)$.
-
-\proof
-Lineární èas s~délkou jehly potøebujeme na~postavení automatu, lineární èas s~délkou sena pak potøebujeme na~samotné vyhledání.
-
-\h{Rabinùv-Karpùv algoritmus}
-
-Nyní si uká¾eme je¹tì jeden algoritmus na~hledání jedné jehly, který nebude mít v~nejhor¹ím pøípadì lineární slo¾itost, ale bude ji mít prùmìrnì. Bude daleko jednodu¹¹í a~uká¾e se, ¾e je v~praxi daleko rychlej¹í. Bude to algoritmus zalo¾ený na~hashování.
-
-
-Pøedstavme si, ¾e máme seno délky $S$ a~jehlu délky $J$, a~vezmìme si nìjakou hashovací funkci, které dáme na~vstup $J$-tici znakù (tedy podslova dlouhá jako jehla). Tato hashovací funkce nám je pak zobrazí do~mno¾iny $\{0,\ldots,N-1\}$ pro nìjaké dost velké~$N$. Jak nám toto pomù¾e pøi hledání jehly? Vezmeme si libovolné \uv{okénko} délky $J$ a~ne¾ budeme zji¹»ovat, zda se v~nìm jehla vyskytuje, tak si spoèítáme hashovací funkci a~porovnáme ji s~hashem jehly. Èili ptáme se, jestli je hash ze sena od~nìjaké pozice $I$ do~pozice $I+J$ roven hashi jehly -- formálnì: $h(\sigma [I: I+J ]) = h(\iota)$. Teprve tehdy, kdy¾ zjistíme, ¾e se hodnota hashovací fce shoduje, zaèneme doopravdy porovnávat øetìzce.
-
-Není to ale nìjaká hloupost? Mù¾e nám vùbec takováto konstrukce pomoci? Není to tak, ¾e na~spoèítání hashovací funkce z~$J$ znakù, potøebujeme tìch $J$ znakù pøeèíst, co¾ je stejnì rychlé, jako rovnou øetìzce porovnávat? Pou¾ijeme trik, který bude spoèívat v~tom, ¾e si zvolíme ¹ikovnou hashovací funkci. Udìláme to tak, abychom ji mohli pøi posunutí \uv {okénka} o~jeden znak doprava v~konstantním èase pøepoèítat. Chceme umìt z~$h(x_1 \dots x_j)$ spoèítat $h(x_2 \dots x_{j+1})$.
-Na~zaèátku si tedy spoèítáme hash jehly a~první $J$-tice znakù sena. Pak ji¾ jenom posouváme \uv {okénko} o~jedna, pøepoèítáme hashovací funkci a~kdy¾ se shoduje s~hashem jehly, tak porovnáme. Budeme pøitom vìøit tomu, ¾e pokud se tam jehla nevyskytuje, pak máme hashovací funkci natolik rovnomìrnou, ¾e pravdìpodobnost toho, ¾e se pøesto strefíme do~hashe jehly, je $1/N$. Jinými slovy jenom v~jednom z~øádovì $N$ pøípadù budeme porovnávat fale¹nì -- tedy provedeme porovnání a~vyjde nám, ¾e výsledek je neshoda. V~prùmìrném pøípadì tedy mù¾eme stlaèit slo¾itost a¾ témìø k~lineární.
-
-Podívejme se teï na~prùmìrnou èasovou slo¾itost. Budeme urèitì potøebovat èas na~projití jehly a~sena. Navíc strávíme nìjaký èas nad fale¹nými porovnáními, kterých bude v~prùmìru na~ka¾dý $N$-tý znak sena jedno porovnání s~jehlou -- tedy $SJ / N$, pøièem¾ $N$ mù¾eme zvolit dost velké na~to, abychom tento èlen dostali pod nìjakou rozumnou konstantu... Nakonec budeme potøebovat jedno porovnání na~ka¾dý opravdový výskyt, èemu¾ se nevyhneme. Pøipoèteme tedy je¹tì $J \cdot$ {\I $\sharp$výskytù}. Dostáváme tedy: $ \O(J+S+SJ/N+J \cdot$ {\I $\sharp$výskytù}).
-
-Zbývá malièkost -- toti¾ kde vzít hashovací funkci, která toto v¹e splòuje. Jednu si uká¾eme. Bude to vlastnì takový hezký polynom:
-$$h(x_1 \dots x_j) := \left(\sum_{I=1}^{J} x_I \cdot p^{J-I}\right) \bmod N.$$
-Jinak zapsáno se tedy jedná o:
-$$(x_1 \cdot p^{J-1} + x_2 \cdot p^{J-2} + \dots + x_J \cdot p^0 ) \bmod N.$$
-Po posunutí okénka o~jedna chceme dostat:
-$$(x_2 \cdot p^{J-1} + x_3 \cdot p^{J-2} + \dots + x_J \cdot p^1 + x_{J+1} \cdot p^0 ) \bmod N.$$
-Kdy¾ se ale podíváme na~èleny tìchto dvou polynomù, zjistíme, ¾e se li¹í jen o~málo. Pùvodní polynom staèí pøenásobit~$p$, odeèíst první èlen s~$x_1$ a~naopak pøièíst chybìjící èlen $x_{J+1}$. Dostáváme tedy:
-$$h(x_2 \dots x_{J+1}) = (p \cdot h(x_1 \dots x_J) - x_1 \cdot p^J + x_{J+1}) \bmod N.$$
-Pøepoèítání hashovací funkce tedy není nic jiného, ne¾ pøenásobení té minulé~$p$, odeètení nìjakého násobku toho znaku, který vypadl z~okénka, a~pøiètení toho znaku, o~který se okénko posunulo. Pokud tedy máme k~dispozici aritmetické operace v~konstantním èase, zvládneme konstantnì pøepoèítávat i~hashovací funkci.
-
-Tato hashovací funkce se dokonce nejen hezky poèítá, ale dokonce se i~opravdu \uv{hezky} chová (tedy \uv{rozumnì} náhodnì), pokud zvolíme vhodné~$p$. To bychom mìli zvolit tak, aby bylo rozhodnì nesoudìlné s~$N$ -- tedy $\<NSD>(p, N) = 1$. Aby se nám navíc dobøe projevilo modulo obsa¾ené v~hashovací funkci, mìlo by být~$p$ relativnì velké (lze dopoèítat, ¾e optimum je mezi $2/3 \cdot N$ a~$3/4 \cdot N$). S~takto zvoleným~$p$ se tato hashovací funkce chová velmi pøíznivì a~v~praxi má celý algoritmus takøka lineární èasovou slo¾itost (prùmìrnou).
-
-\h{Hledání více øetìzcù najednou}
-Nyní si zahrajeme tuté¾ hru, ov¹em v~trochu slo¾itìj¹ích kulisách. Podíváme se na~algoritmus, který si poradí i~s více ne¾ jednou jehlou. 
-Mìjme tedy jehly $\iota_1 \dots \iota_n$, a~jejich délky $J_i = \vert \iota_i \vert $. Dále budeme potøebovat seno $\sigma$ délky $S=\vert \sigma \vert$.
-
-Pøedtím, ne¾ se pustíme do~vlastního vyhledávacího algoritmu, mo¾ná bychom si mìli ujasnit, co vlastnì bude jeho výstupem. U problému hledání jedné jehly to bylo jasné -- byla to nìjaká mno¾ina pozic v~senì, na~kterých zaèínaly výskyty jehly. Jak tomu ale bude zde? Sice bychom také mohli vrátit pouze mno¾inu pozic, ale my budeme chtít malièko víc. Budeme toti¾ chtít vìdìt i~to, která jehla se na~které pozici vyskytuje. Výstup tedy bude vypadat následovnì: $V = \{(i,j)~\vert~\sigma[i:i+J_j]= \iota_j \}$.
-
-Zde se v¹ak skrývá jedna drobná zrada. Budeme se asi muset vzdát nadìje, ¾e najdeme algoritmus, jeho¾ slo¾itost je lineární v~celkové délce v¹ech jehel a~sena. Výstup toti¾ mù¾e být del¹í ne¾ lineární. Mù¾e se nám klidnì stát, ¾e na~jedné pozici v~senì se bude vyskytovat více rùzných jehel -- pokud bude jedna jehla prefixem jiné (co¾ jsme nikde nezakázali), tak máme povinnost ohlásit oba výskyty. Vzhledem k~tomu budeme hledat takový algoritmus, který bude lineární v~délce vstupu plus délce výstupu, co¾ je evidentnì to nejlep¹í, èeho mù¾eme dosáhnout.
-
-Algoritmus, který si nyní uká¾eme, vymysleli nìkdy v~roce 1975 pan Aho a~paní Corasicková. Bude to takové zobecnìní Knuthova-Morrisova-Prattova algoritmu.
-
-\h{Algoritmus Aho-Corasicková}
-
-Opìt se budeme sna¾it sestrojit nìjaký vyhledávací automat a~nìjakým zpùsobem tento automat pou¾ít k~procházení sena. Podívejme se nejprve na~pøíklad. Budeme chtít vyhledávat tato slova: |ara|, |bar|, |arab|, |baraba|, |barbara|. Mìjme tedy tìchto pìt jehel a~rozmysleme si, jak by vypadal nìjaký automat, který by tato slova umìl zatím jenom rozpoznávat. Pro jedno slovo automat vypadal jako cesta, zde u¾ to bude strom. (viz obrázek).
-
-\figure{ara_strom_blank.eps}{Vyhledávací automat -- strom.}{1in}
-
-Navíc budeme muset do~automatu zanést, kde nìjaké slovo konèí. V~pùvodním automatu pro jedno slovo to bylo jednoduché -- ono jedno jediné slovo odpovídalo poslednímu vrcholu cesty. Tady se v¹ak slova mohou vyskytovat vícekrát a~konèit nejenom v~listech ale i~v~nìjakém vnitøním vrcholu (co¾ se stane tehdy, pokud je jedno hledané slovo prefixem jiného hledaného slova). Formálnì to nebudeme dokazovat, ale snadno nahlédneme, ¾e listy stromu odpovídají hledaným slovùm, ale opaènì to neplatí.
-
-\figure{ara_strom_end.eps}{Vyhledávací automat s~konci slov.}{1in}
-
-Dále bychom mìli do~automatu pøidat zpìtné hrany. Jejich definice bude úplnì stejná jako u automatu pro hledání jednoho slova. Jinými slovy z~ka¾dého stavu pùjde zpìtná hrana do~nejdel¹ího vlastního suffixu, který je stavem. Èili kdy¾ budeme mít nìjaké jméno stavu, budeme se ho sna¾it co nejménì (ale alespoò o~znak) zkrátit zleva, abychom zase dostali jméno stavu. Z~koøene -- prázdného stavu -- pak evidentnì ¾ádná zpìtná hrana nepovede.
-
-\figure{ara_strom_final.eps}{Vyhledávací automat se zpìtnými hranami.}{1,25in}
-
-Zbývá nám je¹tì si rozmyslet, jakým zpùsobem bude ná¹ automat hlásit výstup. Opìt smìøujeme k~tomu, aby se automat po~pøeètení nìjakého kusu textu nacházel ve~stavu odpovídajícímu nejdel¹ímu mo¾nému suffixu toho textu. Zatímco u hledání jediné jehly bylo hlá¹ení výskytù jednoduché -- kdykoliv jsme se dostali na~konec \uv{automatové cestièky} tady to bude opìt slo¾itìj¹í.
-
-První, co se nabízí, je vyu¾ít toho, ¾e jsme si oznaèili nìjaké vrcholy, kde hledaná slova konèí. Co tedy zkusit hlásit výskyt tohoto slova v¾dy, kdy¾ pøijdeme do~nìjakého oznaèeného vrcholu? Tento zpùsob v¹ak nefunguje, pokud se uvnitø nìkteré jehly skrývá jehla vnoøená. Napøíklad po~pøeètení slova |bara|, nám ná¹ souèasný automat neøíká, ¾e bychom mìli nìjaké slovo ohlásit, a~pøitom tam evidentì konèí podøetìzec |ara|. Stejnì tak pokud pøeèteme |barbara|, u¾ si nev¹imneme toho, ¾e tam konèí zároveò i~|ara|. Pouhé \uv{hlá¹ení teèek} tedy nefunguje.
-
-Dále si mù¾eme v¹imnout toho, ¾e v¹echna slova, která by se mìla v~daném stavu hlásit, jsou suffixy jména tohoto stavu. Pøitom víme, ¾e zpìtná hrana jméno stavu zkracuje zleva. Tak¾e speciálnì v¹echny suffixy daného stavu, které jsou také stavy, se dají najít tak, ¾e se vydáme po~zpìtných hranách do~koøene. Nabízí se tedy v¾dy projít cestu po~zpìtných hranách a¾ do~koøene a~hlásit v¹echny \uv{teèky}. Tento zpùsob by nám v¹ak celý algoritmus znaènì zpomalil, proto¾e cesta do~koøene mù¾e být relativnì dlouhá, ale teèek na~ní obvykle bude málo.
-
-Mohli bychom také zkusit si pro ka¾dý stav $\beta$ pøedpoèítat mno¾inu $cache(\beta)$, která by obsahovala v¹echna slova, která máme hlásit, kdy¾ se ve~stavu $\beta$ nacházíme. Pokud pak do~tohoto stavu vstoupíme, podíváme se na~tuto mno¾inu a~vypí¹eme v¹e, co v~ní je. Výpis nám bude evidentnì trvat lineárnì k~velikosti mno¾iny, celkovì tedy lineárnì k~velikosti výstupu. Problém je ale ten, ¾e jednotlivé cache mohou být hodnì velké, tak¾e je nestihneme sestrojit v lineárním èase. (Rozmyslete si pøíklad slovníku, kdy se to stane.)
-
-To, co nám ale ji¾ opravdu pomù¾e, bude zavedení zkratek. V¹imli jsme si, ¾e po~zpìtných hranách mù¾eme projít do~koøene a~hlásit v¹echny nalezené teèky. Vadilo nám ale, ¾e se mù¾e stát, ¾e budeme dlouho po~cestì chodit a~pøi tom ¾ádné teèky nenalézat. Zavedeme si proto zkratky k~nejbli¾¹í teèce. 
-
-\s{Definice} (zkratková hrana):
-Budeme mít tedy nìjakou funkci $slovo(\beta) :=$ slovo, které konèí ve~stavu $\beta$ (nebo $\emptyset$, pokud ¾ádné takové slovo není). Dále pak funkci $out(\beta) :=$ nejbli¾¹í vrchol dosa¾itelný po~zpìtných hranách, èili nejdel¹í vlastní suffix stavu $\beta$, v~nìm¾ je definovaná funkce $slovo$. Trochu lid¹tìji øeèeno, ten nejbli¾¹í dosa¾itelný vrchol, ve~kterém je teèka.
-
-Po pøidání tìchto zkratkových hran ji¾ máme reprezentaci, ve~které opravdu umíme v~daném stavu vyjmenovat v¹echna slova, která máme vypsat, a~to v~èase lineárním s~tím, kolik tìch slov je.
-
-\s{Definice:}
-Vyhledávací automat sestává ze stromu dopøedných hran (vrcholy jsou prefixy jehel, hrany odpovídají roz¹íøení o~písmenko), zpìtných hran ($z(\beta) :=$ nejdel¹í vlastní suffix slova $\beta$, který je stavem) a~zkratkových hran.
-
-Automat pak bude na~na¹em pøíkladu vypadat takto (zkratkové hrany jsou znázornìny zelenì):
-
-\figure{ara_strom_zkr.eps}{Vyhledávací automat se zkratkovými hranami.}{1,25in}
-
-Nyní u¾ nám zbývá jenom vlastní algoritmus -- nejdøív popí¹eme algoritmus, který bude hledat pomocí takového automatu, a~potom se pustíme do~toho, jak se takový automat staví.
-
-Nejprve si nadefinujeme, jak vypadá jeden krok automatu. Bude to vlastnì nìjaká funkce, která dostane stav a~písmenko. Ona nás pak pomocí tohoto písmenka posune po~automatu. ($f(\alpha, x)$ bude dopøedná hrana ze stavu $\alpha$ oznaèená písmenem~$x$)
-
-\s{Krok ($\alpha$, $x$):}
-\algo
-\:Dokud $f(\alpha, x) = \emptyset~\&~\alpha \neq \<koøen:>~~\alpha \leftarrow z(\alpha)$.
-\:Pokud $f(\alpha, x) \neq \emptyset:~~\alpha \leftarrow f(\alpha, x)$.
-\:Vrátíme výsledek.
-\endalgo
-
-\s{Hledání:}
-\algo
-\:$\alpha \leftarrow \<koøen>$.
-\:Pro znaky $x$ ze slova $\sigma$:
-\::$\alpha \leftarrow \<Krok>(\alpha, x)$.
-\::$\beta \leftarrow \alpha$
-\::Dokud $\beta \neq \emptyset$:
-\:::Je-li $\<slovo>(\beta) \neq \emptyset$:
-\::::Ohlásíme $\<slovo>(\beta)$.
-\:::$\beta \leftarrow \<out>(\beta)$.
-\endalgo
-
-Algoritmus hledání vlastnì není nic jiného, ne¾ prosté projití po~zelených zkratkových hranách ze stavu $\alpha$, ve~kterém právì jsme, a~ohlá¹ení v¹eho, co po~cestì najdeme.
-
-V ka¾dém okam¾iku se automat nachází ve~stavu, který odpovídá nejdel¹ímu mo¾nému suffixu toho, co jsme u¾ pøeèetli. Dùkaz tohoto invariantu je stejný jako u verze automatu pro hledání pouze jedné jehly, nebo» vychází pouze z~definice zpìtných hran. Podobnì nahlédneme, ¾e èasová slo¾itost vyhledávací procedury je lineární v~délce sena plus to, co spotøebujeme na~hlá¹ení výskytù. Nejprve na~chvíli zapomeneme, ¾e nìjaké výskyty hlásíme a~spoèítáme jenom kroky. Ty mohou vést dopøedu a~zpátky. Krok dopøedu prodlu¾uje jméno stavu o~jedna, krok dozadu zkracuje aspoò o~jedna. Tudí¾ krokù dozadu je maximálnì tolik, co krokù dopøedu a~krokù dopøedu je maximálnì tolik, kolik je délka sena. V¹echny kroky dohromady tedy trvají $\O(S)$. Hlá¹ení výskytù pak trvá $\O(S~+ \vert V \vert)$. Celé hledání tedy trvá lineárnì v~délce vstupu a~výstupu.
-
-Zbývá nám u¾ jen konstrukce automatu. Opìt vyu¾ijeme faktu, ¾e zpìtná hrana ze stavu $\beta$ vede tam, kam by se dostal automat pøi hledání $\beta$ bez prvního písmenka. Tak¾e zase chceme nìco, jako simulovat výpoèet toho automatu na~slovech bez prvního písmenka a~doufat v~to, ¾e si vystaèíme s~tou èástí automatu, kterou jsme u¾ postavili. Tentokrát to v¹ak nemù¾eme dìlat jedno slovo po~druhém, proto¾e zpìtné hrany mohou vést køí¾em mezi jednotlivými vìtvemi automatu. Mohlo by se nám tedy stát, ¾e pøi hledání nìjakého slova potøebujeme zpìtnou hranu, která vede do~jiného slova, které jsme je¹tì nezkonstruovali. Tak¾e tento postup sel¾e. Mù¾eme v¹ak vyu¾ít toho, ¾e ka¾dá zpìtná hrana vede ve~stromu alespoò o~jednu hladinu vý¹. Mù¾eme tak strom konstruovat po~hladinách. Lze si to tedy pøedstavit tak, ¾e paralelnì spustíme vyhledávání v¹ech slov bez prvních písmenek a~v¾dycky udìláme jeden podkrok ka¾dého z~tìch hledání, co¾ nám dá zpìtné hrany z~dal¹ího patra stromu.
-
-\s{Konstrukce automatu:}
-\algo
-\:Zalo¾íme prázdný strom, $r \leftarrow$ jeho koøen.
-\:Vlo¾íme do~stromu slova $\iota_1 \dots \iota_n$, nastavíme $slovo(*)$.
-\:$z(r) \leftarrow \emptyset$, $out(r) \leftarrow \emptyset$.
-\:Zalo¾íme frontu $F$ a~vlo¾íme do~ní syny koøene.
-\:$\forall v~\in F:~~z(v) \leftarrow r, \<out>(v) \leftarrow \emptyset$.
-\:Dokud $F \neq \emptyset$:
-\::Vybereme $u$ z~fronty $F$.
-\::Pro v¹echny syny $v$ vrcholu $u$:
-\:::$q \leftarrow \<Krok>(z(u), \<písmeno na~hranì uv>)$.
-\:::$z(v) \leftarrow q$.
-\:::Pokud $slovo(q) \neq \emptyset$, pak $out(v) \leftarrow q$.
-\::::Jinak $out(v) \leftarrow out(q)$.
-\:::Vlo¾íme $v$ do~fronty $F$.
-\endalgo
-
-To, ¾e tento algoritmus zkonstruuje zpìtné hrany jak má, vyplývá z~toho, ¾e nedìláme nic jiného, ne¾ ¾e spou¹tíme výpoèty po~hladinách na~v¹echna hledaná slova bez prvního písmenka. Stejnì tak to, ¾e dobìhne v~lineárním èase, je takté¾ dùsledkem toho, ¾e efektivnì spou¹tíme v¹echny tyto výpoèty. Jen nìkdy udìláme najednou krok dvou èi více výpoètù (napøíklad |araba| a~|arbara| se poèítají na~zaèátku, dokud jsou stejné, jen jednou). Èasová slo¾itost této konstrukce je tedy men¹í nebo rovna souètu èasových slo¾itostí výpoètù nad v¹emi tìmi slovy. To u¾ ale víme, ¾e je lineární v~celkové délce tìchto slov. Konstrukce automatu tedy trvá nejvý¹e tolik, co hledání v¹ech $\iota_i$, co¾ je $\O(\sum_{i} \iota_i)$.
-
-\s{Vìta:} Algoritmus Aho-Corasicková najde v¹echny výskyty v~èase 
-$$\O\left(\sum_i~\iota_i~+~S~+~\sharp\<výskytù>\right).$$
-
-Je¹tì se na~závìr zamysleme, jak bychom si takový automat ukládali do~pamìti. Urèitì se nám bude hodit si stavy nìjak oèíslovat (tøeba v~poøadí, v~jakém budou vznikat). Potom funkce pro zpìtné a~zkratkové hrany mohou být reprezentované polem indexovaným èíslem stavu. Funkce {\I Slovo}, která øíká, jaké slovo ve~stavu konèí, zase mù¾e být pole indexované stavem, které nám øekne poøadové èíslo slova ve~slovníku. Pro dopøedné hrany v~ka¾dém vrcholu pak mù¾eme mít pole indexované písmenky abecedy, které nám pro ka¾dé písmenko øekne, buï ¾e taková hrana není, nebo nám øekne, kam tato hrana vede. Je vidìt, ¾e takovéto pole se hodí pro pomìrnì malé abecedy. U¾ pro abecedu A-Z~bude velikosti 26 a~z~vìt¹iny bude prázdné, tak¾e bychom plýtvali pamìtí. V praxi se proto èasto pou¾ívá hashovací tabulka. Pøípadnì bychom mohli mít i~jen jednu velkou spoleènou hashovací tabulku, která bude reprezentovat funkci celou, ve~které budou zahashované dvojice (stav, písmenko). Tìchto dvojic je evidentnì tolik, kolik hran stromu, èili lineárnì s~velikostí slovníku, a~je to asi nejkompaktnìj¹í reprezentace.
-
-
-\bye
diff --git a/6-kmp/Makefile b/6-kmp/Makefile
deleted file mode 100644
index 1831d14..0000000
--- a/6-kmp/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=6-kmp
-
-include ../Makerules
diff --git a/6-kmp/ara_strom.eps b/6-kmp/ara_strom.eps
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index 7d054ab..0000000
--- a/6-kmp/ara_strom.eps
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diff --git a/6-kmp/ara_strom_zkr.svg b/6-kmp/ara_strom_zkr.svg
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index 8c3e865..0000000
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-       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
-       x="449.23392"
-       y="138.77879"
-       id="text2450"><tspan
-         sodipodi:role="line"
-         id="tspan2452"
-         x="449.23392"
-         y="138.77879">barbar</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
-       x="522.25323"
-       y="138.77878"
-       id="text2454"><tspan
-         sodipodi:role="line"
-         id="tspan2456"
-         x="522.25323"
-         y="138.77878">barbaro</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
-       x="653.50238"
-       y="138.17416"
-       id="text2458"><tspan
-         sodipodi:role="line"
-         id="tspan2460"
-         x="653.50238"
-         y="138.17416">barbarossa</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
-       x="621.96454"
-       y="139.35719"
-       id="text2466"><tspan
-         sodipodi:role="line"
-         id="tspan2468"
-         x="621.96454"
-         y="139.35719">...</tspan></text>
-    <text
-       xml:space="preserve"
-       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
-       x="37.04578"
-       y="177.70506"
-       id="text2470"><tspan
-         sodipodi:role="line"
-         x="37.04578"
-         y="177.70506"
-         id="tspan2482">b              a              r               b             a              r              o             ...             a  </tspan></text>
-    <path
-       style="fill:none;fill-rule:nonzero;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#Arrow2Lend);fill-opacity:1"
-       d="M 83.509975,164.51929 C 60.542102,190.63893 35.300495,196.8634 5.651051,164.51929"
-       id="path5609"
-       inkscape:connector-type="polyline"
-       sodipodi:nodetypes="cc" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 13.185786,153.96139 L 78.486819,153.96139"
-       id="path7588"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 90.747017,153.9613 L 156.04805,153.9613"
-       id="path9700"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 168.72226,153.9613 L 234.02329,153.9613"
-       id="path9702"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 245.95329,153.96121 L 311.25432,153.96121"
-       id="path9704"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 323.85853,153.9613 L 389.15956,153.9613"
-       id="path9706"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 401.50851,153.9613 L 466.80954,153.9613"
-       id="path9708"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 480.48321,153.9613 L 545.78425,153.9613"
-       id="path9710"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 558.48215,153.9613 L 623.78318,153.9613"
-       id="path9712"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 638.59633,153.9613 L 703.89736,153.9613"
-       id="path9714"
-       inkscape:connector-type="polyline" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 160.62469,178.6356 C 103.11439,214.93588 51.117239,212.64411 3.651051,178.6356"
-       id="path9716"
-       inkscape:connector-type="polyline"
-       sodipodi:nodetypes="cc" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 241.11151,117.89139 C 160.02376,69.180736 82.359446,79.809701 6.2789455,117.89139"
-       id="path9718"
-       inkscape:connector-type="polyline"
-       sodipodi:nodetypes="cc" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 319.79709,118.68308 C 234.48869,66.453676 158.66445,84.678135 84.96452,118.68308"
-       id="path10774"
-       inkscape:connector-type="polyline"
-       sodipodi:nodetypes="cc" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 397.16706,119.82255 C 317.23195,75.067632 238.9209,74.162508 162.33449,119.82255"
-       id="path10776"
-       inkscape:connector-type="polyline"
-       sodipodi:nodetypes="cc" />
-    <path
-       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
-       d="M 473.93389,173.72993 C 393.31902,208.79011 314.69794,213.94352 239.10133,173.72993"
-       id="path10778"
-       inkscape:connector-type="polyline"
-       sodipodi:nodetypes="cc" />
-    <path
-       style="opacity:1;fill:none;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.99999982px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:url(#Arrow2Lend);stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
-       d="M 551.6176,159.60837 C 426.17515,260.43382 -84.115921,225.79475 -0.53486469,163.7889"
-       id="path9778"
-       sodipodi:nodetypes="cc" />
-    <path
-       style="opacity:1;fill:none;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
-       d="M 631.09082,161.52172 C 584.4479,211.71214 440.8137,218.21863 316.61606,220.87291"
-       id="path10874"
-       sodipodi:nodetypes="cc" />
-    <path
-       style="opacity:1;fill:none;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.99999994px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
-       d="M 711.15927,159.63036 C 703.71296,189.73566 643.62165,205.468 570.98179,213.59973 C 531.11935,218.06216 487.47795,220.2357 446.68392,221.23953 C 366.0505,223.2237 296.54156,220.6379 289.32855,222.12506"
-       id="path10880"
-       sodipodi:nodetypes="cssc" />
-  </g>
-</svg>
diff --git a/7-geom/7-geom.mp b/7-geom/7-geom.mp
deleted file mode 100644
index 7375c63..0000000
--- a/7-geom/7-geom.mp
+++ /dev/null
@@ -1,276 +0,0 @@
-input lib
-
-figname("7-geom");
-
-figtag("male_obaly");
-beginfig(1);
-	pickup boldpen;
-	labeloffset:=1cm;
-	pair c,pos; c := (0,0); pos := c;
-	drawemptyvertex(c);
-	label.bot(btex $n=1$ etex, pos);
-	c := (2cm,0);
-	pos := pos + c;
-	pair A[];
-	A[0] := (-0.3cm, -0.2cm)+c; A[1] := (0.2cm, 0.3cm)+c;
-	draw A[0]--A[1];
-	drawemptyvertex(A[0]); drawemptyvertex(A[1]);
-	label.bot(btex $n=2$ etex, pos);
-
-	pos := pos + c;
-	A[2] := (+0.3cm, -0.4cm)+c;
-	for i := 0 upto 2: A[i] := A[i] shifted c; endfor
-	draw A[0]--A[1]--A[2]--cycle;
-	for i := 0 upto 2: drawemptyvertex(A[i]); endfor
-	label.bot(btex $n=3$ etex, pos);
-
-	pos := pos + c;
-	A[3] := (A[0]+A[1]+A[2])/3;
-	for i := 0 upto 3: A[i] := A[i] shifted c; endfor
-	draw A[0]--A[1]--A[2]--cycle;
-	for i := 0 upto 2: drawemptyvertex(A[i]); endfor
-	draw vertex(A[3]);
-	
-	c := (1cm,0);
-	pos := pos + c/2;
-	A[3] := A[1]+(0.3cm,-0.2cm);
-	for i := 0 upto 3: A[i] := A[i] shifted c; endfor
-	draw A[0]--A[1]--A[3]--A[2]--cycle;
-	for i := 0 upto 3: drawemptyvertex(A[i]); endfor
-	label.bot(btex $n=4$ etex, pos);
-endfig;
-
-figtag("pridani_bodu");
-beginfig(2);
-	pair A[],B[],C,shift; shift := (4.5cm,0);
-	A[0] := (-1.7cm,1.1cm);
-	A[1] := (-1.2cm,1.2cm);
-	A[2] := (-0.4cm,1cm);
-	A[3] := (0.2cm,0.2cm);
-	A[4] := (0.4cm,-0.7cm);
-	A[5] := (-0.8cm,-1.3cm);
-	A[6] := (-1.4cm,-1.4cm);
-	B[0] := (-1.1cm, 0.7cm);
-	B[1] := (-0.6cm, 0.1cm);
-	B[2] := (-1.3cm, -0.6cm);
-	C := (1cm, 0.1cm);
-
-	% krok 1
-	pickup boldpen;
-	draw A[0] for i := 1 upto 6: --A[i] endfor;
-	for i := 1 upto 5: drawemptyvertex(A[i]); endfor
-	for i := 0 upto 2: draw vertex(B[i]); endfor
-	draw vertex(C);
-	drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0, -0.1cm) withpen normalpen;
-	for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
-	for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
-	C := C shifted shift;
-
-	% krok 2
-	draw A[0] for i := 1 upto 6: --A[i] endfor;
-	draw A[4]{dir 70}..C;
-	draw A[4]{dir 45}..C;
-	draw C--A[2] dashed evenly withpen normalpen;
-	draw C--A[3] dashed evenly withpen normalpen;
-	for i := 1 upto 5: drawemptyvertex(A[i]); endfor
-	for i := 0 upto 2: draw vertex(B[i]); endfor
-	drawemptyvertex(C);
-	drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0,-0.1cm) withpen normalpen;
-	for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
-	for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
-	C := C shifted shift;
-
-	% krok 3
-	draw for i := 0 upto 2: A[i]-- endfor C for i := 4 upto 6: --A[i] endfor;
-	for i := 1 upto 2: drawemptyvertex(A[i]); endfor
-	for i := 4 upto 5: drawemptyvertex(A[i]); endfor
-	for i := 0 upto 2: draw vertex(B[i]); endfor
-	draw vertex(A[3]);
-	drawemptyvertex(C);
-endfig;
-
-figtag("obalky");
-beginfig(3);
-	labeloffset := 0.2cm;
-	pickup boldpen;
-	pair A[],B[];
-	A[0] := (-7cm, 0cm);
-	A[1] := (-6.2cm, 0.9cm);
-	A[2] := (-4.6cm,1.5cm);
-	A[3] := (-2.4cm,1.8cm);
-	A[4] := (-0.8cm,1.5cm);
-	A[5] := (0.4cm,0.6cm);
-	A[6] := (0.8cm,-0.10cm);
-	A[7] := (-1.6cm,-1.9cm);
-	A[8] := (-4cm,-2.1cm);
-	A[9] := (-6cm, -1.5cm);
-	A[10] := (-7cm, 0cm);
-	
-	B[0] := (-2.2cm, 0.7cm);
-	B[1] := (-1.2cm, 0.1cm);
-	B[2] := (-2.6cm, -0.6cm);
-	B[3] := (-3.6cm, -0.4cm);
-	B[4] := (-3cm, 0.6cm);
-	B[5] := (-2.6cm, 1cm);
-	B[6] := (-1cm, -1.2cm);
-	B[7] := (-6.5cm, 0.2cm);
-	B[8] := (-5cm, 0.8cm);
-	B[9] := (-6cm, -0.6cm);
-	B[10] := (-5cm, -1.2cm);
-	
-	draw createpath(for i := 0 upto 5: A[i]-- endfor A[6]);
-	draw (for i := 6 upto 9: A[i]-- endfor A[10]) dashed evenly;
-	for i := 0 upto 9: drawemptyvertex(A[i]); endfor
-	for i := 0 upto 10: draw vertex(B[i]); endfor
-
-	label(btex \font\myfont=csr10 \myfont horní obálka etex, ((-7cm+0.8cm)/2,2.2cm));
-	label(btex \font\myfont=csr10 \myfont dolní obálka etex, ((-7cm+0.8cm)/2,-2.5cm));
-	label.lft(btex $L$ etex, A[0]);
-	label.rt(btex $P$ etex, A[6]);
-endfig;
-
-figtag("determinant");
-beginfig(4);
-	labeloffset := 0.1cm;
-	pair A[], shift; shift := (4cm,1cm);
-
-	% det(M) > 0
-	A[0] := (-2cm, 0);
-	A[1] := (0,-1cm);
-	A[2] := (1.5cm, 0cm);
-	A[3] := A[0] + A[2] - A[1];
-	
-	fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
-	draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
-	drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
-	drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
-	for i:=0 upto 2: draw vertex(A[i]); endfor
-	label.lft(btex $h_{k-1}$ etex, A[0]);
-	label.bot(btex $h_k$ etex, A[1]);
-	label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
-	label.lrt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
-	label.rt(btex $b$ etex, A[2]);
-	label(btex $\det(M) > 0$ etex, 0.5[A[0],A[2]]);
-
-	% det(M) = 0
-	A[0] := (-1cm, -0.5cm) + shift;
-	A[1] := (0, -1cm) + shift;
-	A[2] := (1cm, -1.5cm) + shift;
-	drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
-	drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
-	for i:=0 upto 2: draw vertex(A[i]); endfor
-	label.lft(btex $h_{k-1}$ etex, A[0]);
-	label.llft(btex $h_k$ etex, A[1]);
-	label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
-	label.llft(btex $\vec v$ etex, 0.5[A[1],A[2]]);
-	label.bot(btex $b$ etex, A[2]);
-	label(btex $\det(M) = 0$ etex, origin) shifted (0,0.3cm) rotated -28 shifted 0.5[A[0], A[2]];
-
-	% det(M) < 0
-	shift := (7.5cm, 1.25cm);
-	A[0] := (-1cm, -0.5cm) + shift;
-	A[1] := (1.5cm, -1cm) + shift;
-	A[2] := (2cm, -2.5cm) + shift;
-	A[3] := A[0] + A[2] - A[1];
-	fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
-	draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
-	drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
-	drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
-	for i:=0 upto 2: draw vertex(A[i]); endfor
-	label.lft(btex $h_{k-1}$ etex, A[0]);
-	label.urt(btex $h_k$ etex, A[1]);
-	label.top(btex $\vec u$ etex, 0.5[A[0],A[1]]);
-	label.rt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
-	label.rt(btex $b$ etex, A[2]);
-	label(btex $\det(M) < 0$ etex, 0.5[A[0],A[2]]);
-endfig;
-
-figtag("rybi_motivace");
-beginfig(5);
-	u := 0.3cm;
-	def draw_fish(expr pos,size,rot) =
-		draw ((-1.3u*size,0){dir 60}..{right}(u*size,-u*size/4)) rotated rot shifted pos;
-		draw ((-1.3u*size,0){dir -50}..{right}(u*size,u*size/4)) rotated rot shifted pos;
-		draw ((u*size,-u*size/4)--(u*size,u*size/4)) rotated rot shifted pos;
-		draw (-1u*size,u*size/15) rotated rot shifted pos withpen pencircle scaled (u/8);
-		for i:=1 upto 3: draw (dirs((u*size,-u*size/4+i*u*size/8), 180, u*size/6)) rotated rot shifted pos; endfor
-	enddef;
-	
-	pair A[],B[];
-	A[0] := (-7cm, 0cm);
-	A[1] := (-6.2cm, 0.9cm);
-	A[2] := (-4.6cm,1.5cm);
-	A[3] := (-2.4cm,1.8cm);
-	A[4] := (-0.8cm,1.5cm);
-	A[5] := (0.4cm,0.6cm);
-	A[6] := (0.8cm,-0.10cm);
-	A[7] := (-1.6cm,-1.9cm);
-	A[8] := (-4cm,-2.1cm);
-	A[9] := (-6cm, -1.5cm);
-	A[10] := (-7cm, 0cm);
-	
-	B[0] := (-2.2cm, 0.7cm);
-	B[1] := (-1.2cm, 0.1cm);
-	B[2] := (-2.6cm, -0.6cm);
-	B[3] := (-3.6cm, -0.4cm);
-	B[4] := (-3cm, 0.6cm);
-	B[5] := (-2.6cm, 1cm);
-	B[6] := (-1cm, -1.2cm);
-	B[7] := (-6.5cm, 0.2cm);
-	B[8] := (-5cm, 0.8cm);
-	B[9] := (-6cm, -0.6cm);
-	B[10] := (-5cm, -1.2cm);
-
-	for i:=0 upto 9: draw_fish(A[i], 1, 0); endfor;
-	for i:=0 upto 10: draw_fish(B[i], 1, 0); endfor;
-	draw createpath(for i:=0 upto 9: A[i]-- endfor cycle) scaled 1.13 shifted (0.4cm,0) withpen boldpen;
-endfig;
-
-figtag("provazkovy_algoritmus");
-beginfig(6);
-	pickup boldpen;
-	pair A[],B[],u; u := (-3cm, 0);
-	for i := 0 upto 3: A[i] := u rotated (-30*i) yscaled 0.7; endfor;
-	A[2] := A[2] + (0,0.1cm);
-	draw for i:=0 upto 2: A[i]-- endfor A[3];
-	drawarrow ((u/2) for i:=1 upto 3: ..u/2 rotated (-30*i) endfor) yscaled 0.7 withpen normalpen;
-	B[0] := (-2cm,0.5cm);
-	B[1] := (-1cm,1.5cm);
-	B[2] := (-0.5cm,0.2cm);
-	for i:=0 upto 2: draw vertex(B[i]); endfor
-
-	path ub; ub := (-20cm,3cm)--(20cm,3cm);
-
-	numeric ang[]; ang[0] = 90; ang[1] = angle(A[1]-A[0]); ang[2] = angle(A[2]-A[1]); ang[3] = angle(A[3]-A[2]);
-	for i:=0 upto 2:
-		draw reverse(dirs(A[i],ang[i],6cm) cutafter ub) withpen normalpen dashed evenly;
-		l := 1cm + (i-1)*0.2cm;
-		drawarrow from(A[i],ang[i],l)..from(A[i],(ang[i]+ang[i+1])/2,l)..from(A[i],ang[i+1],l) withpen normalpen;
-	endfor
-	
-	for i:=0 upto 3: drawemptyvertex(A[i]); endfor;
-endfig;
-
-figtag("naslednik_pres_konvexni_obal");
-beginfig(7);
-	pair A[], C;
-	label.lrt(btex $Q_i$ etex, (1.5cm,-0.8cm));
-	pickup boldpen;
-
-	C := (-4cm,-0.3cm);
-	for i:=0 upto 6:
-		A[i] := (2cm,0) rotated (360*i/7+5) yscaled 0.7;
-		draw vertex(A[i]);
-	endfor;
-	draw for i:=0 upto 6: A[i]-- endfor cycle withpen normalpen;
-	draw C--A[2] dashed evenly;
-
-	draw dirs(C, -140, 0.5cm);
-	drawemptyvertex(A[2]);
-	drawemptyvertex(C);
-	drawdblarrow (fullcircle scaled 2cm rotated (360*2/7-5) yscaled 0.7) cutbefore (origin--(3cm,0) rotated (360*2/7+25)) withpen normalpen;
-	%drawarrow C+(0,0.5cm){dir 60}..A[2]+(0,0.5cm) withpen normalpen;
-	%drawarrow 0.6A[5]{dir 170}..(0.6A[3] rotated -15) withpen normalpen;
-	%drawarrow 0.7A[6]{dir 60}..(0.5A[1] rotated 30) withpen normalpen;
-endfig;
-end
diff --git a/7-geom/7-geom.tex b/7-geom/7-geom.tex
deleted file mode 100644
index d600d11..0000000
--- a/7-geom/7-geom.tex
+++ /dev/null
@@ -1,136 +0,0 @@
-\input lecnotes.tex
-
-\prednaska{7}{Geometrické algoritmy}{(sepsal Pavel Klavík)}
-
-\>Uká¾eme si nìkolik základních algoritmù na øe¹ení geometrických problémù v~rovinì. Proè zrovna v~rovinì? Inu, jednorozmìrné problémy bývají triviální
-a naopak pro vy¹¹í dimenze jsou velice komplikované. Rovina je proto rozumným kompromisem mezi obtí¾ností a zajímavostí.
-
-Celou kapitolou nás bude provázet pohádka ze ¾ivota ledních medvìdù. Pokusíme se vyøe¹it jejich \uv{ka¾dodenní} problémy~\dots
-
-\h{Hledání konvexního obalu}
-
-{\I Daleko na severu ¾ili lední medvìdi. Ve vodách tamního moøe byla hojnost ryb a jak je známo, ryby jsou oblíbenou pochoutkou ledních medvìdù.
-Proto¾e medvìdi z~na¹í pohádky rozhodnì nejsou ledajací a ani chytrost jim neschází, rozhodli se v¹echny ryby pochytat. Znají pøesná místa výskytu
-ryb a rádi by vyrobili obrovskou sí», do které by je v¹echny chytili. Pomozte medvìdùm zjistit, jaký nejmen¹í obvod taková sí» mù¾e mít.}
-
-\figure{7-geom5_rybi_motivace.eps}{Problém ledních mìdvìdù: Jaký je nejmen¹í obvod sítì?}{3in}
-
-Neboli v~øeèi matematické, chceme pro zadanou mno¾inu bodù v~rovinì nalézt její konvexní obal. Co je to konvexní obal? Mno¾ina bodù je {\I konvexní},
-pokud pro ka¾dé dva body obsahuje i celou úseèku mezi nimi. {\I Konvexní obal} je nejmen¹í konvexní podmno¾ina roviny, která obsahuje v¹echny zadané
-body.\foot{Pamatujete si na lineární obaly ve vektorových prostorech? Lineární obal mno¾iny vektorù je nejmen¹í vektorový podprostor, který tyto
-vektory obsahuje. Není náhoda, ¾e tato definice pøipomíná definici konvexního obalu. Na druhou stranu ka¾dý vektor z~lineárního obalu lze vyjádøit
-jako lineární kombinaci daných vektorù. Podobnì platí i pro konvexní obaly, ¾e ka¾dý bod z~obalu je konvexní kombinací daných bodù. Ta se li¹í od
-lineární v~tom, ¾e v¹echny koeficienty jsou v~intervalu $[0,1]$ a navíc souèet v¹ech koeficientù je $1$. Tento algebraický pohled mù¾e mnohé vìci
-zjednodu¹it. Zkuste si dokázat, ¾e obì definice konvexního obalu jsou ekvivalentní.} Z~algoritmického hlediska nás v¹ak bude zajímat jenom jeho
-hranice, kterou budeme dále oznaèovat jako konvexní obal.
-
-Na¹ím úkolem je nalézt konvexní obal koneèné mno¾iny bodù. To je v¾dy konvexní mnohoúhelník, navíc s~vrcholy v~zadaných bodech. Øe¹ením problému tedy
-bude posloupnost bodù, které tvoøí konvexní obal. Pro malé mno¾iny je konvexní obal nakreslen na obrázku, pro více bodù je v¹ak situace mnohem
-slo¾itìj¹í.
-
-\figure{7-geom1_male_obaly.eps}{Konvexní obaly malých mno¾in.}{3in}
-
-Pro jednoduchost budeme pøedpokládat, ¾e v¹echny body mají rùzné $x$-ové souøadnice. Tedy utøídìní bodù zleva doprava je urèené jednoznaènì.\foot{To si
-mù¾eme dovolit pøedpokládat, nebo» se v¹emi body staèí nepatrnì pootoèit. Tím konvexní obal urèitì nezmìníme. Av¹ak jednodu¹¹í øe¹ení je naprogramovat
-tøídìní lexikograficky (druhotnì podle souøadnice $y$) a vyøadit identické body.} Tím máme zaji¹tìné, ¾e existují dva body, nejlevìj¹í a
-nejpravìj¹í, pro které platí následující invariant:
-
-\s{Invariant:} Nejlevìj¹í a nejpravìj¹í body jsou v¾dy v~konvexním obalu.
-
-Algoritmus na nalezení konvexního obalu funguje na následujícím jednoduchém principu, kterému se nìkdy øíká {\I zametání roviny}. Procházíme body
-zleva doprava a postupnì roz¹iøujeme doposud nalezený konvexní obal o~dal¹í body. Na zaèátku bude konvexní obal jediného bodu samotný bod. Na konci
-$k$-tého kroku algoritmu známe konvexní obal prvních $k$ bodù. Kdy¾ algoritmus skonèí, známe hledaný konvexní obal. Podle invariantu musíme v~$k$-tém
-kroku pøidat do obalu $k$-tý nejlevìj¹í bod. Zbývá si jen rozmyslet, jak pøesnì tento bod pøidat.
-
-Pøidání dal¹ího bodu do konvexního obalu funguje, jak je naznaèeno na obrázku. Podle invariantu víme, ¾e bod nejvíc vpravo je souèástí konvexního
-obalu. Za nìj napojíme novì pøidávaný bod. Tím jsme získali nìjaký obal, ale zpravidla nebude konvexní. To lze v¹ak snadno napravit, staèí
-odebírat body, v obou smìrech podél konvexního obalu, tak dlouho, dokud nezískáme konvexní obal. Na pøíkladu z obrázku nemusíme po smìru hodinových
-ruèièek odebrat ani jeden bod, obal je v poøádku. Naopak proti smìru hodinových ruèièek musíme odebrat dokonce dva body.
-
-\figure{7-geom2_pridani_bodu.eps}{Pøidání bodu do konvexního obalu.}{4.5in}
-
-Pro pøípadnou implementaci a rozbor slo¾itosti si nyní popí¹eme algoritmus detailnìji. Aby se lépe popisoval, rozdìlíme si konvexní obal na dvì èásti
-spojující nejlevìj¹í a nejpravìj¹í bod obalu. Budeme jim øíkat {\I horní obálka} a {\I dolní obálka}.
-
-\figure{7-geom3_obalky.eps}{Horní a dolní obálka konvexního obalu.}{3.4in}
-
-Obì obálky jsou lomené èáry, navíc horní obálka poøád zatáèí doprava a dolní naopak doleva. Pro udr¾ování bodù v~obálkách staèí dva zásobníky.
-V~$k$-tém kroku algoritmu pøidáme zvlá¹» $k$-tý bod do horní i dolní obálky. Pøidáním $k$-tého bodu se v¹ak mù¾e poru¹it smìr, ve kterém obálka
-zatáèí. Proto budeme nejprve body z~obálky odebírat a $k$-tý bod pøidáme a¾ ve chvíli, kdy jeho pøidání smìr zatáèení neporu¹í.
-
-\s{Algoritmus:}
-
-\algo
-
-\:Setøídíme body podle $x$-ové souøadnice, oznaème body $b_1, \ldots, b_n$.
-\:Vlo¾íme do horní a dolní obálky bod $b_1$: $H = D = (b_1)$.
-\:Pro ka¾dý dal¹í bod $b = b_2,\ldots,b_n$:
-\::Pøepoèítáme horní obálku:
-\:::Dokud $\vert H\vert \ge 2$, $H = (\ldots, h_{k-1}, h_k)$ a úhel $h_{k-1} h_k b$ je orientovaný doleva:
-\::::Odebereme poslední bod $h_k$ z~obálky $H$.
-\:::Pøidáme bod $b$ do obálky $H$.
-\::Symetricky pøepoèteme dolní obálku (s orientací doprava).
-\: Výsledný obal je tvoøen body v~obálkách $H$ a $D$.
-
-\endalgo
-
-Rozebereme si èasovou slo¾itost algoritmu. Setøídit body podle $x$-ové souøadnice doká¾eme v~èase $\O(n \log n)$. Pøidání dal¹ího bodu do obálek
-trvá lineárnì vzhledem k~poètu odebraných bodù. Zde vyu¾ijeme obvyklý postup: Ka¾dý bod je odebrán nejvý¹e jednou, a tedy v¹echna odebrání trvají
-dohromady $\O(n)$. Konvexní obal doká¾eme sestrojit v~èase $\O(n \log n)$, a pokud bychom mìli seznam bodù ji¾ utøídený, doká¾eme to dokonce v
-$\O(n)$.
-
-\s{Algebraický dodatek:} Existuje jednoduchý postup, jak zjistit orientaci úhlu? Uká¾eme si jeden zalo¾ený na lineární algebøe. Budou se hodit
-vlastnosti determinantu. Absolutní hodnota determinantu je objem rovnobì¾nostìnu urèeného øádkovými vektory matice. Dùle¾itìj¹í v¹ak je, ¾e znaménko
-determinantu urèuje \uv{orientaci} vektorù, zda je levotoèivá èi pravotoèivá. Proto¾e ná¹ problém je rovinný, budeme uva¾ovat determinanty matic $2
-\times 2$.
-
-Uva¾me souøadnicový systém v~rovinì, kde $x$-ová souøadnice roste smìrem doprava a~$y$-ová smìrem nahoru. Chceme zjistit orientaci úhlu $h_{k-1} h_k
-b$. Polo¾me $\vec u = (x_1, y_1)$ jako rozdíl souøadnic $h_k$ a~$h_{k-1}$ a podobnì $\vec v = (x_2, y_2)$ je rozdíl souøadnic $b$ a~$h_k$. Matice $M$
-je definována následovnì:
-$$M = \pmatrix{\vec u \cr \vec v} = \pmatrix {x_1&y_1\cr x_2&y_2}.$$
-Úhel $h_{k-1} h_k b$ je orientován doleva, právì kdy¾ $\det M = x_1y_2 - x_2y_1$ je nezáporný,\foot{Neboli vektory $\vec u$ a $\vec v$ odpovídají
-rozta¾ení a zkosení vektorù báze $\vec x = (1,0)$ a $\vec y = (0,1)$, pro nì¾ je determinant nezáporný.} a spoèítat hodnotu determinantu je jednoduché.
-Mo¾né situace jsou nakresleny na obrázku. Poznamenejme, ¾e k~podobnému vzorci se lze také dostat pøes vektorový souèin vektorù $\vec u$ a $\vec v$.
-
-\figure{7-geom4_determinant.eps}{Jak vypadají determinanty rùzných znamének v~rovinì.}{4.6in}
-
-\s{©lo by to vyøe¹it rychleji?} Také vám vrtá hlavou, zda existují rychlej¹í algoritmy? Na závìr si uká¾eme nìco, co na pøedná¹ce nebylo.\foot{A také
-se nebude zkou¹et.} Nejrychlej¹í známý algoritmus, jeho¾ autorem je T.~Chan, funguje v~èase $\O(n \log h)$, kde $h$ je poèet bodù le¾ících na
-konvexním obalu, a pøitom je pøekvapivì jednoduchý. Zde si naznaèíme, jak tento algoritmus funguje.
-
-Algoritmus pøichází s~následující my¹lenkou. Pøedpokládejme, ¾e bychom znali velikost konvexního obalu $h$. Rozdìlíme body libovolnì do $\lceil {n
-\over h} \rceil$ mno¾in $Q_1, \ldots, Q_k$ tak, ¾e $\vert Q_i \vert \le h$. Pro ka¾dou z~tìchto mno¾in nalezneme konvexní obal pomocí vý¹e popsaného
-algoritmu. To doká¾eme pro jednu v~èase $\O(h \log h)$ a pro v¹echny v~èase $\O(n \log h)$. V druhé fázi spustíme hledání konvexního obalu pomocí
-provázkového algoritmu a pro zrychlení pou¾ijeme pøedpoèítané obaly men¹ích mno¾in. Nejprve popí¹eme jeho my¹lenku. Pou¾ijeme následující pozorování:
-
-\s{Pozorování:} Úseèka spojující dva body $a$ a $b$ le¾í na konvexním obalu, právì kdy¾ v¹echny ostatní body le¾í pouze na jedné její
-stranì.\foot{Formálnì je podmínka následující: Pøímka $ab$ urèuje dvì poloroviny. Úseèka le¾í na konvexním obalu, právì kdy¾ v¹echny body le¾í v jedné
-z polorovin.}
-
-Algoritmu se øíká {\I provázkový}, proto¾e svojí èinností pøipomíná namotávání provázku podél konvexního obalu.  Zaèneme s bodem, který na konvexním
-obalu urèitì le¾í, to je tøeba ten nejlevìj¹í. V ka¾dém kroku nalezneme následující bod po obvodu konvexního obalu. To udìláme napøíklad tak, ¾e
-projdeme v¹echny body a vybereme ten, který svírá nejmen¹í úhel s poslední stranou konvexního obalu. Novì pøidaná úseèka vyhovuje pozorování a proto
-do konvexního obalu patøí. Po $h$ krocích se dostaneme zpìt k nejlevìj¹ímu bodu a výpoèet ukonèíme. V ka¾dém kroku potøebujeme projít v¹echny body a
-vybrat následníka, co¾ doká¾eme v èase $\O(n)$. Celková slo¾itost algoritmu je tedy $\O(n \cdot h)$.
-
-\twofigures{7-geom6_provazkovy_algoritmus.eps}{Provázkový algoritmus.}{1.25in}{7-geom7_naslednik_pres_konvexni_obal.eps}{Hledání kandidáta v pøedpoèítaném obalu.}{2.5in}
-
-Provázkový algoritmus funguje, ale má jednu obrovskou nevýhodu -- je toti¾ ukrutnì pomalý. Ký¾eného zrychlení dosáhneme, pokud pou¾ijeme pøedpoèítané
-konvexní obaly. Ty umo¾ní rychleji hledat následníka. Pro ka¾dou z mno¾in $Q_i$ najdeme zvlá¹» kandidáta a poté z nich vybereme toho nejlep¹ího.
-Mo¾ný kandidát v¾dy le¾í na konvexním obalu mno¾iny $Q_i$. Vyu¾ijeme toho, ¾e body obalu jsou \uv{uspoøádané}, i kdy¾ trochu netypicky do kruhu.
-Kandidáta mù¾eme hledat metodou pùlení intervalu, i kdy¾ detaily jsou malièko slo¾itìj¹í ne¾ je obvyklé. Jak pùlit zjistíme podle smìru zatáèení
-konvexního obalu. Detaily si rozmyslí ètenáø sám.
-
-Èasová slo¾itost pùlení je $\O(\log h)$ pro jednu mno¾inu. Mno¾in je nejvý¹e $\O({n \over h})$, tedy následující bod konvexního obalu nalezneme v èase
-$\O({n \over h} \log h)$. Celý obal nalezneme ve slibovaném èase $\O(n \log h)$. 
-
-Popsanému algoritmu schází jedna dùle¾itá vìc: Ve skuteènosti vìt¹inou neznáme velikost $h$. Budeme proto algoritmus iterovat s~rostoucí hodnotou $h$,
-dokud konvexní obal nesestrojíme. Pokud pøi slepování konvexních obalù zjistíme, ¾e konvexní obal je vìt¹í ne¾ $h$, výpoèet ukonèíme. Zbývá je¹tì
-zvolit, jak rychle má $h$ rùst. Pokud by rostlo moc pomalu, budeme poèítat zbyteènì mnoho fází, naopak pøi rychlém rùstu by nás poslední fáze mohla
-stát pøíli¹ mnoho.
-
-V~$k$-té iteraci polo¾íme $h = 2^{2^k}$. Dostáváme celkovou slo¾itost algoritmu:
-$$\sum_{m=0}^{\O(\log \log h)} \O(n \log 2^{2^m}) = \sum_{m=0}^{\O(\log \log h)} \O(n \cdot 2^m) = \O(n \log h),$$
-kde poslední rovnost dostaneme jako souèet prvních $\O(\log \log h)$ èlenù geometrické øady $\sum 2^m$.
-
-\bye
diff --git a/7-geom/7-geom1_male_obaly.eps b/7-geom/7-geom1_male_obaly.eps
deleted file mode 100644
index 94a501d..0000000
--- a/7-geom/7-geom1_male_obaly.eps
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-1.78256 58.49205 1.99252 58.99881 1.99252 59.52728 curveto closepath stroke
-showpage
-%%EOF
diff --git a/7-geom/7-geom7_naslednik_pres_konvexni_obal.eps b/7-geom/7-geom7_naslednik_pres_konvexni_obal.eps
deleted file mode 100644
index 7a14c41..0000000
--- a/7-geom/7-geom7_naslednik_pres_konvexni_obal.eps
+++ /dev/null
@@ -1,106 +0,0 @@
-%!PS
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diff --git a/7-geom/Makefile b/7-geom/Makefile
deleted file mode 100644
index b06fe95..0000000
--- a/7-geom/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=7-geom
-
-include ../Makerules
diff --git a/7-geom/lib.mp b/7-geom/lib.mp
deleted file mode 100644
index 2536305..0000000
--- a/7-geom/lib.mp
+++ /dev/null
@@ -1,82 +0,0 @@
-% implementation of figure naming and figure transparency
-string name,tag; name := ""; tag := "";
-def updatefigname =
-	if tag="": filenametemplate (name & "%c.eps");
-	else: filenametemplate (name & "%c_" & tag & ".eps");
-	fi;
-enddef;
-	
-def figname(expr n) = name := n; updatefigname; enddef;
-def figtag(expr t) = tag := t; updatefigname; enddef;
-
-picture transparent_picture;
-color transparent_color; transparent_color := 0.9white;
-def drawtransparent(expr num) =
-	transparent_picture := currentpicture;
-endfig;
-
-if tag="": filenametemplate (name & "%c_transparent.eps");
-else: filenametemplate (name & "%c_" & tag & "_transparent.eps");
-fi;
-beginfig(num);
-	draw transparent_picture withcolor transparent_color;
-endfig;
-
-updatefigname;
-enddef;
-
-def from(expr p,d,len) =
-	p+dir(d)*len
-enddef;
-
-def dirs(expr p,d,len) =
-	p--from(p,d,len)
-enddef;
-
-def drawvertices(expr s,n) =
-	for i:=s upto n:
-		draw vertex(PQ[i]);
-	endfor
-enddef;
-
-def drawfvertices(expr s,n,flags) =
-	for i:=s upto n:
-		draw vertex(PQ[i]) flags;
-	endfor
-enddef;
-
-def vertex(expr p) = p withpen pencircle scaled 4pt enddef;
-def drawemptyvertex(expr p) = unfill fullcircle scaled 4pt shifted p; draw fullcircle scaled 4pt shifted p; enddef;
-def drawendpointvertex(expr p) = draw vertex(p) withcolor red; draw fullcircle scaled 6pt shifted p; enddef;
-def createpath(expr p) = shakepath(p, 0.015cm,0.1cm) enddef;
-vardef shakepath(expr p,d,l) =
-	save r,b;
-	path r; r := point(arctime 0 of p) of p;
-	b := -1;
-	for i:=l step l until arclength(p):
-		r := r--(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*d*b;
-		b := -b;
-	endfor
-	r--point(arctime arclength(p) of p) of p
-enddef;
-	
-pen normalpen; normalpen := pencircle scaled 0.6pt;
-pen boldpen; boldpen := pencircle scaled 1.5pt;
-pen bolderpen; bolderpen := pencircle scaled 2pt;
-def dotline = withdots scaled 0.82 withpen boldpen enddef;
-
-vardef unclosedbubblec(expr p,c) =
-	bubblec((p..reverse p..cycle),c)
-enddef;	
-
-vardef bubblec(expr p,c) =
-	save r;
-	path r; r := (point(arctime 0 of p) of p)+dir(angle(direction(arctime 0 of p) of p rotated 90))*c;
-	for i:=0.01cm step 0.025cm until arclength(p):
-		r := r..(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*c;
-	endfor
-	r..(point(arctime arclength(p) of p) of p)+dir(angle(direction(arctime arclength(p) of p) of p rotated 90))*c..cycle
-enddef;
-
-vardef bubble(expr p) = bubblec(p,0.12cm) enddef;
-vardef unclosedbubble(expr p) = unclosedbubblec(p,0.12cm) enddef;
diff --git a/8-geom2/8-geom2.mp b/8-geom2/8-geom2.mp
deleted file mode 100644
index 5a44396..0000000
--- a/8-geom2/8-geom2.mp
+++ /dev/null
@@ -1,137 +0,0 @@
-input lib
-
-figname("8-geom2_");
-figtag("usecky");
-beginfig(1);
-	def drawusecka(expr p,q) = draw vertex(p); draw vertex(q); draw p--q; enddef;
-	pair A[],B[];
-	A0 := origin; B0 := (4cm,1.5cm);
-	A1 := (0.5cm,1.5cm); B1 := (1cm,0);
-	A2 := (2cm,1.3cm); B2 := (3cm, -0.9cm);
-	A3 := (1cm,-1.3cm); B3 := (3.5cm,0);
-	z0 = whatever[A0,B0]; z0 = whatever[A1,B1];
-	z1 = whatever[A0,B0]; z1 = whatever[A2,B2];
-	z2 = whatever[A2,B2]; z2 = whatever[A3,B3];
-	
-	for i:=0 upto 2: fill fullcircle scaled 7pt shifted z[i]; drawemptyvertex(z[i]); endfor
-	for i:=0 upto 3: drawusecka(A[i], B[i]); endfor
-endfig;
-
-figtag("polorovina");
-beginfig(2);
-	pair A,B;
-	A := origin;
-	B := (1.7cm,0.5cm);
-	an := angle(B-A);
-	path p; p := from(.5[A,B],an+90,3cm)--from(.5[A,B],an-90,3cm);
-	fill p--reverse(p shifted (A-0.8[A,B]))--cycle withcolor 0.8white;
-	%fill p{dir (an+180)}..-0.35[A,B]..{dir an}cycle withcolor 0.8white;
-	draw p withpen boldpen;
-
-	pair C; C := point(0.9) of p; 
-	drawarrow from(C,an+180,0.1cm)--from(C,an+180,1cm) withpen boldpen;
-	C := point(0.1) of p; 
-	drawarrow from(C,an+180,0.1cm)--from(C,an+180,1cm) withpen boldpen;
-	label.rt(btex $p$ etex, point(0.15) of p);
-	
-	draw A--B dashed evenly;
-	drawemptyvertex(A); drawemptyvertex(B);
-	label.llft(btex $a$ etex, A);
-	label.urt(btex $b$ etex, B);
-	label(btex $B_a$ etex, A+2(-0.1cm,0.5cm));
-	label(btex $B_b$ etex, B+2(-0.1cm,0.5cm));
-endfig;
-
-figtag("voroneho_diagram");
-beginfig(3);
-	u := 1.35cm;
-	pair A[],B[];
-	A0 := origin; A1 := (-u,-u); A2 := (1.3u,-u); A3 := (1.50821u,0.7u); A4 := (0.6u,0.8u); A5 := (0.2u,1.9u); A6 := (-1.3u,1u);
-	def osa(expr a, b, an,l) = from(.5[a,b], angle(b-a)+an, l) enddef;
-	vardef prusecik_os(expr p,q,r) =
-		save b; pair b;
-		b = whatever[osa(p,q, 90,1cm), osa(p,q,-90,1cm)];
-		b = whatever[osa(q,r, 90,1cm), osa(q,r,-90,1cm)];
-		b
-	enddef;
-	B[0] := prusecik_os(A0,A1,A2); B[1] := prusecik_os(A0,A1,A6); B[2] := prusecik_os(A0,A4,A6); B[3] := prusecik_os(A4,A5,A6);
-	B[4] := prusecik_os(A3,A4,A5); B[5] := prusecik_os(A0,A3,A4); B[6] := prusecik_os(A0,A2,A4);
-	draw osa(A1,A2,-90,2cm)--B[0]--B[1]--osa(A1,A6,90,1.3cm) withpen boldpen;
-	draw B[1]--B[2]--B[3]--osa(A6,A5,90,2cm) withpen boldpen;
-	draw B[3]--B[4]--osa(A3,A5,-90,2.7cm) withpen boldpen;
-	draw B[4]--B[5]--B[2] withpen boldpen;
-	draw B[5]--B[6]--osa(A2,A3,-90,1.3cm) withpen boldpen;
-	draw B[6]--B[0] withpen boldpen;
-	
-	draw A0--A1--A2--A0--A6--A1 dashed evenly;
-	draw A6--A5--A4--A0 dashed evenly; draw A3--A2 dashed evenly;
-	draw A3--A5 dashed evenly; draw A6--A4--A3 dashed evenly;
-
-	for i:=0 upto 6: draw vertex(B[i]); endfor
-	for i:=0 upto 6: drawemptyvertex(A[i]); endfor
-endfig;
-
-figtag("pasy_mnohouhelniku");
-beginfig(4);
-	u := 1.35cm;
-	pair A[],B[];
-	A0 := origin; A1 := (-u,-u); A2 := (1.3u,-u); A3 := (1.50821u,0.7u); A4 := (0.6u,0.8u); A5 := (0.2u,1.9u); A6 := (-1.3u,1u);
-	def osa(expr a, b, an,l) = from(.5[a,b], angle(b-a)+an, l) enddef;
-	vardef prusecik_os(expr p,q,r) =
-		save b; pair b;
-		b = whatever[osa(p,q, 90,1cm), osa(p,q,-90,1cm)];
-		b = whatever[osa(q,r, 90,1cm), osa(q,r,-90,1cm)];
-		b
-	enddef;
-	def drawline(expr p) = draw ((-2.3u,p)--(2.5u,p)) cutbefore (D0--D1) cutafter (D2--D3) enddef;
-	B[0] := prusecik_os(A0,A1,A2); B[1] := prusecik_os(A0,A1,A6); B[2] := prusecik_os(A0,A4,A6); B[3] := prusecik_os(A4,A5,A6);
-	B[4] := prusecik_os(A3,A4,A5); B[5] := prusecik_os(A0,A3,A4); B[6] := prusecik_os(A0,A2,A4);
-	
-	pair C; C := origin; for i:=0 upto 6: C := C + B[i]; endfor C := C/6;
-	pair D[];
-	D0 := C+(-2.25,2.25)*u; D1 := C+(-2.25,-2.25)*u; D2 := C+(2.25,-2.25)*u; D3 := C+(2.25,2.25)*u; 
-	
-	draw B[0]--B[1]--B[2]--B[3]--B[4]--B[5]--B[6]--B[0] withpen boldpen;
-	draw B[5]--B[2] withpen boldpen;
-	pair E; E := 0.6[B0,B4];
-	drawline(ypart(E));
-	drawline(ypart(E)) cutbefore (B2--B5) cutafter (B4--B5) dashed evenly withpen bolderpen;
-	drawemptyvertex(E);
-
-	for i:=0 upto 6: drawline(ypart(B[i])) dashed evenly; endfor
-	for i:=0 upto 6: draw vertex(B[i]); endfor
-endfig;
-
-figtag("upravy_stromu");
-beginfig(5);
-	u := 1cm;
-	draw (0,0.1u)--(2.05u,-2.05u)--(-2.05u,-2.05u)--cycle;
-	pair A[]; A0 := from(origin,-90,0.1u); A1 := from(A0, -70, 0.5u); A2 := from(A1, -110, 0.5u); A3 := from(A2, -80, 0.5u); A4 := from(A3, -120, 0.45u);
-	path p; p := createpath(A0--A1--A2--A3--A4);
-	path q; q := (p scaled 0.93 shifted (-0.15u,-0.15u))--(-1.85u,-1.95u)--cycle; fill q withcolor 0.8white; draw q;
-	q := (p scaled 0.93 shifted (0.15u,-0.15u))--(1.85u,-1.95u)--cycle; fill q withcolor 0.8white; draw q;
-	p := createpath(A0--from(A0,-110,0.1u)--A1--A2--A3--A4); draw p withpen boldpen;
-endfig;
-
-figtag("rychla_perzistence");
-beginfig(6);
-	u := 1cm;
-	def drawtable(expr p, lab, sa, sb) =
-		draw centersquare xscaled 2u yscaled (2u/3) shifted (p+(0,u/3));
-		draw centersquare xscaled 2u yscaled (2u/3) shifted (p-(0,u/3));
-		label(sa, p+(0,u/3));
-		label(sb, p-(0,u/3));
-		label.bot(lab, p-(0,2u/3));
-	enddef;
-
-	pair A[];
-	for i:=0 upto 2: A[i] := (0,2u) rotated 120i; endfor
-	drawtable(A1, btex $v$ etex, btex verze 2 etex, btex verze 1 etex);
-	drawtable(A2, btex $v'$ etex, btex verze 3 etex, "");
-	drawtable(A0, btex $u$ etex, btex verze 2 etex, btex verze 1 etex);
-
-	drawarrow from(A[2]+(0,u/3), 180, 7u/6)--from(A[1]+(0,u/3), 0, 7u/6);
-	drawarrow from(A[0]+(0,-u/3), 180, 7u/6)..{dir -90}from(A[1]+(-3u/4,2u/3), 90, u/6);
-	drawarrow from(A[0]+(0,u/3), 0, 7u/6)..{dir -90}from(A[2]+(3u/4,2u/3), 90, u/6);
-endfig;
-end
diff --git a/8-geom2/8-geom2.tex b/8-geom2/8-geom2.tex
deleted file mode 100644
index 034b395..0000000
--- a/8-geom2/8-geom2.tex
+++ /dev/null
@@ -1,167 +0,0 @@
-\input lecnotes.tex
-
-\prednaska{8}{Geometrie vrací úder}{(sepsal Pavel Klavík)}
-
-\>Kdy¾ s geometrickými problémy poøádnì nezametete, ony vám to vrátí! Ale kdy¾ u¾ zametat, tak urèitì ne pod koberec a místo smetáku pou¾ijte pøímku.
-V této pøedná¹ce nás spolu s dvìma geometrickými problémy samozøejmì èeká pokraèování pohádky o ledních medvìdech.
-
-{\I Medvìdi vyøe¹ili rybí problém a hlad je ji¾ netrápí. Av¹ak na severu ne¾ijí sami, za sousedy mají Eskymáky. Proto¾e je rozhodnì lep¹í se sousedy
-dobøe vycházet, jsou medvìdi a Eskymáci velcí pøátelé. Skoro ka¾dý se se svými pøáteli rád schází. Av¹ak to je musí nejprve nalézt~\dots}
-
-\h{Hledání prùseèíkù úseèek}
-
-Zkusíme nejprve Eskymákùm vyøe¹it lokalizaci ledních medvìdù.
-
-{\I Kdy¾ takový medvìd nemá co na práci, rád se prochází. Na místech, kde se trasy protínají, je zvý¹ená ¹ance, ¾e se dva medvìdi potkají a zapovídají
--- ostatnì co byste èekali od medvìdù. To jsou ta správná místa pro Eskymáka, který chce potkat medvìda. Jenom¾e jak tato køí¾ení najít?}
-
-Pro zjednodu¹ení pøedpokládejme, ¾e medvìdi chodí po úseèkách tam a zpìt. Budeme tedy chtít nalézt v¹echny prùseèíky úseèek v rovinì.
-
-\bigskip
-\centerline{\epsfxsize=1.5in\epsfbox{8-geom2_0_bear.eps}\hskip 4em\epsfxsize2in\epsfbox{8-geom2_1_usecky.eps}}
-\smallskip
-\centerline{Problém Eskymákù: Kde v¹ude se køí¾í medvìdí trasy?}
-\bigskip
-
-Pro $n$ úseèek mù¾e existovat a¾ $\Omega(n^2)$ prùseèíkù.\foot{Zkuste takový pøíklad zkonstruovat.} Tedy optimální slo¾itosti by dosáhl i algoritmus,
-který by pro ka¾dou dvojici úseèek testoval, zda se protínají. Èasovou slo¾itost algoritmu v¹ak posuzujeme i vzhledem k velikosti výstupu $p$. Typické
-rozmístìní úseèek mívá toti¾ prùseèíkù spí¹e pomálu. Pro tento pøípad si uká¾eme podstatnì rychlej¹í algoritmus.
-
-Pro jednodu¹¹í popis pøedpokládejme, ¾e úseèky le¾í v obecné poloze. To znamená, ¾e ¾ádné tøi úseèky se neprotínají v jednom bodì a prùnikem ka¾dých
-dvou úseèek je nejvý¹e jeden bod. Navíc pøedpokládejme, ¾e krajní bod ¾ádné úseèky nele¾í na jiné úseèce a také neexistují vodorovné úseèky. Na závìr si
-uká¾eme, jak se s tìmito pøípady vypoøádat.
-
-Algoritmus funguje na principu zametání roviny, popsaném v minulé pøedná¹ce. Budeme posouvat vodorovnou pøímku odshora dolù. V¾dy, kdy¾ narazíme na
-nový prùnik, ohlásíme jeho výskyt. Samozøejmì spojité posouvání nahradíme diskrétním a pøímku v¾dy posuneme do dal¹ího zajímavého bodu.
-
-Zajímavé události jsou {\I zaèátky úseèek}, {\I konce úseèek} a {\I prùseèíky úseèek}. Po utøídìní známe pro první dva typy událostí poøadí, v jakém
-se objeví. Výskyty prùseèíkù budeme poèítat prùbì¾nì, jinak bychom celý problém nemuseli øe¹it.
-
-V ka¾dém kroku si pamatujeme {\I prùøez} $P$ -- posloupnost úseèek aktuálnì protnutých zametací pøímkou. Tyto úseèky máme utøídìné zleva doprava.  Navíc si
-udr¾ujeme kalendáø $K$ budoucích událostí. Z hlediska prùseèíkù budeme na úseèky nahlí¾et jako na polopøímky. Pro sousední dvojice úseèek si
-udr¾ujeme, zda se jejich smìry nìkde protnou. Algoritmus pro hledání prùnikù úseèek funguje následovnì:
-
-\s{Algoritmus:}
-
-\algo
-
-\:$P \leftarrow \emptyset$.
-\:Do $K$ vlo¾íme zaèátky a konce v¹ech úseèek.
-\:Dokud $K \ne \emptyset$:
-\::Odebereme nejvy¹¹í událost.
-\::Pokud je to zaèátek úseèky, zatøídíme novou úseèku do $P$.
-\::Pokud je to konec úseèky, odebereme úseèku z $P$.
-\::Pokud je to prùseèík, nahlásíme ho a prohodíme úseèky v $P$.
-\::Navíc v¾dy pøepoèítáme prùseèíkové události, v¾dy maximálnì dvì odebereme a dvì nové pøidáme.
-\endalgo
-
-Zbývá rozmyslet si, jaké datové struktury pou¾ijeme, abychom prùseèíky nalezli dostateènì rychle. Pro kalendáø pou¾ijeme napøíklad haldu. Prùøez si
-budeme udr¾ovat ve vyhledávacím stromì. Poznamenejme, ¾e nemusíme znát souøadnice úseèek, staèí znát jejich poøadí, které se mezi jednotlivými
-událostmi nemìní. Pøi pøidávání úseèek procházíme stromem a porovnáváme souøadnice v prùøezu, které prùbì¾nì dopoèítáváme.
-
-Kalendáø obsahuje v¾dy nejvý¹e $\O(n)$ událostí. Podobnì prùøez obsahuje v ka¾dém okam¾iku nejvý¹e $\O(n)$ úseèek.  Jednu událost kalendáøe doká¾eme
-o¹etøit v èase $\O(\log n)$. V¹ech událostí je $\O(n+p)$, a tedy celková slo¾itost algoritmu je $\O((n+p) \log n)$.
-
-Slíbili jsme, ¾e popí¹eme, jak se vypoøádat s vý¹e uvedenými podmínkami na vstup. Události kalendáøe se stejnou $y$-ovou souøadnicí budeme tøídit v
-poøadí zaèátky, prùseèíky a konce úseèek. Tím nahlásíme i prùseèíky krajù úseèek a ani vodorovné úseèky nebudou vadit. Podobnì se není tøeba obávat
-prùseèíkù více úseèek v jednom bodì. Úseèky jdoucí stejným smìrem, jejich¾ prùnik je úseèka, jsou komplikovanìj¹í, ale lze jejich prùseèíky o¹etøit a
-vypsat tøeba souøadnice úseèky tvoøící jejich prùnik.
-
-Na závìr poznamenejme, ¾e Balaban vymyslel efektivnìj¹í algoritmus, který funguje v èase $\O(n \log n + p)$, ale je podstatnì komplikovanìj¹í.
-
-\h{Hledání nejbli¾¹ích bodù a Voroného diagramy}
-
-Nyní se pokusíme vyøe¹it i problém druhé strany -- pomù¾eme medvìdùm nalézt Eskymáky.
-
-{\I Eskymáci tráví vìt¹inu èasu doma, ve svém iglù. Takový medvìd je na své toulce zasnì¾enou krajinou, kdy¾ tu se najednou rozhodne nav¹tívit nìjakého
-Eskymáka. Proto se podívá do své medvìdí mapy a nalezne nejbli¾¹í iglù. Má to ale jeden háèek, iglù jsou spousty a medvìd by dávno usnul, ne¾ by
-nejbli¾¹í objevil.}\foot{Zlí jazykové by øekli, ¾e medvìdi jsou moc líní a nebo v mapách ani èíst neumí!}
-
-Popí¹eme si nejprve, jak vypadá medvìdí mapa. Medvìdí mapa obsahuje celou Arktidu a jsou v ní vyznaèena v¹echna iglù. Navíc obsahuje vyznaèené
-oblasti tvoøené body, které jsou nejblí¾e k jednomu danému iglù. Takovému schématu se øíká {\I Voroného diagram}. Ten pro zadané body $x_1, \ldots, x_n$
-obsahuje rozdìlení roviny na oblasti $B_1, \ldots, B_n$, kde $B_i$ je mno¾ina bodù, které jsou blí¾e k $x_i$ ne¾ k ostatním bodùm $x_j$. Formálnì jsou
-tyto oblasti definovány následovnì:
-$$B_i = \left\{y \in {\bb R}^2\ \vert\ \forall j:\rho(x_i,y) \le \rho(x_j,y)\right\},$$
-kde $\rho(x,y)$ znaèí vzdálenost bodù $x$ a $y$.
-
-Uká¾eme si, ¾e Voroného diagram má pøekvapivì jednoduchou strukturu. Nejprve uva¾me, jak budou vypadat oblasti $B_a$ a $B_b$ pouze pro dva body
-$a$ a $b$, jak je naznaèeno na obrázku. V¹echny body stejnì vzdálené od $a$ i $b$ le¾í na pøímce $p$ -- ose úseèky $ab$. Oblasti $B_a$ a $B_b$
-jsou tedy tvoøeny polorovinami ohranièenými osou $p$. Tedy obecnì tvoøí mno¾ina v¹ech bodù bli¾¹ích k $x_i$ ne¾ k $x_j$ nìjakou polorovinu. Oblast
-$B_i$ obsahuje v¹echny body, které jsou souèasnì bli¾¹í k $x_i$ ne¾ ke v¹em ostatním bodùm $x_j$ -- tedy le¾í ve v¹ech polorovinách souèasnì.
-Ka¾dá z oblastí $B_i$ je tvoøena prùnikem $n-1$ polorovin, tedy je to (mo¾ná neomezený) mnohoúhelník.\foot{Sly¹eli jste u¾ o lineárním programování?
-Jak název vùbec nenapoví, {\I lineární programování} je teorii zabývající se øe¹ením a vlastnostmi soustav lineárních nerovnic. Lineární program je
-popsaný lineární funkcí, kterou chceme maximalizovat za podmínek popsaných soustavou lineárních nerovnic. Ka¾dá nerovnice urèuje poloprostor, ve
-kterém se pøípustná øe¹ení nachází.  Proto¾e pøípustné øe¹ení splòuje v¹echny nerovnice zároveò, je mno¾ina v¹ech pøípustných øe¹ení (mo¾ná neomezený)
-mnohostìn, obecnì ve veliké dimenzi ${\bb R}^d$, kde $d$ je poèet promìnných. Mno¾iny $B_i$ lze snadno popsat jako mno¾iny v¹ech pøípustných øe¹ení
-lineárních programù pomocí vý¹e ukázaných polorovin. Na závìr poznamenejme, ¾e dlouho otevøená otázka, zda lze nalézt optimální øe¹ení lineárního
-programu v polynomiálním èase, byla pozitivnì vyøe¹ena -- je znám polynomiální algoritmus, kterému se øíká {\I metoda vnitøního bodu}. Na druhou
-stranu, pokud chceme najít pøípustné celoèíselné øe¹ení, je úloha NP-úplná a je jednoduché na ni pøevést spoustu optimalizaèních problémù. Dokázat
-NP-tì¾kost není pøíli¹ tì¾ké. Na druhou stranu ukázat, ¾e tento problém le¾í v NP, není vùbec jednoduché.}
-Pøíklad Voroného diagramu je naznaèen na obrázku. Zadané body jsou oznaèeny prázdnými krou¾ky a hranice oblastí $B_i$ jsou vyznaèené èernými èárami.
-
-\twofigures{8-geom2_2_polorovina.eps}{Body bli¾¹í k $a$ ne¾ $b$.}{1.25in}{8-geom2_3_voroneho_diagram.eps}{Voroného diagram.}{2.5in}
-
-Není náhoda, pokud vám hranice oblastí pøipomíná rovinný graf. Jeho vrcholy jsou body, které jsou stejnì vzdálené od alespoò tøí zadaných bodù. Jeho
-stìny jsou oblasti $B_i$. Jeho hrany jsou tvoøeny èástí hranice mezi dvìma oblastmi -- body, které mají dvì oblasti spoleèné. Obecnì prùnik dvou
-oblastí mù¾e být, v závislosti na jejich sousedìní, prázdný, bod, úseèka, polopøímka nebo dokonce celá pøímka.  V dal¹ím textu si pøedstavme, ¾e celý
-Voroného diagram uzavøeme do dostateènì velkého obdélníka,\foot{Pøeci jenom i celá Arktida je omezenì velká.} èím¾ dostaneme omezený rovinný graf.
-
-Poznamenejme, ¾e pøeru¹ované èáry tvoøí hrany duálního rovinného grafu s vrcholy v zadaných bodech. Hrany spojují sousední body na kru¾nicích, které 
-obsahují alespoò tøi ze zadaných bodù. Napøíklad na obrázku dostáváme skoro samé trojúhelníky, proto¾e vìt¹ina kru¾nic obsahuje pøesnì tøi zadané
-body. Av¹ak nalezneme i jeden ètyøúhelník, jeho¾ vrcholy le¾í na jedné kru¾nici.
-
-Zkusíme nyní odhadnout, jak velký je rovinný graf popisující Voroného diagram. Podle slavné Eulerovy formule má ka¾dý rovinný graf nejvý¹e lineárnì
-mnoho vrcholù, hran a stìn -- pro $v$ vrcholù, $e$ hran a $f$ stìn je $e \le 3v-6$ a navíc $v+f = e+2$. Tedy slo¾itost diagramu je lineární vzhledem k
-poètu zadaných bodù $n=f$, $\O(n)$. Navíc Voroného diagram lze zkonstruovat v èase $\O(n \log n)$, napøíklad pomocí zametání roviny nebo metodou
-rozdìl a panuj. Tím se v¹ak zabývat nebudeme,\foot{Pro zvídavé, kteøí nemají zkou¹ku druhý den ráno: Detaily naleznete v zápiscích z pøedloòského
-ADSka.} místo toho si uká¾eme, jak v ji¾ spoèteném Voroného diagramu rychle hledat nejbli¾¹í body.
-
-\h{Lokalizace bodu uvnitø mnohoúhelníkové sítì}
-
-Problém medvìdù je najít v medvìdí mapì co nejrychleji nejbli¾¹í iglù. Máme v rovinì sí» tvoøenou mnohoúhelníky. Chceme pro jednotlivé body rychle
-rozhodovat, do kterého mnohoúhelníku patøí. Na¹e øe¹ení budeme optimalizovat pro jeden pevný rozklad a obrovské mno¾ství rùzných dotazù, které chceme
-co nejrychleji zodpovìdìt.\foot{Pøedstavujme si to tøeba tak, ¾e medvìdùm zprovozníme server. Ten jednou schroustá celou mapu a potom co nejrychleji
-odpovídá na jejich dotazy. Medvìdi tak nemusí v mapách nic hledat, staèí se pøipojit na server a poèkat na odpovìï.} Nejprve pøedzpracujeme zadané
-mnohoúhelníky a vytvoøíme strukturu, která nám umo¾ní rychlé dotazy na jednotlivé body.
-
-Uka¾me si pro zaèátek øe¹ení bez pøedzpracování. Rovinu budeme zametat pøímkou shora dolù. Podobnì jako pøi hledání prùseèíkù úseèek, udr¾ujeme si prùøez
-pøímkou. V¹imnìte si, ¾e tento prùøez se mìní jenom ve vrcholech mnohoúhelníkù. Ve chvíli, kdy narazíme na hledaný bod, podíváme se, do kterého
-intervalu v prùøezu patøí. To nám dá mnohoúhelník, který nahlásíme. Prùøez budeme uchovávat ve vyhledávacím stromì. Takové øe¹ení má slo¾itost $\O(n
-\log n)$ na dotaz, co¾ je hroznì pomalé.
-
-Pøedzpracování bude fungovat následovnì. Jak je naznaèeno na obrázku pøeru¹ovanými èárami, rozøe¾eme si celou rovinu na pásy, bìhem kterých se prùøez
-pøímkou nemìní. Pro ka¾dý z nich si pamatujeme stav stromu popisující, jak vypadal prùøez pøi procházení tímto pásem. Kdy¾ chceme lokalizovat nìjaký bod,
-nejprve pùlením nalezneme pás, ve kterém se nachází. Poté polo¾íme dotaz na pøíslu¹ný strom. Strom procházíme a po cestì si dopoèítáme souøadnice
-prùøezu, a¾ lokalizujeme správný interval v prùøezu. Dotaz doká¾eme zodpovìdìt v èase $\O(\log n)$. Hledaný bod je na obrázku naznaèen prázdným
-koleèkem a nalezený interval v prùøezu je vyta¾ený tuènì.
-
-\figure{8-geom2_4_pasy_mnohouhelniku.eps}{Mnohoúhelníky rozøezané na pásy.}{2.5in}
-
-Jenom¾e na¹e øe¹ení má jeden háèek: Jak zkonstruovat jednotlivé verze stromu dostateènì rychle? K tomu napomohou {\I èásteènì perzistentní} datové
-struktury. Pod perzistencí se myslí, ¾e struktura umo¾òuje uchovávat svoji historii. Èásteènì perzistentní struktury nemohou svoji historii
-modifikovat.
-
-Popí¹eme si, jak vytvoøit perzistentní strom s pamìtí $\O(\log n)$ na zmìnu. Pokud provádíme operaci na stromì, mìní se jenom malá èást stromu.
-Napøíklad pøi vkládání do stromu se mìní jenom prvky na jedné cestièce z koøene do listu (a pøípadnì rotací i na jejím nejbli¾¹ím okolí). Proto si
-ulo¾íme upravenou cestièku a zbytek stromu budeme sdílet s pøedchozí verzí. Na obrázku je vyznaèena cesta, její¾ vrcholy jsou upravovány.  ©edì
-oznaèené podstromy navì¹ené na tuto cestu se nemìní, a proto na nì staèí zkopírovat ukazatele. Mimochodem zmìny ka¾dé operace se slo¾itostí $\O(k)$
-lze zapsat v pamìti $\O(k)$, prostì operace nemá tolik èasu, aby mohla pozmìnit pøíli¹ velikou èást stromu. 
-
-\figure{8-geom2_5_upravy_stromu.eps}{Jedna operace mìní pouze okolí cesty -- navì¹ené podstromy se nemìní.}{2in}
-
-Celková èasová slo¾itost je tedy $\O(n \log n)$ na pøedzpracování Voroného diagramu a vytvoøení persistentního stromu. Kvùli persistenci potøebuje
-toto pøedzpracování pamì» $\O(n \log n)$. Na dotaz spotøebujeme èas $\O(\log n)$, nebo» nejprve vyhledáme pùlením pøíslu¹ný pás a poté polo¾íme dotaz
-na pøíslu¹nou verzi stromu. Rychleji to ani provést nepùjde, nebo» potøebujeme utøídit souøadnice bodù.
-
-\s{Lze to lépe?} Na závìr poznamenejme, ¾e se umí provést vý¹e popsaná persistence vyhledávacího stromu v amortizované pamìti $\O(1)$ na zmìnu. Ve
-struènosti naznaèíme my¹lenku. Pou¾ijeme stromy, které pøi insertu a deletu provádí amortizovanì jenom konstantnì mnoho úprav své struktury. To nám
-napøíklad zaruèí 2-4 stromy z pøedná¹ky a podobnou vlastnost lze dokázat i o èerveno-èerných stromech. Pøi zmìnì potom nebudeme upravovat celou cestu,
-ale upravíme jenom jednotlivé vrcholy, kterých se zmìna týká. Ka¾dý vrchol stromu si v sobì bude pamatovat a¾ dvì své verze. Pokud chceme vytvoøit
-tøetí verzi, vrchol zkopírujeme stranou. To v¹ak mù¾e vyvolat zmìny v jeho rodièích a¾ do koøene. Situace je naznaèena na obrázku. Pøi vytvoøení nové
-verze $3$ pro vrcholu $v$ vytvoøíme jeho kopii $v'$, do které ulo¾íme tuto verzi. Av¹ak musíme také zmìnit rodièe $u$, kterému vytvoøíme novou verzi
-ukazující na $v'$. Abychom dosáhli ký¾ené konstantní pamì»ové slo¾itosti, pomù¾e potenciálový argument -- zmìn se provádí amortizovanì jenom
-konstantnì mnoho. Navíc si pro ka¾dou verzi pamatujeme její koøen, ze kterého máme dotaz spustit.
-
-\figure{8-geom2_6_rychla_perzistence.eps}{Vytvoøení nové verze vrcholu.}{2in}
-
-\bye
diff --git a/8-geom2/8-geom2_0_bear.eps b/8-geom2/8-geom2_0_bear.eps
deleted file mode 100644
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-%%EOF
diff --git a/8-geom2/8-geom2_5_upravy_stromu.eps b/8-geom2/8-geom2_5_upravy_stromu.eps
deleted file mode 100644
index c30d730..0000000
--- a/8-geom2/8-geom2_5_upravy_stromu.eps
+++ /dev/null
@@ -1,135 +0,0 @@
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diff --git a/8-geom2/8-geom2_6_rychla_perzistence.eps b/8-geom2/8-geom2_6_rychla_perzistence.eps
deleted file mode 100644
index 8085675..0000000
--- a/8-geom2/8-geom2_6_rychla_perzistence.eps
+++ /dev/null
@@ -1,106 +0,0 @@
-%!PS
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diff --git a/8-geom2/Makefile b/8-geom2/Makefile
deleted file mode 100644
index 087c424..0000000
--- a/8-geom2/Makefile
+++ /dev/null
@@ -1,3 +0,0 @@
-P=8-geom2
-
-include ../Makerules
diff --git a/8-geom2/lib.mp b/8-geom2/lib.mp
deleted file mode 100644
index cbeedf5..0000000
--- a/8-geom2/lib.mp
+++ /dev/null
@@ -1,84 +0,0 @@
-% implementation of figure naming and figure transparency
-string name,tag; name := ""; tag := "";
-def updatefigname =
-	if tag="": filenametemplate (name & "%c.eps");
-	else: filenametemplate (name & "%c_" & tag & ".eps");
-	fi;
-enddef;
-	
-def figname(expr n) = name := n; updatefigname; enddef;
-def figtag(expr t) = tag := t; updatefigname; enddef;
-
-picture transparent_picture;
-color transparent_color; transparent_color := 0.9white;
-def drawtransparent(expr num) =
-	transparent_picture := currentpicture;
-endfig;
-
-if tag="": filenametemplate (name & "%c_transparent.eps");
-else: filenametemplate (name & "%c_" & tag & "_transparent.eps");
-fi;
-beginfig(num);
-	draw transparent_picture withcolor transparent_color;
-endfig;
-
-updatefigname;
-enddef;
-
-def from(expr p,d,len) =
-	p+dir(d)*len
-enddef;
-
-def dirs(expr p,d,len) =
-	p--from(p,d,len)
-enddef;
-
-def drawvertices(expr s,n) =
-	for i:=s upto n:
-		draw vertex(PQ[i]);
-	endfor
-enddef;
-
-def drawfvertices(expr s,n,flags) =
-	for i:=s upto n:
-		draw vertex(PQ[i]) flags;
-	endfor
-enddef;
-
-path centersquare; centersquare := (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
-
-def vertex(expr p) = p withpen pencircle scaled 4pt enddef;
-def drawemptyvertex(expr p) = unfill fullcircle scaled 4pt shifted p; draw fullcircle scaled 4pt shifted p; enddef;
-def drawendpointvertex(expr p) = draw vertex(p) withcolor red; draw fullcircle scaled 6pt shifted p; enddef;
-def createpath(expr p) = shakepath(p, 0.015cm,0.1cm) enddef;
-vardef shakepath(expr p,d,l) =
-	save r,b;
-	path r; r := point(arctime 0 of p) of p;
-	b := -1;
-	for i:=l step l until arclength(p):
-		r := r--(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*d*b;
-		b := -b;
-	endfor
-	r--point(arctime arclength(p) of p) of p
-enddef;
-	
-pen normalpen; normalpen := pencircle scaled 0.6pt;
-pen boldpen; boldpen := pencircle scaled 1.5pt;
-pen bolderpen; bolderpen := pencircle scaled 2pt;
-def dotline = withdots scaled 0.82 withpen boldpen enddef;
-
-vardef unclosedbubblec(expr p,c) =
-	bubblec((p..reverse p..cycle),c)
-enddef;	
-
-vardef bubblec(expr p,c) =
-	save r;
-	path r; r := (point(arctime 0 of p) of p)+dir(angle(direction(arctime 0 of p) of p rotated 90))*c;
-	for i:=0.01cm step 0.025cm until arclength(p):
-		r := r..(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*c;
-	endfor
-	r..(point(arctime arclength(p) of p) of p)+dir(angle(direction(arctime arclength(p) of p) of p rotated 90))*c..cycle
-enddef;
-
-vardef bubble(expr p) = bubblec(p,0.12cm) enddef;
-vardef unclosedbubble(expr p) = unclosedbubblec(p,0.12cm) enddef;
diff --git a/9-fft/9-fft.tex b/9-fft/9-fft.tex
deleted file mode 100644
index 3340759..0000000
--- a/9-fft/9-fft.tex
+++ /dev/null
@@ -1,392 +0,0 @@
-\def\scharfs{\char"19}
-\input lecnotes.tex
-\def\imply{\Rightarrow}
-\prednaska{9}{Fourierova transformace}{\vbox{\hbox{(K.Jakubec, M.Polák
-	a~G.Ocsovszky,}\hbox{\ V.Tùma, M.Kozák)}}}
-
-Násobení polynomù mù¾e mnohým pøipadat jako pomìrnì (algoritmicky) snadný
-problém. Asi ka¾dého hned napadne \uv{hloupý} algoritmus -- vezmeme
-koeficienty prvního polynomu a~vynásobíme ka¾dý se v¹emi koeficienty druhého
-polynomu a~pøíslu¹nì u~toho seèteme i~exponenty (stejnì jako to dìláme, kdy¾
-násobíme polynomy na~papíøe). Pokud stupeò prvního polynomu je $n$ a~druhého
-$m$, strávíme tím èas $\Omega(mn)$. Pro $m=n$ je to kvadraticky pomalé.
-Na~první pohled se mù¾e zdát, ¾e rychleji to prostì nejde (pøeci musíme
-v¾dy vynásobit \uv{ka¾dý s~ka¾dým}). Ve skuteènosti to ale rychleji fungovat
-mù¾e, ale k~tomu je potøeba znát trochu tajemný algoritmus FFT neboli {\I Fast
-Fourier Transform}.
-
-
-\ss{Trochu algebry na~zaèátek}
-Celé polynomy oznaèujeme velkými písmeny, jednotlivé èleny polynomù pøíslu¹nými
-malými písmeny (pø.: polynom $W$ stupnì $d$ má koeficienty $w_{0}, w_{1},
-w_{2},\ldots, w_{d}$).
-
-Libovolný polynom $P$ stupnì (nejvý¹e) $d$ lze reprezentovat
-jednak jeho koeficienty, tedy èísly $p_{0}, p_{1}, \ldots ,p_{d}$, druhak
-i~pomocí hodnot:
-
-\>{\bf Lemma:} Polynom stupnì nejvý¹e $d$ je jednoznaènì urèeò svými
-hodnotami v~$d+1$ rùzných bodech.
-
-\>{\it Dùkaz:}
-Polynom stupnì $d$ má maximálnì $d$ koøenù (indukcí -- je-li
-$k$ koøenem $P$, pak lze $P$ napsat jako $(x-k)Q$ kde $Q$ je polynom stupnì
-o~jedna men¹í, pøitom polynom stupnì 1 má jediný koøen); uvá¾íme-li
-dva rùzné polynomy $P$ a~$Q$ stupnì $d$ nabývající v~daných bodech stejných
-hodnot, tak $P-Q$ je polynom stupnì maximálnì $d$, ka¾dé
-z $x_{0}\ldots x_{d}$ je koøenem tohoto polynomu $\imply$ spor, polynom stupnì
-$d$ má $d+1$ koøenù $\imply$ $P-Q$ musí být nulový polynom $\imply$ $P=Q$.
-\qed
-\medskip	
-
-Pov¹imnìme si jedné skuteènosti -- máme-li dva polynomy $A$ a~$B$ stupnì $d$
-a~body $x_{0}, \ldots, x_{n}$, dále polynom $C=A \cdot B$ (stupnì $2d$), pak
-platí $C(x_{j}) = A(x_{j}) \cdot B(x_{j})$ pro $j = 0,1,2, \ldots, n$. Toto
-èiní tento druhý zpùsob reprezentace polynomu velice atraktivním pro násobení
--- máme-li $A$ i $B$ reprezentované hodnotami v $n \geq 2d+1$ bodech, pak
-snadno (v $\Theta(n)$) spoèteme takovou reprezentaci $C$.
-Problémem je, ¾e typicky máme polynom zadaný koeficienty, a~ne hodnotami
-v~bodech. Tím pádem potøebujeme nìjaký hodnì rychlý algoritmus (tj.
-rychlej¹í ne¾ kvadratický, jinak bychom si nepomohli oproti hloupému
-algoritmu) na~pøevod polynomu z jedné reprezentace do druhé a~zase zpìt.
-
-\s{Idea, jak by mìl algoritmus pracovat:}
-\algo
-\:Vybereme $n\geq 2d+1$ bodù $x_{0}, x_{1}, \ldots , x_{n-1}$.
-\:V tìchto bodech vyhodnotíme polynomy $A$ a~$B$.
-\:Nyní ji¾ v~lineárním èase získáme hodnoty polynomu $C$ v~tìchto bodech:
-	$C(x_i) = A(x_i)\cdot B(x_i)$
-\:Pøevedeme hodnoty polynomu $C$ na~jeho koeficienty.
-\endalgo
-
-\>Je vidìt, ¾e klíèové jsou kroky 2 a~4.
-Celý trik spoèívá v~chytrém vybrání onìch bodù, ve kterých budeme polynomy
-vyhodnocovat -- zvolí-li se obecná $x_j$, tak se to rychle neumí, pro speciální
-$x_j$ ale uká¾eme, ¾e to rychle jde.
-
-\ss{Vyhodnocení polynomu metodou Rozdìl a~panuj (algoritmus FFT):}
-Mìjme polynom $P$ stupnì $\leq d$ a~chtìjme jej vyhodnotit v~$n$ bodech.
-Vybereme si body tak, aby byly spárované, èili $\pm x_{0}, \pm x_{1},
-\ldots , \pm x_{n/2-1} $. To nám výpoèet urychlí, proto¾e pak se druhé
-mocniny $x_{j}$ shodují s~druhými mocninami $-x_{j}$.
-
-Polynom $P$ rozlo¾íme na~dvì èásti, první obsahuje èleny se sudými exponenty,
-druhá s~lichými:
-$$P(x) = (p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{d-2}x^{d-2}) + (p_{1}x^{1} +
-	p_{3}x^{3} + \ldots + p_{d-1}x^{d-1})$$
-\>se zavedením znaèení:
-$$P_s(t) = p_0t^0 + p_{2}t^{1} + \ldots + p_{d - 2}t^{d - 2\over 2}$$
-$$P_l(t) = p_1t^0 + p_3t^1 + \ldots + p_{d - 1}t^{d - 2\over 2}$$
-
-\>bude $P(x) = P_s(x^{2}) + xP_l(x^{2})$ a~$P(-x) = P_s(x^{2}) -
-xP_l(x^{2})$. Jinak øeèeno, vyhodnocování polynomu $P$ v~$n$ bodech se nám
-smrskne na~vyhodnocení $P_s$ a~$P_l$ v~$n/2$ bodech -- oba jsou polynomy
-stupnì nejvý¹e $d/2$ a~vyhodnocujeme je v~$x^{2}$ (vyu¾íváme
-rovnosti $(x_{i})^{2} = (-x_{i})^{2}$).
-
-\s{Pøíklad:}
-$3 + 4x + 6x^{2} + 2x^{3} + x^{4} + 10x^{5} = (3 + 6x^{2} + x^{4}) + x(4 +
-2x^{2} + 10x^{4})$.
-
-Teï nám ov¹em vyvstane problém s~oním párováním -- druhá mocnina pøece nemù¾e
-být záporná a~tím pádem u¾ v~druhé úrovni rekurze body spárované nebudou.
-Z~tohoto dùvodu musíme pou¾ít komplexní èísla -- tam druhé mocniny záporné býti
-mohou.
-
-% komplex
-\h{Komplexní intermezzo}
-\def\i{{\rm i}}
-\def\\{\hfil\break}
-
-\s{Základní operace}
-
-\itemize\ibull
-\:Definice: ${\bb C} = \{a + b\i \mid a,b \in {\bb R}\}$
-
-\:Sèítání: $(a+b\i)\pm(p+q\i) = (a\pm p) + (b\pm q)\i$. \\
-Pro $\alpha\in{\bb R}$ je $\alpha(a+b\i) = \alpha a + \alpha b\i$.
-
-\:Komplexní sdru¾ení: $\overline{a+b\i} = a-b\i$. \\
-$\overline{\overline x} = x$, $\overline{x\pm y} = \overline{x} \pm
-\overline{y}$, $\overline{x\cdot y} = \overline x \cdot \overline y$, $x
-\cdot \overline x \in {\bb R}$.
-
-\:Absolutní hodnota: $\vert x \vert = \sqrt{x\cdot\overline{x}}$, tak¾e $\vert
-a+b\i \vert = \sqrt{a^2+b^2}$. \\
-Také $\vert \alpha x \vert = \vert \alpha\vert \cdot \vert x \vert$.
-
-\:Dìlení: $x/y = (x\cdot \overline{y}) / (y \cdot \overline{y})$.
-\endlist
-
-\s{Gau{\scharfs}ova rovina a goniometrický tvar}
-
-\itemize\ibull
-\:Komplexním èíslùm pøiøadíme body v~${\bb R}^2$: $a+b\i \leftrightarrow (a,b)$.
-
-\:$\vert x\vert$ je vzdálenost od~bodu $(0,0)$.
-
-\:$\vert x\vert = 1$ pro èísla le¾ící na~jednotkové kru¾nici ({\I komplexní jednotky}). \\
-Pak platí $x=\cos\varphi + \i\sin\varphi$ pro nìjaké $\varphi\in\left[
-0,2\pi \right)$.
-
-\:Pro libovolné $x\in{\bb C}$: $x=\vert x \vert \cdot (\cos\varphi(x) +
-\i\sin\varphi(x))$. \\
-Èíslu $\varphi(x)\in\left[ 0,2\pi \right)$ øíkáme {\I argument}
-èísla~$x$, nìkdy znaèíme $\mathop{\rm arg} x$.
-
-\:Navíc $\varphi({\overline{x}}) = -\varphi(x)$.
-\endlist
-
-\s{Exponenciální tvar}
-
-\itemize\ibull
-\:Eulerova formule: $e^{i\varphi} = \cos\varphi + \i\sin\varphi$.
-
-\:Ka¾dé $x\in{\bb C}$ lze tedy zapsat jako $\vert x\vert \cdot e^{\i\cdot
-\varphi(x)}$.
-
-\:Násobení: $xy = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right) \cdot
-	\left(\vert y\vert\cdot e^{\i\cdot\varphi(y)}\right) = \vert x\vert
-	\cdot \vert y\vert \cdot e^{\i\cdot(\varphi(x) + \varphi(y))}$. \\
-(absolutní hodnoty se násobí, argumenty sèítají)
-
-\:Umocòování: $x^\alpha = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right)^
-	\alpha = {\vert x\vert}^\alpha\cdot e^{\i \alpha \varphi(x)}$.
-
-\:Odmocòování: $\root n\of x = {\vert x\vert}^{1/n} \cdot e^{\i\cdot
-\varphi(x)/n}$. \\
-Pozor -- odmocnina není jednoznaèná: $1^4=(-1)^4=\i^4=(-\i)^4=1$.
-\endlist
-
-\s{Odmocniny z~jednièky}
-
-\itemize\ibull
-\:Je-li nìjaké $x\in{\bb C}$ $n$-tou odmocninou z~jednièky, musí platit:
-$\vert x \vert = 1$, tak¾e $x=e^{\i\varphi}$ pro nìjaké~$\varphi$.
-Proto $x^n = e^{\i\varphi n} = \cos{\varphi n} + \i\sin\varphi n = 1$.
-Platí tedy $\varphi n = 2k\pi$ pro nìjaké $k\in{\bb Z}$.
-
-\:Z~toho plyne: $\varphi = 2k\pi/n$ \\
-(pro $k=0,\ldots,n-1$ dostáváme rùzné $n$-té odmocniny).
-
-\:Obecné odmocòování: $\root n \of x = {\vert x\vert}^{1/n} \cdot e^{\i\varphi
-	(x)/n} \cdot u$, kde $u=\root n\of 1$.
-
-\:Je-li $x$ odmocninou z 1, pak $\overline{x} = x^{-1}$ -- je toti¾ $1 = \vert
-x\cdot \overline{x}\vert = x\cdot \overline{x}$.
-\endlist
-
-\s{Primitivní odmocniny}
-
-\s{Definice:} $x$ je {\I primitivní} $k$-tá odmocnina z 1 $\equiv x^k=1 \land \forall j: 0<j<k \Rightarrow x^j \neq 1$.
-
-\>Tuto definici splòují napøíklad èísla $\omega = e^{2\pi \i / k}$ a $\overline\omega = e^{-2\pi\i/k}$.
-Platí toti¾, ¾e $\omega^j = e^{2\pi\i j/k}$, co¾ je rovno~1 právì tehdy,
-je-li $j$ násobkem~$k$ (jednotlivé mocniny èísla~$\omega$ postupnì obíhají
-jednotkovou kru¾nici). Analogicky pro~$\overline\omega$.
-
-\>Uka¾me si nìkolik pozorování fungujících pro libovolné èíslo~$\omega$,
-které je primitivní $k$-tou odmocninou z~jednièky (nìkdy budeme potøebovat,
-aby navíc $k$ bylo sudé):
-
-\itemize\ibull
-\:Pro $0\leq j<l<k$ je $\omega^j \neq \omega^l$, nebo» $\omega^l /
-\omega^j = \omega^{l-j} \neq 1$, proto¾e $l-j < k$ a $\omega$ je primitivní.
-\:$\omega^{k/2} = -1$, proto¾e $(\omega^{k/2})^2 = 1$, a tedy
-$\omega^{k/2}$ je druhá odmocnina z~1. Takové odmocniny jsou dvì:
-1 a $-1$, ov¹em 1 to být nemù¾e, proto¾e $\omega$ je primitivní.
-\:$\omega^j = - \omega^{k/2 + j}$ -- pøímý dùsledek pøedchozího bodu, pro
-nás ale velice zajímavý: $\omega^0,\omega^1,\ldots,\omega^{k-1}$ jsou po dvou
-spárované.
-\:$\omega^2$ je $k/2$-tá primitivní odmocnina z 1 -- dosazením.
-\endlist
-
-\h{Konec intermezza}
-% end komplex
-
-\bigskip
-
-Vra»me se nyní k algoritmu. Z poslední èásti komplexního intermezza se zdá,
-¾e by nemusel být ¹patný nápad zkusit vyhodnocovat polynom v mocninách
-$n$-té primitivní odmocniny z~jedné (tedy za $x_0,x_1,\ldots,x_{n-1}$
-z~pùvodního algoritmu zvolíme $\omega^0,\omega^1,\ldots,\omega^{n-1}$).
-Aby nám v¹e vycházelo pìknì, zvolíme $n$ jako mocninu dvojky.
-
-\ss{Celý algoritmus bude vypadat takto:}
-\>FFT($P$, $ \omega$)
-
-\>{\sl Vstup:} $p_{0}, \ldots , p_{n-1}$, koeficienty polynomu $P$ stupnì
-nejvý¹e $n-1$, a~$\omega$,
-$n$-tá primitivní odmocina z jedné.
-
-\>{\sl Výstup:} Hodnoty polynomu v~bodech $1, \omega, \omega^{2}, \ldots ,
-\omega^{n - 1}$, èili èísla $P(1), P(\omega), P(\omega^{2}),$ $\ldots ,
-P(\omega^{n - 1})$.
-
-\algo
-\:Pokud $n = 1$, vrátíme $p_{0}$ a~skonèíme.
-\:Jinak rozdìlíme $P$ na èleny se sudými a lichými exponenty (jako v pùvodní
-my¹lence) a~rekurzivnì zavoláme FFT($P_s$, $\omega^{2}$) a~FFT($P_l$,
-$\omega^{2}$) -- $P_l$ i~$P_s$ jsou stupnì max. $n/2-1$, $\omega^2$ je
-$n/2$-tá primitivní odmocnina, a mocniny $\omega^2$ jsou stále po dvou
-spárované ($n$ je mocnina dvojky, a tedy i $n/2$ je sudé; popø. $n=2$ a je to
-zøejmé).
-\:Pro $j = 0, \ldots , n/2 - 1$ spoèítáme:
-
-\:\qquad $P(\omega^{j}) = P_s(\omega^{2j}) + \omega^{j}\cdot P_l(\omega^{2j})$.
-
-\:\qquad $P(\omega^{j+n/2})=P_s(\omega^{2j})-\omega^{j}\cdot P_l(\omega^{2j})$.
-
-\endalgo
-
-
-\s{Èasová slo¾itost:}
-\>$T(n)=2T(n/2) + \Theta(n) \Rightarrow$ slo¾itost $\Theta(n \log n)$, jako
-MergeSort.
-
-
-Máme tedy algoritmus, který pøevede koeficienty polynomu na~hodnoty tohoto
-polynomu v~rùzných bodech. Potøebujeme ale také algoritmus, který doká¾e
-reprezentaci polynomu pomocí hodnot pøevést zpìt na~koeficienty polynomu.
-K~tomu nám pomù¾e podívat se na~ná¹ algoritmus trochu obecnìji.
-
-
-\s{Definice:}
-\>{\I Diskrétní Fourierova transformace} $(DFT)$
-je zobrazení $f: { {\bb C} ^n} \rightarrow { {\bb C} ^n}$, kde $$y=f(x) \equiv
-\forall j \ y_{j} = \sum \limits ^{n-1}_{k=0} x_{k} \cdot \omega ^{jk}$$
-(DFT si lze mimo jiné pøedstavit jako funkci vyhodnocující polynom
-s~koeficienty $x_k$ v~bodech $\omega^j$). Takovéto zobrazení je lineární
-a~tedy popsatelné maticí $\Omega$ s~prvky $\Omega_{jk} = \omega^{jk}$.
-Chceme-li umìt pøevádìt z~hodnot polynomu na koeficienty, zajímá nás inverze
-této matice.
-
-
-\ss{Jak najít inverzní matici?}
-Znaème $\overline{\Omega}$ matici, její¾ prvky
-jsou komplexnì sdru¾ené odpovídajícím prvkùm $\Omega$, a vyu¾ijme následující
-lemma:
-
-\ss{Lemma:}$\quad \Omega\cdot \overline{\Omega} = n\cdot E$.
-
-\proof $$ (\Omega\cdot \overline{\Omega})_{jk} = \sum_{l=0}^{n-1} \omega^{jl}
-\cdot \overline{\omega^{lk}} = \sum \omega^{jl} \cdot \overline{\omega}^{lk} =
-\sum \omega^{jl} \cdot \omega^{-lk} = \sum \omega^{l(j-k)}\hbox{.}$$
-\itemize\ibull
-\:Pokud $j=k$, pak $ \sum \limits ^{n-1}_{l=0} (\omega ^{0}) ^{l} = n$.
-
-\:Pokud $j\neq k$, pou¾ijeme vzoreèek pro souèet geometrické øady
-s kvocientem $\omega ^{(j-k) }$ a~dostaneme ${{\omega^{(j-k)n} -1} \over
-{\omega^{(j-k)} -1}} ={1-1 \over \neq 0} = 0$ (
-$\omega^{j-k} - 1$ je jistì $\neq 0$, nebo» $\omega$ je $n$-tá primitivní
-odmoncina a $j-k<n$).
-\endlist
-\qed
-
-
-\>Na¹li jsme inverzi:
-$\Omega({1 \over n} \overline{\Omega}) = {1 \over n}\Omega \cdot
-\overline{\Omega} = E$, \quad $\Omega^{-1}_{jk} = {1 \over n}\overline{\omega^
-{jk}} = {1 \over n}\omega^{-jk} = {1 \over n} {(\omega^{-1})}^{jk}$, \quad
-(pøipomínáme, $\omega^{-1}$ je $\overline{\omega}$).
-
-Vyhodnocení polynomu lze provést vynásobením $\Omega$, pøevod do pùvodní
-reprezentace vynásobením $\Omega^{-1}$. My jsme si ale v¹imli chytrého
-spárování, a vyhodnocujeme polynom rychleji ne¾ kvadraticky (proto FFT,
-jako¾e {\it fast}, ne jako {\it fuj}). Uvìdomíme-li si, ¾e $\overline \omega =
-\omega^{-1}$ je také $n$-tá primitivní odmocnina z 1 (má akorát
-úhel s opaèným znaménkem), tak mù¾eme stejným trikem vyhodnotit i~zpìtný
-pøevod -- nejprve vyhodnotíme $A$ a $B$ v $\omega^j$, poté pronásobíme
-hodnoty a~dostaneme tak hodnoty polynomu $C=A\cdot B$, a pustíme na nì
-stejný algoritmus s~$\omega^{-1}$ (hodnoty $C$
-vlastnì budou v~algoritmu \uv{koeficienty polynomu}). Nakonec jen získané
-hodnoty vydìlíme $n$ a~máme chtìné koeficienty.
-
-\s{Výsledek:} Pro $n= 2^k$ lze DFT na~${\bb C}^n$ spoèítat v~èase $\Theta(n
-\log n)$ a~DFT$^{-1}$ takté¾.
-
-\s{Dùsledek:}
-Polynomy stupnì $n$ lze násobit v~èase $\Theta(n \log n)$:
-$\Theta(n \log n)$ pro vyhodnocení, $\Theta(n)$ pro vynásobení a~$\Theta(n
-\log n)$ pro pøevedení zpìt.
-
-\s{Pou¾ití FFT:}
-
-\itemize\ibull
-
-\:Zpracování signálu -- rozklad na~siny a~cosiny o~rùzných frekvencích
-$\Rightarrow$ spektrální rozklad.
-\:Komprese dat -- napøíklad formát JPEG.
-\:Násobení dlouhých èísel v~èase $\Theta(n \log n)$.
-\endlist
-
-\s{Paralelní implementace FFT}
-
-\figure{img.eps}{Pøíklad prùbìhu algoritmu na~vstupu velikosti 8}{3in}
-
-Zkusme si prùbìh algoritmu FFT znázornit graficky (podobnì, jako jsme kreslili
-hradlové sítì). Na~levé stranì obrázku se nachází vstupní vektor $x_0,\ldots,x_{n-1}$
-(v~nìjakém poøadí), na~pravé stranì pak výstupní vektor $y_0,\ldots,y_{n-1}$.
-Sledujme chod algoritmu pozpátku: Výstup spoèítáme z~výsledkù \uv{polovièních}
-transformací vektorù $x_0,x_2,\ldots,x_{n-2}$ a $x_1,x_3,\ldots,x_{n-1}$.
-Èerné krou¾ky pøitom odpovídají výpoètu lineární kombinace $a+\omega^kb$,
-kde $a,b$ jsou vstupy krou¾ku a $k$~nìjaké pøirozené èíslo závislé na poloze
-krou¾ku. Ka¾dá z~polovièních transformací se poèítá analogicky z~výsledkù
-transformace velikosti $n/4$ atd. Celkovì výpoèet probíhá v~$\log_2 n$ vrstvách
-po~$\Theta(n)$ operacích.
-
-Jeliko¾ operace na~ka¾dé vrstvì probíhají na sobì nezávisle, mù¾eme je poèítat
-paralelnì. Ná¹ diagram tedy popisuje hradlovou sí» pro paralelní výpoèet FFT
-v~èase $\Theta(\log n\cdot T)$ a prostoru $\O(n\cdot S)$, kde $T$ a~$S$ znaèí
-èasovou a prostorovou slo¾itost výpoètu lineární kombinace dvou komplexních èísel.
-
-\s{Cvièení:} Doka¾te, ¾e permutace vektoru $x_0,\ldots,x_{n-1}$ odpovídá bitovému
-zrcadlení, tedy ¾e na pozici~$b$ shora se vyskytuje prvek $x_d$, kde~$d$ je
-èíslo~$b$ zapsané ve~dvojkové soustavì pozpátku.
-
-\s{Nerekurzivní FFT}
-
-Obvod z~pøedchozího obrázku také mù¾eme vyhodnocovat po~hladinách zleva doprava,
-èím¾ získáme elegantní nerekurzivní algoritmus pro výpoèet FFT v~èase $\Theta(n\log n)$
-a prostoru $\Theta(n)$:
-
-\algo
-\algin $x_0,\ldots,x_{n-1}$
-\:Pro $j=0,\ldots,n-1$ polo¾íme $y_k\= x_{r(k)}$, kde $r$ je funkce bitového zrcadlení.
-\:Pøedpoèítáme tabulku $\omega^0,\omega^1,\ldots,\omega^{n-1}$.
-\:$b\= 1$ \cmt{velikost bloku}
-\:Dokud $b<n$, opakujeme:
-\::Pro $j=0,\ldots,n-1$ s~krokem~$2b$ opakujeme: \cmt{zaèátek bloku}
-\:::Pro $k=0,\ldots,b-1$ opakujeme: \cmt{pozice v~bloku}
-\::::$\alpha\=\omega^{nk/2b}$
-\::::$(y_{j+k},y_{j+k+b}) \= (y_{j+k}+\alpha\cdot y_{j+k+b}, y_{j+k}-\alpha\cdot y_{j+k+b})$.
-\::$b\= 2b$
-\algout $y_0,\ldots,y_{n-1}$
-\endalgo
-
-\s{FFT v~koneèných tìlesech}
-
-Nakonec dodejme, ¾e Fourierovu transformaci lze zavést nejen nad tìlesem
-komplexních èísel, ale i v~nìkterých koneèných tìlesech, pokud zaruèíme
-existenci primitivní $n$-té odmocniny z~jednièky. Napøíklad v~tìlese ${\bb
-Z}_p$ pro prvoèíslo $p=2^k+1$ platí $2^k=-1$, tak¾e $2^{2k}=1$
-a $2^0,2^1,\ldots,2^{2k-1}$ jsou navzájem rùzná, tak¾e èíslo~2 je~primitivní
-$2k$-tá odmocnina z~jedné. To se nám ov¹em nehodí pro algoritmus FFT, jeliko¾
-$2k$ bude málokdy mocnina dvojky.
-
-Zachrání nás ov¹em algebraická vìta, která øíká, ¾e multiplikativní grupa\foot{To je
-mno¾ina v¹ech nenulových prvkù tìlesa s~operací násobení.}
-libovolného koneèného tìlesa je cyklická, tedy ¾e v¹echny nenulové prvky tìlesa lze
-zapsat jako mocniny nìjakého èísla~$g$ (generátoru). Napøíklad pro $p=2^{16}+1=65\,537$
-je jedním takovým generátorem èíslo~$3$. Jeliko¾ mezi èísly $g^1,g^2,\ldots,g^{p-1}$
-se musí ka¾dý nenulový prvek tìlesa vyskytnout právì jednou, je~$g$ primitivní
-$2^k$-tou odmocninou z~jednièky, tak¾e mù¾eme poèítat FFT pro libovolný vektor,
-jeho¾ velikost je mocnina dvojky men¹í nebo rovná $2^k$.\foot{Bli¾¹í prùzkum
-na¹ich úvah o~FFT dokonce odhalí, ¾e není ani potøeba tìleso. Postaèí libovolný
-komutativní okruh, ve~kterém existuje pøíslu¹ná primitivní odmocnina
-z~jednièky, její multiplikativní inverze (ta ov¹em existuje v¾dy, proto¾e
-$\omega^{-1} = \omega^{n-1}$) a multiplikativní inverze èísla~$n$. To nám
-poskytuje je¹tì daleko více volnosti ne¾ tìlesa, ale není snadné takové okruhy
-hledat.}
-
-\bye
diff --git a/9-fft/Makefile b/9-fft/Makefile
deleted file mode 100644
index 0769a79..0000000
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+++ /dev/null
@@ -1,3 +0,0 @@
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diff --git a/9-fft/img.eps b/9-fft/img.eps
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-/rgb2devcmyk{3 copy dup 3 1 roll eq 3 1 roll eq v_keyg 1 eq and and{pop
-pop 1 exch sub 0 0 0 4 -1 roll}{/ucurve where{pop 3{1.0 exch sub 3
-1 roll}repeat 3 copy 2 copy gt{exch}if pop 2 copy gt{exch}if pop dup
-ucurve readcurve exch bcurve readcurve clamp01 3{5 1 roll dup 3 1 roll
-sub clamp01}repeat 5 1 roll pop 4 1 roll ycurve readcurve 4 1 roll
-mcurve readcurve 4 1 roll ccurve readcurve 4 1 roll}{rgb2cmyk}ifelse}ifelse}def
-/rgb2keyG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop}{Max3}ifelse
-1 exch sub bcurve readcurve clamp01}bd/rgb2key{Max3 1 exch sub bcurve
-readcurve clamp01}bd/rgb2cyanG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop
-pop pop 0}{rgb2cyan}ifelse}bd/rgb2cyan{3 copy Max3 1 exch sub ucurve
-readcurve 4 1 roll pop pop 1 exch sub exch sub ccurve readcurve clamp01}bd
-/rgb2magentaG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2magenta}ifelse}bd
-/rgb2magenta{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll pop
-1 exch sub 3 1 roll pop sub mcurve readcurve clamp01}bd/rgb2yellowG{3
-copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2yellow}ifelse}bd
-/rgb2yellow{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll 1 exch
-sub 4 1 roll pop pop sub ycurve readcurve clamp01}bd/rgb2plategray{v_keyg
-0 eq v_plate v_cpky eq{{rgb2key}{rgb2keyG}ifelse}{v_plate v_cpyl eq{{rgb2yellow}{rgb2yellowG}ifelse}{v_plate
-v_cpmg eq{{rgb2magenta}{rgb2magentaG}ifelse}{v_plate v_cpcy eq{{rgb2cyan}{rgb2cyanG}ifelse}{{rgb2key}{rgb2keyG}ifelse}ifelse}ifelse}ifelse}ifelse
-1 exch sub setgray}bd/dc{0 def}bd/aca{/v_cpnone 0 def/v_cpcy 1 def
-/v_cpyl 2 def/v_cpmg 3 def/v_cpky 4 def/v_gseps 0 def/v_keyg 0 def
-/v_plate v_cpnone def/v_mono 0 def/v_wr dc/v_fc dc/v_fm dc/v_fy dc
-/v_fk dc/v_fg dc/v_fr dc/v_fg dc/v_fb dc/v_sc dc/v_sm dc/v_sy dc/v_sk
-dc/v_sg dc/v_sr dc/v_sg dc/v_sb dc/v_sct 0 def/v_fct 0 def/v_ft 0 def
-/v_cxe 0 def/v_cxm 0 def/v_sa -1 def/v_ea -1 def/sR dc/sG dc/sB dc
-/mR dc/mG dc/mB dc/eR dc/eG dc/eB dc/sC dc/sM dc/sY dc/sK dc/eC dc
-/eM dc/eY dc/eK dc/sH dc/sS dc/sV dc/eH dc/eS dc/eV dc/sGy dc/eGy
-dc/mGy dc/ci_datasrc dc/ci_matrix dc/ci_dataleft dc/ci_buf dc/ci_dataofs
-dc/ci_y dc/rciBuf dc/cbslw dc/cmiBuf dc/cPalette dc/cpci_datasrc dc
-/cpci_matrix dc/cpci_bpp dc/cpci_y dc/cpci_sampsleft dc/cpci_nextcol
-dc/cpci_buf dc/startX dc/startY dc/endX dc/endY dc/endX2 dc/endY2 dc
-/fillX dc/urx dc/ury dc/llx dc/lly dc/incD dc/distance dc/slice dc
-/startangle dc/Steps dc/incH dc/incS dc/incV dc/incR dc/incG dc/incB
-dc/incGy dc 0.25 setlinewidth [] 0 setdash 0 setlinejoin 0 setlinecap}bd
-aca/setplategray{v_plate v_cpky eq{1 exch sub setgray pop pop pop}{v_plate
-v_cpyl eq{pop 1 exch sub setgray pop pop}{v_plate v_cpmg eq{pop pop
-1 exch sub setgray pop}{v_plate v_cpcy eq{pop pop pop 1 exch sub setgray}{1
-exch sub setgray pop pop pop}ifelse}ifelse}ifelse}ifelse}bd/setplatecolor{v_plate
-v_cpky eq{1 exch sub 0 0 0 4 -1 roll setcmykcolor pop pop pop}{v_plate
-v_cpyl eq{pop 1 exch sub 0 0 0 4 2 roll setcmykcolor pop pop}{v_plate
-v_cpmg eq{pop pop 1 exch sub 0 0 0 4 1 roll setcmykcolor pop}{v_plate
-v_cpcy eq{pop pop pop 1 exch sub 0 0 0 setcmykcolor}{1 exch sub 0 0
-0 4 -1 roll setcmykcolor pop pop pop}ifelse}ifelse}ifelse}ifelse}bd
-/setcmykcolor where{pop}{/setcmykcolor{cmyk2rgb setrgbcolor}bd}ifelse
-/setlogcmykcolor{v_gseps 1 eq{v_mono 1 eq{1 exch sub setgray pop pop
-pop}{setcmykcolor}ifelse}{v_mono 1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd
-/setlogrgbcolor{v_gseps 1 eq{v_mono 1 eq{rgbtoplategray}{rgb2devcmyk
-setplatecolor}ifelse}{v_mono 1 eq{rgb2gray setgray}{systemdict begin
-setrgbcolor end}ifelse}ifelse}bd/setfillcolor{v_fct 0 eq{v_fc v_fm
-v_fy v_fk setlogcmykcolor}{v_fr v_fg v_fb setlogrgbcolor}ifelse}bd
-/setstrokecolor{v_sct 0 eq{v_sc v_sm v_sy v_sk setlogcmykcolor}{v_sr
-v_sg v_sb setlogrgbcolor}ifelse}bd/setgfillcmyk{v_gseps 1 eq{v_mono
-1 eq{cmyk2rgb rgb2plategray}{cmyk2rgb rgb2devcmyk setplatecolor}ifelse}{v_mono
-1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd/setgfillrgb{v_gseps
-1 eq{v_mono 1 eq{rgb2plategray}{rgb2devcmyk setplatecolor}ifelse}{v_mono
-1 eq{rgb2gray setgray}{systemdict begin setrgbcolor end}ifelse}ifelse}bd
-/setgfillhsb{v_gseps 1 eq{v_mono 1 eq{systemdict begin sethsbcolor
-currentrgbcolor end rgb2plategray}{systemdict begin sethsbcolor currentrgbcolor
-end rgb2devcmyk setplatecolor}ifelse}{v_mono 1 eq{systemdict begin
-sethsbcolor currentgray end setgray}{systemdict begin sethsbcolor end}ifelse}ifelse}bd
-/Max{2 copy lt{exch}if pop}bd/Max3{2 copy lt{exch}if pop 2 copy lt{exch}if
-pop}bd/Min{2 copy gt{exch}if pop}bd/Min3{2 copy gt{exch}if pop 2 copy
-gt{exch}if pop}bd/clamp{3 1 roll Max 2 1 roll Min}bd/clamp01{0 Max
-1 Min}bd/Pythag{dup mul exch dup mul add sqrt}bd/ssc{DeviceRGB setcolorspace
-setcolor}bd/ssg{setgray}bd/p_render{}def/p_count 0 def/vis_flag true
-def/DataString 3 string def/DataSrc{currentfile DataString readhexstring
-pop}bd/DataStr1 1 string def/DataStr2 1 string def/DataStr3 1 string
-def/DataSrc1{DataStr1}bd/DataSrc2{DataStr2}bd/DataSrc3{DataStr3}bd
-/colorimage where{pop/ci{colorimage}bd}{/ci{pop pop/ci_datasrc exch
-def matrix invertmatrix/ci_matrix exch def pop/ci_dataleft 0 def/ci_buf()def
-/ci_dataofs 0 def 0 1 3 -1 roll 1 sub{/ci_y exch def dup 0 1 3 -1
-roll 1 sub{0 1 2{pop ci_dataleft 0 eq{ci_datasrc dup length/ci_dataleft
-exch def/ci_buf exch def/ci_dataofs 0 def}if ci_buf ci_dataofs get
-255 div/ci_dataofs ci_dataofs 1 add def/ci_dataleft ci_dataleft 1 sub
-def}for setrgbcolor dup ci_y 3 -1 roll 1 add ci_y 1 add 4 copy 5 1
-roll 4 2 roll 5 -1 roll 1 1 4{pop ci_matrix transform 8 2 roll}for
-m l l l closepath fill}for}for pop}bd}ifelse/rci{/rciBuf 4 index 3
-index mul 7 add 8 div floor cvi string def{currentfile rciBuf readhexstring
-pop}bind false 3 ci}bd/cbsl{2 eq/cbslL2 xd 5 index/cbslw xd translate
-scale 8 [ 3 index 0 0 5 index 0 0 ] cbslL2{/DataStr1 cbslw string def
-currentfile/ASCII85Decode filter/RunLengthDecode filter DataStr1 readstring
-pop pop/DataStr2 cbslw string def currentfile/ASCII85Decode filter
-/RunLengthDecode filter DataStr2 readstring pop pop/DataStr3 cbslw
-string def currentfile/ASCII85Decode filter/RunLengthDecode filter
-DataStr3 readstring pop pop{DataStr1}bind{DataStr2}bind{DataStr3}bind
-true}{/DataSrc load false}ifelse 3 ci}bd/gbsl{2 eq/gbslL2 xd 5 index
-/gbslw xd translate scale 8 [ 3 index 0 0 5 index 0 0 ] gbslL2{/DataStr1
-gbslw string def currentfile/ASCII85Decode filter/RunLengthDecode filter
-DataStr1 readstring pop pop{DataStr1}bind}{/DataStr1 gbslw string def
-currentfile DataSrc1 readhexstring pop pop{DataStr1}bind}ifelse image}bd
-/cmi{/cmiBuf 4 index 3 index mul 7 add 8 div floor cvi string def{currentfile
-cmiBuf readhexstring pop}bind image}bd/cpal{4 mul string/cPalette exch
-def currentfile cPalette readhexstring pop}bd/cpci{/cpci_datasrc exch
-def matrix invertmatrix/cpci_matrix exch def/cpci_bpp exch def cpci_init
-0 1 3 -1 roll 1 sub{/cpci_y exch def dup cpci_bpp 4 eq{cpci_sampsleft
-1 eq{/cpci_sampsleft 0 def}if}if 0 1 3 -1 roll 1 sub{cpci_nextcol dup
-cpci_y 3 -1 roll 1 add cpci_y 1 add 4 copy 5 1 roll 4 2 roll 5 -1 roll
-1 1 4{pop cpci_matrix transform 8 2 roll}for m l l l closepath fill}for}for
-pop}bd/cpci_init{/cpci_sampsleft 0 def}bd/cpci_buf 1 string def/cpci_nextcol{cpci_bpp
-1 eq{cpci_sampsleft 0 eq{currentfile cpci_buf readhexstring pop pop
-/cpci_sampsleft 8 def}if cpci_buf dup 0 get dup 1 and setgray -1 bitshift
-1 exch put/cpci_sampsleft cpci_sampsleft 1 sub def}{cpci_bpp 4 eq{cpci_sampsleft
-0 eq{currentfile cpci_buf readhexstring pop pop/cpci_sampsleft 2 def}if
-cpci_buf 0 get dup 15 and exch -4 bitshift cpci_buf 0 3 -1 roll put
-/cpci_sampsleft cpci_sampsleft 1 sub def}{currentfile cpci_buf readhexstring
-pop 0 get}ifelse 4 mul dup 2 add cPalette exch get 255 div exch dup
-1 add cPalette exch get 255 div exch cPalette exch get 255 div setrgbcolor}ifelse}bd
-/setup1asciiproc{[ currentfile mystring/readhexstring cvx/pop cvx
-] cvx bind}bd/setup1binaryproc{[ currentfile mystring/readstring cvx
-/pop cvx ] cvx bind}bd level2{save/dontloadlevel1 xd}if/iw 0 def/ih
-0 def/im_save 0 def/setupimageproc 0 def/polarity 0 def/smoothflag
-0 def/mystring 0 def/bpc 0 def/beginimage{/im_save save def dup 0 eq{pop
-/setup1binaryproc}{1 eq{/setup1asciiproc}{(error, can't use level2 data acquisition procs for level1)print
-flush}ifelse}ifelse/setupimageproc exch ld/polarity xd/smoothflag xd
-/imat xd/mystring exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save
-restore}bd/1bitbwcopyimage{1 setgray 0 0 moveto 0 1 rlineto 1 0 rlineto
-0 -1 rlineto closepath fill 0 setgray iw ih polarity imat setupimageproc
-imagemask}bd/1bitcopyimage{setrgbcolor 0 0 moveto 0 1 rlineto 1 0 rlineto
-0 -1 rlineto closepath fill setrgbcolor iw ih polarity imat setupimageproc
-imagemask}bd/1bitmaskimage{setrgbcolor iw ih polarity [iw 0 0 ih 0
-0] setupimageproc imagemask}bd level2{dontloadlevel1 restore}if level2
-not{save/dontloadlevel2 xd}if/setup2asciiproc{currentfile/ASCII85Decode
-filter/RunLengthDecode filter}bd/setup2binaryproc{currentfile/RunLengthDecode
-filter}bd/myimagedict 9 dict dup begin/ImageType 1 def/MultipleDataSource
-false def end def/im_save 0 def/setupimageproc 0 def/polarity 0 def
-/smoothflag 0 def/mystring 0 def/bpc 0 def/ih 0 def/iw 0 def/beginimage{
-/im_save save def dup 2 eq{pop/setup2binaryproc}{dup 3 eq{pop/setup2asciiproc}{0
-eq{/setup1binaryproc}{/setup1asciiproc}ifelse}ifelse}ifelse/setupimageproc
-exch ld{[ 1 0 ]}{[ 0 1 ]}ifelse/polarity xd/smoothflag xd/imat xd/mystring
-exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save restore}bd/1bitbwcopyimage{1
-ssg 0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill
-0 ssg myimagedict dup begin/Width iw def/Height ih def/Decode polarity
-def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
-1 def/Interpolate smoothflag def end imagemask}bd/1bitcopyimage{ssc
-0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill ssc
-myimagedict dup begin/Width iw def/Height ih def/Decode polarity def
-/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
-1 def/Interpolate smoothflag def end imagemask}bd/1bitmaskimage{ssc
-myimagedict dup begin/Width iw def/Height ih def/Decode polarity def
-/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
-1 def/Interpolate smoothflag def end imagemask}bd level2 not{dontloadlevel2
-restore}if
-level2{save/dontloadlevel1 xd}if/startnoload{{/noload save def}if}bd
-/endnoload{{noload restore}if}bd/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}bd
-/ncolors 1 def/colorimage where{pop true}{false}ifelse{/ncolors 0
-statusdict begin/processcolors where{pop pop processcolors}{/deviceinfo
-where{pop deviceinfo/Colors known{pop{deviceinfo/Colors get}}if}if}ifelse
-end def ncolors 0 ne{/colorimage testsystemdict/setcolortransfer testsystemdict
-/currentcolortransfer testsystemdict/currentcmykcolor testsystemdict
-and and and not{/ncolors 0 def}if}if}if ncolors dup 1 ne exch dup 3
-ne exch 4 ne and and{/ncolors 0 def}if ncolors 1 eq dup dup not startnoload{
-/expandbw{expandfactor mul round cvi bwclut exch get 255 div}bd/doclutimage{bwclut
-colorclut pop/bwclut xd bpc dup 8 eq{pop 255}{4 eq{15}{3}ifelse}ifelse
-/expandfactor xd [/expandbw load/exec load dup currenttransfer exch
-] cvx bind settransfer iw ih bpc imat setupimageproc image}bd}if not
-endnoload ncolors dup 3 eq exch 4 eq or dup dup not startnoload{/nullproc{{}}def
-/concatutil{/exec load 7 -1 roll/exec load}bd/defsubclut{1 add getinterval
-def}bd/spconcattransfer{/Dclut exch def/Cclut exch def/Bclut exch def
-/Aclut exch def/ncompute exch ld currentcolortransfer [{Aclut ncompute}concatutil
-] cvx [{Bclut ncompute}concatutil ] cvx [{Cclut ncompute}concatutil
-] cvx [{Dclut ncompute}concatutil ] cvx setcolortransfer}bd/setuprgbcluts{
-/bit3x rgbclut length 3 sub def/bit1x bit3x 3 idiv def/rclut rgbclut
-def/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}bd}if
-not endnoload ncolors 3 eq dup dup not startnoload{/3compute{exch bit3x
-mul round cvi get 255 div}bd/doclutimage{/rgbclut xd pop setuprgbcluts
-/3compute rclut gclut bclut dup spconcattransfer iw ih bpc imat [
-setupimageproc/exec load/dup load dup ] cvx nullproc nullproc true
-3 colorimage}bd}if not endnoload ncolors 4 eq dup dup not startnoload{
-/stuffclut{cmykindex 3 -1 roll put}bd/ftoint{1 exch sub 255 mul round
-cvi}bd/4compute{exch bit4x mul round cvi get 255 div}bd/computecmykclut{setuprgbcluts
-/bit4x rgbclut length 3 idiv 4 mul 4 sub def/cmykclut bit4x 4 add
-string def/cclut cmykclut def/mclut cclut 1 bit4x defsubclut/yclut
-cclut 2 bit4x defsubclut/kclut cclut 3 bit4x defsubclut/cmykindex 0
-def 0 1 bit1x{dup/cmykindex exch bit1x exch sub 4 mul def 3 mul dup
-rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch
-get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint
-yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut}for}bd
-/doclutimage{/rgbclut xd pop invalidcolortable?{computecmykclut}if
-/4compute cclut mclut yclut kclut spconcattransfer iw ih bpc imat
-[ setupimageproc/exec load/dup load dup dup ] cvx nullproc nullproc
-nullproc true 4 colorimage}bd}if not endnoload ncolors 0 eq dup dup
-not startnoload{/lookupandstore{3 mul 3 getinterval putinterval exch
-3 add exch 3 copy}bd/8lookup/lookupandstore ld/4lookup{/byte 1 index
-def -4 bitshift lookupandstore byte 15 and lookupandstore}bd/2lookup{
-/byte 1 index def -6 bitshift lookupandstore byte -4 bitshift 3 and
-lookupandstore byte -2 bitshift 3 and lookupandstore byte 3 and lookupandstore}bd
-/colorexpand{mystringexp 0 rgbclut 3 copy 7 -1 roll/mylookup load
-forall pop pop pop pop pop}bd/createexpandstr{/mystringexp exch mystring
-length mul string def}bd/doclutimage{/rgbclut xd pop/mylookup bpc 8
-eq{3 createexpandstr/8lookup}{bpc 4 eq{6 createexpandstr/4lookup}{12
-createexpandstr/2lookup}ifelse}ifelse ld iw ih bpc imat [ setupimageproc
-/exec load/colorexpand load/exec load] cvx false 3 colorimage}bd}if
-not endnoload/colorimage where{pop true}{false}ifelse dup{/do24image{iw
-ih 8 imat setupimageproc false 3 colorimage}bd}if dup dup startnoload
-not{/rgbtogray{/str xd/len str length def/smlen len 3 idiv def/rstr
-str def/gstr str 1 len 1 sub getinterval def/bstr str 2 len 2 sub getinterval
-def str dup 0 1 smlen 1 sub{dup 3 mul rstr 1 index get .3 mul gstr
-2 index get .59 mul add bstr 3 -1 roll get .11 mul add round cvi put
-dup}for pop 0 smlen getinterval}bd/do24image{iw ih 8 imat [ setupimageproc
-/exec load/rgbtogray load/exec load ] cvx bind image}bd}if endnoload
-/doimage{iw ih 8 imat setupimageproc image}bd level2{dontloadlevel1
-restore}if level2 not{save/dontloadlevel2 xd}if/myappcolorspace/DeviceRGB
-def/rgbclut 0 def/doclutimage{/rgbclut xd pop bpc dup 8 eq{pop 255}{4
-eq{15}{3}ifelse}ifelse/hival xd [/Indexed myappcolorspace hival rgbclut]
-setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode
-[0 hival] def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
-bpc def/Interpolate smoothflag def end image}bd/do24image{myappcolorspace
-setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode
-[0 1 0 1 0 1] def/ImageMatrix imat def/DataSource setupimageproc def
-/BitsPerComponent 8 def/Interpolate smoothflag def end image}bd level2
-not{dontloadlevel2 restore}if
-/NumSteps{dtransform matrix defaultmatrix idtransform Pythag currentscreen
-pop pop 72 exch div div}bd/FindMinSteps{v_ft 4 eq{urx startX sub abs
-llx startX sub abs Max ury startY sub abs lly startY sub abs Max Pythag
-2 3.14159265 mul mul 0}{v_ft 2 eq{endY startY sub endX startX sub Pythag
-endY2 startY sub endX2 startX sub Pythag gt{endY startY sub endX startX
-sub}{endY2 startY sub endX2 startX sub}ifelse}{endY startY sub endX
-startX sub}ifelse}ifelse NumSteps}bd/cxe{/v_cxe exch def}bd/cxm{pop
-/v_cxm exch def}bd/cxmt{pop pop}bd/cxt{pop}bd/S_eoclip{currentflat{{eoclip}stopped{dup
-currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: eoclip]\n)print
-flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
-/S_clip{currentflat{{clip}stopped{dup currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: clip]\n)print
-flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
-/S_eofill{currentflat{{eofill}stopped{dup currentflat exch sub 20
-gt{([Error: PathTooComplex; OffendingCommand: eofill]\n)print flush
-exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
-/gpbbx{pathbbox/ury exch def/urx exch def/lly exch def/llx exch def}bd
-/lineargfill{initgfill{false initgfx/distance endX startX sub endY
-startY sub Pythag def/incD distance Steps div def endY startY sub endX
-startX sub atan newpath llx lly urx ury Bx startX startY translate
-rotate gpbbx eGy ssg newpath llx lly urx ury Bx S_eofill sGy ssg newpath
-llx lly 0 ury Bx S_eofill/fillX 0 def 0 1 Steps 1 sub{stepgfx newpath
-fillX lly fillX incD add dup/fillX exch def ury Bx S_eofill pop}for}if}bd
-/radialgfill{initgfill{false initgfx/distance endX startX sub endY
-startY sub Pythag def/incD distance Steps div def eGy ssg newpath llx
-lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepgfx newpath
-startX startY distance 0 360 arc closepath distance incD add dup/distance
-exch def 0 rmoveto startX startY distance 0 360 arc closepath S_eofill
-pop}for}if}bd/ellipticgfill{initgfill{true initgfx sGy ssg newpath
-llx lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2
-startX sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepgfx
-/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for}if}bd/conicalgfill{initgfill{urx
-startX sub abs llx startX sub abs Max ury startY sub abs lly startY
-sub abs Max Pythag startY endY sub startX endX sub Pythag div/radius
-exch def true initgfx [ endX startX sub endY startY sub startY endY
-sub endX startX sub startX startY ] concat/slice 180 Steps div def
-/startangle 0 def Steps -1 0{pop stepgfx 0 0 moveto 0 0 radius startangle
-neg dup slice add arc fill 0 0 moveto 0 0 radius startangle dup slice
-add dup/startangle exch def arc fill}for}if}bd/initgfill{gpbbx{S_eoclip}{S_clip}ifelse
-startX endX eq startY endY eq and sGy eGy eq or dup{sGy ssg newpath
-llx lly urx ury Bx S_eofill}if not}bd/initgfx{/flag exch def flag{sGy
-eGy/sGy exch def/eGy exch def}if eGy sGy sub abs 256 mul FindMinSteps
-Min 256 Min ceiling 1 Max/Steps exch def eGy sGy sub Steps div/incGy
-exch def sGy/mGy exch def}bd/stepgfx{mGy ssg mGy incGy add/mGy exch
-def}bd/linearfill{initfill{false initfx endX startX sub endY startY
-sub Pythag/distance exch def/incD distance Steps div def endY startY
-sub endX startX sub atan newpath llx lly urx ury Bx startX startY translate
-rotate gpbbx v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse
-newpath llx lly urx ury Bx S_eofill v_cxe 0 eq{sR sG sB setgfillrgb}{sH
-sS sV setgfillhsb}ifelse newpath llx lly 0 ury Bx S_eofill/fillX 0
-def 0 1 Steps 1 sub{stepfx newpath fillX lly fillX incD add dup/fillX
-exch def ury Bx S_eofill pop}for}if}bd/radialfill{initfill{false initfx
-endX startX sub endY startY sub Pythag/distance exch def/incD distance
-Steps div def v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse
-newpath llx lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepfx
-newpath startX startY distance 0 360 arc closepath distance incD add
-dup/distance exch def 0 rmoveto startX startY distance 0 360 arc closepath
-S_eofill pop}for}if}bind def/ellipticalfill{initfill{true initfx v_cxe
-0 eq{sR sG sB setgfillrgb}{sH sS sV setgfillhsb}ifelse newpath llx
-lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2 startX
-sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepfx
-/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for v_cxe 0
-eq{sR sG sB eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def
-/eG exch def/eR exch def}if}if}bd/conicalfill{initfill{urx startX
-sub abs llx startX sub abs Max ury startY sub abs lly startY sub abs
-Max Pythag startY endY sub startX endX sub Pythag div/radius exch def
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-%%PageTrailer
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diff --git a/old/0-demo/0-demo.tex b/old/0-demo/0-demo.tex
new file mode 100644
index 0000000..bc5d7b7
--- /dev/null
+++ b/old/0-demo/0-demo.tex
@@ -0,0 +1,37 @@
+\input ../lecnotes.tex
+
+\prednaska{0}{Ukázka pou¾ití maker}{(zapsal Martin Mare¹)}
+
+% Takhle vypadá nadpis
+\h{Pøedposlední kapitola}
+
+% \> zaøídí, aby se odstavec neodsadil. Pomocí \I se sází kurzíva, \foot vyrobí poznámku pod èarou.
+\>Je svaèveèer. {\I Lysperní jezeleni} se vírnì vrtáèejí v~mokøavì.\foot{Viz Lewis Caroll: Jabberwocky.}
+
+\itemize\ibull
+\:Takhle.
+\:Vypadají.
+\:Odrá¾ky.
+\endlist
+
+\h{Druhá kapitola}
+
+\s{Vìta:} Paøez není strom.
+
+\proof
+Strom je speciálním pøípadem lesa. Les je (podle definice) graf bez kru¾nic.
+Paøez obsahuje alespoò jednu kru¾nici. Proto paøez nemù¾e být les, a~tedy ani
+strom.
+\qed
+
+\s{Algoritmus:} (tøídìní posloupnosti $a_1,\ldots,a_n$ pomocí {\sc Stupidsort}u)
+
+\algo
+\:$\pi \leftarrow \hbox{identická permutace na~mno¾inì $\{1,\ldots,n\}$}$.
+\:Opakuj:
+\::Ovìø, zda je posloupnost $a_{\pi(1)},a_{\pi(2)},\ldots,a_{\pi(n)}$ uspoøádána vzestupnì.
+   Pokud je, vra» ji jako výsledek.
+\::Nahraï $\pi$ jejím lexikografickým následníkem.
+\endalgo
+
+\bye
diff --git a/old/0-demo/Makefile b/old/0-demo/Makefile
new file mode 100644
index 0000000..98def69
--- /dev/null
+++ b/old/0-demo/Makefile
@@ -0,0 +1,3 @@
+P=0-demo
+
+include ../Makerules
diff --git a/old/10-prevody/10-prevody.tex b/old/10-prevody/10-prevody.tex
new file mode 100644
index 0000000..070d102
--- /dev/null
+++ b/old/10-prevody/10-prevody.tex
@@ -0,0 +1,282 @@
+\input lecnotes.tex
+
+\prednaska{10}{Pøevody problémù}{\vbox{\hbox{(zapsali Martin Chytil, Vladimír Kudelas,}\hbox{ Michal Kozák, Vojta Tùma)}}}
+
+\>Na této pøedná¹ce se budeme zabývat rozhodovacími problémy a jejich obtí¾ností.
+Za jednoduché budeme trochu zjednodu¹enì pova¾ovat ty problémy, na~nì¾ známe algoritmus
+pracující v~polynomiálním èase.
+
+\s{Definice:} {\I Rozhodovací problém} je takový problém, jeho¾ výstupem je v¾dy {\sc ano}, nebo {\sc ne}.
+[Formálnì bychom se na~nìj mohli dívat jako na~mno¾inu $L$ vstupù, na~které je odpovìï {\sc ano},
+a místo $L(x)=\hbox{\sc ano}$ psát prostì $x\in L$.]
+
+Vstupy mìjme zakódované jen pomocí nul a jednièek (obecnì je jedno, jaký základ pro soustavu
+kódování zvolíme, pøevody mezi soustavami o nìjakém základu $\neq$ 1 jsou co do velikosti
+zápisu polynomiální). Rozhodovací problém je tedy
+$f:\ \{0,1\}^{\ast} \to \{0,1\}$, to jest funkce z mno¾iny v¹ech øetìzcù jednièek a nul
+do mno¾iny $\{1,0\}$, kde 1 na výstupu znamená {\sc ano}, 0 {\sc ne}.
+
+\s{Pøíklad:} Je dán bipartitní graf $G$ a $k \in {\bb N}$. Existuje v $G$
+párování, které obsahuje alespoò $k$ hran?
+
+To, co bychom ve~vìt¹inì pøípadù chtìli, je samozøejmì nejen zjistit, zda takové párování
+existuje, ale také nìjaké konkrétní najít. V¹imnìme si ale, ¾e kdy¾ umíme rozhodovat
+existenci párování v~polynomiálním èase, mù¾eme ho polynomiálnì rychle i najít:
+
+Mìjme èernou skøíòku (fungující v polynomiálním èase), která odpoví, zda daný
+graf má nebo nemá párování o~$k$ hranách. Odebereme z~grafu libovolnou hranu
+a zeptáme se, jestli i tento nový graf má párovaní velikosti~$k$. Kdy¾ má, pak tato
+hrana nebyla pro existenci párování potøebná, a~tak ji odstraníme. Kdy¾ naopak
+nemá (hrana patøí do ka¾dého párování po¾adované velikosti), tak si danou hranu
+poznamenáme a odebereme nejen ji a její vrcholy, ale také hrany, které do tìchto
+vrcholù vedly. Toto je korektní krok, proto¾e v pùvodním grafu tyto vrcholy
+byly navzájem spárované, a tedy nemohou být spárované s~¾ádnými jinými vrcholy.
+Na~nový graf aplikujeme znovu tentý¾ postup. Výsledkem je mno¾ina hran, které patøí
+do hledaného párování. Hran, a tedy i iterací na¹eho algoritmu, je polynomiálnì
+mnoho a skøíòka funguje v polynomiálním èase, tak¾e celý algoritmus je polynomiální.
+
+A~jak ná¹ rozhodovací problém øe¹it? Nejsnáze tak, ¾e ho pøevedeme
+na~{jiný,\footnote{${}^{\dag}$}{vìrni matfyzáckým vtipùm}} který
+u¾ vyøe¹it umíme. Tento postup jsme (právì u~hledání párování) u¾ pou¾ili
+v~kapitole o~Dinicovì algoritmu. Vytvoøili jsme vhodnou sí», pro kterou
+platilo, ¾e v~ní existuje tok velikosti~$k$ právì tehdy, kdy¾
+v~pùvodním grafu existuje párování velikosti~$k$.
+
+Takovéto pøevody mezi problémy mù¾eme definovat obecnì:
+
+\s{Definice:} Jsou-li $A$, $B$ rozhodovací problémy, pak øíkáme, ¾e $A$ lze {\I
+redukovat} (neboli {\it pøevést}) na $B$ (pí¹eme $A \rightarrow B$) právì tehdy,
+kdy¾ existuje funkce $f$ spoèitatelná v polynomiálním èase taková, ¾e pro $\forall
+x: A(x) = B(f(x))$. V¹imnìme si, ¾e $f$ pracující v polynomiálním èase vstup
+zvìt¹í nejvíce polynomiálnì.
+
+\s{Pozorování:} $A\rightarrow B$ také znamená, ¾e problém~$B$ je alespoò tak tì¾ký
+jako problém~$A$ (tím myslíme, ¾e pokud lze $B$ øe¹it v~polynomiálním èase,
+lze tak øe¹it i~$A$): Nech» problém~$B$ umíme øe¹it v~èase $\O(b^k)$, kde
+$b$ je délka jeho vstupu. Nech» dále funkce $f$ pøevádìjící $A$ na $B$ pracuje
+v~èase $\O(a^\ell)$ pro vstup délky~$a$. Spustíme-li tedy $B(f(x))$ na~nìjaký
+vstup~$x$ problému~$A$, bude mít $f(x)$ délku $\O(a^\ell)$, kde $a=|q|$; tak¾e
+$B(f(x))$ pobì¾í v~èase $\O(a^\ell + (a^\ell)^k) = \O(a^{k\ell})$, co¾ je
+polynomiální v~délce vstupu~$a$.
+
+
+\s{Pozorování:} Pøevoditelnost je
+\itemize\ibull
+\:reflexivní (úlohu mù¾eme pøevést na tu stejnou identickým zobrazením): $A \rightarrow A$,
+\:tranzitivní: Je-li $A \rightarrow B$ funkcí $f$, $B \rightarrow C$ funkcí $g$,
+pak $A \rightarrow C$ slo¾enou funkcí $g \circ f$
+(slo¾ení dvou polynomiálních funkcí je zase polynomiální funkce, jak u¾ jsme zpozorovali
+v~pøedchozím odstavci).
+\endlist
+\>Takovýmto relacím øíkáme kvaziuspoøádání -- nesplòují obecnì antisymetrii, tedy mù¾e nastat
+$A\rightarrow B$ a $B\rightarrow A$. Omezíme-li se v¹ak na tøídy navzájem pøevoditelných
+problémù, dostáváme ji¾ (èásteèné) uspoøádání. Existují i navzájem nepøevoditelné problémy --
+napøíklad problém v¾dy odpovídající 1 a problém v¾dy odpovídající 0.
+Nyní se ji¾ podíváme na nìjaké zajímavé problémy. Obecnì to budou problémy, na které
+polynomiální algoritmus není znám, a vzájemnými pøevody zjistíme ¾e jsou stejnì tì¾ké.
+
+\h{1. problém: SAT}
+\>Splnitelnost (satisfiability) logických formulí, tj. dosazení 1 èi
+0 za promìnné v logické formuli tak, aby formule dala výsledek 1.
+
+\>Zamìøíme se na speciální formu zadání formulí,  {\I konjunktivní normální formu} (CNF),
+	které splòují následující podmínky:
+\itemize\ibull
+\:{\I formule} je zadána pomocí {\I klauzulí}\footnote{${}^{\dag}$}{bez politických konotací} oddìlených $\land$,
+\:ka¾dá {\I klauzule} je slo¾ená z {\I literálù} oddìlených $\lor$,
+\:ka¾dý {\I literál} je buïto promìnná nebo její negace.
+\endlist
+\>Formule mají tedy tvar:
+$$\psi = (\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$
+
+\>{\I Vstup:} Formule $\psi$ v konjunktivní normální formì.
+
+\>{\I Výstup:} $\exists$ dosazení 1 a 0 za promìnné takové, ¾e hodnota formule $\psi(\ldots) = 1$.
+
+
+
+\>Pøevod nìjaké obecné formule $\psi$ na jí ekvivalentní $\chi$ v~CNF mù¾e
+zpùsobit, ¾e $\chi$ je exponenciálnì velká vùèi $\psi$.
+Pozdìji uká¾eme, ¾e lze podniknout pøevod na takovou formuli $\chi'$ v~CNF, která sice není
+ekvivalentní s $\psi$ (pøibydou nám promìnné, a ne ka¾dý roz¹íøený model
+$\psi$ je modelem $\chi'$), ale je splnitelná právì tehdy, kdy¾ je splnitelná $\psi$ -- co¾ nám
+pøesnì staèí -- a je lineárnì velká vùèi $\psi$.
+
+\h{2. problém: 3-SAT}
+\s{Definice:} 3-SAT je takový SAT, v nìm¾ ka¾dá klauzule obsahuje nejvý¹e tøi literály.
+
+\s{Pøevod 3-SAT na SAT:}
+Vstup není potøeba nijak upravovat, 3-SAT splòuje vlastnosti SATu, proto 3-SAT
+$\rightarrow$ SAT (SAT je alespoò tak tì¾ký jako 3-SAT)
+
+\s {Pøevod SAT na 3-SAT:}
+Musíme formuli pøevést tak, abychom neporu¹ili splnitelnost.
+
+\>Trik pro dlouhé klauzule: Ka¾dou \uv{¹patnou} klauzuli
+$$(\alpha \lor \beta) \hbox{, t¾. } \vert\alpha\vert + \vert\beta\vert \ge 4
+,\ \vert\alpha\vert \geq 2,\ \vert\beta\vert\geq 2$$
+pøepí¹eme na: $$(\alpha \lor x) \land (\beta \lor \lnot x),$$
+kde $x$ je nová promìnná (pøi ka¾dém dìlení klauzule {\it jiná}
+nová promìnná).
+%kterou nastavíme tak, abychom neovlivnili splnitelnost formule.
+
+\>Tento trik opakujeme tak dlouho, dokud je to tøeba -- formuli délky $k+l$
+roztrhneme na formule délky $k+1$ a $l+1$. Pokud klauzule pùlíme, dostaneme
+polynomiální èas (strom rekurze má logaritmicky pater -- formule délky alespoò 6
+se nám pøi rozdìlení zmen¹í na dvì instance velikosti maximálnì $2/3$ pùvodní, krat¹í
+formule nás netrápí; na ka¾dém patøe se vykoná tolik co na pøedchozím + $2^{hloubka}$
+za pøidané formule). Velikost výsledné formule je tím pádem polynomiální vùèi pùvodní:
+v ka¾dém kroku se pøidají jen dva literály, tedy celkem {\it èas na pøevod}$\cdot
+2$ nových.
+
+\>Platí-li:
+\itemize\ibull
+\:$\alpha \Rightarrow$ zvolíme $x = 0$ (zajistí splnìní druhé poloviny nové formule),
+\:$\beta \Rightarrow$ zvolíme $x = 1$ (zajistí splnìní první poloviny nové formule),
+\:$\alpha ,\beta / \lnot\alpha ,\lnot\beta \Rightarrow$ zvolíme $x = 0/1$ (je nám to
+	jedno, celkové øe¹ení nám to neovlivní).
+\endlist
+
+Nabízí se otázka, proè mù¾eme pøidanou promìnnou $x$ nastavovat, jak se nám zlíbí.
+Vysvìtlení je prosté -- promìnná $x$ nám pùvodní formuli nijak neovlivní, proto¾e
+se v ní nevyskytuje, proto ji mù¾eme nastavit tak, jak chceme.
+
+\s{Poznámka:} U~3-SAT lze vynutit právì tøi literály, pro krátké klauzule
+pou¾ijeme stejný trik:
+$$(\alpha) \rightarrow (\alpha \vee \alpha) \rightarrow (\alpha \lor x) \land (\alpha \lor \lnot x).$$
+
+\h{3. problém: Hledání nezávislé mno¾iny v grafu}
+
+\>Existuje nezávislá mno¾ina vrcholù z~$G$ velikosti alespoò $k$?
+
+\s{Definice:} {\I Nezávislá mno¾ina} (NzMna) budeme øíkat ka¾dé mno¾inì vrcholù grafu
+takové, ¾e mezi nimi nevede ¾ádná hrana.
+
+\figure{nezmna.eps}{Pøíklad nezávislé mno¾iny}{1in}
+
+\>{\I Vstup:} Neorientovaný graf $G$, $k \in {\bb N}$.
+
+\>{\I Výstup:} $\exists A \subseteq V(G)$, $\vert A \vert \ge k$: $\forall u,v \in A \Rightarrow uv \not\in E(G)$?
+
+\s{Poznámka:} Ka¾dý graf má minimálnì jednu nezávislou mno¾inu, a tou je prázdná mno¾ina. Proto je potøeba zadat i minimální velikost hledané mno¾iny.
+
+\>Uká¾eme, jak na~tento probém pøevést 3-SAT.
+
+\s{Pøevod 3-SAT na NzMna:} Z ka¾dé klauzule vybereme jeden literál, jeho¾ nastavením se klauzuli
+rozhodneme splnit. Samozøejmì tak, abychom v~rùzných klauzulích nevybírali
+konfliktnì, tj.~$x$ a~$\lnot x$.
+
+\s{Pøíklad:}
+$(x \lor y \lor z) \land (x \lor \lnot y \lor \lnot z) \land (\lnot x \lor \lnot y \lor p) $.
+
+\>Pro ka¾dou klauzuli sestrojíme graf (trojúhelník) a pøidáme \uv{konfliktní}
+hrany, tj. $x$ a $\lnot x$. Poèet vrcholù grafu odpovídá poètu literálù ve formuli,
+poèet hran je maximálnì kvadratický a pøevod je tedy polynomiální.
+
+Existuje-li v grafu nezávislá mno¾ina velikosti $k$, pak z~ka¾dého z~$k$ trojúhelníkù
+vybere právì jeden vrchol, a pøitom ¾ádné dva vrcholy nebudou odpovídat literálu a
+jeho negaci -- tedy dostaneme ohodnocení promìnných splòujících alespoò $k$ klauzulí.
+Na druhou stranu, existuje-li ohodnocení $k$ klauzulí, pak pøímo odpovídá nezávislé
+mno¾inì velikosti $k$ (v ka¾dém trojúhelníku zvolíme právì jednu z ohodnocených
+promìnných, nemù¾e se stát ¾e zvolíme vrcholy konfliktní hrany). Ptáme-li se tedy
+na nezávislou mno¾inu velikosti odpovídající
+poètu klauzulí, dostaneme odpovìï {\sc ano} právì tehdy, kdy¾ je formule splnitelná.
+
+Jsou-li ve formuli i klauzule krat¹í ne¾ 3, mù¾eme je buïto prodlou¾it metodou vý¹e
+popsanou; nebo si v grafu necháme dvoj- a jedno-úhelníky, které ¾ádné z na¹ich úvah
+vadit nebudou.
+
+\figure{nezmna_graf.eps}{Ukázka pøevodu 3-SAT na nezávislou mno¾inu}{3in}
+
+\s{Pøevod NzMna na SAT:}
+
+\itemize\ibull
+\:Poøídíme si promìnné $v_1, \ldots, v_n$ odpovídající vrcholùm grafu. Promìnná $v_i$ bude
+  indikovat, zda se $i$-tý vrchol vyskytuje v~nezávislé mno¾inì (tedy pøíslu¹né ohodnocení
+  promìnných bude vlastnì charakteristická funkce nezávislé mno¾iny).
+\:Pro ka¾dou hranu $ij \in E(G)$ pøidáme klauzuli $(\lnot v_i \lor \lnot v_j)$. Tyto klauzule
+  nám ohlídají, ¾e vybraná mno¾ina je vskutku nezávislá.
+\:Je¹tì potøebujeme zkontrolovat, ¾e je mno¾ina dostateènì velká, tak¾e si její prvky
+  oèíslujeme èísly od~1 do~$k$. Oèíslování popí¹eme maticí promìnných $x_{ij}$, pøièem¾
+  $x_{ij}$ bude pravdivá právì tehdy, kdy¾ v~poøadí $i$-tý prvek nezávislé mno¾iny je vrchol~$v_j$
+-- pøidáme tedy klauzule, které nám øeknou, ¾e vybrané do nezávislé mno¾iny jsou právì
+  ty vrcholy, které jsou touto maticí oèíslované: $\forall i,j$, $x_{ij} \Rightarrow v_j$
+  (jen dodejme, ¾e $a\Rightarrow b$ je definované jako $\neg a\vee b$).
+\:Je¹tì potøebujeme zajistit, aby byla v~ka¾dém øádku i sloupci nejvý¹e jedna jednièka:
+  $\forall j,i,i^{'}, i\ne i^{'} : x_{ij} \Rightarrow \lnot x_{i^{'}j}$ a
+  $\forall i,j,j^{'}, j\ne j^{'} : x_{ij} \Rightarrow \lnot x_{ij^{'}}$.
+\:A~nakonec si ohlídáme, aby v~ka¾dém øádku byla alespoò jedna jednièka, klauzulí $\forall i :
+  x_{i1} \lor x_{i2} \lor \ldots \lor x_{in}$.
+\endlist
+Tímto vynutíme NzMnu $\geq k$, co¾ jsme pøesnì chtìli. Takovýto pøevod je zøejmì polynomiální.
+
+\s{Pøíklad matice:} Jako pøíklad pou¾ijeme nezávislou mno¾inu z ukázky nezávislé mno¾iny.
+Nech» jsou vrcholy grafu oèíslované zleva a zeshora. Hledáme nezávislou mno¾inu velikosti $2$.
+Matice pak bude vypadat následovnì:
+$$ \pmatrix{1&0&0&0&0 \cr 0&0&0&1&0}$$
+\s{Vysvìtlení:} Jako první vrchol mno¾iny bude vybrán vrchol $v_1$, proto v prvním
+øádku a v prvním sloupci bude $1$. Jako druhý vrchol mno¾iny bude vybrán
+vrchol $v_4$, proto na druhém øádku a ve ètvrtém sloupci bude $1$. Na ostatních místech bude $0$.
+
+\h{4. problém: Klika}
+
+\>{\I Vstup:} Graf $G, k \in N$.
+
+\>{\I Výstup:} $\exists$ úplný podgraf grafu $G$ na $k$ vrcholech?
+\figure{klika.eps}{Pøíklad kliky}{2in}
+
+\s{Pøevod:} Prohodíme v grafu $G$ hrany a nehrany $\Rightarrow$ (hledání nezávislé mno¾iny $\leftrightarrow$ hledání kliky).
+
+\s{Dùvod:} Pokud existuje úplný graf na $k$ vrcholech, tak v~komplementárním grafu tyto vrcholy nejsou spojeny hranou, tj. tvoøí nezávislou mno¾inu,
+	a naopak.
+
+\figure{doplnek_nm.eps}{Prohození hran a nehran}{2in}
+
+
+\h{5. problém: 3,3-SAT}
+\s{Definice:} 3,3-SAT je speciální pøípad 3-SATu, kde ka¾dá promìnná se vyskytuje v~maximálnì tøech literálech.
+
+\s{Pøevod 3-SAT na 3,3-SAT:}
+Pokud se promìnná $x$ vyskytuje v~$k > 3$ literálech, tak nahradíme výskyty novými promìnnými $x_1, \ldots , x_k$ a pøidáme klauzule:
+$$
+(\lnot x_1 \lor x_2),
+(\lnot x_2 \lor x_3),
+(\lnot x_3 \lor x_4),
+\ldots,
+(\lnot x_{k-1} \lor x_k),
+(\lnot x_k \lor x_1),
+$$
+
+co¾ odpovídá:
+
+$$
+(x_1 \Rightarrow x_2),
+(x_2 \Rightarrow x_3),
+(x_3 \Rightarrow x_4),
+\ldots,
+(x_{k-1} \Rightarrow x_k),
+(x_k \Rightarrow x_1).
+$$
+
+Tímto zaruèíme, ¾e v¹echny nové promìnné budou mít stejnou hodnotu.
+
+Mimochodem, mù¾eme rovnou zaøídit, ¾e ka¾dý literál se vyskytuje nejvíce dvakrát (tedy ¾e
+ka¾dá promìnná se vyskytuje alespoò jednou pozitivnì a alespoò jednou negativnì). Pokud by
+se nìjaká promìnná objevila ve~tøech stejných literálech, mù¾eme na~ni
+také pou¾ít ná¹ trik a nahradit ji tøemi promìnnými. V~nových klauzulích se pak bude
+vyskytovat jak pozitivnì, tak negativnì.
+
+\h{6. problém: 3D párování (3D matching)}
+
+\>{\I Vstup:} Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a mno¾ina kompatibilních trojic (tìch, kteøí se spolu snesou).
+
+\>{\I Výstup:} Perfektní podmno¾ina trojic -- tj. taková podmno¾ina trojic, která obsahuje v¹echna $K$, $H$ a $Z$.
+
+\>Uká¾eme, jak na tento problém pøevést 3,3-SAT (ov¹em to a¾ na dal¹í pøedná¹ce).
+
+\figure{3d_parovani.eps}{Ukázka 3D párování}{3in}
+
+\s{Závìr:} Obrázek ukazuje problémy, jimi¾ jsme se dnes zabývali, a vztahy mezi tìmito problémy.
+\figure{prevody.eps}{Pøevody mezi problémy}{3in}
+
+\bye
diff --git a/old/10-prevody/3d_parovani.eps b/old/10-prevody/3d_parovani.eps
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diff --git a/old/10-prevody/Makefile b/old/10-prevody/Makefile
new file mode 100644
index 0000000..04b5d58
--- /dev/null
+++ b/old/10-prevody/Makefile
@@ -0,0 +1,3 @@
+P=10-prevody
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+include ../Makerules
diff --git a/old/10-prevody/doplnek_nm.eps b/old/10-prevody/doplnek_nm.eps
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index 0000000..431c5c6
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diff --git a/old/11-np/11-np.tex b/old/11-np/11-np.tex
new file mode 100644
index 0000000..c81d913
--- /dev/null
+++ b/old/11-np/11-np.tex
@@ -0,0 +1,334 @@
+\input lecnotes.tex
+
+\prednaska{11}{NP-úplné problémy}{\vbox{\hbox{(zapsali F. Kaèmarik, R. Krivák, D. Remi¹}
+	\hbox{ Michal Kozák, Vojta Tùma)}}}
+
+Dosud jsme zkoumali problémy, které se nás ptaly na to, jestli nìco existuje.
+Napøíklad jsme dostali formuli a problém splnitelnosti se nás ptal, zda
+existuje ohodnocení promìnných takové, ¾e formule platí. Nebo v~pøípadì
+nezávislých mno¾in jsme dostali graf a èíslo $k$ a ptali jsme se, jestli
+v~grafu existuje nezávislá mno¾ina, která obsahuje alespoò~$k$ vrcholù.
+Tyto otázky mìly spoleèné to, ¾e kdy¾ nám nìkdo napovìdìl nìjaký objekt, umìli
+jsme efektivnì øíci, zda je to ten, který hledáme. Napøíklad pokud dostaneme
+ohodnocení promìnných logické formule, staèí jen dosadit a spoèítat, kde
+formule dá \<true> nebo \<false>. Zjistit, ¾e nìjaký objekt je ten, který
+hledáme, umíme efektivnì. Tì¾ké na tom je takový objekt najít. Co¾ vede
+k~definici obecných vyhledávacích problémù, kterým se øíká tøída problémù NP.
+Nadefinujeme si ji poøádnì, ale nejdøíve zaèneme tro¹ièku jednodu¹¹í tøídou.
+
+\s{Definice:} P je {\I tøída rozhodovacích problémù}, které jsou øe¹itelné
+v~polynomiálním èase. Jinak øeèeno, problém
+$L \in {\rm P}  \Leftrightarrow  \exists $ polynom $f$ a~$\exists$ algoritmus~$A$
+takový, ¾e $\forall x: L(x)=A(x)$ a $A(x)$ dobìhne v~èase $\O(f(x))$.
+
+Tøída P tedy odpovídá tomu, o èem jsme se shodli, ¾e umíme efektivnì øe¹it.
+Nadefinujme nyní tøídu NP:
+
+\s{Definice:} NP je {\I tøída rozhodovacích problémù} takových, ¾e $L \in {\rm NP}$ právì tehdy, kdy¾ $\exists $ problém
+$K\in{\rm P}$ a $\exists$  polynom  $g$ takový, ¾e pro
+$\forall x$ platí $L(x)=1  \Leftrightarrow  \exists $ nápovìda $ y: \vert y \vert \leq g(\vert x \vert)$ a souèasnì $K(x,y)=1$.
+
+\s{Pozorování:} Splnitelnost logických formulí je v~NP. Staèí si toti¾ nechat napovìdìt, jak
+ohodnotit jednotlivé promìnné a pak ovìøit, jestli je formule splnìna. Nápovìda je polynomiálnì
+velká (dokonce lineárnì), splnìní zkontrolujeme také v~lineárním èase. Odpovíme tedy ano právì
+tehdy, existuje-li nápovìda, která nás pøesvìdèí, èili pokud je formule splnitelná.
+
+\s{Pozorování:} Tøída P le¾í uvnitø NP.
+V~podstatì øíkáme, ¾e kdy¾ máme problém, který umíme øe¹it v~polynomiálním èase
+bez nápovìdy, tak to zvládneme v~polynomiálním èase i s~nápovìdou.
+
+Problémy z minulé pøedná¹ky jsou v¹echny v NP (napø. pro nezávislou
+mno¾inu je onou nápovìdou pøímo mno¾ina vrcholù deklarující nezávislost),
+o jejich pøíslu¹nosti do P ale nevíme nic.
+Brzy uká¾eme, ¾e to jsou v jistém smyslu nejtì¾¹í problémy v~NP.
+Nadefinujme si:
+
+\s{Definice:} Problém $L$ je NP-{\I tì¾ký} právì tehdy, kdy¾ je na~nìj pøevoditelný
+ka¾dý problém z~NP (viz definici pøevodù z minulé pøedná¹ky).
+
+Rozmyslete si, ¾e pokud umíme øe¹it nìjaký NP-tì¾ký problém v~polynomiálním èase,
+pak umíme vyøe¹it v~polynomiálním èase v¹e v~NP, a tedy ${\rm P}={\rm NP}$.
+
+My se budeme zabývat problémy, které jsou NP-tì¾ké a samotné jsou v~NP. Takovým problémùm se øíká NP-úplné.
+
+\s{Definice:} Problém $L$ je NP-{\I úplný} právì tehdy, kdy¾ $L$ je NP-tì¾ký a $L \in {\rm NP}$.
+
+NP-úplné problémy jsou tedy ve~své podstatì nejtì¾¹í problémy, které le¾í v~NP.
+Kdybychom umìli vyøe¹it nìjaký NP-úplný problém v~polynomiálním èase, pak
+v¹echno v~NP je øe¹itelné v~polynomiálním èase. Bohu¾el to, jestli nìjaký
+NP-úplný problém lze øe¹it v~polynomiálním èase, se neví. Otázka, jestli
+${\rm P}={\rm NP}$, je asi nejznámìj¹í otevøený problém v~celé teoretické
+informatice.
+
+Kde ale nìjaký NP-úplný problém vzít? K~tomu se nám bude velice hodit následující vìta:
+
+\s{Vìta (Cookova):} SAT je NP-úplný.
+
+\>Dùkaz je znaènì technický, pøibli¾nì ho naznaèíme pozdìji. Pøímým dùsledkem
+Cookovy vìty je, ¾e cokoli v~NP je pøevoditelné na SAT.
+K dokazování NP-úplnosti dal¹ích problémù pou¾ijeme následující vìtièku:
+
+\s{Vìtièka:} Pokud problém $L$ je NP-úplný a $L$ se dá pøevést na nìjaký problém $M\in{\rm NP}$, pak $M$ je také NP-úplný.
+
+\proof
+Tuto vìtièku staèí dokázat pro NP-tì¾kost, NP-úplnost plyne okam¾itì z~toho, ¾e
+problémy jsou NP-tì¾ké a le¾í v~NP (podle pøedpokladu).
+
+Víme, ¾e $L$ se dá pøevést na~$M$ nìjakou funkcí~$f$. Jeliko¾ $L$ je NP-úplný,
+pak pro ka¾dý problém $Q\in{\rm NP}$ existuje nìjaká funkce~$g$, která pøevede
+$Q$ na~$L$. Staèí tedy slo¾it funkci~$f$ s~funkcí~$g$, èím¾ získáme funkci pracující
+opìt v~polynomiálním èase, která pøevede~$Q$ na~$M$. Ka¾dý problém z~NP se tedy
+dá pøevést na problém~$M$.
+\qed
+
+\s{Dùsledek:} Cokoliv, na co jsme umìli pøevést SAT, je také NP-úplné.
+Napøíklad nezávislá mno¾ina, rùzné varianty SATu, klika v~grafu~\dots
+
+Jak taková tøída NP vypadá? Pøedstavme si v¹echny problémy tøídy NP, jakoby seøazené
+shora dolu podle obtí¾nosti problémù (tedy navzdor gravitaci), kde porovnání dvou
+problémù urèuje pøevoditelnost (viz obrázek).
+
+\figure{p-np.eps}{Struktura tøídy NP}{2.5cm}
+
+Obecnì mohou nastat dvì situace, proto¾e nevíme, jestli ${\rm P}={\rm NP}$.
+Jestli ano, pak v¹echno je jedna a ta samá tøída. To by bylo v nìkterých
+pøípadech nepraktické, napø. ka¾dá ¹ifra by byla jednodu¹e rozlu¹titelná.
+\foot{Poznámka o ¹ifrách -- libovolnou funkci vyèíslitelnou v polynomiálním
+èase bychom umìli v polynomiálním èase také invertovat.}
+Jestli ne, NP-úplné problémy urèitì nele¾í v P, tak¾e P a NP-úplné problémy
+jsou dvì disjunktní èásti NP. Také se dá dokázat (to dìlat nebudeme, ale je
+dobré to vìdìt), ¾e je¹tì nìco le¾í mezi nimi, tedy ¾e existuje problém, který
+je v~NP, není v~P a není NP-úplný (dokonce je takových problémù nekoneènì mnoho,
+v nekoneènì tøídách).
+
+\s{Katalog NP-úplných problémù}
+
+Uká¾eme si nìkolik základních NP-úplných problémù. O~nìkterých jsme to dokázali
+na~minulé pøedná¹ce, o~dal¹ích si to doká¾eme nyní, zbylým se na~zoubek podíváme
+na~cvièeních.
+
+\itemize\ibull
+\:{\I logické:}
+  \itemize\ibull
+    \:SAT (splnitelnost logických formulí v~CNF)
+    \:3-SAT (ka¾dá klauzule obsahuje max.~3 literály)
+    \:3,3-SAT (a navíc ka¾dá promìnná se vyskytuje nejvý¹e tøikrát)
+    \:SAT pro obecné formule (nejen CNF)
+    \:Obvodový SAT (není to formule, ale obvod)
+  \endlist
+\:{\I grafové:}
+  \itemize\ibull
+    \:Nezávislá mno¾ina (existuje mno¾ina alespoò~$k$ vrcholù taková, ¾e ¾ádné dva nejsou propojeny hranou?)
+    \:Klika (existuje úplný podgraf na~$k$ vrcholech?)
+    \:3D párování (tøi mno¾iny se zadanými trojicemi, existuje taková mno¾ina disjunktních trojic, ve~které jsou v¹echny prvky?)
+    \:Barvení grafu (lze obarvit vrcholy $k$~barvami tak, aby vrcholy stejné barvy nebyly nikdy spojeny hranou? NP-úplné u¾ pro~$k=3$)
+    \:Hamiltonovská cesta (cesta obsahující v¹echny vrcholy [právì jednou])
+    \:Hamiltonovská kru¾nice (kru¾nice, která nav¹tíví v¹echny vrcholy [právì jednou])
+  \endlist
+\:{\I èíselné:}
+  \itemize\ibull
+    \:Batoh (nejjednodu¹¹í verze: dána mno¾ina èísel, zjistit, zda existuje podmno¾ina se zadaným souètem)
+	\:Batoh -- optimalizace (podobnì jako u pøedchozího problému, ale místo mno¾iny èísel máme mno¾inu
+		pøedmìtù s vahami a cenami, chceme co nejdra¾¹í podmno¾inu její¾ váha nepøesáhne zadanou kapacitu
+		batohu)
+    \:Loupe¾níci (lze rozdìlit danou mno¾inu èísel na~dvì podmno¾iny se stejným souètem)
+    \:$Ax=b$ (soustava celoèíslených lineárních rovnic; $x_i$ mohou být pouze 0 nebo 1; NP-úplné i pokud $A_{ij}\in\{0,1\}$ a $b_i\in\{0,1\}$)
+    \:Celoèíselné lineární programování (existuje vektor nezáporných celoèísených $x$ takový, ¾e $Ax \leq b$ ?)
+  \endlist
+\endlist
+
+Nyní si uká¾eme, jak pøevést SAT na nìjaký problém. Kdy¾ chceme ukázat, ¾e na
+nìco se dá pøevést SAT, potøebujeme obvykle dvì vìci:
+konstrukci, která bude simulovat promìnné, tedy nìco, co nabývá dvou stavù
+\<true>/\<false>; a nìco, co bude reprezentovat klauzule a umí zaøídit, aby
+ka¾dá klauzule byla splnìna alespoò jednou promìnnou.
+Roz¹iøme tedy ná¹ katalog problémù o následující:
+\h{3D párování (3D matching)}
+
+\>{\I Vstup:} Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a
+mno¾ina $T$ kompatibilních trojic (tìch, kteøí se spolu snesou),
+	tj. $T \subseteq K\times H\times Z$.
+
+\>{\I Výstup:} Perfektní podmno¾ina trojic $P\subseteq K\times H \times Z$ --
+	tj. taková podmno¾ina trojic, ¾e $(\forall k\in K\ \exists !p\in P: k\in p)
+	\wedge(\forall h\in H\ \exists !p\in P: h\in p)
+	\wedge(\forall z\in Z\ \exists !p\in P: z\in p)$ -- tedy ka¾dý byl vybrán
+	právì jednou.
+
+
+\h{ Uká¾eme, jak na 3D-párování pøevést 3,3-SAT }
+
+Uva¾ujme takovouto konfiguraci:
+
+\fig{3d.eps}{4cm}
+
+\>4 zvíøátka, 2 kluci, 2 dívky a 4 trojice, které oznaèíme $A, B, C, D$.
+Je¹tì pøedpokládáme, ¾e zvíøátka
+ se mohou úèastnit nìjakých jiných trojic, ale
+tito ètyøi lidé se vyskytují pouze v~tìchto ètyøech trojicích a~nikde jinde.
+V¹imneme si, ¾e existují právì dvì mo¾nosti, jak tento obrázek spárovat.
+Abychom spárovali kluka $k_1$, tak si musíme vybrat $A$ nebo $B$. Kdy¾ si
+vybereme $A$, $k_1$ i $d_2$ u¾ jsou spárovaní tak¾e si nesmíme vybrat $B$ ani
+$D$. Pak jediná mo¾nost, jak spárovat $d_1$ a~$k_2$ je $C$. Jedna mo¾nost je
+tedy vybrat si $A$ a $C$ a jeliko¾ je obrázek symetrický, tak kdy¾ vybereme
+místo $A$ trojici $B$, dostaneme $B$ a~$D$. V¾dy si tedy vybereme dvì protìj¹í
+trojice v~obrázku.
+
+Tyto konfigurace budeme pou¾ívat k~reprezentaci promìnných. Pro ka¾dou
+promìnnou si nakreslíme takový obrázek a~to, ¾e $A$ bude spárované s~$C$, bude
+odpovídat tomu, ¾e $x=1$, a~spárování $B$ a~$D$ bude odpovídat
+ $x=0$. Pokud jsme
+pou¾ili $A$ a~$C$, zvíøata se sudými èísly, tj. $z_2$ a~$z_4$, horní a~dolní,
+jsou nespárovaná a~pokud jsme pou¾ili $B$ a~$D$, zvíøátka $z_1$ a~$z_3$ zùstala
+nespárovaná. Pøes tato nespárovaná zvíøátka mù¾eme pøedávat informaci, jestli
+promìnná $x$ má hodnotu \<true> nebo \<false> do dal¹ích èástí vstupu.
+
+Zbývá vymyslet, jak reprezentovat klauzule. Klauzule jsou trojice popø. dvojice
+literálù, napø. $\kappa = (x \lor y \lor \lnot r)$, kde
+potøebujeme zajistit, aby $x$ bylo nastavené na $1$ nebo $y$ bylo nastavené
+na~$1$ nebo $r$ na $0$.
+
+\fig{klauzule.eps}{4cm}
+
+\>Pro takovouto klauzuli si poøídíme dvojici kluk-dívka, kteøí budou figurovat
+ve tøech trojicích se tøemi rùznými zvíøátky, co¾ mají být volná zvíøátka
+z~obrázkù pro pøíslu¹né promìnné (podle toho, má-li se promìnná vyskytnout
+s negací nebo ne). A~zaøídíme to tak, aby ka¾dé zvíøátko bylo
+pou¾ité maximálnì v~jedné takové trojici, co¾ jde proto, ¾e ka¾dý literál se
+vyskytuje maximálnì dvakrát a~pro ka¾dý literál máme dvì volná zvíøátka,
+z~èeho¾ plyne, ¾e zvíøátek je dost pro v¹echny klauzule. Pro dvojice se postupuje
+obdobnì.
+
+Je¹tì nám ale urèitì zbude $2p-k$ zvíøátek, kde $p$ je poèet promìnných, $k$
+poèet klauzulí -- ka¾dá promìnná vyrobí 4 zvíøátka, klauzule zba¹tí jedno
+a samotné ohodnocení 2 zvíøátka -- tak pøidáme je¹tì $2p-k$ párù
+kluk-dìvèe, kteøí milují
+v¹echna zvíøátka, a~ti vytvoøí zbývající trojice.
+
+Pokud formule byla splnitelná, pak ze splòujícího ohodnocení mù¾eme vyrobit
+párování s~na¹í konstrukcí. Obrázek pro ka¾dou promìnnou spárujeme podle
+ohodnocení, tj. promìnná je $0$ nebo $1$ a~pro ka¾dou klauzuli si vybereme
+nìkterou z~promìnných, kterými je ta klauzule splnìna. Funguje to ale
+i~opaènì: Kdy¾ nám nìkdo dá párovaní v~na¹í konstrukci, pak z nìho doká¾eme
+vyrobit splòující ohodnocení dané formule. Podíváme se, v~jakém stavu je
+promìnná, a~to je v¹echno. Z~toho, ¾e jsou správnì spárované klauzule, u¾
+okam¾itì víme, ¾e jsou v¹echny splnìné.
+
+Zbývá ovìøit, ¾e na¹e redukce funguje v~polynomiálním èase. Pro ka¾dou klauzuli
+spotøebujeme konstantnì mnoho èasu, $2p-k$ je také polynomiálnì mnoho a~kdy¾ v¹e
+seèteme, máme polynomiální èas vzhledem k~velikosti vstupní formule. Tím je
+pøevod hotový a~mù¾eme 3D-párování zaøadit mezi NP-úplné problémy.
+
+
+%RK
+
+
+\h{Náznak dùkazu Cookovy vìty}
+
+Abychom mohli budovat teorii NP-úplnosti, potøebujeme alespoò jeden problém,
+o kterém doká¾eme, ¾e je NP-úplný, z definice. Cookova vìta øíká o NP-úplnosti
+SAT-u, ale nám se to hodí dokázat o tro¹ku jiném problému -- {\I obvodovém SAT-u}.
+
+\>{\I Obvodový SAT} je splnitelnost, která nepracuje s~formulemi, ale s~booleovskými
+obvody (hradlovými sítìmi). Ka¾dá formule se dá pøepsat do booleovského obvodu,
+který ji poèítá, tak¾e dává smysl zavést splnitelnost i pro obvody. Na¹e obvody
+budou mít nìjaké vstupy a~jenom jeden výstup. Budeme se ptát, jestli se vstupy
+tohoto obvodu dají nastavit tak, abychom na výstupu dostali \<true>.
+
+\>Nejprve doká¾eme NP-úplnost {\I obvodového SAT-u} a~pak uká¾eme, ¾e se dá
+pøevést na obyèejný SAT v~CNF. Tím bude dùkaz Cookovy vìty hotový. Zaènìme
+s pomocným lemmatem.
+
+\s{Lemma:} Nech» $L$ je problém v P. Potom existuje polynom $p$ a algoritmus,
+který pro $\forall n \ge 0$ spoète v èase $p(n)$ hradlovou sí» $Bn$ s $n$ vstupy
+a 1 výstupem takovou, ¾e $\forall x \in \{ 0, 1 \}^n : Bn(x) = L(x)$.
+
+\proof
+Náznakem. Na základì zku¹eností z Principù poèítaèù intuitivnì chápeme poèítaèe
+jako nìjaké slo¾ité booleovské obvody, jejich¾ stav se mìní v~èase. Uva¾me tedy
+nìjaký problém $L \in {\rm P}$ a polynomiální algoritmus, který ho øe¹í. Pro vstup
+velikosti~$n$ dobìhne v~èase~$T$ polynomiálním v~$n$ a spotøebuje $\O(T)$ bunìk pamìti.
+Staèí nám tedy \uv{poèítaè s~pamìtí velkou $\O(T)$}, co¾ je nìjaký booleovský obvod
+velikosti polynomiální v~$T$, a~tedy i v~$n$. Vývoj v~èase o¹etøíme tak, ¾e sestrojíme~$T$
+kopií tohoto obvodu, ka¾dá z~nich bude odpovídat jednomu kroku výpoètu a bude
+propojena s~\uv{minulou} a \uv{budoucí} kopií. Tím sestrojíme booleovský obvod,
+který bude øe¹it problém~$L$ pro vstupy velikosti~$n$ a bude polynomiálnì velký
+vzhledem k~$n$.
+
+\s{Poznámka:}
+Pro dùkaz následující vìty si dovolíme drobnou úpravu v~definici tøídy NP.
+Budeme chtít, aby nápovìda
+mìla pevnou velikost, závislou pouze na~velikosti vstupu (tedy: $\vert y \vert
+= g(\vert x \vert)$ namísto $\vert y \vert \le g(\vert x \vert)$). Proè je taková
+úprava BÚNO? Jistì si dovedete pøedstavit,
+¾e pùvodní nápovìdu doplníme na po¾adovanou délku nìjakými \uv{mezerami}, které
+program ignoruje. (Tedy upravíme program tak, aby mu nevadilo, ¾e dostane na
+konci nápovìdy nìjak kódované mezery.)
+
+\s{Vìta:} Obvodový SAT je NP-úplný.
+
+\proof
+Máme tedy nìjaký problém $L$ z~NP a~chceme dokázat, ¾e $L$ se dá pøevést
+na~obvodový SAT (tj. NP-tì¾kost). Kdy¾ nám nìkdo pøedlo¾í nìjaký vstup $x$
+(chápeme jako posloupnost $x_1, x_2, \ldots, x_n$),
+spoèítáme velikost nápovìdy $g(n)$. Víme, ¾e kontrolní
+algoritmus~$K$ (který kontroluje, zda nápovìda je správnì) je v~P. Vyu¾ijeme
+pøedchozí lemma, abychom získali obvod, který pro konkrétní velikost vstupu
+$x$ poèítá to, co kontrolní algoritmus $K$. Na vstupu tohoto obvodu bude $x$
+(vstup problému $L$) a~nápovìda~$y$. Na výstupu nám øekne, jestli je nápovìda
+správná. Velikost vstupu tohoto obvodu bude tedy $p(g(n))$, co¾ je také polynom.
+
+\fig{kobvod.eps}{2.3cm}
+
+\>V tomto obvodu zafixujeme vstup $x$ (na místa vstupu dosadíme konkrétní hodnoty z $x$). Tím získáme obvod, jeho¾ vstup je jen $y$ a~ptáme se, zda za $y$ mù¾eme dosadit nìjaké hodnoty tak, aby na výstupu bylo \<true>. Jinými slovy, ptáme se, zda je tento obvod splnitelný.
+
+\>Pro libovolný problém z~NP tak doká¾eme sestrojit funkci, která pro ka¾dý vstup~$x$ v~polynomiálním èase vytvoøí obvod, který je splnitelný pravì tehdy, kdy¾ odpovìï tohoto problému na vstup $x$ má být \<true>. Tedy libovolný problém z~NP se dá
+v~polynomiálním èase pøevést na obvodový SAT.
+
+\>Obvodový SAT je v NP triviálnì -- staèí si nechat poradit vstup, sí»
+topologicky setøídit a v~tomto poøadí poèítat hodnoty hradel.
+\qed
+
+\s{Lemma:} Obvodový SAT se dá pøevést na 3-SAT.
+
+\proof
+Budeme postupnì budovat formuli v~konjunktivní normální formì. Ka¾dý booleovský obvod se dá pøevést v~polynomiálním èase na~ekvivalentní obvod, ve~kterém se vyskytují jen hradla {\sc and} a {\sc not}, tak¾e staèí najít klauzule odpovídající tìmto hradlùm. Pro ka¾dé hradlo v~obvodu zavedeme novou promìnnou popisující jeho výstup. Pøidáme klauzule, které nám kontrolují, ¾e toto hradlo máme ohodnocené konzistentnì.
+
+\>{\I Pøevod hradla \sc not}: na vstupu hradla budeme mít nìjakou promìnnou $x$ (která pøi¹la buïto pøímo ze~vstupu toho celého obvodu nebo je to promìnná, která vznikla na výstupu nìjakého hradla) a na výstupu promìnnou $y$. Pøidáme klauzule, které nám zaruèí, ¾e jedna promìnná bude negací té druhé:
+$$\matrix{ (x \lor y), \cr
+  (\neg{x} \lor \neg{y}). \cr }
+  \hskip 0.2\hsize
+\vcenter{\hbox{\epsfxsize=0.7cm\epsfbox{not.eps}}}
+$$
+
+\>{\I Pøevod hradla \sc and}: Hradlo má vstupy $x, y$ a~výstup $z$. Potøebujeme pøidat klauzule, které nám popisují, jak se má hradlo {\sc and} chovat. Tyto vztahy pøepí¹eme do~konjunktivní normální formy:
+$$
+\left. \matrix{
+  x\ \&\ y \Rightarrow z \cr
+  \neg{x} \Rightarrow \neg{z} \cr
+  \neg{y} \Rightarrow \neg{z} \cr
+}
+\ \quad
+ \right\}
+\quad
+\matrix{
+ (z \lor \neg{x} \lor \neg{y}) \cr
+ (\neg{z} \lor x)              \cr
+ (\neg{z} \lor y)              \cr
+ }
+ \hskip 0.1\hsize
+\vcenter{\hbox{\epsfxsize=0.7cm\epsfbox{and.eps}}}
+$$
+
+\>Kdy¾ chceme pøevádìt obvodový SAT na 3-SAT, obvod nejdøíve pøelo¾íme na takový, ve~kterém jsou jen hradla {\sc and} a~{\sc not}, a~pak hradla tohoto obvodu pøelo¾íme na klauzule. Formule vzniklá z~takovýchto klauzulí je splnitelná pravì tehdy, kdy¾ je splnitelný daný obvod. Pøevod pracuje v polynomiálním èase.
+\qed
+
+\s{Poznámka:}
+Kdy¾ jsme zavádìli SAT, omezili jsme se jen na formule, které jsou
+v~konjunktivní normální formì (CNF). Teï u¾ víme, ¾e splnitelnost obecné
+booleovské formule doká¾eme pøevést na obvodovou splnitelnost a tu pak
+pøevést na 3-SAT. Opaèný pøevod je samozøejmì triviální, tak¾e obecný SAT
+je ve~skuteènosti ekvivalentní s~na¹ím \uv{standardním} SATem pro CNF.
+
+\bye
+
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diff --git a/old/12-apx/12-apx.tex b/old/12-apx/12-apx.tex
new file mode 100644
index 0000000..d2583f3
--- /dev/null
+++ b/old/12-apx/12-apx.tex
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+\input lecnotes.tex
+\prednaska{12}{Aproximaèní algoritmy}{\vbox{\hbox{(F. Ha¹ko, J. Menda, M. Mare¹,}
+	\hbox{ Michal Kozák, Vojta Tùma)}}}
+
+\>Na~minulých pøedná¹kách jsme se zabývali rùznými tì¾kými rozhodovacími
+problémy. Tato se zabývá postupy, jak se v~praxi vypoøádat s~øe¹ením tìchto
+problémù.
+
+\h{Co dìlat, kdy¾ potkáme NP-úplný problém}
+\algo
+\:Nepanikaøit.
+\:Spokojit se s~málem.
+\:Rozmyslet, jestli opravdu potøebujeme obecný algoritmus. Mnohdy potøebujeme pouze
+speciálnìj¹í pøípady, které mohou být øe¹itelné v~polynomiálním èase.
+\:Spokojit se s~pøibli¾ným øe¹ením, (pou¾ít aproximaèní algoritmus).
+\:Pou¾ít heuristiku -- napøíklad genetické algoritmy nebo randomizované algoritmy.
+Velmi pomoci mù¾e i jen výhodnìj¹í poøadí pøi~prohledávání èi oøezávání nìkterých
+napohled nesmyslných vìtví výpoètu.
+\endalgo
+
+\h{První zpùsob: Speciální pøípad}
+
+\>Èasto si vystaèíme s~vyøe¹ením speciálního pøípadu NP-úplného problému, který
+le¾í v~$P$. Napøíklad pøi øe¹ení grafové úlohy nám mù¾e staèit øe¹ení
+pro~speciální druh grafù (stromy, bipartitní grafy, \dots). Barvení grafu je lehké
+napø. pro~dvì barvy èi pro intervalové grafy. 2-SAT, jako speciální pøípad SATu,
+se dá øe¹it v~lineárním èase.
+
+\>Uká¾eme si dva takové pøípady (budeme øe¹ení hledat, nejen rozhodovat, zda existuje)
+
+\s{Problém: Maximální nezávislá mno¾ina ve~stromì (ne rozhodovací)}
+
+\>{\I Vstup:} Zakoøenìný strom~$T$.
+
+\>{\I Výstup:} Maximální (co do~poètu vrcholù) nezávislá mno¾ina vrcholù~$M$~v~$T$.
+
+\>BÚNO mù¾eme pøedpokládat, ¾e v~$M$ jsou v¹echny listy $T$. Pokud by nìkterý
+list $l$ v~$M$ nebyl, tak se podíváme na~jeho otce:
+\itemize\ibull
+\:Pokud otec není v~$M$, tak list $l$ pøidáme do~$M$, èím¾ se nezávislost
+mno¾iny zachovala a velikost stoupla o~1.
+\:Pokud tam otec je, tak ho z~$M$ vyjmeme a na~místo nìho vlo¾íme $l$.
+Nezávislost ani velikost $M$ se nezmìnily.
+\endlist
+\>Nyní listy spolu s~jejich otci z~$T$ odebereme a postup opakujeme. $T$ se
+mù¾e rozpadnout na~les, ale to nevadí $\to$ tentý¾ postup aplikujeme na~v¹echny stromy v~lese.
+
+\s{Algoritmus:}
+\>MaxNz$(T)$
+\algo
+\:Polo¾íme $L$:=$\{$listy stromu $T\}$.
+\:Polo¾íme $O$:=$\{$otcové vrcholù z~$L\}$.
+\:Vrátíme $L \cup$ MaxNz$(T\setminus(O \cup L))$.
+\endalgo
+\>{\I Poznámka:} Toto doká¾eme naprogramovat v~$\O(n)$ (udr¾ujeme si frontu listù).
+
+\s{Problém: Batoh}
+
+\>Je daná mno¾ina $n$~pøedmìtù s~hmotnostmi $h_1,\ldots,h_n$
+a cenami $c_1,\ldots,c_n$ a~batoh, který unese hmostnost~$H$. Najdìte takovou
+podmno¾inu pøedmìtù, jejich¾ celková hmotnost je maximálnì $H$ a celková cena
+je maximální mo¾ná.
+
+\>Tento problém je zobecnìním problému batohu z~minulé pøedná¹ky dvìma smìry:
+Jednak místo rozhodovacího problému øe¹íme optimalizaèní, jednak pøedmìty
+mají ceny (pøedchozí verze odpovídala tomu, ¾e ceny jsou rovny hmotnostem).
+Uká¾eme si algoritmus pro øe¹ení tohoto obecného problému, jeho¾ èasová
+slo¾itost bude polynomiální v~poètu pøedmìtù~$n$ a souètu v¹ech cen~$C=\sum_i
+c_i$.
+
+\>Pou¾ijeme dynamické programování. Pøedstavme si problém omezený na~prvních~$k$
+pøedmìtù. Oznaème si $A_k(c)$ (kde $0\le c\le C$) minimální hmotnost
+podmno¾iny, její¾ cena je právì~$c$. Tato $A_k$ spoèteme indukcí podle~$k$:
+Pro $k=0$ je urèitì $A_0(0)=0$, $A_0(c)=infty$ pro $c>0$. Pokud ji¾ známe
+$A_{k-1}$, spoèítáme $A_k$ následovnì: $A_k(c)$ odpovídá nìjaké podmno¾inì
+pøedmìtù z~$1,\ldots,k$. V~této podmno¾inì jsme buïto $k$-tý pøedmìt nepou¾ili
+(a pak je $A_k(c)=A_{k-1}(c)$), nebo pou¾ili a tehdy bude $A_k(c) =
+A_{k-1}(c-c_k) + h_k$ (to samozøejmì jen pokud $c\ge c_k$). Z~tìchto dvou
+mo¾ností si vybereme tu, která dává mno¾inu s~men¹í hmotností. Tedy:
+$$
+A_k(c) = \min (A_{k-1}(c), A_{k-1}(c-c_k) + h_k).
+$$
+Tímto zpùsobem v~èase $\O(C)$ spoèteme $A_k(c)$ pro fixní $k$ a v¹echna $c$,
+v~èase $\O(nC)$ pak v¹echny $A_k(c)$.
+
+\>Podle $A_n$ snadno nalezneme maximální cenu mno¾iny, která se vejde do~batohu.
+To bude nejvìt¹í~$c^*$, pro nì¾ je $A_n(c^*) \le H$. Jeho nalezení nás stojí
+èas $\O(C)$.
+
+\>A~jak zjistit, které pøedmìty do~nalezené mno¾iny patøí? Upravíme algoritmus,
+aby si pro ka¾dé $A_k(c)$ pamatoval $B_k(c)$, co¾ bude index posledního pøedmìtu,
+který jsme do~pøíslu¹né mno¾iny pøidali. Pro nalezené $c^*$ tedy bude $i=B_n(c^*)$
+poslední pøedmìt v~nalezené mno¾inì, $i'=B_{i-1}(c^*-c_i)$ ten pøedposlední
+a tak dále. Takto v~èase $\O(n)$ rekonstruujeme celou mno¾inu od~posledního
+prvku k~prvnímu.
+
+\>Ukázali jsme tedy algoritmus s~èasovou slo¾itostí $\O(nC)$, který vyøe¹í
+problém batohu. Jeho slo¾itost není polynomem ve~velikosti vstupu ($C$~mù¾e
+být a¾ exponenciálnì velké vzhledem k~velikosti vstupu), ale pouze ve~velikosti
+èísel na~vstupu. Takovým algoritmùm se øíká {\I pseudopolynomiální.} Ani takové
+algoritmy ale nejsou k dispozici pro v¹echny problémy (napø. u problému obchodního
+cestujícího nám vùbec nepomù¾e, ¾e váhy hran budou malá èísla).
+
+\s{Verze bez cen:} Na verzi s~cenami rovnými hmotnostem se dá pou¾ít
+i jiný algoritmus zalo¾ený na~dynamickém programování: poèítáme mno¾iny
+$Z_k$ obsahující v¹echny hmotnosti men¹í ne¾~$H$, kterých nabývá
+nìjaká podmno¾ina prvních~$k$ prvkù. Pøitom $Z_0=\{0\}$, $Z_k$
+spoèteme ze~$Z_{k-1}$ --- udr¾ujme si $Z_{k-1}$ jako setøídìný spojový seznam,
+výpoèet dal¹ího seznamu udìláme slitím dvou seznamù $Z_{k-1}$ a $Z_{k-1}$ se
+v¹emi prvky zvý¹enými o hmotnost $k$ zahazujíce duplicitní a pøíli¹ velké hodnoty ---
+a ze~$Z_n$ vyèteme výsledek. V¹echny tyto mno¾iny
+mají nejvý¹e $H$ prvkù, tak¾e celková èasová slo¾itost algoritmu je~$\O(nH)$.
+
+\h{Druhý zpùsob: Aproximace}
+
+\>V pøedcházejících problémech jsme se zamìøili na~speciální pøípady. Obèas v¹ak
+takové ¹tìstí nemáme a musíme vyøe¹it celý NP-úplný problém. Mù¾eme si v¹ak
+pomoct tím, ¾e se ho nebudeme sna¾it vyøe¹it optimálnì -- namísto optimálního
+øe¹ení najdeme nìjaké, které je nejvý¹e $c$-krát hor¹í pro nìjakou konstantu $c$.
+
+\s{Problém: Obchodní cestující}
+
+\>{\I Vstup:} Neorientovaný graf~$G$, ka¾dá hrana
+je ohodnocená funkcí $w: E(G)\rightarrow {\bb R }^+_0$.
+
+\>{\I Výstup:} Hamiltonovská kru¾nice (obsahující v¹echny vrcholy grafu), a~to ta nejkrat¹í
+(podle ohodnocení).
+
+\>Tento problém je hned na~první pohled nároèný -- u¾ sama existence
+hamiltonovské kru¾nice je NP-úplná. Najdeme aproximaèní algoritmus nejprve za pøedpokladu,
+¾e vrcholy splòují trojúhelníkovou nerovnost (tj. $\forall x,y,z \in V: w(xz)\le
+w(xy)+w(yz)$), potom uká¾eme, ¾e v úplnì obecném pøípadé by samotná existence
+aproximaèního algoritmu implikovala ${\rm P=NP }$.
+
+\>{\I a) trojúhelníková nerovnost:}
+
+Existuje pìkný algoritmus, který najde hamiltonovskou kru¾nici o délce $\leq
+2\cdot opt$, kde $opt$ je délka nejkrat¹í hamiltonovské kru¾nice.
+Vedle pøedpokladu trojúhelníkové
+nerovnosti budeme potøebovat, aby ná¹ graf byl úplný. Souhrnnì mù¾eme
+pøedpokládat, ¾e úlohu øe¹íme v nìjakém metrickém protoru, ve kterém jsou obì
+podmínky podle definice splnìny.
+
+Najdeme nejmen¹í kostru grafu a obchodnímu cestujícímu poradíme, a» jde po~ní -- kostru
+zakoøeníme a projdeme jako strom do hloubky, pøièem¾ se zastavíme a¾ v koøeni po projití
+v¹ech vrcholù. Problém v¹ak je, ¾e prùchod po kostøe obsahuje
+nìkteré vrcholy i hrany vícekrát, a proto musíme nahradit nepovolené vracení se.
+Máme-li na nìjaký vrchol vstoupit podruhé, prostì ho ignorujeme a pøesuneme se
+rovnou na dal¹í nenav¹tívený -- dovolit si to mù¾eme, graf je úplný a obsahuje
+hrany mezi v¹emi dvojicemi vrcholù
+(jinak øeèeno, poøadí vrcholù kru¾nice bude preorder výpis prùchodem do hloubky).
+Pokud platí trojúhelníková nerovnost, tak si tìmito zkratkami neu¹kodíme.
+Nech» minimální kostra má váhu~$T$. Pokud bychom pro¹li celou kostru, bude mít
+sled váhu~$2T$ (ka¾dou hranou kostry jsme ¹li tam a zpátky), a pøeskakování
+vrcholù celkovou váhu nezvìt¹uje (pøi pøeskoku
+nahradíme cestu $xyz$ jedinou hranou $xz$, pøièem¾ z trojúhelníkové nerovnosti
+máme $xz \leq xy + xz$), tak¾e váha nalezené
+hamiltonovské kru¾nice bude také nanejvý¹ $2T$.
+
+Kdy¾ máme hamiltonovskou kru¾nici $C$ a z~ní vy¹krtneme hranu, dostaneme kostru
+grafu~$G$ s~váhou men¹í ne¾ $C$ -- ale ka¾dá kostra je alespoò tak tì¾ká
+jako minimální kostra $T$. Tedy optimální hamiltonovská kru¾nice je urèitì tì¾¹í
+ne¾ minimální kostra $T$. Kdy¾ tyto dvì nerovnosti slo¾íme
+dohromady, algoritmus nám vrátí hamiltonovskou kru¾nici $T'$ s~váhou nanejvý¹
+dvojnásobnou vzhledem k optimální hamiltonovské kru¾nici ($T' \leq 2T < 2C$). Takovéto
+algoritmy se nazývají {\I 2-aproximaèní}, kdy¾ øe¹ení je maximálnì dvojnásobné
+od~optimálního.\foot{Hezkým trikem se v obecných metrických prostorech umí
+$1{,}5$-aproximace. Ve~nìkterých metrických prostorech (tøeba v euklidovské
+rovinì) se aproximaèní pomìr dá dokonce srazit na
+libovolnì blízko k 1. Zaplatíme ale na èase -- èím pøesnìj¹í výsledek
+po algoritmu chceme, tím déle to bude trvat.}
+
+\>{\I b) bez~trojúhelníkové nerovnosti:}
+
+Zde se budeme naopak sna¾it ukázat, ¾e ¾ádný polynomiální aproximaèní
+algoritmus neexistuje.
+
+\s{Vìta:} Pokud pro~libovolné~$\varepsilon>0$ existuje polynomiální
+$(1+\varepsilon)$-aproximaèní algoritmus pro~problém obchodního cestujícího bez~trojúhelníkové nerovnosti, tak ${\rm P = NP }$.
+
+\proof Uká¾eme, ¾e v~takovém pøípadì doká¾eme v~polynomiálním èase zjistit,
+zda v grafu existuje hamiltonovská kru¾nice.
+
+\>Dostali jsme graf~$G$, ve~kterém hledáme hamiltonovskou kru¾nici. Doplníme
+$G$ na~úplný graf~$G'$ a~váhy hran~$G'$ nastavíme takto:
+\itemize\ibull
+\: $w(e) = 1$, kdy¾ $e \in E(G)$
+\: $w(e) = c \gg 1$, kdy¾ $e \not\in E(G)$
+\endlist
+\>Konstantu $c$ potøebujeme zvolit tak velkou, abychom jasnì poznali, jestli
+je ka¾dá hrana z nalezené hamiltonovské kru¾nice hranou grafu $G$ (pokud by
+nebyla, bude kru¾nice obsahovat aspoò jednu hranu s váhou $c$, která vy¾ene
+souèet poznatelnì vysoko). Pokud existuje hamiltonovská kru¾nice v~$G'$ slo¾ená jen
+z~hran, které byly
+pùvodnì v~$G$, pak optimální øe¹ení bude mít váhu~$n$, jinak bude urèitì
+minimálnì $n-1+c$. Kdy¾ máme aproximaèní algoritmus s~pomìrem~$1+\varepsilon$,
+musí tedy být
+$$
+\eqalign{
+(1+\varepsilon)\cdot n &< n-1+c \cr
+\varepsilon n+1 &< c
+}
+$$
+\>Kdyby takový algoritmus existoval, máme polynomiální algoritmus
+na~hamiltonovskou kru¾nici.
+\qed
+
+\s{Poznámka:} O existenci pseudopolynomiálního algoritmu
+platí analogická vìta, a doká¾e se analogicky -- existující hrany budou
+mít váhu 1, neexistující váhu 2.
+
+\h{Aproximaèní schéma pro problém batohu}
+
+Ji¾ víme, jak optimalizaèní verzi problému batohu vyøe¹it v~èase $\O(nC)$,
+pokud jsou hmotnosti i ceny na~vstupu pøirozená èísla a $C$ je souèet v¹ech cen.
+Jak si poradit, pokud je~$C$ obrovské? Kdybychom mìli ¹tìstí a v¹echny
+ceny byly dìlitelné nìjakým èíslem~$p$, mohli bychom je tímto èíslem
+vydìlit. Tím bychom dostali zadání s~men¹ími èísly, jeho¾ øe¹ením by byla
+stejná mno¾ina pøedmìtù jako u~zadání pùvodního.
+
+Kdy¾ nám ¹tìstí pøát nebude, mù¾eme pøesto zkusit ceny vydìlit a výsledky
+nìjak zaokrouhlit. Øe¹ení nové úlohy pak sice nebude pøesnì odpovídat optimálnímu
+øe¹ení té pùvodní, ale kdy¾ nastavíme parametry správnì, bude alespoò jeho dobrou aproximací.
+
+\s{Základní my¹lenka:}
+
+Oznaèíme si $c_{max}$ maximum z~cen~$c_i$. Zvolíme si nìjaké pøirozené èíslo~$M < c_{max}$
+a zobrazíme interval cen $[0, c_{max}]$ na $[0,M]$ (tedy ka¾dou cenu znásobíme
+$M/c_{max}$).
+Jak jsme tím zkreslili výsledek? V¹imnìme si, ¾e efekt je stejný, jako kdybychom jednotlivé
+ceny zaokrouhlili na~násobky èísla $c_{max}/M$ (prvky z intervalu
+$[i\cdot c_{max}/M,(i+1)\cdot c_{max}/M)$ se zobrazí na stejný prvek). Ka¾dé $c_i$ jsme tím
+tedy zmìnili o~nejvý¹e $c_{max}/M$, celkovou cenu libovolné podmno¾iny pøedmìtù pak
+nejvý¹e o~$n\cdot c_{max}/M$. Teï si je¹tì v¹imnìme, ¾e pokud ze~zadání odstraníme
+pøedmìty, které se samy nevejdou do~batohu, má optimální øe¹ení pùvodní úlohy cenu $OPT\ge c_{max}$,
+tak¾e chyba v~souètu je nejvý¹e $n\cdot OPT/M$. Má-li tato chyba být shora omezena
+$\varepsilon\cdot OPT$, musíme zvolit $M\ge n/\varepsilon$.\foot{Pøipomìòme, ¾e toto je¹tì není dùkaz, nebo» velkoryse pøehlí¾íme chyby dané zaokrouhlováním. Dùkaz provedeme ní¾e.}
+
+\s{Algoritmus:}
+\algo
+\:Odstraníme ze~vstupu v¹echny pøedmìty tì¾¹í ne¾~$H$.
+\:Spoèítáme $c_{max}=\max_i c_i$ a zvolíme $M=\lceil n/\varepsilon\rceil$.
+\:Kvantujeme ceny: $\forall i: \hat{c}_i \leftarrow \lfloor c_i \cdot M/c_{max} \rfloor$.
+\:Vyøe¹íme dynamickým programováním problém batohu pro upravené ceny $\hat{c}_1, \ldots, \hat{c}_n$
+a pùvodní hmotnosti i kapacitu batohu.
+\:Vybereme stejné pøedmìty, jaké pou¾ilo optimální øe¹ení kvantovaného zadání.
+\endalgo
+
+\>Kroky 1--3 a 5 jistì zvládneme v~èase $\O(n)$. Krok~4 øe¹í problém batohu
+se souètem cen $\hat{C}\le nM = \O(n^2/\varepsilon)$, co¾ stihne v~èase $\O(n\hat{C})=\O(n^3/\varepsilon)$.
+Zbývá dokázat, ¾e výsledek na¹eho algoritmu má opravdu relativní chybu nejvý¹e~$\varepsilon$.
+
+Nejprve si rozmyslíme, jakou cenu budou mít pøedmìty které daly optimální øe¹ení
+v pùvodním zadání (tedy mají v pùvodním zadání dohromady cenu $OPT$),
+kdy¾ jejich ceny nakvantujeme (mno¾inu indexù tìchto pøedmìtù si oznaèíme~$Y$):
+$$
+\eqalign{
+\widehat{OPT} &= \sum_{i\in Y} \hat{c}_i =
+\sum_i \left\lfloor c_i\cdot {M\over c_{max}} \right\rfloor \ge
+\sum_i \left( c_i\cdot {M\over c_{max}} - 1 \right) \ge \cr
+&\ge
+\biggl(\sum_i c_i \cdot {M\over c_{max}}\biggr) - n =
+OPT \cdot {M\over c_{max}} - n.
+}
+$$
+Nyní spoèítejme, jak dopadne optimální øe¹ení~$Q$ nakvantovaného problému pøi pøepoètu
+na~pùvodní ceny (to je výsledek na¹eho algoritmu):
+$$
+\eqalign{
+ALG &= \sum_{i\in Q} c_i \ge
+\sum_i \hat{c}_i \cdot {c_{max}\over M} =
+\biggl(\sum_i \hat{c}_i\biggr) \cdot {c_{max}\over M} \ge^*
+\widehat{OPT} \cdot {c_{max}\over M}.
+}
+$$
+Nerovnost $\ge^*$ platí proto, ¾e $\sum_{i\in Q} \hat{c}_i$ je optimální øe¹ení
+kvantované úlohy, zatímco $\sum_{i\in Y} \hat{c}_i$ je nìjaké dal¹í øe¹ení té¾e úlohy,
+které nemù¾e být lep¹í. Teï u¾ staèí slo¾it obì nerovnosti a dosadit za~$M$:
+$$
+\eqalign{
+ALG &\ge \biggl( { OPT \cdot M\over c_{max}} - n\biggr) \cdot {c_{max}\over M} \ge
+OPT - {n\cdot c_{max}\over n / \varepsilon} \ge OPT - \varepsilon c_{max} \ge \cr
+&\ge OPT - \varepsilon OPT = (1-\varepsilon)\cdot OPT.
+}
+$$
+Algoritmus tedy v¾dy vydá øe¹ení, které je nejvý¹e $(1-\varepsilon)$-krát hor¹í ne¾ optimum,
+a~doká¾e to pro libovolné~$\varepsilon$ v~èase polynomiálním v~$n$. Takovému algoritmu øíkáme
+{\I polynomiální aproximaèní schéma} (jinak té¾ PTAS\foot{Polynomial-Time Approximation Scheme}).
+V~na¹em pøípadì je dokonce slo¾itost polynomiální i v~závislosti na~$1/\varepsilon$, tak¾e
+schéma je {\I plnì polynomiální} (øeèené té¾ FPTAS\foot{Fully Polynomial-Time Approximation
+Scheme}). U nìkterých problémù se stává, ¾e aproximaèní schéma závisí na
+$1/\varepsilon$ exponenciálnì, co¾ tak pøíjemné není. Shròme, co jsme zjistili, do následující vìty:
+
+\s{Vìta:}
+Existuje algoritmus, který pro ka¾dé $\varepsilon > 0$ nalezne
+{\I $(1 - \varepsilon)$-aproximaci} problému batohu s $n$ pøedmìty v èase
+$\O(n^3/\varepsilon)$.
+
+\bye
diff --git a/old/12-apx/Makefile b/old/12-apx/Makefile
new file mode 100644
index 0000000..11f9d14
--- /dev/null
+++ b/old/12-apx/Makefile
@@ -0,0 +1,3 @@
+P=12-apx
+
+include ../Makerules
diff --git a/old/2-dinic/2-dinic.tex b/old/2-dinic/2-dinic.tex
new file mode 100644
index 0000000..244427c
--- /dev/null
+++ b/old/2-dinic/2-dinic.tex
@@ -0,0 +1,223 @@
+\input lecnotes.tex
+
+\prednaska{2}{Dinicùv algoritmus}{(zapsala Markéta Popelová)}
+
+
+Na~minulé pøedná¹ce jsme si~ukázali Fordùv-Fulkersonùv algoritmus. Tento algoritmus hledal maximální tok tak, ¾e zaèal s~tokem nulovým a~postupnì ho zvìt¹oval. Pro~ka¾dé zvìt¹ení potøeboval v~síti najít cestu, na~které mají v¹echny hrany kladnou rezervu (po takovéto cestì mù¾eme poslat více, ne¾ po~ní aktuálnì teèe). Ukázali jsme, ¾e pokud takováto cesta existuje, jde tok vylep¹it (zvìt¹it). Zároveò pokud tok jde vylep¹it, pak takováto cesta existuje. Dokázali jsme, ¾e pro~racionální kapacity je algoritmus koneèný a~najde maximální tok.
+
+Fordùv-Fulkersonùv algoritmus má ov¹em znaèné nevýhody. Funguje pouze pro~racionální kapacity a~je pomìrnì pomalý. Nyní si~uká¾eme jiný algoritmus, který nevylep¹uje tok pomocí cest, ale pomocí tokù\dots Budeme k~tomu potøebovat sí» rezerv.
+
+\s{Definice:} {\I Sí» rezerv} k~toku~$f$ v~síti $S=(V,E,z,s,c)$ je sí» $R=(S,f)=(V,E,z,s,r)$, kde~$r(e)$ je rezerva hrany~$e$ v~toku~$f$.
+
+\s{Konvence:} Pro~hranu~$e$ znaèí~$\overleftarrow{e}$ hranu opaènou. Napø. pokud $e=uv$, tak $\overleftarrow{e}=vu$.
+
+Je dùle¾ité si~uvìdomit, ¾e sí» rezerv je závislá jak na~pùvodní síti~$S$, tak na~nìjakém toku~$f$ v~síti~$S$. Sí» rezerv~$R$ se~pak od sítì~$S$ li¹í pouze kapacitami -- sí»~$R$ má jako kapacitu hrany rezervu hrany v pùvodní síti. Pro~pøipomenutí: rezervu hrany~$e$ v~síti $S=(V,E,z,s,c)$ s~tokem~$f$ jsme si~definovali jako $r(e)=c(e) - f(e) + f(\overleftarrow{e})$.
+
+Ne¾ si~uká¾eme samotný algoritmus, doká¾eme si~následující lemma.
+
+\s{Lemma:} Pro~ka¾dý tok~$f$ v~síti~$S$ a~pro~ka¾dý tok~$g$ v~síti $R=(S,f)$ lze v~èase $\O(m)$ nalézt tok~$f'$ v~síti~$S$ takový, ¾e $\vert f' \vert = \vert f \vert + \vert g \vert$.
+
+\proof
+
+Dùkaz rozdìlíme do~tøí krokù. V~prvním kroku si~uká¾eme, jak budeme tok~$f'$ v~síti~$S$ konstruovat. V~druhém kroku doká¾eme, ¾e takto zkonstruované~$f'$ je opravdu tok. A~nakonec uká¾eme, ¾e splòuje po¾adovanou vlastnost, tedy ¾e jeho velikost je souèet velikostí tokù~$f$ a~$g$.
+
+\>{\it 1. konstrukce~$f'$}
+
+\noindent
+Pro ka¾dou dvojici hran~$e, \overleftarrow{e}$ urèíme~$f'(e)$ a~$f'(\overleftarrow{e})$ následovnì:
+
+\itemize\ibull
+\:Pokud~$g(e) = g(\overleftarrow{e}) = 0$, pak nastavme:
+	\itemize\ibull
+	\:$f'(e) := f(e)$,
+	\:$f'(\overleftarrow{e}) := f(\overleftarrow{e})$.
+	\endlist
+	
+\:Pokud~$g(e) > 0$ a~$g(\overleftarrow{e}) = 0$, pak polo¾me:
+	\itemize\ibull
+	\:$\varepsilon := \min (g(e), f(\overleftarrow{e}))$,
+	\:$f'(e) := f(e) + g(e) - \varepsilon$,
+	\:$f'(\overleftarrow{e}) := f(\overleftarrow{e}) - \varepsilon$.
+	\endlist
+
+\:Pøípad~$g(e) = 0$ a~$g(\overleftarrow{e}) > 0$ vyøe¹íme obdobnì.
+
+\:Pokud~$g(e) > 0$ a~$g(\overleftarrow{e}) > 0$, pak odeèteme od toku~$g$ cirkulaci po cyklu tvoøeném hranami~$e$ a $\overleftarrow{e}$:
+	\itemize\ibull
+	\:$\delta := \min (g(e),g(\overleftarrow{e}))$,
+	\:$g'(e) := g(e) - \delta$,
+	\:$g'(\overleftarrow{e}) := g(\overleftarrow{e}) - \delta$. 	
+	\endlist
+
+	Tok $g'$ nyní spadá pod nìkterý z~pøedchozích pøípadù, které u¾ umíme vyøe¹it.
+	
+\endlist
+
+\>{\it 2. $f'$ je tok}
+
+\numlist{\ndotted}
+
+\:Nejdøíve ovìøme první podmínku: $\forall e \in E: 0 \leq f(e) \leq c(e)$. Vezmìme libovolnou hranu~$e \in E$. Podle toho, co teèe po~hranách~$e$ a~$\overleftarrow{e}$ v~toku~$g$, jsme rozdìlili konstrukci toku na~tøi pøípady:
+
+	\numlist{\ndotted}	
+	
+	\:Pokud po~hranách~$e$ a~$\overleftarrow{e}$ netekl ¾ádný tok~$g$, pak jsme nastavili $f'(e) := f(e)$ a~$f'(\overleftarrow{e}) := f(\overleftarrow{e})$. Tedy pokud~$f$ dodr¾oval kapacity, tak pro~$f'$ musí platit to samé.
+
+
+	\:Pokud po~hranì~$e$ tekl tok~$g$ nenulový a~po opaèné nulový, tak jsme zvolili: $f'(e) := f(e) + g(e) - \varepsilon$. Víme, ¾e jsme si~$\varepsilon$ vybrali tak, ¾e $\varepsilon \leq g(e)$. Proto $f'(e) \geq 0$.
+
+	Teï ovìøme, ¾e $f'(e) \leq c(e)$. V~pøípadì, ¾e $\varepsilon = g(e)$, tak $f'(e) = f(e) \leq c(e)$. V~opaèném pøípadì platí, ¾e $\varepsilon = f(\overleftarrow{e})$. Pak ov¹em
+	$$f'(e) = f(e) + g(e) - f(\overleftarrow{e}) \leq $$
+	$$\leq f(e) + \left[ c(e) - f(e) + f(\overleftarrow{e}) \right] - f(\overleftarrow{e}) = c(e).$$
+	Vyu¾ili jsme, ¾e~$g$ je tok v~síti rezerv, tedy $g(e) \leq  c(e) - f(e) + f(\overleftarrow{e})$.
+
+	Pro tok $f'(\overleftarrow{e})$ platí, ¾e $\varepsilon \leq f(\overleftarrow{e})$. Proto $f'(\overleftarrow{e}) = f(\overleftarrow{e}) - \varepsilon \geq 0$. Zároveò $f'(\overleftarrow{e}) \leq f(\overleftarrow{e}) \leq c(\overleftarrow{e})$.
+
+	Tím jsme dokázali, ¾e~$f'(e)$ i~$f'(\overleftarrow{e})$ dodr¾ují kapacity.
+
+	\:V posledním pøípadì tekl po~obou hranách kladný tok~$g$. Men¹í tok z~$g(e)$ a~$g(\overleftarrow{e})$ jsme vynulovali a~od vìt¹ího odeèetli ten men¹í. Tok~$g'(e)$ a~$g'(\overleftarrow{e})$ tedy zùstal korektní a~tok~$f'$ u¾ konstruujeme podle pøedchozího pøípadu.
+
+	\endlist
+
+\:Teï musíme je¹tì dokázat, ¾e nový tok neporu¹uje Kirchhoffovy zákony: $$\forall v~\in V \setminus \{z,s\}: f'^\Delta(v)=0.$$
+	% neboli $$\forall v~\in V \setminus \{z,s\}: \sum_{u: uv \in E}{f'(uv)}=\sum_{u: vu \in E}{f'(vu)}.$$
+
+	Vezmìme si~libovolnou hranu~$e = uv \in E$. Uvìdomme si, ¾e pøi~pøechodu z~$f(e)$ na~$f'(e)$ a~z~$f(\overleftarrow{e})$ na~$f'(\overleftarrow{e})$ bylo:
+	\itemize\idot
+	\:$f^\Delta(u)$ sní¾eno o~$g(e)$
+	\:$f^\Delta(v)$ zvý¹eno o~$g(e)$.
+	\endlist
+	Seèteme-li úpravy na v¹ech hranách, dostaneme: $$f'^\Delta(v) = f^\Delta(v) + \sum_{u:uv \in E} g(uv) - \sum_{u:vu \in E} g(vu) =$$ $$= f^\Delta(v) + g^+(v) - g^-(v) = f^\Delta(v) + g^\Delta(v).$$
+
+	Jeliko¾~$f$ byl tok, tak $f^\Delta(v) = 0$ a jeliko¾~$g$ byl tok, tak $g^\Delta(v) = 0$. Proto $f'^\Delta(v) = f^\Delta(v) + g^\Delta(v) = 0$.
+	
+\endlist
+
+Tím jsme dokázali, ¾e~$f'$ je tok v~síti~$S$.
+
+\>{\it 3. $\vert f' \vert = \vert f \vert + \vert g \vert$}
+
+Pou¾ijme vztah pro souèet pøebytkù z pøedchozího kroku:
+$$\vert f' \vert = f'^\Delta(s) = f^\Delta(s) + g^\Delta(s) = \vert f \vert + \vert g \vert.$$
+\qed
+
+
+Pro algoritmus budeme potøebovat vybírat kvalitní toky~$g$ v~síti rezerv. Pokud se~nám to bude daøit, bude se~tok~$f'$ rychle zvìt¹ovat, a¾ bychom mohli dojít k~maximálnímu toku. Nejlépe by se~nám hodily co nejvìt¹í toky v~síti rezerv. Kdybychom si~dali za cíl najít v¾dy maximální tok v~síti rezerv, výsledek by byl sice krásný (dostali bychom tak rovnou i~maximální tok v~pùvodní síti), ale problém hledání maximálního toku bychom pouze pøenesli na~jinou sí». Na¹e po¾adavky na~tento tok budou tedy takové, aby byl dostateènì velký, ale abychom bìhem jeho hledání nestrávili moc èasu. Podívejme se, jak se~s~tímto problémem vyrovná {\I Dinicùv algoritmus}. Nejdøíve si~ale zadefinujme nìkolik pojmù.
+
+\s{Definice:} Tok~$f$ je {\I blokující}, jestli¾e pro~ka¾dou orientovanou cestu~$P$ ze~$z$ do~$s$ existuje hrana~$e \in P$ taková, ¾e $f(e) = c(e)$.
+
+\s{Definice:} Sí» je {\I vrstevnatá (proèi¹tìná)}, kdy¾ v¹echny vrcholy a~hrany le¾í na~nejkrat¹ích cestách ze~$z$ do~$s$.
+
+Dinicùv algoritmus zaèíná s~nulovým tokem. Potom v¾dy podle toku~$f$ sestrojí sí» rezerv a~v~ní vyma¾e hrany s~nulovou rezervou. Pokud v~této promazané síti rezerv neexistuje cesta ze~zdroje do~stoku, tak skonèí a~prohlásí tok~$f$ za maximální. Jinak proèistí sí» rezerv tak, aby se~z ní stala vrstevnatá sí» (rozdìlí vrcholy do~vrstev podle vzdálenosti od zdroje a~odstraní pøebyteèné hrany). Ve~vrstevnaté síti najde blokující tok, pomocí nìho¾ zlep¹í tok~$f$. Pak opìt pokraèuje sestrojením sítì rezerv na~tomto vylep¹eném toku~$f$ atd.
+
+\figure{dinic-cistasit.eps}{Proèi¹tìná sí» rozdìlená do~vrstev}{0.4\hsize}
+
+\s{Algoritmus (Dinicùv)}
+
+\algo
+\:$f \leftarrow$ nulový tok.
+\:Sestrojíme sí» rezerv~$R$ a~sma¾eme $e: r(e) = 0$.
+\:$l \leftarrow$ délka nejkrat¹í cesty ze~$z$ do~$s$ v~$R$.
+\:Pokud $l = \infty$, zastavíme se~a vrátíme~$f$.
+\:Proèistíme sí» $R \rightarrow$ sí»~$C$.
+\:$g \leftarrow$ blokující tok v~$C$.
+\:Zlep¹íme tok~$f$ pomocí~$g$.
+\:GOTO 2.
+\endalgo
+
+\s{Pozorování:} Pokud se~algoritmus zastaví, vydá maximální tok.
+
+\proof
+Víme, ¾e~$f$ je stále korektní tok (jediné, jak ho mìníme je pøièítání toku~$g$, co¾ je, jak jsme si~dokázali, \uv{ne¹kodná operace}). Jakmile neexistuje cesta ze~$z$ do~$s$ v~$R$, tak je $f$ maximální tok, nebo» v~tuto dobu by se~zastavil (a vydal maximální tok) i~Fordùv-Fulkersonùv algoritmus, který je korektní.
+\qed
+
+A teï je¹tì musíme ujasnit, jak budeme èistit sí» rezerv a~vybírat blokující tok~$g$.
+
+\s{Algoritmus proèi¹tìní sítì rezerv}
+
+\algo
+\:Rozdìlíme vrcholy do~vrstev podle vzdálenosti od~$z$.
+\:Odstraníme vrstvy za~$s$ (tedy vrcholy, které jsou od~$z$ vzdálenìj¹í ne¾~$s$), hrany do~minulých vrstev a~hrany uvnitø vrstev.
+\:Odstraníme \uv{slepé ulièky}, tedy vrcholy s~$\deg^{out}(v) = 0$, a~to opakovanì pomocí fronty. (Nejdøíve zaøadíme do~fronty v¹echny vrcholy s~ $\deg^{out}(v) = 0$. Pak dokud není fronta prázdná, v¾dy vybereme vrchol~$v$ z~fronty, odstraníme~$v$ a~v¹echny hrany~$uv$. Pro~ka¾dý takový vrchol~$u$ zkontrolujeme, zda se~tím nesní¾il výstupní stupeò vrcholu~$u$ na~nulu ($\deg^{out}(u) = 0$). Pokud sní¾il, tak ho zaøadíme do~fronty.)
+\endalgo
+
+\figure{dinic-neprocistenasit.eps}{Neproèi¹tìná sí». Obsahuje zpìtné hrany, hrany uvnitø vrstvy a~slepé ulièky.}{0.45\hsize}
+
+Hledání blokujícího toku zaèneme s~tokem nulovým. Pak vezmeme v¾dy orientovanou cestu ze~zdroje do~stoku v~síti~$C$. V~této cestì najdeme hranu s~nejni¾¹í hodnotou výrazu $r(e) - g(e)$ (neboli $c(e) - f(e)$ v~pùvodní síti). Tuto hodnotu oznaèíme~$\varepsilon$. Pak ke~ v¹em hranám na~této cestì pøièteme~$\varepsilon$. Pokud tok~$g$ na~nìjaké hranì dosáhne kapacity hrany, co¾ je zde~$r(e)$, tak hranu vyma¾eme. Následnì sí» doèistíme, aby splòovala podmínky vrstevnaté sítì. A~pokud je¹tì existuje nìjaká orientovaná cesta ze~zdroje do~stoku, tak opìt pokraèujeme s~touto cestou.
+
+\s{Algoritmus hledání blokujícího toku}
+
+\algo
+\:$g \leftarrow$ nulový tok.
+\:Dokud existuje orientovaná cesta~$P$ ze~$z$ do~$s$ v~$C$, opakuj:
+\::$\varepsilon \leftarrow \min_{e \in P} (r(e) - g(e))$.
+\::Pro~$\forall e \in P: g(e) \leftarrow g(e) + \varepsilon$.
+\:::Pokud $g(e) = r(e)$, sma¾eme~$e$ z~$C$.
+\::Doèistíme sí» zase pomocí fronty.
+\endalgo
+
+\s{Èasová slo¾itost} Rozeberme si~jednotlivé kroky algoritmu.
+
+\numlist{\ndotted}
+\:Inicializace toku~$f$ \dots $\O(m)$.
+\:Sestrojení sítì rezerv a~smazání hran s~nulovou rezervou \dots $\O(m + n)$.
+\:Najití nejkrat¹í cesty (prohledáváním do~¹íøky) \dots $\O(m + n)$.
+\:Zkontrolování délky nejkrat¹í cesty \dots $\O(1)$.
+\:Proèi¹tìní sítì \dots $\O(m + n)$.
+	\numlist{\ndotted}
+	\:Rozdìlení vrcholù do~vrstev -- provedlo ji¾ prohledávání do~¹íøky \dots $\O(1)$.
+	\:Odstranìní nìkterých hran \dots $\O(m + n)$.
+	\:Odstranìní \uv{slepých ulièek} pomocí fronty -- ka¾dou hranu odstraníme nejvý¹e jedenkrát, ka¾dý vrchol se~dostane do~fronty nejvý¹e jedenkrát \dots $\O(m + n)$.
+	\endlist
+\:Najití blokujícího toku~$g$ \dots $\O(m \cdot n)$.
+	\numlist{\ndotted}
+	\:Inicializace toku~$g$ \dots $\O(m)$.
+	\:Najití orientované cesty v~proèi¹tìné síti rezerv (staèí vzít libovolnou cestu ze~zdroje, nebo» ka¾dá z~nich v~této síti vede do~stoku) \dots $\O(n)$.
+	\:Výbìr minima z~výrazu $r(e) - g(e)$ pøes v¹echny hrany cesty -- ta mù¾e být dlouhá nejvý¹e~$n$ \dots $\O(n)$.
+	\:Pøepoèítání v¹ech hran cesty \dots $\O(n)$.
+	\:Smazání hran cesty, jejich¾ tok~$g(e)$ se~zvý¹il na~hodnotu~$r(e)$ \dots $\O(n)$.
+	\:Doèi¹»ování vyøe¹me zvlá¹».
+	\endlist
+	
+	Vnitøní cyklus (kroky 2 a¾ 6) provedeme nejvý¹e~$m$ krát, nebo» pøi~ka¾dém prùchodu vyma¾eme alespoò jednu hranu (tak jsme si~volili~$\varepsilon$).
+	
+	Èi¹tìní bìhem celého hledání blokujícího toku~$g$ v~proèi¹tìné síti rezerv trvá dohromady $\O(m + n)$, nebo» ka¾dou hranu a~vrchol sma¾eme nejvý¹e jedenkrát.
+	
+	Najití blokujícího toku bude tedy trvat $\O(m \cdot n + (m + n)) = \O(m \cdot n)$.
+	
+\:Zlep¹ení toku~$f$ pomocí toku~$g$ \dots $\O(m)$.
+\:Skok na~2. krok \dots $\O(1)$.
+\endlist
+
+Zbývá nám jen urèit, kolikrát projdeme vnìj¹ím cyklem (fází). Doká¾eme si~lemma, ¾e hodnota~$l$ vzroste mezi prùchody vnìj¹ím cyklem (fázemi) alespoò o~1. Z~toho plyne, ¾e vnìj¹ím cyklem mù¾eme projít nejvý¹e $n$-krát, nebo» cesta v síti na~$n$ vrcholech mù¾e být dlouhá nejvý¹e $n$.
+
+Uvìdomme si, ¾e uvnitø vnìj¹ího cyklu pøevládá èlen $\O(m \cdot n)$, tak¾e celková èasová slo¾itost bude $\O(n^2 \cdot m)$.
+
+\s{Lemma:} Hodnota~$l$ (délka nejkrat¹í cesty ze~$z$ do~$s$ v~proèi¹tìné síti) vzroste mezi fázemi alespoò o~1.
+
+\proof
+
+Podíváme se~na~prùbìh jednoho prùchodu vnìj¹ím cyklem. Délku aktuálnì nejkrat¹í cesty ze~zdroje do~stoku oznaème~$l$. V¹echny pùvodní cesty délky~$l$ se~bìhem prùchodu zaruèenì nasytí, proto¾e tok~$g$ je blokující. Musíme v¹ak dokázat, ¾e nemohou vzniknout ¾ádné nové cesty délky~$l$ nebo men¹í. V~síti rezerv toti¾ mohou hrany nejen
+ubývat, ale i~pøibývat: pokud po¹leme tok po~hranì, po~které je¹tì nic neteklo, tak v~protismìru z~dosud nulové rezervy vyrobíme nenulovou. Rozmysleme si~tedy, jaké hrany mohou pøibývat.
+
+Hrany mohou pøibývat jen tehdy, kdy¾ jsme po~opaèné hranì nìco poslali. Ale my nìco posíláme po~hranách pouze z~vrstvy do~té následující. Hrany tedy pøibývají do~minulé vrstvy.
+
+Vznikem nových hran by proto mohly vzniknout nové cesty ze~zdroje do~stoku, které pou¾ívají zpìtné hrany. Jen¾e cesta ze~zdroje do~stoku, která pou¾ije zpìtnou hranu, musí alespoò jednou skoèit o~vrstvu zpìt a~nikdy nemù¾e skoèit o~více ne¾ jednu vrstvu dopøedu, a~proto je její délka alespoò $l+2$. Pokud cesta novou zpìtnou hranu nepou¾ije, má buï délku~$> l$, co¾ je v~poøádku, nebo má délku~$= l$, pak je zablokovaná.
+
+Tím je lemma dokázáno.
+\qed
+
+\figure{dinic-cestashranouzpet.eps}{Cesta u¾ívající novou zpìtnou hranu}{0.4\hsize}
+
+V¹echna dokázaná tvrzení mù¾eme shrnout do~následující vìty:
+
+\s{Vìta:} Dinicùv algoritmus najde maximální tok v~èase $\O(n^2\cdot m)$.
+
+\s{Poznámka:} Algoritmus se~chová hezky na~sítích s~malými celoèíselnými kapacitami, ale kupodivu i~na~rùzných jiných sítích. Èasto se~pou¾ívá, nebo» se~chová efektivnì. A~je mnoho zpùsobù, jak ho je¹tì vylep¹ovat, èi odhadovat ni¾¹í slo¾itost na~speciálních sítích.
+
+\s{Poznámka:} Algoritmus nevy¾aduje racionální kapacity! Dal¹í z~dùvodù, proè maximální tok existuje i~v~síti s~iracionálními kapacitami.
+
+
+%k,s,v,na,do,ke,pro,pøi,a,u,i,po,
+
+\bye
diff --git a/old/2-dinic/Makefile b/old/2-dinic/Makefile
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+++ b/old/2-dinic/Makefile
@@ -0,0 +1,3 @@
+P=2-dinic
+
+include ../Makerules
diff --git a/old/2-dinic/dinic-cestashranouzpet.eps b/old/2-dinic/dinic-cestashranouzpet.eps
new file mode 100644
index 0000000..5633e61
--- /dev/null
+++ b/old/2-dinic/dinic-cestashranouzpet.eps
@@ -0,0 +1,472 @@
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diff --git a/old/3-goldberg/3-goldberg.tex b/old/3-goldberg/3-goldberg.tex
new file mode 100644
index 0000000..70544d4
--- /dev/null
+++ b/old/3-goldberg/3-goldberg.tex
@@ -0,0 +1,283 @@
+\input lecnotes.tex
+
+\prednaska{3}{Goldbergùv algoritmus}{(zapsala Markéta Popelová)}
+
+Pøedstavíme si~nový algoritmus pro~hledání maximálního toku v~síti, který se~uká¾e být stejnì dobrý jako {\I Dinicùv algoritmus} ($\O(MN^{2})$) a~po~nìkolika vylep¹eních bude i~lep¹í. Nejdøíve si~pøipomeòme definice, které budeme potøebovat:
+
+\s{Definice:} Mìjme sí» $S=(V,E,z,s,c)$, tok~$f$ a~libovolný vrchol~$v$. Pak $f^{\Delta}(v)$ nazýváme {\I pøebytek} ve~vrcholu~$v$ a~definujeme ho takto: $$f^{\Delta}(v):=\sum_{uv \in E}{f(uv)} - \sum_{vu \in E}{f(vu)}.$$ Pøebytek ve~vrcholu~$v$ je tedy souèet v¹eho, co do~vrcholu~$v$ pøiteèe, minus souèet v¹eho, co z~$v$ odteèe.
+
+\s{Definice:} Dále pro~libovolnou hranu~$uv \in E$ definujeme její {\I rezervu} následovnì:
+$$r(uv) = c(uv) - f(uv) + f(vu).$$ Rezerva hrany znaèí, co je¹tì je mo¾no po~této hranì poslat.
+
+\s{Poznámka:} Dále budeme oznaèovat písmenem~$N$ poèet vrcholù a~$M$ poèet hran, tedy~$N = \vert V \vert$ a~$M = \vert E \vert$.
+
+Goldbergùv algoritmus na~rozdíl od~Dinicova algoritmu zaèíná s~ohodnocením hran, které pravdìpodobnì není tokem (budeme ho nazývat {\I vlna}), a~postupnì ho zmen¹uje a¾ na~korektní tok.
+
+\s{Definice:} Funkce $f:E \rightarrow {\bb R}_{0}^{+}$ je {\I vlna} v~síti~$(V, E, z, s, c)$ tehdy, kdy¾ jsou splnìny následující dvì podmínky:
+	\numlist\ndotted
+	\:$\forall e \in E : f(e) \leq c(e)$ (vlna na hranì nepøekroèí kapacitu hrany)
+	\:$ \forall v \in V \setminus \{z, s\} : f^{\Delta}(v) \geq 0$ (pøebytek ve vrcholu je nezáporný).
+	\endlist
+
+\s{Pozorování:} Ka¾dý tok~$f$ je také vlna, ale opaènì to obvykle platit nemusí.
+
+\s{Operace:} {\I Pøevedení pøebytku}
+
+Algoritmus bude potøebovat pøevádìt pøebytky z~vrcholu~$u$ na~sousední vrchol~$v$. Mìjme hranu~$uv$ s~kladnou rezervou $r(uv) > 0$ a~kladným pøebytkem ve~vrcholu~$u$: $f^\Delta(u) > 0$. Èást pøebytku budeme chtít poslat z~vrcholu~$u$ do~vrcholu~$v$. Vezmeme $\delta := \min (f^\Delta(u), r(uv))$ a~po~hranì~$uv$ po¹leme tok o velikosti~$\delta$. Výsledná situace bude vypadat následovnì:
+	\itemize\ibull
+	\:$f'^\Delta(u) = f^\Delta(u) - \delta$.
+	\:$f'^\Delta(v) = f^\Delta(v) + \delta$.
+	\:$r'(uv) = r(uv) - \delta$.
+	\:$r'(vu) = r(vu) + \delta$.
+	\endlist
+	
+Kdybychom ov¹em nepøidali ¾ádnou jinou podmínku, ná¹ algoritmus by se~mohl krásnì zacyklit (napø. posílat pøebytek z~$u$ do~$v$ a~zase zpátky). Abychom se~tomu vyhnuli, zavedeme {\I vý¹ku vrcholu} $h: V \to {\bb N}$ a~dovolíme pøevádìt pøebytek pouze z~vy¹¹ího vrcholu~$u$ na~ni¾¹í $v$: $h(u) > h(v)$.
+
+\s{Shrnutí:} Podmínky pro~pøevedení pøebytku po~hranì $uv \in E$:
+	\numlist\ndotted
+	\:Ve~vrcholu~$u$ je nenulový pøebytek: $f^{\Delta}(u) > 0$.
+	\:Vrchol~$u$ je vý¹ ne¾ vrchol~$v$: $h(u) > h(v)$.
+	\:Hrana~$uv$ má nenulovou rezervu: $r(uv)>0$.
+	\endlist
+
+
+\s{Operace:} Pro~vrchol~$u \in V$ definujme {\I zvednutí vrcholu}:
+Pokud bìhem výpoètu narazíme ve~vrcholu~$u$ na~pøebytek, který nelze nikam pøevést, zvìt¹íme vý¹ku vrcholu~$u$ o~jednièku, tj. $h(u) \leftarrow h(u)+1$.
+
+
+\s{Algoritmus (Goldbergùv)}
+
+\algo
+\:$\forall v \in V: h(v)\leftarrow 0$ (v¹em vrcholùm nastavíme poèáteèní vý¹ku nula) a~$h(z)\leftarrow N$ (zdroj zvedneme do~vý¹ky~$N$).
+\:$\forall e \in E: f(e)\leftarrow 0$ (po~hranách nejdøíve nenecháme protékat nic) a~$\forall zu \in E : f(zu)\leftarrow c(zu)$ (ze~zdroje pustíme maximální mo¾nou vlnu).
+\:Dokud $\exists u \in V \setminus \{z,s\}: f^{\Delta}(u)>0$:
+\::Pokud $\exists v \in V: uv \in E,~r(uv)>0$ a~$h(u)>h(v)$, pak pøevedeme pøebytek po~hranì z~$u$ do~$v$.
+\::V~opaèném pøípadì zvedneme $u$:~$h(u) \leftarrow h(u) + 1$.
+\:Vrátíme tok~$f$ jako výsledek.
+\endalgo
+
+\noindent
+Nyní bude následovat nìkolik lemmat a~invariantù, jimi¾ doká¾eme správnost a~èasovou slo¾itost Goldbergova algoritmu.
+
+\s{Invariant A (základní):}
+	\numlist \ndotted
+	\:Funkce~$f$ je v~ka¾dém kroku algoritmu vlna.
+	\:$h(v)$ nikdy neklesá pro~¾ádné~$v$.
+	\:$h(z)=N$ a~$h(s)=0$ po~celou dobu bìhu algoritmu.
+	\endlist
+
+\proof Indukcí dle poètu prùchodù cyklem (3. -- 5. krok algoritmu).
+
+Na zaèátku je v¹e v~poøádku ($f$ je nulová funkce, pøebytky v¹ech vrcholù jsou nezáporné, tedy~$f$ je vlna, $h(z)=N$ a~$h(s)=0$). V~prùbìhu se~tyto hodnoty mìní pouze pøi:
+	\itemize\ibull
+	\:Pøevedení po~hranì~$uv$: Po hranì~$uv$ se~nepo¹le více ne¾ její rezerva. Pøebytek~$u$ se~sní¾í, ale nejménì na~nulu. Pøebytek~$v$ se~zvý¹í. Tedy~$f$ zùstává vlnou. Vý¹ky se~nemìní.
+	\:Zvednutí vrcholu~$u$: Mìní pouze vý¹ky -- a~to vrcholù rùzných od zdroje èi stoku -- a~pouze se zvy¹ují.
+	\qeditem
+	\endlist
+
+\s{Invariant S (o~Spádu):} Neexistuje hrana $uv \in E: r(uv)>0$ \& $h(u) > h(v)+1$ (s~kladnou rezervou a~spádem vìt¹ím ne¾ jedna).
+
+\proof Indukcí dle bìhu algoritmu.
+
+Na zaèátku mají v¹echny hrany ze~zdroje rezervu nulovou a~v¹echny ostatní vedou mezi vrcholy s~vý¹kou 0. V~prùbìhu by se~tento invariant mohl pokazit pouze dvìma zpùsoby:
+	\itemize\ibull
+	\:Zvednutím vrcholu~$u$, ze~kterého vede hrana~$uv$ s~kladnou rezervou a~spádem 1. Tento pøípad nemù¾e nastat, nebo» hranu zvedáme pouze tehdy, kdy¾ neexistuje vrchol~$v$ takový, ¾e hrana~$uv$ má kladnou rezervu a~spád alespoò 1. Takový vrchol v~na¹em pøípadì existuje, proto se~místo zvednutí vrcholu~$u$ po¹le pøebytek po~hranì~$uv$.
+	\:Zvìt¹ením rezervy hrany se~spádem vìt¹ím ne¾ 1. Toto také nemù¾e nastat, nebo» rezervu bychom mohli zvìt¹it jedinì tak, ¾e bychom poslali nìco v~protismìru -- a~to nesmíme, jeliko¾ bychom poslali pøebytek z~ni¾¹ího vrcholu na~vy¹¹í.
+	\qeditem
+	\endlist
+
+\s{Definice:} Cestu~$P$ nazveme {\I nenasycenou}, pokud v¹echny její hrany mají kladnou rezervu. Neboli $\forall e \in P: r(e) > 0$.
+
+\s{Lemma K (o~Korektnosti):} Kdy¾ se~algoritmus zastaví, je~$f$ maximální tok.
+
+\proof Dùkaz rozlo¾me do~dvou krokù. Nejdøíve uká¾eme, ¾e~$f$ je tok, a~pak jeho maximalitu.
+
+	\numlist\ndotted
+	\:Nech» se~algoritmus zastavil. Pak nemohl existovat ¾ádný vrchol~$v$ (kromì zdroje a~stoku) s~kladným pøebytkem. Tedy $\forall v \in V~\setminus \{z,s\}: f^\Delta(v) = 0$. (Víme ji¾, ¾e~$f$ je po~celou dobu vlna, tak¾e pøebytek nemù¾e být nikdy záporný.) V~tom pøípadì splòuje~$f$ podmínky toku.
+	\:Pro spor pøedpokládejme, ¾e tok~$f$ není maximální. Pak existuje nenasycená cesta ze~zdroje do~stoku. Vezmìme si~libovolnou takovou cestu. Zdroj je stále ve~vý¹ce~$N$ a~spotøebiè ve~vý¹ce 0 (viz invariant A). Tato cesta tedy pøekonává vý¹ku~$N$, ale mù¾e mít nejvý¹e~$N-1$ hran. Proto existuje alespoò jedna hrana se~spádem alespoò 2. Tato hrana tedy nemù¾e mít kladnou rezervu (viz invariant S). Tato cesta proto nemù¾e být zlep¹ující, co¾ je spor. Tím jsme dokázali, ¾e~$f$ je nutnì maximální tok.
+	\qeditem
+	\endlist
+	
+\s{Lemma C (Cesta):} Mìjme vrchol $v \in V$. Pokud $f^{\Delta}(v) > 0$, pak existuje nenasycená cesta z~vrcholu~$v$ do~zdroje.
+
+\proof
+Pro vrchol~$v \in V$ s $f^{\Delta}(v) > 0$ definujme mno¾inu $A := \{ u \in V : \exists$ nenasycená cesta z~$v$ do~$u \}$.
+
+Seètìme pøebytky ve~v¹ech vrcholech mno¾iny~$A$. Pøebytek ka¾dého vrcholu se~spoèítá jako souèet tokù do~nìj vstupujících minus souèet tokù z~nìj vystupujících. V¹echny hrany, jejich¾ oba vrcholy le¾í v~$A$, se~jednou pøiètou a~jednou odeètou. Proto nás budou zajímat pouze hrany mezi~$A$ a~$V \setminus A$.
+
+ $$\sum_{u \in A}f^{\Delta}(u) = \underbrace{ \sum_{ab \in E \cap ( (V \setminus A) \times A )} f(ab) }\limits_{=0} -  \underbrace{ \sum_{ab \in E \cap (  A \times (V \setminus A) )} f(ab) }\limits_{\geq 0}~\leq~0.$$
+
+Uka¾me si, proè je první svorka rovna nule. Mìjme vrcholy $a \in V \setminus A$ a~$b \in A$ takové, ¾e $ab\in E$. O~nich víme, ¾e $r(ba) = 0$ (jinak by~$a$ patøilo do~$A$) $\Rightarrow f(ba) = c(ba) \Rightarrow f(ab)=0$. Proto do~$A$ nic nepøitéká.
+
+\figure{Goldberg01.eps}{Obrázek k dùkazu lemmatu C}{0.2\hsize}
+
+Proè je druhá svorka nezáporná, je zøejmé, nebo» tok na~hranì je v¾dy nezáporný a~souèet nezáporných èísel je nezáporné èíslo.
+
+Proto $\sum_{u \in A}{f^\Delta(u) \le 0}$. Zároveò v¹ak v~$A$ je aspoò jeden vrchol s~kladným pøebytkem, toti¾~$v$, proto v~$A$ musí být také vrchol se~záporným pøebytkem -- a~jediný takový je zdroj. Tím je dokázáno, ¾e $z \in A$, tedy ¾e vede nenasycená cesta z~vrcholu~$v$ do~zdroje.
+\qed
+
+\s{Invariant V (Vý¹ka):} $\forall v \in V$ platí $h(v)\leq 2N$.
+
+\proof
+Kdyby existoval vrchol~$v$ s~vý¹kou $h(v) > 2N$, tak by musel být nìkdy zvednut z~vý¹ky~$2N$. Tehdy musel mít kladný pøebytek $f^\Delta(v)>0$ (jinak by nemohl být zvednut). Dle lemmatu C musela existovat nenasycená cesta z~$v$ do~zdroje. Tato cesta mìla spád alespoò~$N$, ale mohla mít nejvý¹e~$N-1$ hran (jinak by to nebyla cesta v~síti na~$N$ vrcholech). Tudí¾ musela na~této cestì existovat hrana se~spádem alespoò 2, co¾ je spor s~invariantem S (nebo» v¹echny hrany této cesty mají z~definice nenasycené cesty kladné rezervy).
+\qed
+
+\s{Lemma Z (poèet Zvednutí):} Poèet v¹ech zvednutí je maximálnì~$2N^{2}$.
+
+\proof
+Staèí si~uvìdomit, ¾e ka¾dý vrchol mù¾eme zvednout maximálnì~$2N$-krát a~vrcholù je~$N$.
+\qed
+
+Teï nám je¹tì zbývá urèit poèet provedených pøevedení. Bude se~nám hodit, kdy¾ pøevedení rozdìlíme na~dva druhy:
+
+\s{Definice:} Øekneme, ¾e pøevedení je {\I nasycené}, pokud po~pøevodu rezerva na~hranì~$uv$ klesla na~nulu, tedy $r(uv)=0$. V~opaèném pøípadì je {\I nenasycené}, a~tehdy urèitì klesne pøebytek ve~vrcholu~$u$ na~nulu, tedy $f^{\Delta}(u) = 0$ (pøi~nasyceném pøevedení se~to~ale mù¾e stát také).
+
+\s{Lemma S (naSycená pøevedení):} Poèet v¹ech nasycených pøevedení je nejvý¹~$NM$.
+
+\proof
+Pro ka¾dou hranu~$uv$ spoèítejme poèet nasycených pøevedení (tedy takových pøevedení, ¾e po~nich klesne rezerva hrany na~nulu). Abychom dvakrát nasycenì pøevedli pøebytek (nebo jeho èást) z~vrcholu~$u$ do~vrcholu~$v$, tak jsme museli~$u$ mezitím alespoò dvakrát zvednout:
+
+Po~prvním nasyceném pøevedení z~vrcholu~$u$ do~vrcholu~$v$ se~vynulovala rezerva hrany~$uv$. Uvìdomme si, ¾e pøi~této operaci muselo být~$u$ vý¹e ne¾~$v$, a~dokonce víme, ¾e bylo vý¹e pøesnì o~1 (viz lemma~S). Po~této hranì tedy nemù¾eme u¾~nic více pøevést. Aby do¹lo k~druhému nasycenému pøevedení z~$u$ do~$v$, musíme nejprve opìt zvý¹it rezervu hrany~$uv$. Jediný zpùsob, jak toho lze dosáhnout, je pøevést èást pøebytku z~$v$ zpátky do~$u$. K~tomu se~musí~$v$ dostat (alespoò o~1) vý¹e ne¾~$u$. Po~pøelití bude rezerva~$uv$ opìt kladná. A~abychom provedli nasycené pøevedení znovu ve~smìru z~$u$ do~$v$, musíme zase~$u$ dostat (alespoò o~1) vý¹e ne¾~$v$. Proto musíme~$u$ alespoò o~2 zvednout -- nejprve na~úroveò~$v$ a~pak je¹tì o~1 vý¹e.
+
+
+Ukázali jsme si~tedy, ¾e mezi ka¾dými dvìma nasycenými pøevedeními jsme vrchol~$u$ zvedli alespoò dvakrát. Nicménì libovolnou hranu mù¾eme zvednout nejvý¹e~$2N$-krát (viz invariant V). V¹ech hran je~$M$, tudí¾ poèet v¹ech nasycených pøevedení je nejvý¹e~$NM$.
+\qed
+
+\s{Lemma N (Nenasycená pøevedení):} Poèet v¹ech nenasycených pøevedení je~$\O(N^2M)$.
+
+\proof
+Dùkaz provedeme pomocí potenciálové metody -- nadefinujme si~následující funkci jako potenciál:
+ $$ \Phi := \sum_{\scriptstyle{v: f^{\Delta}(v) > 0} \atop \scriptstyle{v \ne z,s}} h(v). $$
+Nyní se~podívejme, jak se~ná¹ potenciál bìhem algoritmu vyvíjí a~jaké má vlastnosti:
+
+	\itemize\ibull
+	\:Na poèátku je $ \Phi = 0 $.
+	\:Bìhem celého algoritmu je $ \Phi \ge 0 $, nebo» je souètem nezáporných èlenù.
+	\:Zvednutí vrcholu zvý¹í $\Phi$ o~jednièku (Aby byl vrchol zvednut, musel mít kladný pøebytek $\Rightarrow$ vrchol do~sumy ji¾ pøispíval, teï jen pøispìje èíslem o 1 vy¹¹ím.). Ji¾ víme, ¾e za~celý prùbìh algoritmu je v¹ech zvednutí maximálnì~$2N^2$, proto zvedáním vrcholù zvý¹íme potenciál dohromady nejvý¹e o~$2N^2$.
+	\:Nasycené pøevedení zvý¹í~$\Phi$ nejvý¹e o~$2N$, proto¾e buï po~pøevodu hranou~$uv$ v~$u$ zùstal nìjaký pøebytek, tak¾e se~mohl potenciál zvý¹it nejvý¹e o~$h(v) \leq 2N$, nebo je pøebytek v~$u$ po~pøevodu nulový a~potenciál se~dokonce o~jedna sní¾il. Za~celý prùbìh tak dojde k~maximálnì~$NM$ takovýmto pøevedením, díky nim¾ se~potenciál zvý¹í maximálnì o~$2N^2M$.
+	\:Koneènì kdy¾ pøevádíme po~hranì~$uv$ nenasycenì, tak od~potenciálu urèitì odeèteme vý¹ku vrcholu~$u$ (nebo» se~vynuluje pøebytek ve~vrcholu~$u$) a~mo¾ná pøièteme vý¹ku vrcholu~$v$. Jen¾e $h(v) = h(u) - 1$, a~proto nenasycené pøevedení potenciál v¾dy sní¾í alespoò o~jedna.
+	\endlist
+
+\>Z~tohoto rozboru chování potenciálu~$\Phi$ v~prùbìhu algoritmu získáváme, ¾e poèet v¹ech nenasycených pøevedení mù¾e být nejvý¹e $2N^2 + 2N^2M$, co¾ je $\O(N^2M)$.
+\qed
+
+\s{Implementace:}
+
+Budeme si~pamatovat seznam~$P$ v¹ech vrcholù~$v \ne z,s$ s~kladným pøebytkem. Neboli
+$$P = \{ v \in V \setminus \{z,s\} ~\vert~ f^{\Delta}(v) > 0 \}.$$
+Kdy¾ mìníme pøebytek nìjakého vrcholu, mù¾eme ná¹ seznam v~konstantním èase aktualizovat (napø. tak, ¾e si~ka¾dý vrchol pamatuje pozici, na~které v~seznamu~$P$ je). V~konstantním èase také umíme odpovìdìt, zda existuje nìjaký vrchol s~pøebytkem.
+
+Dále si~pro ka¾dý vrchol~$u \in V$ budeme pamatovat~$L(u)$-seznam hran~$uv \in E$ takových, které vedou dolù (mají spád alespoò 1) a~kladnou rezervu. Neboli
+$$L(u) = \{ uv \in E ~\vert~ v \in V,~ r(uv) > 0,~ h(v) < h(u)\}.$$
+Díky tomu mù¾eme pøistupovat k~patøièným sousedùm~$u$ v~èase $\O(1)$, stejnì jako pøidávat hrany do~$L(u)$, resp. je mazat. Opìt ka¾dá hrana si~bude pamatovat pozici, na~které se~nachází v~seznamu~$L$.
+
+\s{Rozbor èasové slo¾itosti algoritmu:}
+
+\numlist\ndotted
+\:Inicializace vý¹ek \dots\ $\O(N)$.
+\:Inicializace vlny~$f$ \dots\ $\O(M)$.
+\:Výbìr vrcholu~$u$ s~kladným pøebytkem -- vezmeme první vrchol v~$P$ \dots\ $\O(1)$.
+\:Výbìr vrcholu~$v$, do~kterého vede z~$u$ hrana s~kladnou rezervou a~který je ní¾e ne¾~$u$ -- vezmeme první hranu z~$L(u)$ \dots\ $\O(1)$.
+	
+	Pøevedení pøebytku: \dots\ $\O(1)$.
+		\itemize\idot
+		\:Nasycené pøevedení \dots\ $\O(1)$.
+			\itemize\idot
+			\:Rezerva hrany~$uv$ klesne na~nulu $\Rightarrow$ hrana~$uv$ vypadne z~$L(u)$ \dots\ $\O(1)$.
+			\:Pøebytek vrcholu~$v$ se~zvý¹í $\Rightarrow$ pokud je¹tì nebyl v~seznamu~$P$, tak se~tam pøidá \dots\ $\O(1)$.
+			\:Pøebytek vrcholu~$u$ mo¾ná také klesne na~nulu $\Rightarrow$ pak by vrchol~$u$ vypadnul z~$P$ \dots\ $\O(1)$.
+			\endlist
+		\:Nenasycené pøevedení \dots\ $\O(1)$.
+			\itemize\idot
+			\:Rezerva hrany~$uv$ zùstane nezáporná $\Rightarrow$ hrana~$uv$ zùstane v~$L(u)$ \dots\ $\O(1)$.
+			\:Vynuluje se~pøebytek vrcholu~$u$~$\Rightarrow$ vrchol $u$ vypadne z~$P$ \dots~$\O(1)$.
+			\:Pøebytek vrcholu~$v$ se~zvý¹í~$\Rightarrow$ pokud je¹tì nebyl v~seznamu~$P$, tak se~tam pøidá \dots\ $\O(1)$.
+			\endlist
+		\endlist
+\:Zvednutí vrcholu~$u$ \dots $\O(N)$.
+
+Musíme obejít v¹echny hrany do~$u$ a~z~$u$, kterých je nejvý¹e~$2N-2$, porovnat
+vý¹ky a~pøípadnì tyto hrany~$uv$ odebrat ze~seznamu~$L(v)$ resp. pøidat
+do~$L(u)$. Abychom pro~odebrání hrany~$uv$ ze~seznamu~$L(v)$ nemuseli procházet
+celý seznam, budeme si~$\forall u \in V$ pamatovat je¹tì $L^{-1}(u) := $ seznam
+ukazatelù na~hrany~$uv$ v~seznamech~$L(v)$.
+
+\endlist
+
+Vidíme, ¾e ka¾dé zvednutí je sice drahé, ale je jich zase pomìrnì málo. Naopak pøevádìní pøebytkù je èastá operace, tak¾e je výhodné, ¾e trvá konstantní èas.
+
+\s{Shrnutí:}
+
+\itemize\ibull
+\:V¹ech zvednutí je $\O(N^2)$ (viz lemma Z), ka¾dé trvá $\O(N)$ \dots\ $\O(N^3).$
+\:V¹ech nasycených pøevedení je $\O(NM)$ (viz lemma S), ka¾dé trvá $\O(1)$ \dots\ $\O(NM).$
+\:V¹ech nenasycených pøevedení je $\O(N^2M)$ (viz lemma N), ka¾dé trvá $\O(1)$ \dots\ $\O(N^2M).$
+\endlist
+
+Dohromady má tedy Goldbergùv algoritmus èasovou slo¾itost $\O(N^2M)$. Vidíme, ¾e u¾ v~tomto obecném pøípadì to není hor¹í ne¾ Dinicùv algoritmus. Pøí¹tì si~uká¾eme, ¾e mù¾e mít i~mnohem lep¹í. Nejdøíve ale zformulujme v¹echna dokázaná tvrzení do~následující vìty:
+
+\s{Vìta:} Goldbergùv algoritmus najde maximální tok v~èase $\O(N^2M)$.
+
+\s{Pozorování:} Pokud bychom volili v¾dy nejvy¹¹í z~vrcholù s~pøebytkem, tak by se~mohl algoritmus chovat lépe. Podívejme se~na~to pozornìji a~vylep¹ený Goldebrgùv algoritmus oznaème G'.
+
+\s{Algoritmus (Vylep¹ený Goldbergùv algoritmus)}
+
+\algo
+\:$\forall v \in V: h(v)\leftarrow 0$ (v¹em vrcholùm nastavíme poèáteèní vý¹ku nula) a~$h(z)\leftarrow N$ (zdroj zvedneme do~vý¹ky~$N$).
+\:$\forall e \in E: f(e)\leftarrow 0$ (po~hranách nejdøíve nenecháme protékat nic) a~$\forall zu \in E : f(zu)\leftarrow c(zu)$ (ze~zdroje pustíme maximální mo¾nou vlnu).
+\:Dokud $\exists u \in V \setminus \{z,s\}: f^{\Delta}(u)>0$:
+\::Vybereme z~vrcholù s~pøebytkem ten s~nejvy¹¹í vý¹kou, oznaèíme ho~$u$.
+\:::Pokud $\exists v \in V: uv \in E,~r(uv)>0$ a~$h(u)>h(v)$, pak pøevedeme pøebytek po~hranì z~$u$ do~$v$.
+\:::V~opaèném pøípadì zvedneme $u$:~$h(u) \leftarrow h(u) + 1$.
+\:Vrátíme tok~$f$ jako výsledek.
+\endalgo
+
+Rozmysleme si, o~kolik bude vylep¹ený algoritmus G' lep¹í ne¾ ten pùvodní. Ten pùvodní mìl èasovou slo¾itost $\O(N^2M)$ a~pøevládal èlen, který odpovídal nenasyceným pøevedením. Zkusme tedy právì poèet nenasycených pøevedení odhadnout ve~vylep¹eném algoritmu o~nìco tìsnìji.
+
+\s{Lemma N' (Nenasycená pøevedení):} Algoritmus G' provede~$\O(N^3)$ nenasycených pøevedení.
+
+\proof
+Dokazovat budeme opìt pomocí potenciálové metody. Zadefinujme si~potenciál {\I nejvy¹¹í hladinu s~pøebytkem}:
+$$H := \max \{ h(v) \mid v \neq z,s ~\&~ f^\Delta(v) > 0\}.$$
+Rozdìlíme bìh algoritmu na~{\I fáze}. Ka¾dá fáze konèí tím, ¾e~se~$H$ zmìní. Jak se~mù¾e zmìnit? Buï se~$H$ zvý¹í, co¾ znamená, ¾e~nìjaký vrchol s~pøebytkem v~nejvy¹¹í hladinì byl o~1 zvednut, nebo se~$H$ sní¾í. My víme, ¾e zvednutí je v~celém algoritmu $\O(N^2)$. Zároveò si~mù¾eme uvìdomit, ¾e~$H$ je nezáporný potenciál, kdy sní¾ení i~zvý¹ení ho zmìní o~1, tedy poèet sní¾ení bude stejný jako poèet zvý¹ení, a~proto obojího je~$\O(N^2)$. Tudí¾ poèet fází je také~$\O(N^2)$.
+
+Je dùle¾ité, ¾e~bìhem jedné fáze provedeme nejvý¹e jedno nenasycené pøevedení z~ka¾dého vrcholu. Po~ka¾dém nenasyceném pøevedení po~hranì $uv$ se~toti¾ vynuluje pøebytek v~$u$ a~aby se~provedlo dal¹í nenasycené pøevedení z~vrcholu~$u$, muselo by nejdøíve být co~pøevádìt. Muselo by tedy do~$u$ nìco pøitéci. My ale víme, ¾e pøevádíme pouze shora dolù a~$u$ je v~nejvy¹¹í hladinì (to zajistí právì to vylep¹ení algoritmu), tedy nejdøíve by musel být nìjaký jiný vrchol zvednut. Tím by se~ale zmìnilo~$H$ a~skonèila by tato fáze.
+
+Proto poèet v¹ech nenasycených pøevedení bìhem jedné fáze je nejvý¹e~$N$. A ji¾ jsme dokázali, ¾e~fází je~$\O(N^2)$. Tedy poèet v¹ech nenasycených pøevedení je~$\O(N^3)$.
+\qed
+
+Tento odhad je hezký, ale stále není tìsný a~algoritmus se~chová lépe. Doka¾me si~je¹tì jeden tìsnìj¹í odhad na~poèet nenasycených pøevedení.
+
+\s{Lemma N'' (Nenasycená pøevedení):} Poèet nenasycených pøevedení je~$\O(N^2 \sqrt{M})$.
+
+\s{Poznámka:} Tato èasová slo¾itost je výhodná napøíklad pro~øídké grafy. Ty mají toti¾ pomìrnì malý poèet hran.
+
+\proof
+Rozdìlme si~fáze na~dva druhy: laciné a~drahé podle toho, kolik se~v~nich provede nenasycených pøevedení. Zvolme si~nìjaké nezáporné~$K$. Zatím nebudeme urèovat jeho hodnotu. Uvidíme, ¾e~èasová slo¾itost algoritmu bude závislá na~tomto parametru~$K$. Proto jeho hodnotu zvolíme a¾ pozdìji a~to tak, aby byla slo¾itost co nejni¾¹í.
+
+{\I Laciné fáze} budou ty, bìhem nich¾ se~provede nejvý¹e~$K$ nenasycených pøevedení. {\I Drahé fáze} budou ty ostatní, tedy takové, bìhem nich¾ se~provede více jak~$K$ nenasycených pøevedení.
+
+Teï potøebujeme odhadnout, kolik nás budou stát oba typy fází. Zaènìme s~tìmi jednodu¹¹ími -- s~lacinými. Víme, ¾e~v¹ech fází je~$\O(N^2)$. Tìch laciných bude tedy urèitì také~$O(N^2)$. Nenasycených pøevedení se~bìhem jedné laciné fáze provede nejvíce~$K$. Tedy celkem se~bìhem laciných fází provede~$\O(N^2K)$ nenasycených pøevedení.
+
+Pro~poèet nenasycených pøevedení v~drahých fázích si~zaveïme nový potenciál definovaný následovnì:
+$$\Phi := \sum_{\scriptstyle{v \ne z,s} \atop \scriptstyle{f^{\Delta}(v) \ne 0}} {p(v) \over K},$$
+kde~$p(v)$ je poèet takových vrcholù~$u$, které nejsou vý¹e ne¾~$v$. Neboli
+$$p(v) = \vert \{ u \in V \mid h(u) \leq h(v) \} \vert.$$
+Tedy platí, ¾e~$p(v)$ je v¾dy nezáporné a~nejvý¹e má hodnotu~$N$. Dále víme, ¾e~$\Phi$ bude v¾dy nezáporné (nebo» je to souèet nezáporných èlenù) a~nejvý¹e bude nabývat hodnoty~$N^2 \over K$. Rozmysleme si, jak nám ovlivní tento potenciál na¹e tøi operace:
+\itemize\ibull
+\:{\bf Zvednutí}: Za~ka¾dý zvednutý vrchol pøibude nejvý¹e~$N \over K$ (tento vrchol mù¾e být nadzvednut nejvý¹e nad~v¹echny ostatní vrcholy) a~mo¾ná nìco ubude (napø. kdy¾ vrchol vyzvedneme na~úroveò k~ostatním).
+\:{\bf Nasycené pøevedení} po~hranì $uv$: Mù¾e vynulovat pøebytek ve~vrcholu~$u$, pak se~$\Phi$ sní¾í. Mù¾e zvý¹it pøebytek ve~$v$ z~nuly, pak se~$\Phi$ zvý¹í. Ale nejvý¹e se~zvý¹í o~$N \over K$, nebo» do~$\Phi$ pøibude jen jeden sèítanec za~vrchol $v$ a~ten pøispìje nejvý¹e hodnotou~$N \over K$ (pod ním mù¾e být nejvíce~$N$ vrcholù).
+\:{\bf Nenasycená pøevedení} po~hranì $uv$ v~drahých fázích: Tato operace vynuluje pøebytek v~$u$, tedy~$\Phi$ klesne alespoò o~$p(u) \over K$. Zároveò mù¾e zvý¹it pøebytek ve~$v$ z~nuly, ale~$\Phi$ stoupne nejvý¹e o~$p(v) \over K$. Celkem tedy~$\Phi$ klesne alespoò o~$p(u) - p(v) \over K$.
+\endlist
+Uvìdomme si, ¾e~pokud pøevádíme po~hranì~$uv$, tak platí, ¾e~$h(u) = h(v) + 1$. Pak~$p(u) - p(v)$ je pøesnì poèet vrcholù na~hladinì~$H$. Tìch je alespoò tolik, kolik je nenasycených pøevedení bìhem jedné fáze (to jsme dokázali ji¾ v~lemmatu N'), a~my jsme si~zadefinovali, ¾e v~drahé fázi je poèet nenasycených pøevedení alespoò~$K$. Tedy~$p(u) - p(v) > K$. Proto bìhem jednoho nenasyceného pøevedení~$\Phi$ klesne alespoò o~${K \over K} = 1$. Nenasycená pøevedení potenciál nezvy¹ují.
+
+Potenciál~$\Phi$ se~mù¾e zvìt¹it pouze pøi~operacích zvednutí a~nasycené pøevedení. Zvednutí se~provede celkem~$\O(N^2)$ a~ka¾dé zvý¹í potenciál nejvý¹e o~$N \over K$. Nasycených pøevedení se provede celkem~$\O(NM)$ a~ka¾dé zvý¹í potenciál takté¾ nejvý¹e o~$N \over K$. Celkem se~tedy~$\Phi$ zvý¹í nejvý¹e o
+$${N \over K} \O(N^2) + {N \over K} \O(NM) = \O \left({N^3 \over K} + {N^2M \over K}\right).$$
+
+Teï vyu¾ijeme toho, ¾e~$\Phi$ je nezáporný potenciál, tedy kdy¾ ka¾dé nenasycené pøevdení v~drahé fázi sní¾í~$\Phi$ alespoò o~1, tak v¹ech nenasycených pøevdení v~drahých fázích je~$\O({N^3 \over K} + {N^2M \over K})$. U¾ jsme ukázali, ¾e~nenasycených pøevední v~laciných fázích je~$\O(N^2K)$. Proto celkem v¹ech nenasycených pøevedení je
+$$\O \left(N^2K + {N^3 \over K} + {N^2M \over K} \right) = \O \left(N^2K + {N^2M \over K} \right)$$
+(nebo» pro~souvislé grafy platí, ¾e~$M \geq N \Rightarrow N^2M \geq N^3$). A~my chceme, aby jich bylo co nejménì. Tato funkce má minimum tehdy, kdy¾ $N^2K = {N^2M \over K}$, èili $K = \sqrt{M}$.
+
+Proto v¹ech nenasycených pøevedení je	$\O(N^2\sqrt{M})$.
+\qed
+\bye
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diff --git a/old/3-goldberg/Makefile b/old/3-goldberg/Makefile
new file mode 100644
index 0000000..6330e4c
--- /dev/null
+++ b/old/3-goldberg/Makefile
@@ -0,0 +1,3 @@
+P=3-goldberg
+
+include ../Makerules
diff --git a/old/4-hradla/4-hradla.tex b/old/4-hradla/4-hradla.tex
new file mode 100644
index 0000000..ccb0c50
--- /dev/null
+++ b/old/4-hradla/4-hradla.tex
@@ -0,0 +1,139 @@
+\input ../lecnotes.tex
+
+\prednaska{4}{Hradlové sítì}{(zapsal: Petr Jankovský)}
+
+\def\land{\mathbin{\&}}
+
+Výkon poèítaèù nelze zvy¹ovat donekoneèna a pøesto¾e ji¾ pìkných pár let platí,
+¾e se jejich rychlost s~èasem exponenciálnì zvìt¹uje, jednou urèitì narazíme
+pøinejmen¹ím na~fyzikální limity.
+
+Kdy¾ tedy nemù¾eme zvy¹ovat rychlost jednoho procesoru, jak poèítat rychleji?
+Øe¹ením by mohlo být poøídit si procesorù víc. U¾~dnes na~bì¾ném PéCéèku máme
+k~dispozici vícejádrové procesory, díky nim¾ mù¾eme vyu¾ít pararelní poèítání
+a úlohu øe¹it tak, ¾e práci ¹ikovnì rozdìlíme mezi procesory (èi jádra) a
+zamìstnáme je pøi~výpoètu v¹echny.
+
+My se podíváme na~abstraktní výpoèetní model, který je je¹tì paralelnìj¹í.
+Techniky, které si uká¾eme na~tomto modelu, se v¹ak dají pøekvapivì vyu¾ít i~pøi~reálném
+paralelizování na~nìkolika málo procesorech. Konec koncù i proto, ¾e vnitøní
+architektura procesoru se na¹emu modelu velmi podobá. Budeme se zabývat 
+jednoduchým modelem paralelního poèítaèe, toti¾ hradlovou sítí.
+
+\h{Hradlové sítì}
+
+\s{Definice:} {\I Hradlo} je prvek, který umí vyhodnocovat nìjakou funkci
+nad~koneènou abecedou $\Sigma$.
+
+Obecnì se na~hradlo díváme jako na~funkci $f: {\Sigma}^{k} \rightarrow \Sigma$, která dostane $k$ vstupù
+a~vrátí jeden výstup, pøièem¾ hodnoty, nad~kterými pracuje, budou z~nìjaké koneèné
+abecedy -- tedy z~nìjaké koneèné mno¾iny symbolù $\Sigma$. Písmenku $k$ zde øíkáme {\I arita
+hradla}.
+
+\s{Pøíklad:} Èasto studujeme hradla booleovská (pracující nad abecedou $\{0,1\}$), která poèítají logické funkce.
+
+Z~nich nejèastìji potkáme:
+
+\itemize\ibull
+\:nulární: to jsou konstanty (FALSE=0, TRUE=1),
+\:unární: napø. negace (znaèíme~$\lnot$),
+\:binární: logický souèin ({\sc and},~$\land$), souèet ({\sc or},~$\lor$), ...
+\endlist
+
+\>Hradla kreslíme tøeba následovnì:
+
+\figure{hradlo_and.eps}{Binární hradlo provádící logickou operaci {\sc and}.}{1in}
+
+Jednotlivá hradla mù¾eme navzájem urèitým zpùsobem propojovat a vytváøet
+z nich {\I hradlové sítì}. Pokud pou¾íváme pouze booleovská hradla, øíkáme takto vytvoøeným
+sítím {\I booleovské obvody}. Pokud pracujeme s~operacemi nad nìjakou obecnìj¹í (ale koneènou) 
+mno¾inou symbolù (abecedou), nazývají se {\I kombinaèní obvody.}
+
+Ka¾dá hradlová sí» má nìjaké vstupy, nìjaké výstupy a uvnitø jsou propojovaná
+hradla. Ka¾dý vstup hradla je pøipojen buïto na~nìkterý ze~vstupù sítì nebo
+na~výstup jiného hradla. Výstupy hradel mohou být propojeny na~vstupy dal¹ích
+hradel (mohou se vìtvit), nebo na výstupy sítì. Pøitom máme zakázáno vytváøet
+cykly. 
+
+Ne¾ si øekneme formální definici, podívejme se na obrázek.
+
+\figure{hradlova_sit.eps}{Hradlová sí» -- tøívstupová verze funkce {\I majorita}.}{3in}
+
+Obrazek znázoròuje hradlovou sí», která poèítá, zda je alespoò na~dvou ze~vstupù
+jednièka. Pojïme si ale {\I hradlovou sí»} definovat formálnì.
+
+\s{Definice:} {\I Hradlová sí»} je urèena:
+\itemize\ibull
+\:{\I abecedou} $\Sigma$ (to je nìjaká koneèná mno¾ina symbolù, obvykle $\Sigma=\{0,1\}$);
+\:po dvou disjunktními koneènými mno¾inami $I$~({\I vstupy}), $O$~({\I výstupy}) \hfil\break a~$H$~({\I hradla});
+\:acyklickým orientovaným multigrafem~$(V,E)$, kde~$V = I \cup O \cup H$;
+\:zobrazením~$F$, které ka¾dému hradlu $h\in H$ pøiøadí nìjakou funkci~$F(h):
+  \Sigma^{a(h)} \rightarrow \Sigma$, co¾ je funkce, kterou toto hradlo vykonává.
+  Èíslu $a(h)$ øíkáme {\I arita} hradla~$h$;
+\:zobrazením~$z: E \rightarrow {\bb N}$, které ka¾dé hranì vedoucí do~nìjakého
+  hradla pøiøazuje nìkterý ze vstupù tohoto hradla.
+\endlist
+
+\>Pøitom jsou splnìny následující podmínky:
+
+\itemize\ibull
+\:$\forall i \in I: \deg^{in}(i)=0$ (do~vstupù nic nevede);
+\:$\forall o \in O: \deg^{in}(o)=1~\land~\deg^{out}(o)=0$ (z~výstupù nic nevede a do~ka¾dého vede právì jedna hrana);
+\:$\forall h \in H: \deg^{in}(v)=a(v)$ (do~ka¾dého hradla vede tolik hran, kolik je jeho arita);
+\:$\forall h \in H~\forall j: 1\le j\le a(h)$ existuje právì jedena hrana~$e$ taková, ¾e~$e$ konèí v~$h$ a~$z(e)=j$, 
+  (v¹echny vstupy hradel jsou zapojeny).
+\endlist
+
+\s{Pozorování:} Kdybychom pøipustili hradla s~libovolnì vysokým poètem vstupù, mohli
+bychom libovolný problém se vstupem délky~$n$ vyøe¹it jedním hradlem o~$n$~vstupech,
+kterému bychom pøiøadili funkci, která by na¹i úlohu rovnou vyøe¹ila. Tento model
+v¹ak není ani realistický, ani pìkný. Proto pøijmìme omezení, ¾e~arity v¹ech hradel
+budou omezeny nìjakou pevnou konstantou $k$ (uká¾e se, ¾e nám bude staèit dvojka
+a~vystaèíme si tedy pouze s nulárními, unárními a binárními hradly). 
+Následující obrázky ukazují, jak hradla o~více vstupech nahradit dvouvstupovými:
+
+\twofigures{hradlo_ternor.eps}{Trojvstupové hradlo \sc or.}{0.5in}{hradlo_ternbior.eps}{Jeho nahrazení 2-vstupovými hradly.}{0.6in}
+
+
+Nyní bychom je¹tì mìli definovat, co taková hradlová sí» vlastnì poèítá a~jak
+její výpoèet probíhá.
+
+\s{Definice:} {\I Výpoèet sítì} probíhá po~{\I taktech.} V nultém taktu jsou definovány pouze 
+hodnoty na~vstupech sítì a na~výstupech hradel arity 0. Mù¾eme si to pøedstavit
+tak, ¾e na~zaèátku nemá ¾ádné hradlo definovánu výstupní hodnotu (a¾ na ji¾ 
+zmínìná hradla nulární). V~ka¾dém dal¹ím taktu pak vydají výstup hradla, která
+na~konci minulého taktu mìla definovány v¹echny hodnoty na vstupech. Jakmile
+budou po~nìjakém koneèném poètu taktù definované i hodnoty v¹ech výstupù, sí»
+se zastaví a~vydá výsledek.
+
+\s{Pozorování:} Proto¾e je sí» acyklická, je jasné, ¾e jakmile jednou nìjaké hradlo vydá
+výstup, tak se tento výstup bìhem dal¹ího výpoètu sítì ji¾ nezmìní.
+
+\figure{vypocet_site.eps}{Výpoèet hradlové sítì.}{6cm}
+
+\>Podle toho, jak sí» poèítá, si ji mù¾eme rozdìlit na~vrstvy:
+
+\s{Definice:} {\I $i$-tá vrstva $S_i$} obsahuje právì takové vrcholy~$v$, pro~které nejdel¹í cesta ze~vstupù
+sítì do $v$ má délku rovnou~$i$.
+
+\s{Pozorování:} V¹imnìme si, ¾e v $i$-tém taktu vydají hodnoty právì hradla z $S_i$.
+ 
+Dává tedy smysl prohlásit za~{\I èasovou slo¾itost} sítì poèet
+jejích vrstev. Podobnì {\I prostorovou slo¾itost} definujeme jako poèet hradel v~síti.
+
+\s{Pøíklad:} Sestrojme sí», která zjistí, zda se mezi jejími~$n$ vstupy
+vyskytuje alespoò jedna jednièka.
+
+\>{\I První øe¹ení:} zapojíme hradla za~sebe (sériovì). Èasová i prostorová
+slo¾itost odpovídají~$\Theta(n)$. Zde ov¹em vùbec nevyu¾íváme toho, ¾e by mohlo poèítat více
+hradel souèasnì.
+
+\figure{hloupy_or.eps}{Hradlová sí», která zjistí, zdali je na vstupu alespoò jedna jednièka.}{0.7in}
+
+\>{\I Druhé øe¹ení:} Hradla budeme spojovat do~dvojic, pak výsledky z~tìchto dvojic opìt
+do~dvojic a tak dále. Díky paralelnímu zapojení dosáhneme èasové slo¾itosti $\Theta(\log n)$,
+prostorová slo¾itost zùstane lineární.
+
+\figure{chytry_or.eps}{Chytøej¹í øe¹ení stejného problému pro vstup velikosti 16.}{3in}
+
+\bye
diff --git a/old/4-hradla/Makefile b/old/4-hradla/Makefile
new file mode 100644
index 0000000..65a25ae
--- /dev/null
+++ b/old/4-hradla/Makefile
@@ -0,0 +1,3 @@
+P=4-hradla
+
+include ../Makerules
diff --git a/old/4-hradla/chytry_or.eps b/old/4-hradla/chytry_or.eps
new file mode 100644
index 0000000..3d88cff
--- /dev/null
+++ b/old/4-hradla/chytry_or.eps
@@ -0,0 +1,2324 @@
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diff --git a/old/5-addsort/1_9_deleni_bloku.eps b/old/5-addsort/1_9_deleni_bloku.eps
new file mode 100644
index 0000000..2c96b1c
--- /dev/null
+++ b/old/5-addsort/1_9_deleni_bloku.eps
@@ -0,0 +1,3225 @@
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: Mayura Draw, Version 4.3
+%%Title: 1_9_deleni_bloku.ai
+%%CreationDate: Mon Oct 29 23:24:27 2007
+%%BoundingBox: -9 -9 276 157
+%%EndComments
+%%BeginProlog
+%%BeginResource: procset MayuraDraw_ops
+%%Version: 4.3
+%%Copyright: (c) 1993-2003 Mayura Software
+/PDXDict 100 dict def
+PDXDict begin
+% width height matrix proc key cache
+% definepattern -\> font
+/definepattern { %def
+  7 dict begin
+    /FontDict 9 dict def
+    FontDict begin
+      /cache exch def
+      /key exch def
+      /proc exch cvx def
+      /mtx exch matrix invertmatrix def
+      /height exch def
+      /width exch def
+      /ctm matrix currentmatrix def
+      /ptm matrix identmatrix def
+      /str
+      (xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx)
+      def
+    end
+    /FontBBox [ %def
+      0 0 FontDict /width get
+      FontDict /height get
+    ] def
+    /FontMatrix FontDict /mtx get def
+    /Encoding StandardEncoding def
+    /FontType 3 def
+    /BuildChar { %def
+      pop begin
+      FontDict begin
+        width 0 cache { %ifelse
+          0 0 width height setcachedevice
+        }{ %else
+          setcharwidth
+        } ifelse
+        0 0 moveto width 0 lineto
+        width height lineto 0 height lineto
+        closepath clip newpath
+        gsave proc grestore
+      end end
+    } def
+    FontDict /key get currentdict definefont
+  end
+} bind def
+
+% dict patternpath -
+% dict matrix patternpath -
+/patternpath { %def
+  dup type /dicttype eq { %ifelse
+    begin FontDict /ctm get setmatrix
+  }{ %else
+    exch begin FontDict /ctm get setmatrix
+    concat
+  } ifelse
+  currentdict setfont
+  FontDict begin
+    FontMatrix concat
+    width 0 dtransform
+    round width div exch round width div exch
+    0 height dtransform
+    round height div exch
+    round height div exch
+    0 0 transform round exch round exch
+    ptm astore setmatrix
+
+    pathbbox
+    height div ceiling height mul 4 1 roll
+    width div ceiling width mul 4 1 roll
+    height div floor height mul 4 1 roll
+    width div floor width mul 4 1 roll
+
+    2 index sub height div ceiling cvi exch
+    3 index sub width div ceiling cvi exch
+    4 2 roll moveto
+
+    FontMatrix ptm invertmatrix pop
+    { %repeat
+      gsave
+        ptm concat
+        dup str length idiv { %repeat
+          str show
+        } repeat
+        dup str length mod str exch
+        0 exch getinterval show
+      grestore
+      0 height rmoveto
+    } repeat
+    pop
+  end end
+} bind def
+
+% dict patternfill -
+% dict matrix patternfill -
+/patternfill { %def
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+    eoclip patternpath
+  grestore
+  newpath
+} bind def
+
+/img { %def
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+  /imgh exch def
+  /imgw exch def
+  concat
+  imgw imgh 8
+  [imgw 0 0 imgh neg 0 imgh]
+  /colorstr 768 string def
+  /colorimage where {
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+    { currentfile colorstr readhexstring pop }
+    false 3 colorimage
+  }{
+    /graystr 256 string def
+    {
+      currentfile colorstr readhexstring pop
+      length 3 idiv
+      dup 1 sub 0 1 3 -1 roll
+      {
+        graystr exch
+        colorstr 1 index 3 mul get 30 mul
+        colorstr 2 index 3 mul 1 add get 59 mul
+        colorstr 3 index 3 mul 2 add get 11 mul
+        add add 100 idiv
+        put
+      } for
+      graystr 0 3 -1 roll getinterval
+    } image
+  } ifelse
+  grestore
+} bind def
+
+/arrowhead {
+  gsave
+    [] 0 setdash
+    strokeC strokeM strokeY strokeK setcmykcolor
+    2 copy moveto
+    4 2 roll exch 4 -1 roll exch
+    sub 3 1 roll sub
+    exch atan rotate dup scale
+    arrowtype
+    dup 0 eq {
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+      closepath fill
+    } if
+    dup 1 eq {
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+      closepath fill
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+    } if
+    dup 3 eq {
+      -6 0 rmoveto -1 2 rlineto 7 -2 rlineto -7 -2 rlineto
+      closepath fill
+    } if
+    dup 4 eq {
+      -9 0 rmoveto 0 3 rlineto 9 -3 rlineto -9 -3 rlineto
+      closepath fill
+    } if
+    dup 5 eq {
+      currentpoint newpath 3 0 360 arc
+      closepath fill
+    } if
+    dup 6 eq {
+      2.5 2.5 rmoveto 0 -5 rlineto -5 0 rlineto 0 5 rlineto
+      closepath fill
+    } if
+    pop
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+} bind def
+
+/setcmykcolor where { %ifelse
+  pop
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+  /setcmykcolor {
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+     yellow black add dup 1 gt { pop 1 } if 1 exch sub
+     setrgbcolor
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+} ifelse
+
+/RE { %def
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+  currentdict dup length dict begin
+    { %forall
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+      /Encoding Encoding 256 array copy def
+      0 exch { %forall
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+          Encoding 2 index 2 index put
+          pop 1 add
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+          exch pop
+        } ifelse
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+  currentdict dup end end
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+} bind def
+
+/spacecount { %def
+  0 exch
+  ( ) { %loop
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diff --git a/old/5-addsort/5-addsort.tex b/old/5-addsort/5-addsort.tex
new file mode 100644
index 0000000..36b35cc
--- /dev/null
+++ b/old/5-addsort/5-addsort.tex
@@ -0,0 +1,312 @@
+\input ../lecnotes.tex
+
+\prednaska{5}{Paralelní sèítání, bitonické tøídìní}{(zapsal: Petr Jankovský)}
+
+Minule jsme si zavedli paralelní výpoèetní model, ve kterém si nyní nìco naprogramujeme \dots
+
+\h{Sèítání binárních èísel}
+
+Mìjme dvì èísla $x$ a $y$ zapsané ve~dvojkové soustavì. Jejich èíslice oznaème
+$x_{n-1}\ldots x_0$ a $y_{n-1}\ldots y_0$, kde $i$-tý øád má váhu $2^i$. Nyní budeme chtít tato èísla
+seèíst.
+
+K tomuto úèelu se~ihned nabízí pou¾ít starý dobrý \uv{¹kolní algoritmus}, který
+funguje ve~dvojkové soustavì stejnì dobøe jako v~desítkové. Zkrátka sèítáme èísla
+zprava doleva. V¾dy seèteme pøíslu¹né èíslice pod~sebou a~pøièteme pøenos z~ni¾¹ího
+øádu. Tím dostaneme jednu èíslici výsledku a~pøenos do~vy¹¹ího øádu. Formálnì
+bychom to mohli zapsat tøeba takto:
+$$
+z_i=x_i \oplus y_i \oplus c_i,
+$$
+kde $z_i$ je $i$-tá èíslice souètu, $\oplus$ znaèí operaci {\sc xor} (souèet modulo~2) a~$c_i$ je {\I pøenos}
+z~$(i-1)$-ního øádu do~$i$-tého. Pøenos pøitom nastane tehdy, pokud se~nám potkají
+dvì jednièky pod~sebou, nebo kdy¾ se~vyskytne alespoò jedna jednièka a~k~tomu
+pøenos z~ni¾¹ího øádu. Jinými slovy tehdy, kdy¾ ze~tøí xorovaných èíslic jsou alespoò
+dvì jednièky (pomocí obvodu pro majoritu z minulé pøedná¹ky lehce zkonstruujeme):
+$$
+\eqalign{
+c_{0} &= 0, \cr
+c_{i+1} &= (x_i~\&~y_i)\lor(x_i~\&~c_i)\lor(y_i~\&~c_i).\cr
+}
+$$
+
+Takovéto sèítání sice perfektnì funguje, nicménì je bohu¾el pomìrnì pomalé.
+Pokud bychom stavìli hradlovou sí» podle tohoto pøedpisu, byla by slo¾ená z~nìjakých
+podsítí (\uv{krabièek}), které budou mít na~vstupu $x_i$, $y_i$ a~$c_i$ a jejich výstupem
+bude $z_i$ a~$c_{i+1}$. 
+
+\figure{hloupe_scitani.eps}{Sèítání ¹kolním algoritmem.}{1.5in}
+
+V¹imìme si, ¾e~ka¾dá krabièka závisí na~výstupu té pøedcházející. Jednotlivé
+krabièky tedy musí urèitì le¾et na~rùzných hladinách. Celkovì bychom museli pou¾ít
+$\Theta{(n)}$ hladin a~jeliko¾ je ka¾dá krabièka konstantnì velká, také $\Theta{(n)}$ hradel. To dává
+lineární èasovou i~prostorovou slo¾itost, èili oproti sekvenènímu algoritmu jsme si nepomohli.
+
+Zamysleme se nad tím, jak by se proces sèítání mohl zrychlit.
+
+\h{Pøenosy v~blocích}
+Jediné, co nás pøi sèítání brzdí, jsou pøenosy mezi jednotlivými øády. Ka¾dý øád,
+aby~vydal souèet, musí poèkat na~to, a¾~dopoèítají v¹echny pøedcházející øády.
+Teprve pak se~toti¾ dozví pøenos. Kdybychom ov¹em pøenosy dokázali spoèítat
+nìjakým zpùsobem paralelnì, máme vyhráno. Jakmile známe v¹echny pøenosy, souèet
+u¾~zvládneme dopoèítat na~konstantnì mnoho hladin -- tedy v~konstantním èase.
+
+Podívejme se na~libovolný {\I blok} v~na¹em souètu. Tak budeme øíkat èíslùm
+$x_j\ldots x_i$ a $y_j\ldots y_i$ v~nìjakém intervalu indexù $\left<i,j\right>$. Pøenos $c_{j+1}$ vystupující z~tohoto bloku závisí mimo hodnot sèítancù u¾ pouze na~pøenosu $c_{i}$, který do bloku vstupuje.
+
+\figure{blok_scitani.eps}{Blok souètu.}{3in}
+
+Z tohoto pohledu se mù¾eme na blok také dívat jako na nìjakou funkci, která
+dostane jednobitový vstup a vydá jednobitový výstup. To je nám pøíjemné, nebo» 
+takových funkcí existují jenom ètyøi typy:
+\numlist\ndotted
+\:$f(x) = 0$,  ~~~~(0)
+\:$f(x) = 1$,  ~~~~(1)
+\:$f(x) = x$,  ~~~~($<$ -- kopírování)
+\:$f(x) = \neg{x}$.
+\endlist
+Jak se za~chvíli uká¾e, poslední pøípad, kdy by~nìjaký blok pøedával opaèný
+pøenos, ne¾ do~nìj vstupuje, navíc nikdy nemù¾e nastat. Pojïme si to rozmyslet.
+Jednobitové bloky se chovají velice jednodu¹e:
+
+\figure{bloky_1bit.eps}{Tabulka triviálních blokù.}{1.1in}
+
+Z prvního bloku evidentnì v¾dy vyleze 0, a»~do~nìj vstoupí jakýkoli pøenos.
+Poslední blok naopak sám o~sobì pøenos vytváøí, a»~ji¾ do~nìj vleze jakýkoliv.
+Bloky prostøední se chovají stejnì, a~to tak, ¾e~samy o~sobì ¾ádný pøenos nevytvoøí,
+ale~pokud do~nich nìjaký pøijde, tak~také odejde.
+
+Mìjme nyní nìjaký vìt¹í blok~$C$ slo¾ený ze~dvou men¹ích podblokù $A$ a~$B$, jejich¾
+chování u¾ známe. Z~toho mù¾eme odvodit, jak se chová celý blok:
+
+\figure{tabulka_skladani_bloku.eps}{Skládání chování blokù.}{1.3in}
+
+Pokud vy¹¹í blok pøenos pohlcuje, pak a»~se u¾~ni¾¹í blok chová jakkoli, slo¾ení
+tìchto blokù musí v¾dy pohlcovat. V~prvním øádku tabulky jsou tudí¾ nuly. Analogicky,
+pokud vy¹¹í blok generuje pøenos, tak~ten ni¾¹í na~tom nic nezmìní. V~druhém
+øádku tabulky jsou tedy samé jednièky. Zajímavìj¹í pøípad nastává, pokud vy¹¹í blok
+kopíruje -- tehdy zále¾í èistì na~chování ni¾¹ího bloku.
+
+V¹imnìme si, ¾e~skládání chování blokù je vlastnì úplnì obyèejné skládání
+funkcí. Nyní bychom mohli prohlásit, ¾e~budeme poèítat nad~tøíprvkovou abecedou,
+a~¾e~celou tabulku doká¾eme spoèítat jedním jediným hradlem. Pojïmì si v¹ak
+rozmyslet, jak~bychom takovouto vìc popsali èistì binárnì. Jak tedy tyto tøi stavy
+popisovat pouze nìkolika bity?
+
+Evidentnì nám k tomuto binárnímu zakódování tøí stavù budou staèit bity dva.
+Oznaème si je jako $p$ a $q$. Tato dvojice mù¾e nabývat hned ètyø mo¾ných hodnot,
+kterým pøiøadíme tøi mo¾ná chování bloku. Toto kódování mù¾eme zvolit zcela
+libovolnì, ale pokud si ho zvolíme ¹ikovnì, u¹etøíme si dále práci pøi kompozici.
+Zvolme si tedy kódování takto:
+
+\itemize\ibull
+\:$(1,*) = <$,
+\:$(0,0) = 0$,
+\:$(0,1) = 1$
+\endlist
+
+Tomu, ¾e blok kopíruje, odpovídá dvojice $p = 1$; $q =$ \<cokoliv>. V~ostatních
+pøípadech bude~$p$ nulové a~$q$ nám bude øíkat, co je na~výstupu pøíslu¹ného bloku.
+Jinými slovy $p = 0$ znamená, ¾e funkce je konstanta, pøièem¾ $q$ øíká jaká; naproti
+tomu $p = 1$~znamená, ¾e funkce je identita, a»~u¾~je $q$ cokoli.
+
+Pojïme si nyní ukázat, jak bude celé skládání blokù vypadat. Rozmysleme si,
+kdy je~$p$ celého dvojbloku jednièkové, tedy kdy celý dvojblok kopíruje. To nastane
+tehdy, pokud kopírují obì jeho èásti, a tedy  $p = p_a~\&~p_b$. Dále $q$ bude rovno jednièce,
+pokud $q = (\neg{p_a}~\&~q_a) \lor (p_a~\&~q_b)$.
+
+Skládání chování blokù lze tedy popsat buï ternárnì -- tabulkou, ale lze to
+i~binárnì vý¹e uvedeným pøedpisem.
+
+Nyní si tedy mù¾eme dopøedu vypoèítat chování bloku velikosti jedna, poté
+ z~nich skládáním blokù velikosti dva, dál velikosti ètyøi, osm, atd \dots
+
+\h{Paralelní sèítání}
+
+Paralelní algoritmus na~sèítání u¾~zkonstruujeme pomìrnì snadno. Bez újmy
+na~obecnosti budeme pøedpokládat, ¾e~poèet bitù vstupních èísel je mocnina dvojky,
+jinak si vstup doplníme nulami, co¾ výsedný èas bìhu algoritmu zhor¹í maximálnì
+konstanta-krát.
+
+\algo
+\:Spoèteme chování blokù velikosti~1. ($\O(1)$ hladin)
+\:Postupnì poèítáme chování blokù\foot{myslíme \uv{pøirozenì zarovnané} bloky, tedy takové, jejich¾ poloha je násobkem velikosti} velikosti 2, 4, 8, ..., $2^k$. 
+  ($\O(\log n)$ hladin, na~nich¾ se skládají bloky)
+\:$c_0 \leftarrow 0$ (pøenos do nejni¾¹ího øádu je v¾dy 0)
+\:Urèíme $c_n$ podle $c_0$ a chování (jediného) bloku velikosti~$n$.
+\:Postupnì dopoèítáme pøenosy na~hranicích dìlitelných $2^k$ \uv{zahu¹»ováním}:
+  jakmile víme $c_{2^k}$, mù¾eme dopoèítat $c_{2^k+2^{k-1}}$ podle
+  chování bloku $\left< 2^k, 2^k+2^{k-1}\right>$. ($\O(\log n)$ hladin,
+  na~nich¾ se dosazuje)
+\:Seèteme: $\forall i: z_i = x_i \oplus y_i \oplus c_i$.
+\endalgo
+
+\figure{1_9_deleni_bloku.eps}{Výpoèet pøenosu.}{2.5in}
+
+Algoritmus pracuje v~èase $\O(\log n)$. Hradel je pou¾ito lineárnì: na~jednotlivých
+hladinách kroku~2 poèet hradel exponenciálnì klesá od~$n$ k~1, na~hladinách kroku~5
+exponenciálnì stoupá od~1 k~$n$, tak¾e dohromady se seète na~$\Theta(n)$.
+
+\h{Tøídìní}
+
+Nyní se podíváme na~druhý problém, a~to na~problém tøídìní. Ji¾ známe pomìrnì efektivní sekvenèní algoritmy, které dovedou tøídit v~èase $\O(n\log n)$. Byli bychom jistì rádi, kdybychom to zvládli je¹tì rychleji. Pojïme se podívat, zda by nám v tom nepomohlo problém paralelizovat.
+
+Budeme pøi tom pracovat ve~výpoèetním modelu, kterému se øíká komparátorová sí». Ta je postavená z~hradel, kterým se øíká komparátory.
+
+\s{Definice:} {\I Komparátorová sí»} je kombinaèní obvod, jeho¾ hradla jsou
+komparátory.
+
+\figure{sortnet.0}{Komparátor}{0.7in}
+
+Komparátor umí porovnat dvì hodnoty a~rozhodnout, která z~nich je vìt¹í
+a~která men¹í. Nevrací v¹ak booleovský výsledek jako bì¾né hradlo, ale má dva
+výstupy, pøièem¾ na~jednom vrátí men¹í ze~vstupních hodnot a~na~druhém vìt¹í.
+
+Výstupy komparátorù se nevìtví. Nemù¾eme tedy jeden výstup \uv{rozdvojit} a~pøipojit ho na~dva vstupy. (Vìtvení by dokonce ani nemìlo smysl, proto¾e zatímco rozdvojit bychom mohli, slouèit u¾~ne. Pokud tedy chceme, aby sí» mìla $n$~vstupù i~$n$~výstupù, rozdvojení stejnì nesmíme provést, i kdybychom jej mìli povolené.)
+
+\s{Pøíklad:} {\sl Bubble sort}
+
+Obrázek Bubble 1 ilustruje pou¾ití komparátorù pro tøídìní Bubble sortem.
+©ipky pøedstavují jednotlivé komparátory. Výpoèet v¹ak je¹tì mù¾eme vylep¹it.
+
+\twofigures{sortnet.1}{Bubble 1}{143pt}{sortnet.2}{Bubble 2}{143pt}
+
+Sna¾íme se výpoèet co nejvíce paralelizovat (viz obrázek Bubble 2).
+Jak je vidìt, komparátory na sebe nemusejí èekat. Tím mù¾eme výpoèet urychlit a místo èasu $\Theta{(n^2)}$ docílit èasové slo¾itosti $\Theta{(n)}$. V obou pøípadech je zachován kvadratický prostor.
+
+Nyní si uká¾eme je¹tì rychlej¹í tøídící algoritmus. Pùjdeme na nìj v¹ak trochu \uv{od lesa}. Nejdøíve vymyslíme sí», která bude umìt tøídit jenom nìco -- toti¾ bitonické posloupnosti. Bez újmy na obecnosti budeme pøedpokládat, ¾e ka¾dé dva prvky na vstupu jsou navzájem rùzné.
+
+\medskip
+\s{Definice:} Øekneme, ¾e posloupnost $x_0,\dots,x_{n-1} $ je {\I èistì bitonická} právì tehdy, kdy¾
+pro nìjaké $x_k, k\in\{0, \dots, n-1\} $ platí, ¾e~v¹echny prvky pøed ním (vèetnì jeho samotného)
+tvoøí rostoucí poslopnost, kde¾to prvky stojící za~ním tvoøí poslopnost klesající.
+Formálnì zapsáno musí platit, ¾e:
+$$x_0\leq x_1\leq \dots \leq x_k \geq x_{k+1}\geq\dots \geq x_{n-1}.$$
+
+\s{Definice:} Posloupnost $x_0 \dots x_{n-1}$ je {\I bitonická}, právì kdy¾ $\exists~j\in \{0,\dots ,n-1\}$, pro
+které je rotace pùvodní posloupnosti o $j$ prvkù, tedy posloupnost
+$$x_j,x_{(j+1) \bmod n},\dots, x_{(j+n-1) \bmod n},$$ èistì bitonická.
+
+\s{Definice:} {\I Separátor $S_n$} je sí», ve které jsou v¾dy~$i$-tý a~$(i+{n/2})$-tý prvek vstupu
+(pro $i=0,\dots, {n/2}-1$) propojeny komparátorem. Minimum se pak stane~$i$-tým,
+maximum  $(i+{n/2})$-tým prvkem výstupu.
+\figure{sortnet.3}{$(y_i, y_{i+{n/2}}) = \<CMP>(x_i, x_{i+{n/2}})$} {300pt}
+
+\s{Lemma:} Pokud separátor dostane na~vstupu bitonickou posloupnost, pak jeho výstup $y_0, \dots, y_{n-1}$
+splòuje:
+
+(i) $y_0,\dots, y_{n/2 -1}$ a~$y_{n/2},\dots, y_{n-1}$ jsou 
+bitonické posloupnosti,
+
+(ii) Pro v¹echna $i,j< {n/2}$ platí $y_i < y_{j + {n/2}}$.
+
+Separátor nám tedy jednu bitonickou posloupnost na~vstupu rozdìlí na~dvì
+bitonické posloupnosti, pøièem¾ navíc ka¾dý prvek první posloupnosti ($y_0,\dots, y_{n/2 -1}$)
+je men¹í nebo roven prvkùm druhé posloupnosti ($y_{n/2}, \dots, y_{n-1}$).
+
+\>Ne¾ pøistoupíme k dùkazu lemmatu, uka¾me si, k èemu se nám bude hodit.
+
+
+\s{Definice:} {\I Bitonická tøídièka $B_n$} je obvod sestavený ze separátorù, který dostane-li na vstupu bitonickou posloupnost délky $n$ (BÚNO konstruujeme tøídièku pro $n=2^k$), vydá setøídìnou zadanou posloupnost délky $n$. 
+
+Tøídièka dostane na~vstupu bitonickou posloupnost. Separátor~$S_n$ ji pak dle lemmatu rozdìlí na~dvì bitonické posloupnosti, kdy je ka¾dý prvek z~první posloupnosti men¹í ne¾ libovolný prvek z~druhé. Tyto poloviny pak dal¹í separátory rozdìlí na~ètvrtiny, ..., a¾~na~konci zbudou pouze jednoduché posloupnosti délky jedna (zjevnì setøídìné), které mezi sebou mají po¾adovanou nerovnost -- tedy ka¾dá posloupnost (nebo spí¹e prvek) nalevo je $\leq$ ne¾ prvek napravo od~nìj.
+
+\centerline{\epsfbox{sortnet.5}}
+
+Jak je vidìt, bitonická tøídièka nám libovolnou bitonickou posloupnost délky~$n$
+setøídí na~$\Theta(\log n)$ hladin.
+
+Nyní se dá odvodit, ¾e pokud umíme tøídit bitonické posloupnosti, umíme setøídit v¹echno.
+Vzpomeòme si na~tøídìní sléváním -- Merge sort. To funguje tak, ¾e zaène s~jednoprvkovými posloupnostmi, které jsou evidentnì setøídìné, a~poté v¾dy dvì setøídìné posloupnosti slévá do~jedné. Kdybychom nyní umìli paralelnì slévat, mohli bychom vytvoøit i~paralelní Merge sort. Jinými slovy, potøebujeme dvì rostoucí posloupnosti nìjak efektivnì slít do~jedné setøídìné. Uvìdomme si, ¾e to zvládneme jednodu¹e -- staèí druhou z~posloupností obrátit a~\uv{pøilepit} za~první, èím¾ vznikne bitonická posloupnost, kterou poté mù¾eme setøídit na¹í bitonickou tøídièkou.
+
+\s{Pøíklad:} {\sl Merge sort}
+
+Bitonická tøídièka se tedy dá pou¾ít ke~slévání setøídìných posloupností. 
+Uka¾me si, jak s~její pomocí sestavíme souèástky {\I slévaèky} $M_n$:
+
+Setøídìné posloupnosti $x_0,\dots, x_{n-1}$ a~$y_0,\dots, y_{n-1}$ spojíme do jedné bitonické
+posloupnosti $x_0,\dots, x_{n-1},y_{n-1},\dots, y_0$. Z~této posloupnosti vytvoøíme pomocí bitonické tøídièky
+$B_{2n}$ setøídìnou posloupnost. Vytvoøíme tedy blok $M_{2n}$, který se ov¹em sestává de~facto pouze z~bloku $B_{2n}$, jeho¾ druhá polovina vstupù je zapojena v~obráceném poøadí.
+\figure{sortnet.6}{Paralelní MergeSort.}{3in}
+
+Nyní se pokusme odhadnout èasovou slo¾itost. Ná¹ MergeSort bude mít øádovì hloubku blokù $\log n$.
+V ka¾dém bloku $M_n$ je navíc ukryta bitonická tøídièka s takté¾ logaritmickou hloubkou.
+Celková hloubka tedy bude $\log 2 + \log4 + \dots + \log 2^k + \dots + \log n$.
+Po seètení nakonec dostáváme výslednou èasovou slo¾itost $\Theta (\log^2 n)$.
+
+Dodejme je¹tì, ¾e existuje i~tøídicí algoritmus, kterému staèí jen ${\O}(\log n)$ hladin.
+Jeho multiplikativní konstanta je v¹ak pøíli¹ veliká, tak¾e je v~praxi nepou¾itelný.
+
+Vra»me se nyní k dùkazu lemmatu, který jsme na pøedminulé stránce vynechali.
+
+\h{Dùkaz Lemmatu:}
+(i) Nejprve nahlédneme, ¾e lemma platí, je-li vstupem èistì bitonická
+posloupnost. Dále BÚNO pøedpokládejme, ¾e vrchol posloupnosti je v~první polovinì (kdyby
+byl vrchol za~polovinou, staèilo by zrcadlovì obrátit posloupnost i~komparátory a~øe¹ili
+bychom stejný problém). Nyní si definujme $k := \min j: x_j > x_{j+n/2}$. (Pokud
+by~takové~$k$ neexistovalo, znamenalo by~to, ¾e vstupní posloupnost je monotónní.
+Separátor by tedy nic nedìlal a~pouze zkopíroval vstup na~výstup, co¾ jistì lemma splòuje.) Nyní si v¹imìme,
+¾e jakmile jednou zaène platit, ¾e prvek na~levé stranì je men¹í ne¾ na~pravé, bude
+nám tato relace platit a¾~do~konce. Oznaème vrchol vstupní posloupnosti jako~$x_m$.
+Pak~$k$ bude jistì men¹í ne¾~$m$ a~$k+{n/2}$ bude vìt¹í ne¾~$m$. Mezi~$k$ a~$m$ je tedy
+vstupní posloupnost neklesající, mezi $k+{n/2}$ a~$n-1$ nerostoucí.
+
+Do pozice~$k$ tedy separátor bude pouze kopírovat vstup na~výstup, od~pozice~$k$
+dál u¾~jen prohazuje. Pro ka¾dé~$i$, $(k\leq i\leq {n/2}-1)$ se prvky
+$x_i$ a~$x_{i+{n/2}}$ prohodí. Úsek mezi~$k$ a~${n/2}-1$ tedy
+nahradíme nerostoucí posloupností, první polovina výstupu tedy bude
+(dokonce èistì) bitonická. Podobnì úsek $k+{n/2}$ a¾~$n-1$ nahradíme èistì
+bitonickou posloupností. Obì poloviny tedy budou bitonické.
+
+Dostaneme-li na vstupu obecnou bitonickou posloupnost, pøedstavíme si,
+¾e je to èistì bitonická posloupnost zrotovaná o~$r$ prvkù (BÚNO
+doprava). Zjistíme, ¾e v~komparátorech se porovnávají tyté¾ prvky,
+jako kdyby zrotovaná nebyla. Výstup se od výstupu èistì bitonické
+posloupnosti zrotovaného o~$r$ bude li¹it prohozením úsekù $x_0$ a¾
+$x_{r-1}$ a~$x_{n/2}$ a¾ $x_{{n/2}+r-1}$. Obì výstupní
+posloupnosti tedy budou zrotované o~$r$ prvkù, ale na jejich
+bitoniènosti se nic nezmìní.
+
+(ii) Z dùkazu (i) pro èistì bitonickou posloupnost víme, ¾e $y_0\dots y_{n/2-1}$ je èistì bitonická a bude rovna $x_0\dots x_{k-1},x_{k+n/2}\dots x_{n-1}$ pro vhodné $k$ a navíc bude mít maximum v $x_{k-1}$ nebo $x_k+{n/2}$. Mezi tìmito body ov¹em ve vstupní posloupnosti urèitì nele¾el ¾ádný $x_i$ men¹í ne¾ $x_k-1$ nebo $x_k+{n/2}$ (jak je vidìt z obrázku) a posloupnost  $x_k \dots x_{k-1+{n/2}}$ je rovna $y_{n/2}\dots y_{n-1}$. Pro obecné bitonické posloupnosti uká¾eme stejnì jako v (i).
+\qed
+
+\medskip
+
+\centerline{\epsfbox{sortnet.7}}
+
+
+\h{Paralelní násobení}
+
+Podobnì jako u~sèítání si vzpomeneme na~¹kolní algoritmus -- tentokráte v¹ak pro násobení.
+To fungovalo tak, ¾e~jsme si jedno ze~dvou binárních èísel na~vstupu (øíkejme mu tøeba $x$) $n$-krát posouvali. Tam kde pak byly v~èísle~$y$ jednièky, pøíslu¹né kopie $x$ jsme seèetli. Jinými slovy tedy násobení umíme pøevést na~nìjaké posuny (ty lze realizovat pouze \uv{pøedrátováním} -- nic nás nestojí), násobení jedním bitem (co¾ je 
+{\sc and} ) a~nakonec potøebujeme výsledných~$n$ èísel seèíst.
+\figure{skolni_scitani.eps}{©kolní sèítání.}{2in}
+
+Jak nyní seèíst $n$ $n$-bitových èísel..? Nabízí se vyu¾ít osvìdèený \uv{stromeèek} -- sèítat dvojice èísel, výsledky pak opìt po~dvojicích seèíst, a¾ na~konci vyjde jediný výsledek.
+\figure{stromecek.eps}{Stromeèek}{1.4in}
+
+Toto øe¹ení by v¹ak vedlo na~èasovou slo¾itost $\Theta (\log^2 n)$, nebo» sèítat umíme v~èase $\Theta (\log n)$. To je sice dle na¹ich mìøítek
+docela efektivní, ale pøekvapivì to jde je¹tì lépe -- toti¾ na~$\Theta (\log n)$ hladin. Této 
+slo¾itosti dosáhneme malým trikem.
+
+Vymyslíme obvod konstantní hloubky, který na~vstupu dostane tøi èísla. Odpoví pak dvìma èísly
+takovými, ¾e budou mít stejný souèet jako pùvodní tøi èísla. Jinými slovy pomocí
+tohoto obvodu budeme umìt seètení tøí èísel pøevést na seètení dvou èísel.
+
+\figure{obvod.eps}{$x+y+z = p+q$}{0.3in}
+
+V¹imnìme si, ¾e~kdy¾ sèítáme tøi bity, mù¾e být pøenos do~vy¹¹ího øádu nula èi~jednièka. Vezmeme si tedy bity $x_i$, $y_i$, $z_i$ a~ty seèteme. To nám dá dvoubitový výsledek, pøièem¾ ni¾¹í bit z~tohoto výsledku po¹leme do~èísla~$p$, vy¹¹í do~èísla~$q$.
+
+\figure{obvod_real.eps}{Redukování sèítání}{1.2in}
+
+Toto zredukování sèítání nám nyní umo¾ní opìt stavit strom, by» o malièko slo¾itìj¹í.
+
+\figure{sl_stromecek.eps}{Slo¾itìj¹í stromeèek}{0.9in}
+
+Pokud jsme mìli na~zaèátku $n$ èísel, po~první redukci nám jich zbývá jen $2/3 \cdot n$ a~obecnì v~$k$-té hladiné $(2/3)^k \cdot n$. Znamená to, ¾e èísel nám ubývá exponenciálnì, tak¾e poèet hladin bude logaritmický. Redukující obvod je pøi tom jen konstantnì hluboký, tak¾e celé redukování zvládneme v~èase $\Theta (\log n)$. Na~konci Slo¾itìj¹ího stromeèku pak máme umístìnou jednu obyèejnou sèítaèku, která zbývající dvì èísla seète v~logaritmickém èase.
+
+Seètení v¹ech $n$ èísel tedy zabere $\Theta (\log n)$ hladin.
+
+Kdy¾ se nyní vrátíme k~násobení, zbývá nám vyøe¹it posouvání a~{\sc and}ování. Uvìdomme si, ¾e to je plnì paralelení a~zvládneme ho za~konstantnì mnoho hladin. Celé násobení tedy zvládneme v~logaritmickém èase.
+
+\bye
diff --git a/old/5-addsort/Makefile b/old/5-addsort/Makefile
new file mode 100644
index 0000000..4f7be80
--- /dev/null
+++ b/old/5-addsort/Makefile
@@ -0,0 +1,3 @@
+P=5-addsort
+
+include ../Makerules
diff --git a/old/5-addsort/blok_scitani.eps b/old/5-addsort/blok_scitani.eps
new file mode 100644
index 0000000..225a323
--- /dev/null
+++ b/old/5-addsort/blok_scitani.eps
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diff --git a/old/5-addsort/sortnet.mp b/old/5-addsort/sortnet.mp
new file mode 100644
index 0000000..081f9f4
--- /dev/null
+++ b/old/5-addsort/sortnet.mp
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+%defaultfont:="csr12";
+%verbatimtex \input twelvecs etex
+u:=1 ;
+ahlength:=u*3;
+
+beginfig(0);
+v:=u*7mm;
+
+z12=(1v,1v);
+z13=(2v,1v);
+z21=(0v,2v);
+z22=(1v,2v);
+z23=(2v,2v);
+z24=(3v,2v);
+z31=(0v,4v);
+z32=(1v,4v);
+z33=(2v,4v);
+z34=(3v,4v);
+z42=(1v,5v);
+z43=(2v,5v);
+
+pickup pencircle scaled 0.4pt;
+draw(z21--z22--z23--z24--z34--z33--z32--z31--z21);
+drawarrow(z22--z12);
+drawarrow(z23--z13);
+drawarrow(z42--z32);
+drawarrow(z43--z33);
+
+label.bot(btex \strut a etex,z32);
+label.bot(btex \strut b etex,z33);
+label.top(btex min etex,z22);
+label.top(btex max etex,z23);
+
+
+endfig;
+
+beginfig(1);
+v:=u*5mm;
+z10=(0v,0v);
+z20=(2v,0v);
+z30=(4v,0v);
+z40=(6v,0v);
+z50=(8v,0v);
+
+z11=(0v,1v);
+z12=(0v,2v);
+z13=(0v,3v);
+z14=(0v,4v);
+z15=(0v,5v);
+z16=(0v,6v);
+z17=(0v,7v);
+z18=(0v,8v);
+z19=(0v,9v);
+z110=(0v,10v);
+
+z21=(2v,1v);
+z22=(2v,2v);
+z23=(2v,3v);
+z24=(2v,4v);
+z25=(2v,5v);
+z26=(2v,6v);
+z27=(2v,7v);
+z28=(2v,8v);
+z29=(2v,9v);
+z210=(2v,10v);
+
+z31=(4v,1v);
+z32=(4v,2v);
+z33=(4v,3v);
+z34=(4v,4v);
+z35=(4v,5v);
+z36=(4v,6v);
+z37=(4v,7v);
+z38=(4v,8v);
+z39=(4v,9v);
+z310=(4v,10v);
+
+z41=(6v,1v);
+z42=(6v,2v);
+z43=(6v,3v);
+z44=(6v,4v);
+z45=(6v,5v);
+z46=(6v,6v);
+z47=(6v,7v);
+z48=(6v,8v);
+z49=(6v,9v);
+z410=(6v,10v);
+
+z51=(8v,1v);
+z52=(8v,2v);
+z53=(8v,3v);
+z54=(8v,4v);
+z55=(8v,5v);
+z56=(8v,6v);
+z57=(8v,7v);
+z58=(8v,8v);
+z59=(8v,9v);
+z510=(8v,10v);
+
+z111=(0v,11v);
+z211=(2v,11v);
+z311=(4v,11v);
+z411=(6v,11v);
+z511=(8v,11v);
+
+z1015=(-1v,1.5v);
+z1615=(9v,1.5v);
+z1035=(-1v,3.5v);
+z1635=(9v,3.5v);
+z1065=(-1v,6.5v);
+z1665=(9v,6.5v);
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z111);
+draw(z20--z211);
+draw(z30--z311);
+draw(z40--z411);
+draw(z50--z511);
+drawarrow(z11--z21);
+drawarrow(z13--z23);
+drawarrow(z16--z26);
+drawarrow(z110--z210);
+drawarrow(z22--z32);
+drawarrow(z25--z35);
+drawarrow(z29--z39);
+drawarrow(z34--z44);
+drawarrow(z38--z48);
+drawarrow(z47--z57);
+
+pickup pencircle scaled 0.7pt;
+draw(z1015--z1615) dashed withdots scaled 0.7;
+draw(z1035--z1635) dashed withdots scaled 0.7;
+draw(z1065--z1665) dashed withdots scaled 0.7;
+
+label.top(btex x1 etex,z111);
+label.top(btex x2 etex,z211);
+label.top(btex x3 etex,z311);
+label.top(btex x4 etex,z411);
+label.top(btex x5 etex,z511);
+
+endfig;
+
+beginfig(2);
+v:=u*5mm;
+z13=(0v,3v);
+z23=(2v,3v);
+z33=(4v,3v);
+z43=(6v,3v);
+z53=(8v,3v);
+
+z14=(0v,4v);
+z15=(0v,5v);
+z16=(0v,6v);
+z17=(0v,7v);
+z18=(0v,8v);
+z19=(0v,9v);
+z110=(0v,10v);
+
+z24=(2v,4v);
+z25=(2v,5v);
+z26=(2v,6v);
+z27=(2v,7v);
+z28=(2v,8v);
+z29=(2v,9v);
+z210=(2v,10v);
+
+z34=(4v,4v);
+z35=(4v,5v);
+z36=(4v,6v);
+z37=(4v,7v);
+z38=(4v,8v);
+z39=(4v,9v);
+z310=(4v,10v);
+
+z44=(6v,4v);
+z45=(6v,5v);
+z46=(6v,6v);
+z47=(6v,7v);
+z48=(6v,8v);
+z49=(6v,9v);
+z410=(6v,10v);
+
+z54=(8v,4v);
+z55=(8v,5v);
+z56=(8v,6v);
+z57=(8v,7v);
+z58=(8v,8v);
+z59=(8v,9v);
+z510=(8v,10v);
+
+z111=(0v,11v);
+z211=(2v,11v);
+z311=(4v,11v);
+z411=(6v,11v);
+z511=(8v,11v);
+
+pickup pencircle scaled 0.4pt;
+draw(z13--z111);
+draw(z23--z211);
+draw(z33--z311);
+draw(z43--z411);
+draw(z53--z511);
+drawarrow(z14--z24);
+drawarrow(z16--z26);
+drawarrow(z18--z28);
+drawarrow(z110--z210);
+drawarrow(z25--z35);
+drawarrow(z27--z37);
+drawarrow(z29--z39);
+drawarrow(z36--z46);
+drawarrow(z38--z48);
+drawarrow(z47--z57);
+
+label.top(btex x1 etex,z111);
+label.top(btex x2 etex,z211);
+label.top(btex x3 etex,z311);
+label.top(btex x4 etex,z411);
+label.top(btex x5 etex,z511);
+
+endfig;
+
+
+beginfig(3);
+v:=u*5mm;
+
+z10=(0v,0v);
+z15=(0v,5v);
+z16=(0v,6v);
+
+z20=(2v,0v);
+z24=(2v,4v);
+z26=(2v,6v);
+
+z30=(4v,0v);
+z33=(4v,3v);
+z36=(4v,6v);
+
+z40=(6v,0v);
+z42=(6v,2v);
+z46=(6v,6v);
+
+z50=(8v,0v);
+z51=(8v,1v);
+z56=(8v,6v);
+
+z60=(10v,0v);
+z65=(10v,5v);
+z66=(10v,6v);
+
+z70=(12v,0v);
+z74=(12v,4v);
+z76=(12v,6v);
+
+z80=(14v,0v);
+z83=(14v,3v);
+z86=(14v,6v);
+
+z90=(16v,0v);
+z92=(16v,2v);
+z96=(16v,6v);
+
+z100=(18v,0v);
+z101=(18v,1v);
+z106=(18v,6v);
+
+z110=(20v,0v);
+z116=(20v,6v);
+
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z16);
+draw(z20--z26);
+draw(z30--z36);
+draw(z40--z46);
+draw(z50--z56);
+draw(z60--z66);
+draw(z70--z76);
+draw(z80--z86);
+draw(z90--z96);
+draw(z100--z106);
+drawarrow(z15--z65);
+drawarrow(z24--z74);
+drawarrow(z33--z83);
+drawarrow(z42--z92);
+drawarrow(z51--z101);
+
+label.top(btex $x_0$ etex,z16);
+label.top(btex $x_1$ etex,z26);
+label.top(btex $x_2$ etex,z36);
+label.top(btex \dots etex,z66);
+label.top(btex $x_{n-2}$ etex,z96);
+label.top(btex $x_{n-1}$ etex,z106);
+
+endfig;
+
+
+
+beginfig(4);
+v:=u*5mm;
+
+z1075=(7.5v,0v);
+
+z10=(0v,1v);
+z175=(7.5v,1v);
+z115=(15v,1v);
+
+z20=(0v,2v);
+z235=(3.5v,2v);
+z211=(11.5v,2v);
+z215=(15v,2v);
+
+z30=(0v,3v);
+z335=(3.5v,3v);
+z37=(7v,3v);
+z38=(8v,3v);
+z311=(11.5v,3v);
+z315=(15v,3v);
+
+z40=(0v,4v);
+z415=(1.5v,4v);
+z455=(5.5v,4v);
+z47=(7v,4v);
+z48=(8v,4v);
+z495=(9.5v,4v);
+z413=(13.5v,4v);
+z416=(15v,4v);
+
+z50=(0v,5v);
+z515=(1.5v,5v);
+z53=(3v,5v);
+z54=(4v,5v);
+z555=(5.5v,5v);
+z57=(7v,5v);
+z58=(8v,5v);
+z595=(9.5v,5v);
+z511=(11v,5v);
+z512=(12v,5v);
+z513=(13.5v,5v);
+z516=(15v,5v);
+
+z60=(0v,6v);
+z61=(1v,6v);
+z62=(2v,6v);
+z63=(3v,6v);
+z64=(4v,6v);
+z65=(5v,6v);
+z66=(6v,6v);
+z67=(7v,6v);
+z68=(8v,6v);
+z69=(9v,6v);
+z610=(10v,6v);
+z611=(11v,6v);
+z612=(12v,6v);
+z613=(13v,6v);
+z614=(14v,6v);
+z615=(15v,6v);
+
+z71=(1v,7v);
+z72=(2v,7v);
+z75=(5v,7v);
+z76=(6v,7v);
+z79=(9v,7v);
+z710=(10v,7v);
+z713=(13v,7v);
+z714=(14v,7v);
+
+
+
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z115--z215--z20--cycle);
+draw(z30--z37--z47--z40--cycle);
+draw(z38--z315--z416--z48--cycle);
+draw(z50--z53--z63--z60--cycle);
+draw(z54--z57--z67--z64--cycle);
+draw(z58--z511--z611--z68--cycle);
+draw(z512--z516--z615--z612--cycle);
+drawarrow(z71--z61);
+drawarrow(z72--z62);
+drawarrow(z75--z65);
+drawarrow(z76--z66);
+drawarrow(z79--z69);
+drawarrow(z710--z610);
+drawarrow(z713--z613);
+drawarrow(z714--z614);
+drawarrow(z515--z415);
+drawarrow(z555--z455);
+drawarrow(z595--z495);
+drawarrow(z513--z413);
+drawarrow(z335--z235);
+drawarrow(z311--z211);
+drawarrow(z175--z1075);
+
+endfig;
+
+
+beginfig(5);
+v:=u*5mm;
+
+z075=(7.5v,8v);
+
+z10=(0v,7v);
+z175=(7.5v,7v);
+z115=(15v,7v);
+
+z20=(0v,6v);
+z235=(3.5v,6v);
+z211=(11.5v,6v);
+z215=(15v,6v);
+
+z30=(0v,5v);
+z335=(3.5v,5v);
+z37=(7v,5v);
+z38=(8v,5v);
+z311=(11.5v,5v);
+z315=(15v,5v);
+
+z40=(0v,4v);
+z415=(1.5v,4v);
+z455=(5.5v,4v);
+z47=(7v,4v);
+z48=(8v,4v);
+z495=(9.5v,4v);
+z413=(13.5v,4v);
+z416=(15v,4v);
+
+z50=(0v,3v);
+z515=(1.5v,3v);
+z53=(3v,3v);
+z54=(4v,3v);
+z555=(5.5v,3v);
+z57=(7v,3v);
+z58=(8v,3v);
+z595=(9.5v,3v);
+z511=(11v,3v);
+z512=(12v,3v);
+z513=(13.5v,3v);
+z516=(15v,3v);
+
+z60=(0v,2v);
+z61=(1v,2v);
+z62=(2v,2v);
+z63=(3v,2v);
+z64=(4v,2v);
+z65=(5v,2v);
+z66=(6v,2v);
+z67=(7v,2v);
+z68=(8v,2v);
+z69=(9v,2v);
+z610=(10v,2v);
+z611=(11v,2v);
+z612=(12v,2v);
+z613=(13v,2v);
+z614=(14v,2v);
+z615=(15v,2v);
+
+% ve skutecnosti dle znaceni 
+% by melo byt z6* ale uz 
+% obsazeno
+z815=(1.5v,2v);
+z855=(5.5v,2v);
+z895=(9.5v,2v);
+z813=(13.5v,2v);
+
+z715=(1.5v,1v);
+z755=(5.5v,1v);
+z795=(9.5v,1v);
+z713=(13.5v,1v);
+
+z9=(4v,0v);
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z115--z215--z20--cycle);
+draw(z30--z37--z47--z40--cycle);
+draw(z38--z315--z416--z48--cycle);
+draw(z50--z53--z63--z60--cycle);
+draw(z54--z57--z67--z64--cycle);
+draw(z58--z511--z611--z68--cycle);
+draw(z512--z516--z615--z612--cycle);
+drawarrow(z075--z175);
+drawarrow(z415--z515);
+drawarrow(z455--z555);
+drawarrow(z495--z595);
+drawarrow(z413--z513);
+drawarrow(z235--z335);
+drawarrow(z211--z311);
+drawarrow(z815--z715);
+drawarrow(z855--z755);
+drawarrow(z895--z795);
+drawarrow(z813--z713);
+
+label.llft(btex $n$ etex,z075);
+label.bot(btex $S_n$ etex,z175);
+label.bot(btex $S_{n\over 2}$ etex,z335);
+label.bot(btex $S_{n\over 2}$ etex,z311);
+label.bot(btex $S_{n\over 4}$ etex,z515);
+label.bot(btex $S_{n\over 4}$ etex,z555);
+label.bot(btex $S_{n\over 4}$ etex,z595);
+label.bot(btex $S_{n\over 4}$ etex,z513);
+label.rt(btex Bitonick\'a t\v r\'\i di\v cka $B_{n}$ etex,z9);
+
+endfig;
+
+
+
+beginfig(6);
+v:=u*5mm;
+
+z1075=(7.5v,0v);
+
+z10=(0v,1v);
+z175=(7.5v,1v);
+z115=(15v,1v);
+
+z20=(0v,2v);
+z235=(3.5v,2v);
+z211=(11.5v,2v);
+z215=(15v,2v);
+
+z30=(0v,3v);
+z335=(3.5v,3v);
+z37=(7v,3v);
+z38=(8v,3v);
+z311=(11.5v,3v);
+z315=(15v,3v);
+
+z40=(0v,4v);
+z415=(1.5v,4v);
+z455=(5.5v,4v);
+z47=(7v,4v);
+z48=(8v,4v);
+z495=(9.5v,4v);
+z413=(13.5v,4v);
+z416=(15v,4v);
+
+z50=(0v,5v);
+z515=(1.5v,5v);
+z53=(3v,5v);
+z54=(4v,5v);
+z555=(5.5v,5v);
+z57=(7v,5v);
+z58=(8v,5v);
+z595=(9.5v,5v);
+z511=(11v,5v);
+z512=(12v,5v);
+z513=(13.5v,5v);
+z516=(15v,5v);
+
+z60=(0v,6v);
+z61=(1v,6v);
+z62=(2v,6v);
+z63=(3v,6v);
+z64=(4v,6v);
+z65=(5v,6v);
+z66=(6v,6v);
+z67=(7v,6v);
+z68=(8v,6v);
+z69=(9v,6v);
+z610=(10v,6v);
+z611=(11v,6v);
+z612=(12v,6v);
+z613=(13v,6v);
+z614=(14v,6v);
+z615=(15v,6v);
+
+z71=(1v,7v);
+z72=(2v,7v);
+z75=(5v,7v);
+z76=(6v,7v);
+z79=(9v,7v);
+z710=(10v,7v);
+z713=(13v,7v);
+z714=(14v,7v);
+
+
+
+
+pickup pencircle scaled 0.4pt;
+draw(z10--z115--z215--z20--cycle);
+draw(z30--z37--z47--z40--cycle);
+draw(z38--z315--z416--z48--cycle);
+draw(z50--z53--z63--z60--cycle);
+draw(z54--z57--z67--z64--cycle);
+draw(z58--z511--z611--z68--cycle);
+draw(z512--z516--z615--z612--cycle);
+drawarrow(z71--z61);
+drawarrow(z72--z62);
+drawarrow(z75--z65);
+drawarrow(z76--z66);
+drawarrow(z79--z69);
+drawarrow(z710--z610);
+drawarrow(z713--z613);
+drawarrow(z714--z614);
+drawarrow(z515--z415);
+drawarrow(z555--z455);
+drawarrow(z595--z495);
+drawarrow(z513--z413);
+drawarrow(z335--z235);
+drawarrow(z311--z211);
+drawarrow(z175--z1075);
+
+label.top(btex $M_8$ etex,z175);
+label.top(btex $M_4$ etex,z335);
+label.top(btex $M_4$ etex,z311);
+label.top(btex $M_2$ etex,z515);
+label.top(btex $M_2$ etex,z555);
+label.top(btex $M_2$ etex,z595);
+label.top(btex $M_2$ etex,z513);
+
+endfig;
+
+
+beginfig(7);
+v:=u*7mm;
+
+z12=(1v,2v);
+z13=(1v,3v);
+z16=(1v,6v);
+z356=(3.5v,6v);
+z40=(5v,1v);
+z42=(5v,2v);
+z47=(5v,6.5v);
+z72=(9v,2v);
+z74=(9v,4v);
+z76=(9v,6v);
+
+z100=(0.5v,0.5v);
+z101=(1.5v,0.5v);
+
+z0=whatever[z13,z356];
+z1=whatever[z356,z74];
+z1=z0+4v*right;
+z2=whatever[z12,z72];
+z2=z0+whatever*down;
+z3=whatever[z12,z72];
+z3=z1+whatever*down;
+z4=z356+4v*right;
+z5=whatever[z72,z76];
+z5=whatever[z4,z1+4v*right];
+z6=whatever[z40,z47];
+z6=z74+4v*left;
+
+
+pickup pencircle scaled 0.4pt;
+draw(z16--z12--z72--z76);
+draw(z13--z356--z74);
+draw(z40--z47) dashed evenly;
+draw(z1--z4) dashed withdots scaled 0.7;
+draw(z4--z5) dashed withdots scaled 0.7;
+draw(z0--z6) dashed withdots scaled 0.7;
+draw(z0--z2) dashed evenly;
+draw(z1--z3) dashed evenly;
+
+draw(z100--z101) dashed withdots scaled 0.7;
+
+pickup pencircle scaled 3pt;
+drawdot(z0);
+drawdot(z1);
+
+label.bot(btex \strut 0 etex,z12);
+label.bot(btex $k$ etex,z2);
+label.llft(btex \strut ${n\over 2} - 1$ etex,z42);
+label.bot(btex \strut $k+{n\over 2}$ etex,z3);
+label.bot(btex \strut $n-1$ etex,z72);
+label.rt(btex posloupnost prohozen\'a separ\'atorem etex,z101);
+
+endfig;
+
+
+
+
+
+
+
+end;
diff --git a/old/5-addsort/sortnet.mpx b/old/5-addsort/sortnet.mpx
new file mode 100644
index 0000000..9814d6d
--- /dev/null
+++ b/old/5-addsort/sortnet.mpx
@@ -0,0 +1,548 @@
+% Written by DVItoMP, Version 0.64/color (Web2C 7.5.4)
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("a",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,-3.4869)--(4.9813,-3.4869)--
+ (4.9813,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("b",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,-3.4869)--(5.5348,-3.4869)--
+ (5.5348,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("min",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(16.6045,0.0000)--
+ (16.6045,6.6536)--(0,6.6536)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("max",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(18.5416,0.0000)--
+ (18.5416,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x1",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x2",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x3",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x4",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x5",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x1",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x2",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x3",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x4",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("x5",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(10.2394,0.0000)--
+ (10.2394,6.4204)--(0,6.4204)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("0",_n2,1.00000,5.6939,-1.4944,);
+setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
+ (10.1633,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("1",_n2,1.00000,5.6939,-1.4944,);
+setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
+ (10.1633,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,5.6939,-1.4944,);
+setbounds _p to (0,-1.4944)--(10.1633,-1.4944)--
+ (10.1633,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s(":",_n1,1.00000,0.0000,0.0000,);
+_s(":",_n1,1.00000,4.4278,0.0000,);
+_s(":",_n1,1.00000,8.8556,0.0000,);
+setbounds _p to (0,0.0000)--(13.2834,0.0000)--
+ (13.2834,1.0516)--(0,1.0516)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,5.6939,-1.4944,);
+_n4="cmsy7";
+_s(char0,_n4,1.00000,10.6188,-1.4944,);
+_n2="cmr7";
+_s("2",_n2,1.00000,16.8454,-1.4944,);
+setbounds _p to (0,-2.3246)--(21.3148,-2.3246)--
+ (21.3148,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("x",_n1,1.00000,0.0000,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,5.6939,-1.4944,);
+_n4="cmsy7";
+_s(char0,_n4,1.00000,10.6188,-1.4944,);
+_n2="cmr7";
+_s("1",_n2,1.00000,16.8454,-1.4944,);
+setbounds _p to (0,-2.3246)--(21.3148,-2.3246)--
+ (21.3148,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("n",_n1,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(5.9799,0.0000)--
+ (5.9799,4.2895)--(0,4.2895)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,6.1090,-1.4944,);
+setbounds _p to (0,-1.4944)--(11.5320,-1.4944)--
+ (11.5320,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("2",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("2",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("4",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("4",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("4",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("S",_n1,1.00000,0.0000,0.0000,);
+_n5="cmmi5";
+_s("n",_n5,1.00000,7.3045,1.1829,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,);
+_n6="cmr5";
+_s("4",_n6,1.00000,7.8033,-3.8949,);
+setbounds _p to (0,-3.8949)--(13.3857,-3.8949)--
+ (13.3857,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("Bitonic",_n0,1.00000,0.0000,0.0000,);
+_s("k"&char19,_n0,1.00000,31.1333,0.0000,);
+_s("a",_n0,1.00000,36.3914,0.0000,);
+_s("t",_n0,1.00000,44.6936,0.0000,);
+_s(char20,_n0,1.00000,48.0283,0.0000,);
+_s("r",_n0,1.00000,48.5680,0.0000,);
+_s(char19,_n0,1.00000,51.3631,0.0000,);
+_s(char16&"di",_n0,1.00000,52.4700,0.0000,);
+_s(char20,_n0,1.00000,63.2629,0.0000,);
+_s("ck",_n0,1.00000,63.5397,0.0000,);
+_s("a",_n0,1.00000,72.6721,0.0000,);
+_n1="cmmi10";
+_s("B",_n1,1.00000,80.9743,0.0000,);
+_n3="cmmi7";
+_s("n",_n3,1.00000,88.5311,-1.4944,);
+setbounds _p to (0,-1.4944)--(93.9541,-1.4944)--
+ (93.9541,6.9185)--(0,6.9185)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("8",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("4",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("4",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("M",_n1,1.00000,0.0000,0.0000,);
+_n2="cmr7";
+_s("2",_n2,1.00000,9.6652,-1.4944,);
+setbounds _p to (0,-1.4944)--(14.1345,-1.4944)--
+ (14.1345,6.8078)--(0,6.8078)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n0="cmr10";
+_s("0",_n0,1.00000,0.0000,0.0000,);
+setbounds _p to (0,-3.4869)--(4.9813,-3.4869)--
+ (4.9813,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
+_s("k",_n1,1.00000,0.0000,0.0000,);
+setbounds _p to (0,0.0000)--(5.5002,0.0000)--
+ (5.5002,6.9185)--(0,6.9185)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n3="cmmi7";
+_s("n",_n3,1.00000,1.1955,3.9226,);
+interim linecap:=0;
+vardef _r(expr _a,_w)(text _t) =
+  addto _p doublepath _a withpen pencircle scaled _w _t enddef;
+_r((1.1955,2.4907)..(6.1204,2.4907), 0.3985,);
+_n2="cmr7";
+_s("2",_n2,1.00000,1.6723,-3.4355,);
+_n7="cmsy10";
+_s(char0,_n7,1.00000,9.5298,0.0000,);
+_n0="cmr10";
+_s("1",_n0,1.00000,19.4924,0.0000,);
+setbounds _p to (0,-3.4869)--(24.4737,-3.4869)--
+ (24.4737,8.4682)--(0,8.4682)--cycle;
+_p endgroup
+mpxbreak
+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+  addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
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diff --git a/old/6-kmp/6-kmp.tex b/old/6-kmp/6-kmp.tex
new file mode 100644
index 0000000..041c06a
--- /dev/null
+++ b/old/6-kmp/6-kmp.tex
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+\input lecnotes.tex
+
+\prednaska{6}{Vyhledávání v~textu}{(zapsal: Petr Jankovský)}
+
+Nyní se budeme vìnovat následujícímu problému: v~textu délky $S$ (senì) budeme chtít najít v¹echny výskyty hledaného slova délky $J$ (jehly). Nejprve se podívejme na~jeden primitivní algoritmus, který nefunguje. Je ale zajímavé rozmyslet si, proè.
+
+\h{Hloupý algoritmus} 
+Zaèneme prvním písmenkem hledaného slova a~budeme postupnì procházet text, a¾ najdeme první výskyt poèáteèního písmenka. Poté budeme testovat, zda souhlasí i~písmenka dal¹í. Pokud nastane neshoda, v~hledaném slovì se vrátíme na~zaèátek a~v~textu pokraèujeme znakem, ve~kterém neshoda nastala. Podívejme se na~pøíklad.
+
+\s{Pøíklad:} Budeme hledat slovo |jehla| v~textu |jevkupcejejehla|. Vezmeme si tedy první písmenko |j| v~hledaném slovì a~zjistíme, ¾e v~textu se nachází hned na~zaèátku. Vezmeme tedy dal¹í písmenko |e|, které se vyskytuje jako druhé i~v~textu. Pøi tøetím písmenku ale narazíme na~neshodu. V~tuto chvíli tedy zresetujeme a~opìt hledáme výskyt písmenka |j|, tentokrát v¹ak a¾ od~tøetího písmene v~textu. Takto postupujeme postupnì dál, a¾ narazíme na~dal¹í |je|, které ov¹em není následováno písmenem~|h|, tudí¾ opìt zresetujeme a~nakonec najdeme shodu s~celým hledaným øetìzcem. V~tomto pøípadì tedy algoritmus na¹el hledané slovo.
+
+Tento algoritmus v¹ak zjevnì mù¾e hanebnì selhat. Mù¾e se stát, ¾e zaèneme porovnávat, a¾ v~jednu chvíli narazíme na~neshodu. Celý tento kus tedy pøeskoèíme. Pøi tom se ale v~tomto kusu textu mohl vyskytovat nìjaký pøekrývající se výskyt hledané \uv{jehly}. Hledejme napøíklad øetìzec |kokos| v~textu |clanekokokosu|. Algoritmus tedy zaène porovnávat. Ve~chvíli kdy najde prefix |koko| a~na~vstupu dostane~|k|, dochází k~neshodì. Proto algoritmus zresetuje a~pokraèuje v~hledání od~tohoto znaku. Najde sice je¹tì výskyt |ko|, ov¹em s~dal¹ím písmenkem |s| ji¾ dochází k~neshodì a~algotimus sel¾e. Nesprávnì se toti¾ \uv{upnul} na~první nalezené |koko| a~s~dal¹ím~|k| pak \uv{zahodil} i~správný zaèátek.
+
+Máme tedy algoritmus, který i~kdy¾ je ¹patnì, tak funguje urèitì kdykoli se první písmenko hledaného slova v~tomto slovì u¾ nikde jinde nevyskytuje -- co¾ |jehla| splòovala, ale |kokos| u¾ ne.
+
+{\I Hloupý algoritmus} se na~ka¾dé písmenko textu podívá jednou, tudí¾ èasová slo¾itost bude lineární s~délkou textu ve~kterém hledáme -- tedy $\O(S)$.
+
+\h{Pomalý algoritmus}
+Zkusíme algoritmus vylep¹it tak, aby fungoval správnì: pokud nastane nìjaká neshoda, vrátíme se zpátky tìsnì za~zaèátek toho, kdy se nám to zaèalo shodovat. To je ov¹em vlastnì skoro toté¾, jako brát postupnì v¹echny mo¾né zaèátky v~\uv{senì} a~pro ka¾dý z~nìj ovìøit, jestli se tam \uv{jehla} nachází èi nikoliv.
+
+Tento algoritmus evidentnì funguje. Bì¾í v¹ak v~èase: $S$ mo¾ných zaèátkù, krát èas potøebný na~jedno porovnání (zda se na~dané pozici nenachází \uv{jehla}), co¾ nám mù¾e trvat a¾ $J$. Proto je èasová slo¾itost $\O(SJ)$. V praxi bude algoritmus èasto rychlej¹í, proto¾e typicky velmi brzo zjistíme, ¾e se øetìzce neshodují, ale je mo¾né vymyslet vstup, kde bude potøeba porovnání opravdu tolik.
+
+Nyní se pokusme najít takový algoritmus, který by byl tak rychlý, jako {\I Hloupý algoritmus}, ale chytrý, jako ten {\I Pomalý}.
+
+\h{Chytrý algoritmus}
+Ne¾ vlastní algoritmus vybudujeme, zkusíme se cestou nauèit pøemý¹let o~øetìzcích obèas trochu pøekrouceným zpùsobem. Podívejme se na~je¹tì jeden pøíklad.
+
+\s{Pøíklad:} Vezmìme si napøíklad staré italské pøízvisko |barbarossa|, které znamená Rudovous. Pøedstavme si, ¾e takovéto slovo hledáme v~nìjakém textu, který zaèíná |barbar|. Víme, ¾e a¾ sem se nám hledaný øetìzec shodoval. Øeknìme, ¾e dal¹í písmenko textu se shodovat pøestane -- místo |o| naèteme napøíklad opìt |b|. {\I Hloupý algoritmus} by velil vrátit se k~|a| a~od~nìj hledat dál. Uvìdomme si ale, ¾e kdy¾ se vracíme z~|barbar| do~|arbar| (tedy øetìzce, který ji¾ známe), mù¾eme si pøedpoèítat, jak dopadne hledání, kdy¾ ho pustíme na~nìj. V~pøedpoèítaném bychom tedy chtìli ukládat, ¾e kdy¾ máme øetìzec |arbar|, tak |ar| a~|r| nám do~hledaného nepasuje a~a¾~|bar| se bude shodovat. Tedy místo toho, abychom spustili nové hledání od~|a|, mù¾eme ho spustit a¾~od~|b|. Co víc, my dokonce víme, jak dopadne to -- pokud toti¾ nastane neshoda po~pøeètení |barbar|, je to stejné, jako kdybychom pøeèetli pouze |bar|, na~které se (pùvodne neshodující se) |b| u¾ navázat dá. Kdyby se nedalo navázat ani tam, tak bychom opìt zkracovali... Nejen, ¾e tedy víme, kam se máme vrátit, ale víme dokonce i~to, co tam najdeme. 
+
+My¹lenka, ke které míøíme, je pøedpoèítat si nìjakou tabulku, která nám bude øíkat, jak se máme pøi hledání vracet a~jak to dopadne, a~pak u¾ jenom prohlédávat s~pou¾itím této tabulky. 
+
+Aby se nám o~tìchto algoritmech lépe mluvilo a~pøedev¹ím psalo, pojïme si povìdìt nìkolik definic.
+ 
+\s{Definice:}
+\itemize\ibull
+\:{\I Abeceda $\Sigma$} je koneèná mno¾ina znakù\foot{Mù¾eme pøi tom jít a¾~do~extrémù. Pøíkladem extrémních abeced je binární abeceda slo¾ená pouze z~nul a~jednièek. Pøíklad z~druhého konce (který rádi dìlají lingvisti) je abeceda, která má jako abecedu v¹echna èeská slova. V¹echny èeské vìty, pak nejsou nic jiného, ne¾ slova nad touto abecedou. Pou¾itá abeceda tedy mù¾e být i~relativnì obrovská. Dal¹ím takovým pøíkladem mù¾e být Unicode. Pro na¹e potøeby ale zatím budeme pøedpokládat, ¾e abeceda je nejen konstantnì velká, ale i~rozumnì malá. Budeme si moci tedy dovolit napøíklad indexovat pole znakem abecedy (kdybychom nemohli, tak bychom místo pole pou¾ili napøíklad hashovací tabulku, èi nìco podobného\dots).}, ze~kterých tvoøíme text, øetìzce, slova. 
+
+
+\:{\I $\Sigma^*$} je mno¾ina v¹ech slov nad abecedou $\Sigma$. Èili mno¾ina v¹ech neprázdných koneèných posloupností znakù ze $\Sigma$.
+\endlist
+\s{Znaèení:}
+Aby se nám nepletlo znaèení, budeme rozli¹ovat promìnné pro slova, promìnné pro písmenka a~promìnné pro èísla.
+
+\itemize\ibull
+\:{\I Slova} budeme znaèit malými písmenky øecké abecedy $\alpha$, $\beta$, ... .
+\:$\iota$ bude oznaèovat \uv{jehlu}
+\:$\sigma$ bude oznaèovat \uv{seno}
+\:{\I Znaky} oznaèíme malými písmeny latinky $a$, $b$, \dots .
+\:{\I Èísla} budeme znaèit velkými písmeny $A$, $B$, \dots .
+\:{\I Délka slova} $\vert \alpha  \vert$ pro $\alpha \in \Sigma^*$ je poèet jeho znakù.
+\:{\I Prázdné slovo} znaèíme písmenem $\varepsilon$, $\vert \varepsilon \vert = 0$.
+\:{\I Zøetìzení} $\alpha\beta$ vznikne zapsáním slov $\alpha$ a~$\beta$ za sebe. Platí $\vert \alpha\beta  \vert=\vert \alpha \vert+\vert \beta \vert$, $\alpha\varepsilon=\varepsilon\alpha=\alpha$.
+\:$\alpha[k]$ je $k$-tý znak slova $\alpha$, indexujeme od~$0$.
+\:$\alpha[k:l]$ je podslovo zaèínající $k$-tým znakem a~$l$-tý znak je první, který v~nìm není. Jedná se tedy o~podslovo skládající se z~$\alpha[k]$,$\alpha[k+1]$,\dots,$\alpha[l-1]$. Platí tedy: $\alpha[k:k]=\varepsilon$, $\alpha[k:k+1]=\alpha[k]$. Jednu (èi obì) meze mù¾eme i~vynechat, tento zápis pak bude znamenat buï \uv{od zaèátku slova a¾ nìkam}, nebo \uv{odnìkud a¾ do~konce}:
+%TODO - zaøadit pod pøedchozí bod
+\:$\alpha[:k]$ je {\I prefix} obsahující prvních $k$ znakù slova $\alpha$ ($\alpha[0]$,\dots,$\alpha[k-1]$).
+\:$\alpha[k:]$ je {\I suffix} obsahující znaky slova $\alpha$ poèínaje $k$-tým znakem a¾ do~konce.
+\:$\alpha[:] = \alpha$
+\endlist
+
+
+V¹imnìme si, ¾e prázdné slovo je prefixem, suffixem i~podslovem jakéhokoliv slova vèetnì sebe sama.
+Ka¾dé slovo je také prefixem, suffixem i~podslovem sebe sama. To se ne v¾dy hodí. Nìkdy budeme chtít øíct, ¾e nìjaké slovo je {\I vlastním} prefixem nebo suffixem. To bude znamenat, ¾e to nebude celé slovo.
+
+\> $\alpha$ je {\I vlastní prefix} slova $\beta \equiv \alpha$ je prefix $\beta~\&~\alpha \neq \beta$.
+
+\h{Vyhledávací automat (Knuth, Morris, Pratt)}
+{\I Vyhledávací automat} bude graf, jeho¾ vrcholùm budeme øíkat {\I stavy}. Jejich jména budou prefixy hledaného slova a~hrany budou odpovídat tomu, jak jeden prefix mù¾eme získat z~pøedchozího prefixu pøidáním jednoho písmene. Poèáteèní stav je prázdné slovo $\varepsilon$ a~koncový je celá $\iota$. Dopøedné hrany grafu budou popisovat pøechod mezi stavy ve~smyslu zvìt¹ení délky jména stavu (dopøedná funkce $h(\alpha)$, urèující znak na~dopøedné hranì z~$\alpha$). Zpìtné hrany grafu budou popisovat pøechod (zpìtná funkce $z(\alpha)$) mezi stavem $\alpha$ a~nejdel¹ím vlastním suffixem $\alpha$, který je prefixem $\iota$, kdy¾ nastane neshoda.
+
+\figure{barb.eps}{Vyhledávací automat.}{4.1in}
+
+\s{Hledej($\sigma$):}
+\algo
+\:$\alpha \leftarrow \varepsilon$.
+\:Pro $x\in\sigma$ postupnì:
+\:$\indent$Dokud $h(\alpha) \neq x~\&~\alpha \neq \varepsilon : \alpha \leftarrow z(\alpha)$. 
+\:$\indent$Pokud $h(\alpha) = x: \alpha \leftarrow \alpha x$.
+\:$\indent$Pokud $\alpha = \iota$, ohlásíme výskyt.
+\endalgo
+
+\>Vstupem je $\iota$, hledané slovo (jehla) délky $J=\vert \iota \vert$ a~$\sigma$, text (seno) délky $S=\vert \sigma \vert$.
+\>Výstupem jsou v¹echny výskyty hledaného slova $\iota$ v~textu $\sigma$, tedy mno¾ina $\left\{ k \mid \sigma[k:k+J]=\iota \right\}$
+
+Pojïme nyní dokázat, ¾e tento algoritmus správnì ohlásí v¹echny výskyty.
+
+\s{Definice}: $\alpha(\tau) := $ stav automatu po~pøeètení $\tau$
+
+\s{Invariant:} Pokud algoritmus pøeète nìjaký vstup, nachází se ve~stavu, který je nejdel¹ím suffixem pøeèteného vstupu, který je nìjakým stavem.
+$\alpha(\tau) =$ nejdel¹í stav (nejdel¹í prefix jehly), který je suffixem $\tau$ (pøeèteného vstupu).
+
+Pojïme si rozmyslet, ¾e z~tohoto invariantu ihned plyne, ¾e algoritmus najde to, co má. Kdykoli toti¾ ohlásí nìjaký výskyt, tak tam tento výskyt opravdu je. Kdykoli pak má nìjaký výskyt ohlásit, tak se v~této situaci jako suffix toho právì pøeèteného textu vyskytuje hledané slovo, pøièem¾ hledané slovo je urèitì stav a~zároveò nejdel¹í ze v¹ech existujících stavù. Tak¾e invariant opravdu øíká, ¾e jsme právì v~koncovém stavu a~algoritmus nám tedy ohlásí výskyt.
+
+\proof {\I (invariantu)}
+Indukcí podle kroku algoritmu. Na~zaèátku pro prázdný naètený vstup invariant triviálnì platí, tedy prázdný suffix $\tau$ je prefixem $\iota$. V~kroku $n$ máme naètený vstup $\tau$ a~k~nìmu pøipojíme znak $x$. Invariant nám øíká, ¾e nejdel¹í stav, který je suffixem, je nejdel¹í suffix, který je stavem. Nyní se ptáme, jaký je nejdel¹í stav, který se dá \uv{napasovat} na~konec øetìzce $\tau x$. Kdykoli v¹ak takovýto suffix máme, tak z~nìj mù¾eme $x$ na~konci odebrat, èím¾ dostaneme suffix slova $\tau$.
+
+\>Tedy: pokud $\beta$ je neprázdným suffixem slova $\tau x$, pak $\beta = \gamma x$, kde $\gamma$ je suffix $\tau$.
+
+Suffix, který máme sestrojit, tedy vznikne z~nìjakého suffixu slova $\tau$ pøipsáním~$x$. Chceme najít nejdel¹í suffix slova $\tau x$, který je stavem, tak¾e chceme najít i~nejdel¹í suffix pùvodního slova $\tau$, za který se dá pøidat $x$ tak, aby vy¹lo jméno stavu. Staèí tedy u¾ jen \uv{probírat} suffixy slova $\tau$ od~nejdel¹ího po~nejkrat¹í, zkou¹et k~nim pøidávat $x$ a~a¾ to pùjde, tak jsme na¹li nejdel¹í suffix $\tau x$. Pøesnì toto ov¹em algoritmus dìlá, nebo» zpìtná funkce mu v¾dy øekne nejbli¾¹í krat¹í suffix, který je stavem. Pokud pak nemù¾eme $x$ pøidat ani do~$\varepsilon$, pak je øe¹ením prázdný suffix. Algoritmus tedy funguje. \qed
+
+Nyní pojïmì zkoumat to, jak je ve~skuteènosti ná¹ algoritmus rychlý. K tomu bychom si ale nejdøív mìli øíct, jak pøesnì budeme automat reprezentovat. V~algoritmu vystupují nìjaká porovnávání stavù, pøièem¾ není úplnì jasné, jak zaøídit, aby v¹e trvalo konstantnì dlouho. Vyjde nám to ale docela snadno. K reprezentaci automatu nám toti¾ budou staèit pouze dvì pole.
+
+\s{Reprezentace automatu:}
+Oèíslujeme si stavy délkami pøíslu¹ných prefixù, tedy èísly $0 \dots J$. Poté je¹tì potøebujeme nìjakým zpùsobem zakódovat dopøedné a~zpìtné hrany. Vzhledem k~tomu, ¾e z~ka¾dého vrcholu vede v¾dy nejvý¹e jedna dopøedná a~nejvý¹e jedna zpìtná, tak nám evidentnì staèí pamatovat si pro ka¾dý typ hran pouze jedno èíslo na~vrchol. Budeme mít tedy nìjaké pole dopøedných hran, které nám pro ka¾dý stav øekne, jakým písmenkem je nadepsaná dopøedná hrana ze stavu $I$ do~$I+1$. To jsou ale pøesnì písmenka jehly, tak¾e si staèí pamatovat jehlu samotnou. Èili z~$I$ do~$I+1$ vede hrana nadepsaná $\iota [I]$. Pro zpìtné hrany pak budeme potøebovat pole $Z$, které nám pro stav $I$ øekne èíslo stavu, do~kterého vede zpìtná hrana. Tedy $Z[I]$ je cíl zpìtné hrany ze stavu $I$.
+S~touto reprezentací ji¾ doká¾eme na¹i hledací proceduru pøímoèaøe pøepsat tak, aby sahala pouze do~tìchto dvou polí:
+\algo
+\:$I \leftarrow 0$.
+\:Pro znaky $x$ z~textu:
+\:$\indent$Dokud $\iota[I] \neq x~\&~I \neq 0: I \leftarrow Z[I]$.
+\:$\indent$Pokud $\iota[I] = x$, pak $I \leftarrow I + 1$.
+\:$\indent$Pokud $I = J$, ohlásíme výskyt.
+\endalgo
+
+Zatím se v~algoritmu je¹tì skrývá drobná chyba -- toti¾ algoritmus se obèas zeptá na~dopøednou hranu z~posledního stavu. Pokud jsme právì ohlásili výskyt (jsme tedy v~posledním stavu) a~pøijde nìjaký dal¹í znak, algoritmus se ptá, zda je roven tomu, co je na~dopøedné hranì z~posledního stavu. Ta ale ov¹em neexistuje. Jednodu¹e to ale napravíme tak, ¾e si pøidáme fiktivní hranu, na~které se vyskytuje nìjaké \uv{nepísmenko} -- nìco, co se nerovná ¾ádnému jinému písmenku. Zajistíme tak, ¾e se po~této hranì nikdy nevydáme. Dodefinujeme tedy $\iota[J]$ odli¹nì od~v¹ech znakù.\foot{V jazyce C se toto dodefinování provede vlastnì zadarmo, nebo» ka¾dý øetìzec je v~nìm ukonèen znakem s~kódem nula, který se ve~vstupu nevyskytne\dots Algoritmus bude tedy fungovat i~bez tohoto dodefinování. V jiných jazycích je ale tøeba na~nìj nezapomenout!}
+
+\s{Lemma:} Funkce {\I Hledej} bì¾í v~èase $\O(S)$.
+
+\proof
+Funkce {\I Hledej} chodí po~dopøedných a~zpìtných hranách. Dopøedných hran projdeme urèitì maximálnì tolik, kolik je délka sena. Pro ka¾dý znak pøeètený ze sena toti¾ jdeme nejvý¹e jednou po~dopøedné hranì. Se zpìtnými hranami se to má tak, ¾e na~jeden pøeètený znak z~textu se mù¾eme po~zpìtné hranì vracet maximálnì $J$-krát. Z~tohoto by nám v¹ak vy¹la slo¾itost $\O(JS)$, èím¾ bychom si nepomohli. Zachrání nás ale pøímoèarý potenciál. Uvìdomme si, ¾e chùze po~dopøedné hranì zvý¹í $I$ o~jedna a~chùze po~zpìtné hranì $I$ sní¾í alespoò o~jedna. Vzhledem k~tomu, ¾e $I$ není nikdy záporné a~na~zaèátku je nulové, zjistíme, ¾e krokù zpìt mù¾e být maximálnì tolik, kolik krokù dopøedu. Èasová slo¾itost hledání je tedy lineární vzhledem k~délce sena. \qed
+
+Nyní nám zbývá na~první pohled malièkost -- toti¾ zkonstruovat automat. Zkonstruovat dopøedné hrany zvládneme zjevnì snadno, jsou toti¾ explicitnì popsané hledaným slovem. Tì¾¹í u¾ to bude pro hrany zpìtné. Vyu¾ijeme k~tomu následující pozorování:
+
+\s{Pozorování:}
+Pøedstavme si, ¾e automat u¾ máme hotový a~tím, ¾e budeme sledovat jeho chování, chceme zjistit, jak v~nìm vedou zpìtné hrany.
+Vezmìme si nìjaký stav~$\beta$. To, kam z~nìj vede zpìtná hrana zjistíme tak, ¾e spustíme automat na~øetìzec $\beta$~bez prvního písmenka a~stav, ve~kterém se automat zastaví, je pøesnì ten, kam má vést i~zpìtná hrana z~$\beta$. Jinými slovy víme, ¾e $z(\beta) = \alpha (\beta[1:])$. 
+Proè takováto vìc funguje? V¹imìme si, ¾e definice $z$ a~to, co nám o~$\alpha$ øíká invariant, je témìø toto¾né -- $z(\beta)$ je nejdel¹í vlastní suffix $\beta$, který je stavem, $\alpha(\beta)$ je nejdel¹í suffix $\beta$, který je stavem. Jediná odli¹nost je v~tom, ¾e definice $z$ narozdíl od~definice $\alpha$ zakazuje nevlastní suffixy. Jak nyní vylouèit suffix $\beta$, který by byl roven $\beta$ samotné? Zkrátíme $\beta$ o~první znak. Tím pádem v¹echny suffixy $\beta$ bez prvního znaku jsou stejné jako v¹echny vlastní suffixy $\beta$.
+
+K èemu je toto pozorování dobré? Rozmysleme si, ¾e pomocí nìj u¾ doká¾eme zkonstruovat zpìtné hrany. Není to ale trochu divné, kdy¾ pøi simulování automatu na~øetìzec bez prvního znaku u¾ zpìtné hrany potøebujeme? Není. Za chvíli zjistíme, ¾e takto mù¾eme zji¹»ovat zpìtné hrany postupnì -- a~to tak, ¾e pou¾íváme v¾dy jenom ty, které jsme u¾ sestrojili.
+ 
+Takovémuhle pøístupu, kdy pøi konstruování chtìného u¾ pou¾íváme to, co chceme sestrojit, ale pouze ten kousek, který ji¾ máme hotový, se v~angliètinì øíká {\I bootstrapping}\foot{Z~tohoto slova vzniklo i~{\I bootování} poèítaèù, kdy operaèní systém v~podstatì zavádí sám sebe. Bootstrap znamená èesky ¹truple -- tedy oèko na~konci boty, které slou¾í k~usnadnìní nazouvání. A~jak souvisí ¹truple s~algoritmem? To se zase musíme vrátit k~pøíbìhùm o~baronu Prá¹ilovi, mezi nimi¾ je i~ten, ve~kterém baron Prá¹il vypráví o~tom, jak sám sebe vytáhl z~ba¾iny za ¹truple. Stejnì tak i~my budeme algoritmus konstruovat tím, ¾e se budeme sami vytahovat za ¹truple, tedy bootstrappovat.}.
+V¹imnìme si, ¾e pøi výpoètu se vstupem $\beta$ projde automat jenom prvních $\vert \beta  \vert$ stavù. Automat se evidentnì nemù¾e dostat dál, proto¾e na~ka¾dý krok dopøedu (doprava) spotøebuje písmenko $\beta$. Tak¾e krokù doprava je maximálnì tolik, kolik je  $\vert \beta  \vert$. Jinými slovy kdybychom ji¾ mìli zkonstruované zpìtné hrany pro prvních  $\vert \beta  \vert$ stavù (tedy $0 \dots \vert \beta  \vert - 1$), tak pøi tomto výpoètu, který potøebujeme na~zkonstruování zpìtné hrany z~$\beta$, je¹tì tuto zpìtnou hranu nemù¾eme potøebovat. Vystaèíme si s~tìmi, které ji¾ máme zkonstruované.
+
+Nabízí se tedy zaèít zpìtnou hranou z~prvního znaku (která vede evidentnì do~$\varepsilon$), pak postupnì brát dal¹í stavy a~pro ka¾dý z~nich si spoèítat, kdy spustíme automat na~jméno stavu bez prvního znaku a~tím získáme dal¹í zpìtnou hranu. Toto funguje, ale je to kvadratické \dots. Máme toti¾ $J$ stavù a~pro ka¾dý z~nich nám automat bì¾í v~èase a¾ lineárním s~$J$. Jak z~toho ven?
+
+Z~prvního stavu povede zpìtná funkce do~$\varepsilon$. Pro dal¹í stavy chceme spoèítat zpìtnou funkci. Z~druhého stavu $\iota[0:2]$ tedy automat spustíme na~$\iota[1:2]$, dále pak na~$\iota[1:3]$, $\iota[1:4]$, atd. Ty øetìzce, pro které potøebujeme spo¹tìt automat, abychom dostali zpìtné hrany, jsou tedy ve~skuteènosti takové, ¾e ka¾dý dal¹í dostaneme roz¹íøením pøedchozího o~jeden znak. To jsou ale pøesnì ty stavy, kterými projde automat pøi zpracovávání øetìzce $\iota$ od~prvního znaku dál. Jedním prùchodem automatu nad jehlou bez prvního písmenka se tím pádem rovnou dozvíme v¹echny údaje, které potøebujeme.
+Z~pøedchozího pozorování plyne, ¾e nikdy nebudeme potøebovat zpìtnou hranu, kterou jsme je¹tì nezkonstruovali a~jeliko¾ víme, ¾e jedno prohledání trvá lineárnì s~délkou toho, v~èem hledáme, tak toto celé pobì¾í v~lineárním èase. Dostaneme tedy následující algoritmus:
+
+\s{Konstrukce zpìtné funkce:}
+\algo
+\:$Z[0] \leftarrow ?$, $Z[1] \leftarrow 0$.
+\:$I \leftarrow 0$.
+\:Pro $k = 2 \dots J$:
+\::$I \leftarrow \<Krok>( I , \iota [k])$.
+\::$Z[k] \leftarrow I$.
+\endalgo
+
+Zaèínáme tím, ¾e nastavíme zpìtnou hranu z~prvních dvou stavù, pøièem¾ $z[0]$ je nedefinované, proto¾e tuto zpìtnou hranu nikdy nepou¾íváme. Dále postupnì simulujeme výpoèet automatu nad slovem bez prvního znaku a~po ka¾dém kroku se dozvíme novou zpìtnou hranu. {\I Krokem} automatu pak není nic jiného ne¾ vnitøek (3. a~4. bod) na¹í hledací procedury. To, kam jsme se dostali, pak zaznamenáme jako zpìtnou funkci z~$k$.
+Èili pou¹tíme automat na~jehlu bez prvního písmenka, provedeme v¾dy jeden krok automatu (pøes dal¹í písmenko jehly) a~zapamatujeme si, jakou zpìtnou funkci jsme zrovna dostali. Díky pozorováním navíc víme, ¾e zpìtné hrany konstruujeme správnì, nikdy nepou¾ijeme zpìtnou hranu, kterou jsme je¹tì nesestrojili a~víme i~to, ¾e celou konstrukci zvládneme v~lineárním èase s~délkou jehly.
+
+\s{Vìta:} Algoritmus KMP najde v¹echny výskyty v~èase $O(J+S)$.
+
+\proof
+Lineární èas s~délkou jehly potøebujeme na~postavení automatu, lineární èas s~délkou sena pak potøebujeme na~samotné vyhledání.
+
+\h{Rabinùv-Karpùv algoritmus}
+
+Nyní si uká¾eme je¹tì jeden algoritmus na~hledání jedné jehly, který nebude mít v~nejhor¹ím pøípadì lineární slo¾itost, ale bude ji mít prùmìrnì. Bude daleko jednodu¹¹í a~uká¾e se, ¾e je v~praxi daleko rychlej¹í. Bude to algoritmus zalo¾ený na~hashování.
+
+
+Pøedstavme si, ¾e máme seno délky $S$ a~jehlu délky $J$, a~vezmìme si nìjakou hashovací funkci, které dáme na~vstup $J$-tici znakù (tedy podslova dlouhá jako jehla). Tato hashovací funkce nám je pak zobrazí do~mno¾iny $\{0,\ldots,N-1\}$ pro nìjaké dost velké~$N$. Jak nám toto pomù¾e pøi hledání jehly? Vezmeme si libovolné \uv{okénko} délky $J$ a~ne¾ budeme zji¹»ovat, zda se v~nìm jehla vyskytuje, tak si spoèítáme hashovací funkci a~porovnáme ji s~hashem jehly. Èili ptáme se, jestli je hash ze sena od~nìjaké pozice $I$ do~pozice $I+J$ roven hashi jehly -- formálnì: $h(\sigma [I: I+J ]) = h(\iota)$. Teprve tehdy, kdy¾ zjistíme, ¾e se hodnota hashovací fce shoduje, zaèneme doopravdy porovnávat øetìzce.
+
+Není to ale nìjaká hloupost? Mù¾e nám vùbec takováto konstrukce pomoci? Není to tak, ¾e na~spoèítání hashovací funkce z~$J$ znakù, potøebujeme tìch $J$ znakù pøeèíst, co¾ je stejnì rychlé, jako rovnou øetìzce porovnávat? Pou¾ijeme trik, který bude spoèívat v~tom, ¾e si zvolíme ¹ikovnou hashovací funkci. Udìláme to tak, abychom ji mohli pøi posunutí \uv {okénka} o~jeden znak doprava v~konstantním èase pøepoèítat. Chceme umìt z~$h(x_1 \dots x_j)$ spoèítat $h(x_2 \dots x_{j+1})$.
+Na~zaèátku si tedy spoèítáme hash jehly a~první $J$-tice znakù sena. Pak ji¾ jenom posouváme \uv {okénko} o~jedna, pøepoèítáme hashovací funkci a~kdy¾ se shoduje s~hashem jehly, tak porovnáme. Budeme pøitom vìøit tomu, ¾e pokud se tam jehla nevyskytuje, pak máme hashovací funkci natolik rovnomìrnou, ¾e pravdìpodobnost toho, ¾e se pøesto strefíme do~hashe jehly, je $1/N$. Jinými slovy jenom v~jednom z~øádovì $N$ pøípadù budeme porovnávat fale¹nì -- tedy provedeme porovnání a~vyjde nám, ¾e výsledek je neshoda. V~prùmìrném pøípadì tedy mù¾eme stlaèit slo¾itost a¾ témìø k~lineární.
+
+Podívejme se teï na~prùmìrnou èasovou slo¾itost. Budeme urèitì potøebovat èas na~projití jehly a~sena. Navíc strávíme nìjaký èas nad fale¹nými porovnáními, kterých bude v~prùmìru na~ka¾dý $N$-tý znak sena jedno porovnání s~jehlou -- tedy $SJ / N$, pøièem¾ $N$ mù¾eme zvolit dost velké na~to, abychom tento èlen dostali pod nìjakou rozumnou konstantu... Nakonec budeme potøebovat jedno porovnání na~ka¾dý opravdový výskyt, èemu¾ se nevyhneme. Pøipoèteme tedy je¹tì $J \cdot$ {\I $\sharp$výskytù}. Dostáváme tedy: $ \O(J+S+SJ/N+J \cdot$ {\I $\sharp$výskytù}).
+
+Zbývá malièkost -- toti¾ kde vzít hashovací funkci, která toto v¹e splòuje. Jednu si uká¾eme. Bude to vlastnì takový hezký polynom:
+$$h(x_1 \dots x_j) := \left(\sum_{I=1}^{J} x_I \cdot p^{J-I}\right) \bmod N.$$
+Jinak zapsáno se tedy jedná o:
+$$(x_1 \cdot p^{J-1} + x_2 \cdot p^{J-2} + \dots + x_J \cdot p^0 ) \bmod N.$$
+Po posunutí okénka o~jedna chceme dostat:
+$$(x_2 \cdot p^{J-1} + x_3 \cdot p^{J-2} + \dots + x_J \cdot p^1 + x_{J+1} \cdot p^0 ) \bmod N.$$
+Kdy¾ se ale podíváme na~èleny tìchto dvou polynomù, zjistíme, ¾e se li¹í jen o~málo. Pùvodní polynom staèí pøenásobit~$p$, odeèíst první èlen s~$x_1$ a~naopak pøièíst chybìjící èlen $x_{J+1}$. Dostáváme tedy:
+$$h(x_2 \dots x_{J+1}) = (p \cdot h(x_1 \dots x_J) - x_1 \cdot p^J + x_{J+1}) \bmod N.$$
+Pøepoèítání hashovací funkce tedy není nic jiného, ne¾ pøenásobení té minulé~$p$, odeètení nìjakého násobku toho znaku, který vypadl z~okénka, a~pøiètení toho znaku, o~který se okénko posunulo. Pokud tedy máme k~dispozici aritmetické operace v~konstantním èase, zvládneme konstantnì pøepoèítávat i~hashovací funkci.
+
+Tato hashovací funkce se dokonce nejen hezky poèítá, ale dokonce se i~opravdu \uv{hezky} chová (tedy \uv{rozumnì} náhodnì), pokud zvolíme vhodné~$p$. To bychom mìli zvolit tak, aby bylo rozhodnì nesoudìlné s~$N$ -- tedy $\<NSD>(p, N) = 1$. Aby se nám navíc dobøe projevilo modulo obsa¾ené v~hashovací funkci, mìlo by být~$p$ relativnì velké (lze dopoèítat, ¾e optimum je mezi $2/3 \cdot N$ a~$3/4 \cdot N$). S~takto zvoleným~$p$ se tato hashovací funkce chová velmi pøíznivì a~v~praxi má celý algoritmus takøka lineární èasovou slo¾itost (prùmìrnou).
+
+\h{Hledání více øetìzcù najednou}
+Nyní si zahrajeme tuté¾ hru, ov¹em v~trochu slo¾itìj¹ích kulisách. Podíváme se na~algoritmus, který si poradí i~s více ne¾ jednou jehlou. 
+Mìjme tedy jehly $\iota_1 \dots \iota_n$, a~jejich délky $J_i = \vert \iota_i \vert $. Dále budeme potøebovat seno $\sigma$ délky $S=\vert \sigma \vert$.
+
+Pøedtím, ne¾ se pustíme do~vlastního vyhledávacího algoritmu, mo¾ná bychom si mìli ujasnit, co vlastnì bude jeho výstupem. U problému hledání jedné jehly to bylo jasné -- byla to nìjaká mno¾ina pozic v~senì, na~kterých zaèínaly výskyty jehly. Jak tomu ale bude zde? Sice bychom také mohli vrátit pouze mno¾inu pozic, ale my budeme chtít malièko víc. Budeme toti¾ chtít vìdìt i~to, která jehla se na~které pozici vyskytuje. Výstup tedy bude vypadat následovnì: $V = \{(i,j)~\vert~\sigma[i:i+J_j]= \iota_j \}$.
+
+Zde se v¹ak skrývá jedna drobná zrada. Budeme se asi muset vzdát nadìje, ¾e najdeme algoritmus, jeho¾ slo¾itost je lineární v~celkové délce v¹ech jehel a~sena. Výstup toti¾ mù¾e být del¹í ne¾ lineární. Mù¾e se nám klidnì stát, ¾e na~jedné pozici v~senì se bude vyskytovat více rùzných jehel -- pokud bude jedna jehla prefixem jiné (co¾ jsme nikde nezakázali), tak máme povinnost ohlásit oba výskyty. Vzhledem k~tomu budeme hledat takový algoritmus, který bude lineární v~délce vstupu plus délce výstupu, co¾ je evidentnì to nejlep¹í, èeho mù¾eme dosáhnout.
+
+Algoritmus, který si nyní uká¾eme, vymysleli nìkdy v~roce 1975 pan Aho a~paní Corasicková. Bude to takové zobecnìní Knuthova-Morrisova-Prattova algoritmu.
+
+\h{Algoritmus Aho-Corasicková}
+
+Opìt se budeme sna¾it sestrojit nìjaký vyhledávací automat a~nìjakým zpùsobem tento automat pou¾ít k~procházení sena. Podívejme se nejprve na~pøíklad. Budeme chtít vyhledávat tato slova: |ara|, |bar|, |arab|, |baraba|, |barbara|. Mìjme tedy tìchto pìt jehel a~rozmysleme si, jak by vypadal nìjaký automat, který by tato slova umìl zatím jenom rozpoznávat. Pro jedno slovo automat vypadal jako cesta, zde u¾ to bude strom. (viz obrázek).
+
+\figure{ara_strom_blank.eps}{Vyhledávací automat -- strom.}{1in}
+
+Navíc budeme muset do~automatu zanést, kde nìjaké slovo konèí. V~pùvodním automatu pro jedno slovo to bylo jednoduché -- ono jedno jediné slovo odpovídalo poslednímu vrcholu cesty. Tady se v¹ak slova mohou vyskytovat vícekrát a~konèit nejenom v~listech ale i~v~nìjakém vnitøním vrcholu (co¾ se stane tehdy, pokud je jedno hledané slovo prefixem jiného hledaného slova). Formálnì to nebudeme dokazovat, ale snadno nahlédneme, ¾e listy stromu odpovídají hledaným slovùm, ale opaènì to neplatí.
+
+\figure{ara_strom_end.eps}{Vyhledávací automat s~konci slov.}{1in}
+
+Dále bychom mìli do~automatu pøidat zpìtné hrany. Jejich definice bude úplnì stejná jako u automatu pro hledání jednoho slova. Jinými slovy z~ka¾dého stavu pùjde zpìtná hrana do~nejdel¹ího vlastního suffixu, který je stavem. Èili kdy¾ budeme mít nìjaké jméno stavu, budeme se ho sna¾it co nejménì (ale alespoò o~znak) zkrátit zleva, abychom zase dostali jméno stavu. Z~koøene -- prázdného stavu -- pak evidentnì ¾ádná zpìtná hrana nepovede.
+
+\figure{ara_strom_final.eps}{Vyhledávací automat se zpìtnými hranami.}{1,25in}
+
+Zbývá nám je¹tì si rozmyslet, jakým zpùsobem bude ná¹ automat hlásit výstup. Opìt smìøujeme k~tomu, aby se automat po~pøeètení nìjakého kusu textu nacházel ve~stavu odpovídajícímu nejdel¹ímu mo¾nému suffixu toho textu. Zatímco u hledání jediné jehly bylo hlá¹ení výskytù jednoduché -- kdykoliv jsme se dostali na~konec \uv{automatové cestièky} tady to bude opìt slo¾itìj¹í.
+
+První, co se nabízí, je vyu¾ít toho, ¾e jsme si oznaèili nìjaké vrcholy, kde hledaná slova konèí. Co tedy zkusit hlásit výskyt tohoto slova v¾dy, kdy¾ pøijdeme do~nìjakého oznaèeného vrcholu? Tento zpùsob v¹ak nefunguje, pokud se uvnitø nìkteré jehly skrývá jehla vnoøená. Napøíklad po~pøeètení slova |bara|, nám ná¹ souèasný automat neøíká, ¾e bychom mìli nìjaké slovo ohlásit, a~pøitom tam evidentì konèí podøetìzec |ara|. Stejnì tak pokud pøeèteme |barbara|, u¾ si nev¹imneme toho, ¾e tam konèí zároveò i~|ara|. Pouhé \uv{hlá¹ení teèek} tedy nefunguje.
+
+Dále si mù¾eme v¹imnout toho, ¾e v¹echna slova, která by se mìla v~daném stavu hlásit, jsou suffixy jména tohoto stavu. Pøitom víme, ¾e zpìtná hrana jméno stavu zkracuje zleva. Tak¾e speciálnì v¹echny suffixy daného stavu, které jsou také stavy, se dají najít tak, ¾e se vydáme po~zpìtných hranách do~koøene. Nabízí se tedy v¾dy projít cestu po~zpìtných hranách a¾ do~koøene a~hlásit v¹echny \uv{teèky}. Tento zpùsob by nám v¹ak celý algoritmus znaènì zpomalil, proto¾e cesta do~koøene mù¾e být relativnì dlouhá, ale teèek na~ní obvykle bude málo.
+
+Mohli bychom také zkusit si pro ka¾dý stav $\beta$ pøedpoèítat mno¾inu $cache(\beta)$, která by obsahovala v¹echna slova, která máme hlásit, kdy¾ se ve~stavu $\beta$ nacházíme. Pokud pak do~tohoto stavu vstoupíme, podíváme se na~tuto mno¾inu a~vypí¹eme v¹e, co v~ní je. Výpis nám bude evidentnì trvat lineárnì k~velikosti mno¾iny, celkovì tedy lineárnì k~velikosti výstupu. Problém je ale ten, ¾e jednotlivé cache mohou být hodnì velké, tak¾e je nestihneme sestrojit v lineárním èase. (Rozmyslete si pøíklad slovníku, kdy se to stane.)
+
+To, co nám ale ji¾ opravdu pomù¾e, bude zavedení zkratek. V¹imli jsme si, ¾e po~zpìtných hranách mù¾eme projít do~koøene a~hlásit v¹echny nalezené teèky. Vadilo nám ale, ¾e se mù¾e stát, ¾e budeme dlouho po~cestì chodit a~pøi tom ¾ádné teèky nenalézat. Zavedeme si proto zkratky k~nejbli¾¹í teèce. 
+
+\s{Definice} (zkratková hrana):
+Budeme mít tedy nìjakou funkci $slovo(\beta) :=$ slovo, které konèí ve~stavu $\beta$ (nebo $\emptyset$, pokud ¾ádné takové slovo není). Dále pak funkci $out(\beta) :=$ nejbli¾¹í vrchol dosa¾itelný po~zpìtných hranách, èili nejdel¹í vlastní suffix stavu $\beta$, v~nìm¾ je definovaná funkce $slovo$. Trochu lid¹tìji øeèeno, ten nejbli¾¹í dosa¾itelný vrchol, ve~kterém je teèka.
+
+Po pøidání tìchto zkratkových hran ji¾ máme reprezentaci, ve~které opravdu umíme v~daném stavu vyjmenovat v¹echna slova, která máme vypsat, a~to v~èase lineárním s~tím, kolik tìch slov je.
+
+\s{Definice:}
+Vyhledávací automat sestává ze stromu dopøedných hran (vrcholy jsou prefixy jehel, hrany odpovídají roz¹íøení o~písmenko), zpìtných hran ($z(\beta) :=$ nejdel¹í vlastní suffix slova $\beta$, který je stavem) a~zkratkových hran.
+
+Automat pak bude na~na¹em pøíkladu vypadat takto (zkratkové hrany jsou znázornìny zelenì):
+
+\figure{ara_strom_zkr.eps}{Vyhledávací automat se zkratkovými hranami.}{1,25in}
+
+Nyní u¾ nám zbývá jenom vlastní algoritmus -- nejdøív popí¹eme algoritmus, který bude hledat pomocí takového automatu, a~potom se pustíme do~toho, jak se takový automat staví.
+
+Nejprve si nadefinujeme, jak vypadá jeden krok automatu. Bude to vlastnì nìjaká funkce, která dostane stav a~písmenko. Ona nás pak pomocí tohoto písmenka posune po~automatu. ($f(\alpha, x)$ bude dopøedná hrana ze stavu $\alpha$ oznaèená písmenem~$x$)
+
+\s{Krok ($\alpha$, $x$):}
+\algo
+\:Dokud $f(\alpha, x) = \emptyset~\&~\alpha \neq \<koøen:>~~\alpha \leftarrow z(\alpha)$.
+\:Pokud $f(\alpha, x) \neq \emptyset:~~\alpha \leftarrow f(\alpha, x)$.
+\:Vrátíme výsledek.
+\endalgo
+
+\s{Hledání:}
+\algo
+\:$\alpha \leftarrow \<koøen>$.
+\:Pro znaky $x$ ze slova $\sigma$:
+\::$\alpha \leftarrow \<Krok>(\alpha, x)$.
+\::$\beta \leftarrow \alpha$
+\::Dokud $\beta \neq \emptyset$:
+\:::Je-li $\<slovo>(\beta) \neq \emptyset$:
+\::::Ohlásíme $\<slovo>(\beta)$.
+\:::$\beta \leftarrow \<out>(\beta)$.
+\endalgo
+
+Algoritmus hledání vlastnì není nic jiného, ne¾ prosté projití po~zelených zkratkových hranách ze stavu $\alpha$, ve~kterém právì jsme, a~ohlá¹ení v¹eho, co po~cestì najdeme.
+
+V ka¾dém okam¾iku se automat nachází ve~stavu, který odpovídá nejdel¹ímu mo¾nému suffixu toho, co jsme u¾ pøeèetli. Dùkaz tohoto invariantu je stejný jako u verze automatu pro hledání pouze jedné jehly, nebo» vychází pouze z~definice zpìtných hran. Podobnì nahlédneme, ¾e èasová slo¾itost vyhledávací procedury je lineární v~délce sena plus to, co spotøebujeme na~hlá¹ení výskytù. Nejprve na~chvíli zapomeneme, ¾e nìjaké výskyty hlásíme a~spoèítáme jenom kroky. Ty mohou vést dopøedu a~zpátky. Krok dopøedu prodlu¾uje jméno stavu o~jedna, krok dozadu zkracuje aspoò o~jedna. Tudí¾ krokù dozadu je maximálnì tolik, co krokù dopøedu a~krokù dopøedu je maximálnì tolik, kolik je délka sena. V¹echny kroky dohromady tedy trvají $\O(S)$. Hlá¹ení výskytù pak trvá $\O(S~+ \vert V \vert)$. Celé hledání tedy trvá lineárnì v~délce vstupu a~výstupu.
+
+Zbývá nám u¾ jen konstrukce automatu. Opìt vyu¾ijeme faktu, ¾e zpìtná hrana ze stavu $\beta$ vede tam, kam by se dostal automat pøi hledání $\beta$ bez prvního písmenka. Tak¾e zase chceme nìco, jako simulovat výpoèet toho automatu na~slovech bez prvního písmenka a~doufat v~to, ¾e si vystaèíme s~tou èástí automatu, kterou jsme u¾ postavili. Tentokrát to v¹ak nemù¾eme dìlat jedno slovo po~druhém, proto¾e zpìtné hrany mohou vést køí¾em mezi jednotlivými vìtvemi automatu. Mohlo by se nám tedy stát, ¾e pøi hledání nìjakého slova potøebujeme zpìtnou hranu, která vede do~jiného slova, které jsme je¹tì nezkonstruovali. Tak¾e tento postup sel¾e. Mù¾eme v¹ak vyu¾ít toho, ¾e ka¾dá zpìtná hrana vede ve~stromu alespoò o~jednu hladinu vý¹. Mù¾eme tak strom konstruovat po~hladinách. Lze si to tedy pøedstavit tak, ¾e paralelnì spustíme vyhledávání v¹ech slov bez prvních písmenek a~v¾dycky udìláme jeden podkrok ka¾dého z~tìch hledání, co¾ nám dá zpìtné hrany z~dal¹ího patra stromu.
+
+\s{Konstrukce automatu:}
+\algo
+\:Zalo¾íme prázdný strom, $r \leftarrow$ jeho koøen.
+\:Vlo¾íme do~stromu slova $\iota_1 \dots \iota_n$, nastavíme $slovo(*)$.
+\:$z(r) \leftarrow \emptyset$, $out(r) \leftarrow \emptyset$.
+\:Zalo¾íme frontu $F$ a~vlo¾íme do~ní syny koøene.
+\:$\forall v~\in F:~~z(v) \leftarrow r, \<out>(v) \leftarrow \emptyset$.
+\:Dokud $F \neq \emptyset$:
+\::Vybereme $u$ z~fronty $F$.
+\::Pro v¹echny syny $v$ vrcholu $u$:
+\:::$q \leftarrow \<Krok>(z(u), \<písmeno na~hranì uv>)$.
+\:::$z(v) \leftarrow q$.
+\:::Pokud $slovo(q) \neq \emptyset$, pak $out(v) \leftarrow q$.
+\::::Jinak $out(v) \leftarrow out(q)$.
+\:::Vlo¾íme $v$ do~fronty $F$.
+\endalgo
+
+To, ¾e tento algoritmus zkonstruuje zpìtné hrany jak má, vyplývá z~toho, ¾e nedìláme nic jiného, ne¾ ¾e spou¹tíme výpoèty po~hladinách na~v¹echna hledaná slova bez prvního písmenka. Stejnì tak to, ¾e dobìhne v~lineárním èase, je takté¾ dùsledkem toho, ¾e efektivnì spou¹tíme v¹echny tyto výpoèty. Jen nìkdy udìláme najednou krok dvou èi více výpoètù (napøíklad |araba| a~|arbara| se poèítají na~zaèátku, dokud jsou stejné, jen jednou). Èasová slo¾itost této konstrukce je tedy men¹í nebo rovna souètu èasových slo¾itostí výpoètù nad v¹emi tìmi slovy. To u¾ ale víme, ¾e je lineární v~celkové délce tìchto slov. Konstrukce automatu tedy trvá nejvý¹e tolik, co hledání v¹ech $\iota_i$, co¾ je $\O(\sum_{i} \iota_i)$.
+
+\s{Vìta:} Algoritmus Aho-Corasicková najde v¹echny výskyty v~èase 
+$$\O\left(\sum_i~\iota_i~+~S~+~\sharp\<výskytù>\right).$$
+
+Je¹tì se na~závìr zamysleme, jak bychom si takový automat ukládali do~pamìti. Urèitì se nám bude hodit si stavy nìjak oèíslovat (tøeba v~poøadí, v~jakém budou vznikat). Potom funkce pro zpìtné a~zkratkové hrany mohou být reprezentované polem indexovaným èíslem stavu. Funkce {\I Slovo}, která øíká, jaké slovo ve~stavu konèí, zase mù¾e být pole indexované stavem, které nám øekne poøadové èíslo slova ve~slovníku. Pro dopøedné hrany v~ka¾dém vrcholu pak mù¾eme mít pole indexované písmenky abecedy, které nám pro ka¾dé písmenko øekne, buï ¾e taková hrana není, nebo nám øekne, kam tato hrana vede. Je vidìt, ¾e takovéto pole se hodí pro pomìrnì malé abecedy. U¾ pro abecedu A-Z~bude velikosti 26 a~z~vìt¹iny bude prázdné, tak¾e bychom plýtvali pamìtí. V praxi se proto èasto pou¾ívá hashovací tabulka. Pøípadnì bychom mohli mít i~jen jednu velkou spoleènou hashovací tabulku, která bude reprezentovat funkci celou, ve~které budou zahashované dvojice (stav, písmenko). Tìchto dvojic je evidentnì tolik, kolik hran stromu, èili lineárnì s~velikostí slovníku, a~je to asi nejkompaktnìj¹í reprezentace.
+
+
+\bye
diff --git a/old/6-kmp/Makefile b/old/6-kmp/Makefile
new file mode 100644
index 0000000..1831d14
--- /dev/null
+++ b/old/6-kmp/Makefile
@@ -0,0 +1,3 @@
+P=6-kmp
+
+include ../Makerules
diff --git a/old/6-kmp/ara_strom.eps b/old/6-kmp/ara_strom.eps
new file mode 100644
index 0000000..7d054ab
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+       id="text2446"><tspan
+         sodipodi:role="line"
+         id="tspan2448"
+         x="374.26086"
+         y="138.77878">barba</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
+       x="449.23392"
+       y="138.77879"
+       id="text2450"><tspan
+         sodipodi:role="line"
+         id="tspan2452"
+         x="449.23392"
+         y="138.77879">barbar</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
+       x="522.25323"
+       y="138.77878"
+       id="text2454"><tspan
+         sodipodi:role="line"
+         id="tspan2456"
+         x="522.25323"
+         y="138.77878">barbaro</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
+       x="653.50238"
+       y="138.17416"
+       id="text2458"><tspan
+         sodipodi:role="line"
+         id="tspan2460"
+         x="653.50238"
+         y="138.17416">barbarossa</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
+       x="621.96454"
+       y="139.35719"
+       id="text2466"><tspan
+         sodipodi:role="line"
+         id="tspan2468"
+         x="621.96454"
+         y="139.35719">...</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-size:18px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Arial;-inkscape-font-specification:Arial"
+       x="37.04578"
+       y="177.70506"
+       id="text2470"><tspan
+         sodipodi:role="line"
+         x="37.04578"
+         y="177.70506"
+         id="tspan2482">b              a              r               b             a              r              o             ...             a  </tspan></text>
+    <path
+       style="fill:none;fill-rule:nonzero;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#Arrow2Lend);fill-opacity:1"
+       d="M 83.509975,164.51929 C 60.542102,190.63893 35.300495,196.8634 5.651051,164.51929"
+       id="path5609"
+       inkscape:connector-type="polyline"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 13.185786,153.96139 L 78.486819,153.96139"
+       id="path7588"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 90.747017,153.9613 L 156.04805,153.9613"
+       id="path9700"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 168.72226,153.9613 L 234.02329,153.9613"
+       id="path9702"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 245.95329,153.96121 L 311.25432,153.96121"
+       id="path9704"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 323.85853,153.9613 L 389.15956,153.9613"
+       id="path9706"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 401.50851,153.9613 L 466.80954,153.9613"
+       id="path9708"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 480.48321,153.9613 L 545.78425,153.9613"
+       id="path9710"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 558.48215,153.9613 L 623.78318,153.9613"
+       id="path9712"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-start:none;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 638.59633,153.9613 L 703.89736,153.9613"
+       id="path9714"
+       inkscape:connector-type="polyline" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 160.62469,178.6356 C 103.11439,214.93588 51.117239,212.64411 3.651051,178.6356"
+       id="path9716"
+       inkscape:connector-type="polyline"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 241.11151,117.89139 C 160.02376,69.180736 82.359446,79.809701 6.2789455,117.89139"
+       id="path9718"
+       inkscape:connector-type="polyline"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 319.79709,118.68308 C 234.48869,66.453676 158.66445,84.678135 84.96452,118.68308"
+       id="path10774"
+       inkscape:connector-type="polyline"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 397.16706,119.82255 C 317.23195,75.067632 238.9209,74.162508 162.33449,119.82255"
+       id="path10776"
+       inkscape:connector-type="polyline"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker-end:url(#Arrow2Lend);stroke-opacity:1"
+       d="M 473.93389,173.72993 C 393.31902,208.79011 314.69794,213.94352 239.10133,173.72993"
+       id="path10778"
+       inkscape:connector-type="polyline"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="opacity:1;fill:none;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.99999982px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:url(#Arrow2Lend);stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       d="M 551.6176,159.60837 C 426.17515,260.43382 -84.115921,225.79475 -0.53486469,163.7889"
+       id="path9778"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="opacity:1;fill:none;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       d="M 631.09082,161.52172 C 584.4479,211.71214 440.8137,218.21863 316.61606,220.87291"
+       id="path10874"
+       sodipodi:nodetypes="cc" />
+    <path
+       style="opacity:1;fill:none;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.99999994px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate"
+       d="M 711.15927,159.63036 C 703.71296,189.73566 643.62165,205.468 570.98179,213.59973 C 531.11935,218.06216 487.47795,220.2357 446.68392,221.23953 C 366.0505,223.2237 296.54156,220.6379 289.32855,222.12506"
+       id="path10880"
+       sodipodi:nodetypes="cssc" />
+  </g>
+</svg>
diff --git a/old/7-geom/7-geom.mp b/old/7-geom/7-geom.mp
new file mode 100644
index 0000000..7375c63
--- /dev/null
+++ b/old/7-geom/7-geom.mp
@@ -0,0 +1,276 @@
+input lib
+
+figname("7-geom");
+
+figtag("male_obaly");
+beginfig(1);
+	pickup boldpen;
+	labeloffset:=1cm;
+	pair c,pos; c := (0,0); pos := c;
+	drawemptyvertex(c);
+	label.bot(btex $n=1$ etex, pos);
+	c := (2cm,0);
+	pos := pos + c;
+	pair A[];
+	A[0] := (-0.3cm, -0.2cm)+c; A[1] := (0.2cm, 0.3cm)+c;
+	draw A[0]--A[1];
+	drawemptyvertex(A[0]); drawemptyvertex(A[1]);
+	label.bot(btex $n=2$ etex, pos);
+
+	pos := pos + c;
+	A[2] := (+0.3cm, -0.4cm)+c;
+	for i := 0 upto 2: A[i] := A[i] shifted c; endfor
+	draw A[0]--A[1]--A[2]--cycle;
+	for i := 0 upto 2: drawemptyvertex(A[i]); endfor
+	label.bot(btex $n=3$ etex, pos);
+
+	pos := pos + c;
+	A[3] := (A[0]+A[1]+A[2])/3;
+	for i := 0 upto 3: A[i] := A[i] shifted c; endfor
+	draw A[0]--A[1]--A[2]--cycle;
+	for i := 0 upto 2: drawemptyvertex(A[i]); endfor
+	draw vertex(A[3]);
+	
+	c := (1cm,0);
+	pos := pos + c/2;
+	A[3] := A[1]+(0.3cm,-0.2cm);
+	for i := 0 upto 3: A[i] := A[i] shifted c; endfor
+	draw A[0]--A[1]--A[3]--A[2]--cycle;
+	for i := 0 upto 3: drawemptyvertex(A[i]); endfor
+	label.bot(btex $n=4$ etex, pos);
+endfig;
+
+figtag("pridani_bodu");
+beginfig(2);
+	pair A[],B[],C,shift; shift := (4.5cm,0);
+	A[0] := (-1.7cm,1.1cm);
+	A[1] := (-1.2cm,1.2cm);
+	A[2] := (-0.4cm,1cm);
+	A[3] := (0.2cm,0.2cm);
+	A[4] := (0.4cm,-0.7cm);
+	A[5] := (-0.8cm,-1.3cm);
+	A[6] := (-1.4cm,-1.4cm);
+	B[0] := (-1.1cm, 0.7cm);
+	B[1] := (-0.6cm, 0.1cm);
+	B[2] := (-1.3cm, -0.6cm);
+	C := (1cm, 0.1cm);
+
+	% krok 1
+	pickup boldpen;
+	draw A[0] for i := 1 upto 6: --A[i] endfor;
+	for i := 1 upto 5: drawemptyvertex(A[i]); endfor
+	for i := 0 upto 2: draw vertex(B[i]); endfor
+	draw vertex(C);
+	drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0, -0.1cm) withpen normalpen;
+	for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
+	for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
+	C := C shifted shift;
+
+	% krok 2
+	draw A[0] for i := 1 upto 6: --A[i] endfor;
+	draw A[4]{dir 70}..C;
+	draw A[4]{dir 45}..C;
+	draw C--A[2] dashed evenly withpen normalpen;
+	draw C--A[3] dashed evenly withpen normalpen;
+	for i := 1 upto 5: drawemptyvertex(A[i]); endfor
+	for i := 0 upto 2: draw vertex(B[i]); endfor
+	drawemptyvertex(C);
+	drawarrow (C+(0.5cm,0)--C+(1.5cm,0)) shifted (0,-0.1cm) withpen normalpen;
+	for i:=0 upto 6: A[i] := A[i] shifted shift; endfor
+	for i:=0 upto 2: B[i] := B[i] shifted shift; endfor
+	C := C shifted shift;
+
+	% krok 3
+	draw for i := 0 upto 2: A[i]-- endfor C for i := 4 upto 6: --A[i] endfor;
+	for i := 1 upto 2: drawemptyvertex(A[i]); endfor
+	for i := 4 upto 5: drawemptyvertex(A[i]); endfor
+	for i := 0 upto 2: draw vertex(B[i]); endfor
+	draw vertex(A[3]);
+	drawemptyvertex(C);
+endfig;
+
+figtag("obalky");
+beginfig(3);
+	labeloffset := 0.2cm;
+	pickup boldpen;
+	pair A[],B[];
+	A[0] := (-7cm, 0cm);
+	A[1] := (-6.2cm, 0.9cm);
+	A[2] := (-4.6cm,1.5cm);
+	A[3] := (-2.4cm,1.8cm);
+	A[4] := (-0.8cm,1.5cm);
+	A[5] := (0.4cm,0.6cm);
+	A[6] := (0.8cm,-0.10cm);
+	A[7] := (-1.6cm,-1.9cm);
+	A[8] := (-4cm,-2.1cm);
+	A[9] := (-6cm, -1.5cm);
+	A[10] := (-7cm, 0cm);
+	
+	B[0] := (-2.2cm, 0.7cm);
+	B[1] := (-1.2cm, 0.1cm);
+	B[2] := (-2.6cm, -0.6cm);
+	B[3] := (-3.6cm, -0.4cm);
+	B[4] := (-3cm, 0.6cm);
+	B[5] := (-2.6cm, 1cm);
+	B[6] := (-1cm, -1.2cm);
+	B[7] := (-6.5cm, 0.2cm);
+	B[8] := (-5cm, 0.8cm);
+	B[9] := (-6cm, -0.6cm);
+	B[10] := (-5cm, -1.2cm);
+	
+	draw createpath(for i := 0 upto 5: A[i]-- endfor A[6]);
+	draw (for i := 6 upto 9: A[i]-- endfor A[10]) dashed evenly;
+	for i := 0 upto 9: drawemptyvertex(A[i]); endfor
+	for i := 0 upto 10: draw vertex(B[i]); endfor
+
+	label(btex \font\myfont=csr10 \myfont horní obálka etex, ((-7cm+0.8cm)/2,2.2cm));
+	label(btex \font\myfont=csr10 \myfont dolní obálka etex, ((-7cm+0.8cm)/2,-2.5cm));
+	label.lft(btex $L$ etex, A[0]);
+	label.rt(btex $P$ etex, A[6]);
+endfig;
+
+figtag("determinant");
+beginfig(4);
+	labeloffset := 0.1cm;
+	pair A[], shift; shift := (4cm,1cm);
+
+	% det(M) > 0
+	A[0] := (-2cm, 0);
+	A[1] := (0,-1cm);
+	A[2] := (1.5cm, 0cm);
+	A[3] := A[0] + A[2] - A[1];
+	
+	fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
+	draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
+	drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+	drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+	for i:=0 upto 2: draw vertex(A[i]); endfor
+	label.lft(btex $h_{k-1}$ etex, A[0]);
+	label.bot(btex $h_k$ etex, A[1]);
+	label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+	label.lrt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+	label.rt(btex $b$ etex, A[2]);
+	label(btex $\det(M) > 0$ etex, 0.5[A[0],A[2]]);
+
+	% det(M) = 0
+	A[0] := (-1cm, -0.5cm) + shift;
+	A[1] := (0, -1cm) + shift;
+	A[2] := (1cm, -1.5cm) + shift;
+	drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+	drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+	for i:=0 upto 2: draw vertex(A[i]); endfor
+	label.lft(btex $h_{k-1}$ etex, A[0]);
+	label.llft(btex $h_k$ etex, A[1]);
+	label.llft(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+	label.llft(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+	label.bot(btex $b$ etex, A[2]);
+	label(btex $\det(M) = 0$ etex, origin) shifted (0,0.3cm) rotated -28 shifted 0.5[A[0], A[2]];
+
+	% det(M) < 0
+	shift := (7.5cm, 1.25cm);
+	A[0] := (-1cm, -0.5cm) + shift;
+	A[1] := (1.5cm, -1cm) + shift;
+	A[2] := (2cm, -2.5cm) + shift;
+	A[3] := A[0] + A[2] - A[1];
+	fill A[0]--A[1]--A[2]--A[3]--cycle withcolor 0.8white;
+	draw A[0]--A[1]--A[2]--A[3]--cycle dashed evenly;
+	drawarrow A[0]--0.96[A[0],A[1]] withpen boldpen;
+	drawarrow A[1]--0.96[A[1],A[2]] withpen boldpen;
+	for i:=0 upto 2: draw vertex(A[i]); endfor
+	label.lft(btex $h_{k-1}$ etex, A[0]);
+	label.urt(btex $h_k$ etex, A[1]);
+	label.top(btex $\vec u$ etex, 0.5[A[0],A[1]]);
+	label.rt(btex $\vec v$ etex, 0.5[A[1],A[2]]);
+	label.rt(btex $b$ etex, A[2]);
+	label(btex $\det(M) < 0$ etex, 0.5[A[0],A[2]]);
+endfig;
+
+figtag("rybi_motivace");
+beginfig(5);
+	u := 0.3cm;
+	def draw_fish(expr pos,size,rot) =
+		draw ((-1.3u*size,0){dir 60}..{right}(u*size,-u*size/4)) rotated rot shifted pos;
+		draw ((-1.3u*size,0){dir -50}..{right}(u*size,u*size/4)) rotated rot shifted pos;
+		draw ((u*size,-u*size/4)--(u*size,u*size/4)) rotated rot shifted pos;
+		draw (-1u*size,u*size/15) rotated rot shifted pos withpen pencircle scaled (u/8);
+		for i:=1 upto 3: draw (dirs((u*size,-u*size/4+i*u*size/8), 180, u*size/6)) rotated rot shifted pos; endfor
+	enddef;
+	
+	pair A[],B[];
+	A[0] := (-7cm, 0cm);
+	A[1] := (-6.2cm, 0.9cm);
+	A[2] := (-4.6cm,1.5cm);
+	A[3] := (-2.4cm,1.8cm);
+	A[4] := (-0.8cm,1.5cm);
+	A[5] := (0.4cm,0.6cm);
+	A[6] := (0.8cm,-0.10cm);
+	A[7] := (-1.6cm,-1.9cm);
+	A[8] := (-4cm,-2.1cm);
+	A[9] := (-6cm, -1.5cm);
+	A[10] := (-7cm, 0cm);
+	
+	B[0] := (-2.2cm, 0.7cm);
+	B[1] := (-1.2cm, 0.1cm);
+	B[2] := (-2.6cm, -0.6cm);
+	B[3] := (-3.6cm, -0.4cm);
+	B[4] := (-3cm, 0.6cm);
+	B[5] := (-2.6cm, 1cm);
+	B[6] := (-1cm, -1.2cm);
+	B[7] := (-6.5cm, 0.2cm);
+	B[8] := (-5cm, 0.8cm);
+	B[9] := (-6cm, -0.6cm);
+	B[10] := (-5cm, -1.2cm);
+
+	for i:=0 upto 9: draw_fish(A[i], 1, 0); endfor;
+	for i:=0 upto 10: draw_fish(B[i], 1, 0); endfor;
+	draw createpath(for i:=0 upto 9: A[i]-- endfor cycle) scaled 1.13 shifted (0.4cm,0) withpen boldpen;
+endfig;
+
+figtag("provazkovy_algoritmus");
+beginfig(6);
+	pickup boldpen;
+	pair A[],B[],u; u := (-3cm, 0);
+	for i := 0 upto 3: A[i] := u rotated (-30*i) yscaled 0.7; endfor;
+	A[2] := A[2] + (0,0.1cm);
+	draw for i:=0 upto 2: A[i]-- endfor A[3];
+	drawarrow ((u/2) for i:=1 upto 3: ..u/2 rotated (-30*i) endfor) yscaled 0.7 withpen normalpen;
+	B[0] := (-2cm,0.5cm);
+	B[1] := (-1cm,1.5cm);
+	B[2] := (-0.5cm,0.2cm);
+	for i:=0 upto 2: draw vertex(B[i]); endfor
+
+	path ub; ub := (-20cm,3cm)--(20cm,3cm);
+
+	numeric ang[]; ang[0] = 90; ang[1] = angle(A[1]-A[0]); ang[2] = angle(A[2]-A[1]); ang[3] = angle(A[3]-A[2]);
+	for i:=0 upto 2:
+		draw reverse(dirs(A[i],ang[i],6cm) cutafter ub) withpen normalpen dashed evenly;
+		l := 1cm + (i-1)*0.2cm;
+		drawarrow from(A[i],ang[i],l)..from(A[i],(ang[i]+ang[i+1])/2,l)..from(A[i],ang[i+1],l) withpen normalpen;
+	endfor
+	
+	for i:=0 upto 3: drawemptyvertex(A[i]); endfor;
+endfig;
+
+figtag("naslednik_pres_konvexni_obal");
+beginfig(7);
+	pair A[], C;
+	label.lrt(btex $Q_i$ etex, (1.5cm,-0.8cm));
+	pickup boldpen;
+
+	C := (-4cm,-0.3cm);
+	for i:=0 upto 6:
+		A[i] := (2cm,0) rotated (360*i/7+5) yscaled 0.7;
+		draw vertex(A[i]);
+	endfor;
+	draw for i:=0 upto 6: A[i]-- endfor cycle withpen normalpen;
+	draw C--A[2] dashed evenly;
+
+	draw dirs(C, -140, 0.5cm);
+	drawemptyvertex(A[2]);
+	drawemptyvertex(C);
+	drawdblarrow (fullcircle scaled 2cm rotated (360*2/7-5) yscaled 0.7) cutbefore (origin--(3cm,0) rotated (360*2/7+25)) withpen normalpen;
+	%drawarrow C+(0,0.5cm){dir 60}..A[2]+(0,0.5cm) withpen normalpen;
+	%drawarrow 0.6A[5]{dir 170}..(0.6A[3] rotated -15) withpen normalpen;
+	%drawarrow 0.7A[6]{dir 60}..(0.5A[1] rotated 30) withpen normalpen;
+endfig;
+end
diff --git a/old/7-geom/7-geom.tex b/old/7-geom/7-geom.tex
new file mode 100644
index 0000000..d600d11
--- /dev/null
+++ b/old/7-geom/7-geom.tex
@@ -0,0 +1,136 @@
+\input lecnotes.tex
+
+\prednaska{7}{Geometrické algoritmy}{(sepsal Pavel Klavík)}
+
+\>Uká¾eme si nìkolik základních algoritmù na øe¹ení geometrických problémù v~rovinì. Proè zrovna v~rovinì? Inu, jednorozmìrné problémy bývají triviální
+a naopak pro vy¹¹í dimenze jsou velice komplikované. Rovina je proto rozumným kompromisem mezi obtí¾ností a zajímavostí.
+
+Celou kapitolou nás bude provázet pohádka ze ¾ivota ledních medvìdù. Pokusíme se vyøe¹it jejich \uv{ka¾dodenní} problémy~\dots
+
+\h{Hledání konvexního obalu}
+
+{\I Daleko na severu ¾ili lední medvìdi. Ve vodách tamního moøe byla hojnost ryb a jak je známo, ryby jsou oblíbenou pochoutkou ledních medvìdù.
+Proto¾e medvìdi z~na¹í pohádky rozhodnì nejsou ledajací a ani chytrost jim neschází, rozhodli se v¹echny ryby pochytat. Znají pøesná místa výskytu
+ryb a rádi by vyrobili obrovskou sí», do které by je v¹echny chytili. Pomozte medvìdùm zjistit, jaký nejmen¹í obvod taková sí» mù¾e mít.}
+
+\figure{7-geom5_rybi_motivace.eps}{Problém ledních mìdvìdù: Jaký je nejmen¹í obvod sítì?}{3in}
+
+Neboli v~øeèi matematické, chceme pro zadanou mno¾inu bodù v~rovinì nalézt její konvexní obal. Co je to konvexní obal? Mno¾ina bodù je {\I konvexní},
+pokud pro ka¾dé dva body obsahuje i celou úseèku mezi nimi. {\I Konvexní obal} je nejmen¹í konvexní podmno¾ina roviny, která obsahuje v¹echny zadané
+body.\foot{Pamatujete si na lineární obaly ve vektorových prostorech? Lineární obal mno¾iny vektorù je nejmen¹í vektorový podprostor, který tyto
+vektory obsahuje. Není náhoda, ¾e tato definice pøipomíná definici konvexního obalu. Na druhou stranu ka¾dý vektor z~lineárního obalu lze vyjádøit
+jako lineární kombinaci daných vektorù. Podobnì platí i pro konvexní obaly, ¾e ka¾dý bod z~obalu je konvexní kombinací daných bodù. Ta se li¹í od
+lineární v~tom, ¾e v¹echny koeficienty jsou v~intervalu $[0,1]$ a navíc souèet v¹ech koeficientù je $1$. Tento algebraický pohled mù¾e mnohé vìci
+zjednodu¹it. Zkuste si dokázat, ¾e obì definice konvexního obalu jsou ekvivalentní.} Z~algoritmického hlediska nás v¹ak bude zajímat jenom jeho
+hranice, kterou budeme dále oznaèovat jako konvexní obal.
+
+Na¹ím úkolem je nalézt konvexní obal koneèné mno¾iny bodù. To je v¾dy konvexní mnohoúhelník, navíc s~vrcholy v~zadaných bodech. Øe¹ením problému tedy
+bude posloupnost bodù, které tvoøí konvexní obal. Pro malé mno¾iny je konvexní obal nakreslen na obrázku, pro více bodù je v¹ak situace mnohem
+slo¾itìj¹í.
+
+\figure{7-geom1_male_obaly.eps}{Konvexní obaly malých mno¾in.}{3in}
+
+Pro jednoduchost budeme pøedpokládat, ¾e v¹echny body mají rùzné $x$-ové souøadnice. Tedy utøídìní bodù zleva doprava je urèené jednoznaènì.\foot{To si
+mù¾eme dovolit pøedpokládat, nebo» se v¹emi body staèí nepatrnì pootoèit. Tím konvexní obal urèitì nezmìníme. Av¹ak jednodu¹¹í øe¹ení je naprogramovat
+tøídìní lexikograficky (druhotnì podle souøadnice $y$) a vyøadit identické body.} Tím máme zaji¹tìné, ¾e existují dva body, nejlevìj¹í a
+nejpravìj¹í, pro které platí následující invariant:
+
+\s{Invariant:} Nejlevìj¹í a nejpravìj¹í body jsou v¾dy v~konvexním obalu.
+
+Algoritmus na nalezení konvexního obalu funguje na následujícím jednoduchém principu, kterému se nìkdy øíká {\I zametání roviny}. Procházíme body
+zleva doprava a postupnì roz¹iøujeme doposud nalezený konvexní obal o~dal¹í body. Na zaèátku bude konvexní obal jediného bodu samotný bod. Na konci
+$k$-tého kroku algoritmu známe konvexní obal prvních $k$ bodù. Kdy¾ algoritmus skonèí, známe hledaný konvexní obal. Podle invariantu musíme v~$k$-tém
+kroku pøidat do obalu $k$-tý nejlevìj¹í bod. Zbývá si jen rozmyslet, jak pøesnì tento bod pøidat.
+
+Pøidání dal¹ího bodu do konvexního obalu funguje, jak je naznaèeno na obrázku. Podle invariantu víme, ¾e bod nejvíc vpravo je souèástí konvexního
+obalu. Za nìj napojíme novì pøidávaný bod. Tím jsme získali nìjaký obal, ale zpravidla nebude konvexní. To lze v¹ak snadno napravit, staèí
+odebírat body, v obou smìrech podél konvexního obalu, tak dlouho, dokud nezískáme konvexní obal. Na pøíkladu z obrázku nemusíme po smìru hodinových
+ruèièek odebrat ani jeden bod, obal je v poøádku. Naopak proti smìru hodinových ruèièek musíme odebrat dokonce dva body.
+
+\figure{7-geom2_pridani_bodu.eps}{Pøidání bodu do konvexního obalu.}{4.5in}
+
+Pro pøípadnou implementaci a rozbor slo¾itosti si nyní popí¹eme algoritmus detailnìji. Aby se lépe popisoval, rozdìlíme si konvexní obal na dvì èásti
+spojující nejlevìj¹í a nejpravìj¹í bod obalu. Budeme jim øíkat {\I horní obálka} a {\I dolní obálka}.
+
+\figure{7-geom3_obalky.eps}{Horní a dolní obálka konvexního obalu.}{3.4in}
+
+Obì obálky jsou lomené èáry, navíc horní obálka poøád zatáèí doprava a dolní naopak doleva. Pro udr¾ování bodù v~obálkách staèí dva zásobníky.
+V~$k$-tém kroku algoritmu pøidáme zvlá¹» $k$-tý bod do horní i dolní obálky. Pøidáním $k$-tého bodu se v¹ak mù¾e poru¹it smìr, ve kterém obálka
+zatáèí. Proto budeme nejprve body z~obálky odebírat a $k$-tý bod pøidáme a¾ ve chvíli, kdy jeho pøidání smìr zatáèení neporu¹í.
+
+\s{Algoritmus:}
+
+\algo
+
+\:Setøídíme body podle $x$-ové souøadnice, oznaème body $b_1, \ldots, b_n$.
+\:Vlo¾íme do horní a dolní obálky bod $b_1$: $H = D = (b_1)$.
+\:Pro ka¾dý dal¹í bod $b = b_2,\ldots,b_n$:
+\::Pøepoèítáme horní obálku:
+\:::Dokud $\vert H\vert \ge 2$, $H = (\ldots, h_{k-1}, h_k)$ a úhel $h_{k-1} h_k b$ je orientovaný doleva:
+\::::Odebereme poslední bod $h_k$ z~obálky $H$.
+\:::Pøidáme bod $b$ do obálky $H$.
+\::Symetricky pøepoèteme dolní obálku (s orientací doprava).
+\: Výsledný obal je tvoøen body v~obálkách $H$ a $D$.
+
+\endalgo
+
+Rozebereme si èasovou slo¾itost algoritmu. Setøídit body podle $x$-ové souøadnice doká¾eme v~èase $\O(n \log n)$. Pøidání dal¹ího bodu do obálek
+trvá lineárnì vzhledem k~poètu odebraných bodù. Zde vyu¾ijeme obvyklý postup: Ka¾dý bod je odebrán nejvý¹e jednou, a tedy v¹echna odebrání trvají
+dohromady $\O(n)$. Konvexní obal doká¾eme sestrojit v~èase $\O(n \log n)$, a pokud bychom mìli seznam bodù ji¾ utøídený, doká¾eme to dokonce v
+$\O(n)$.
+
+\s{Algebraický dodatek:} Existuje jednoduchý postup, jak zjistit orientaci úhlu? Uká¾eme si jeden zalo¾ený na lineární algebøe. Budou se hodit
+vlastnosti determinantu. Absolutní hodnota determinantu je objem rovnobì¾nostìnu urèeného øádkovými vektory matice. Dùle¾itìj¹í v¹ak je, ¾e znaménko
+determinantu urèuje \uv{orientaci} vektorù, zda je levotoèivá èi pravotoèivá. Proto¾e ná¹ problém je rovinný, budeme uva¾ovat determinanty matic $2
+\times 2$.
+
+Uva¾me souøadnicový systém v~rovinì, kde $x$-ová souøadnice roste smìrem doprava a~$y$-ová smìrem nahoru. Chceme zjistit orientaci úhlu $h_{k-1} h_k
+b$. Polo¾me $\vec u = (x_1, y_1)$ jako rozdíl souøadnic $h_k$ a~$h_{k-1}$ a podobnì $\vec v = (x_2, y_2)$ je rozdíl souøadnic $b$ a~$h_k$. Matice $M$
+je definována následovnì:
+$$M = \pmatrix{\vec u \cr \vec v} = \pmatrix {x_1&y_1\cr x_2&y_2}.$$
+Úhel $h_{k-1} h_k b$ je orientován doleva, právì kdy¾ $\det M = x_1y_2 - x_2y_1$ je nezáporný,\foot{Neboli vektory $\vec u$ a $\vec v$ odpovídají
+rozta¾ení a zkosení vektorù báze $\vec x = (1,0)$ a $\vec y = (0,1)$, pro nì¾ je determinant nezáporný.} a spoèítat hodnotu determinantu je jednoduché.
+Mo¾né situace jsou nakresleny na obrázku. Poznamenejme, ¾e k~podobnému vzorci se lze také dostat pøes vektorový souèin vektorù $\vec u$ a $\vec v$.
+
+\figure{7-geom4_determinant.eps}{Jak vypadají determinanty rùzných znamének v~rovinì.}{4.6in}
+
+\s{©lo by to vyøe¹it rychleji?} Také vám vrtá hlavou, zda existují rychlej¹í algoritmy? Na závìr si uká¾eme nìco, co na pøedná¹ce nebylo.\foot{A také
+se nebude zkou¹et.} Nejrychlej¹í známý algoritmus, jeho¾ autorem je T.~Chan, funguje v~èase $\O(n \log h)$, kde $h$ je poèet bodù le¾ících na
+konvexním obalu, a pøitom je pøekvapivì jednoduchý. Zde si naznaèíme, jak tento algoritmus funguje.
+
+Algoritmus pøichází s~následující my¹lenkou. Pøedpokládejme, ¾e bychom znali velikost konvexního obalu $h$. Rozdìlíme body libovolnì do $\lceil {n
+\over h} \rceil$ mno¾in $Q_1, \ldots, Q_k$ tak, ¾e $\vert Q_i \vert \le h$. Pro ka¾dou z~tìchto mno¾in nalezneme konvexní obal pomocí vý¹e popsaného
+algoritmu. To doká¾eme pro jednu v~èase $\O(h \log h)$ a pro v¹echny v~èase $\O(n \log h)$. V druhé fázi spustíme hledání konvexního obalu pomocí
+provázkového algoritmu a pro zrychlení pou¾ijeme pøedpoèítané obaly men¹ích mno¾in. Nejprve popí¹eme jeho my¹lenku. Pou¾ijeme následující pozorování:
+
+\s{Pozorování:} Úseèka spojující dva body $a$ a $b$ le¾í na konvexním obalu, právì kdy¾ v¹echny ostatní body le¾í pouze na jedné její
+stranì.\foot{Formálnì je podmínka následující: Pøímka $ab$ urèuje dvì poloroviny. Úseèka le¾í na konvexním obalu, právì kdy¾ v¹echny body le¾í v jedné
+z polorovin.}
+
+Algoritmu se øíká {\I provázkový}, proto¾e svojí èinností pøipomíná namotávání provázku podél konvexního obalu.  Zaèneme s bodem, který na konvexním
+obalu urèitì le¾í, to je tøeba ten nejlevìj¹í. V ka¾dém kroku nalezneme následující bod po obvodu konvexního obalu. To udìláme napøíklad tak, ¾e
+projdeme v¹echny body a vybereme ten, který svírá nejmen¹í úhel s poslední stranou konvexního obalu. Novì pøidaná úseèka vyhovuje pozorování a proto
+do konvexního obalu patøí. Po $h$ krocích se dostaneme zpìt k nejlevìj¹ímu bodu a výpoèet ukonèíme. V ka¾dém kroku potøebujeme projít v¹echny body a
+vybrat následníka, co¾ doká¾eme v èase $\O(n)$. Celková slo¾itost algoritmu je tedy $\O(n \cdot h)$.
+
+\twofigures{7-geom6_provazkovy_algoritmus.eps}{Provázkový algoritmus.}{1.25in}{7-geom7_naslednik_pres_konvexni_obal.eps}{Hledání kandidáta v pøedpoèítaném obalu.}{2.5in}
+
+Provázkový algoritmus funguje, ale má jednu obrovskou nevýhodu -- je toti¾ ukrutnì pomalý. Ký¾eného zrychlení dosáhneme, pokud pou¾ijeme pøedpoèítané
+konvexní obaly. Ty umo¾ní rychleji hledat následníka. Pro ka¾dou z mno¾in $Q_i$ najdeme zvlá¹» kandidáta a poté z nich vybereme toho nejlep¹ího.
+Mo¾ný kandidát v¾dy le¾í na konvexním obalu mno¾iny $Q_i$. Vyu¾ijeme toho, ¾e body obalu jsou \uv{uspoøádané}, i kdy¾ trochu netypicky do kruhu.
+Kandidáta mù¾eme hledat metodou pùlení intervalu, i kdy¾ detaily jsou malièko slo¾itìj¹í ne¾ je obvyklé. Jak pùlit zjistíme podle smìru zatáèení
+konvexního obalu. Detaily si rozmyslí ètenáø sám.
+
+Èasová slo¾itost pùlení je $\O(\log h)$ pro jednu mno¾inu. Mno¾in je nejvý¹e $\O({n \over h})$, tedy následující bod konvexního obalu nalezneme v èase
+$\O({n \over h} \log h)$. Celý obal nalezneme ve slibovaném èase $\O(n \log h)$. 
+
+Popsanému algoritmu schází jedna dùle¾itá vìc: Ve skuteènosti vìt¹inou neznáme velikost $h$. Budeme proto algoritmus iterovat s~rostoucí hodnotou $h$,
+dokud konvexní obal nesestrojíme. Pokud pøi slepování konvexních obalù zjistíme, ¾e konvexní obal je vìt¹í ne¾ $h$, výpoèet ukonèíme. Zbývá je¹tì
+zvolit, jak rychle má $h$ rùst. Pokud by rostlo moc pomalu, budeme poèítat zbyteènì mnoho fází, naopak pøi rychlém rùstu by nás poslední fáze mohla
+stát pøíli¹ mnoho.
+
+V~$k$-té iteraci polo¾íme $h = 2^{2^k}$. Dostáváme celkovou slo¾itost algoritmu:
+$$\sum_{m=0}^{\O(\log \log h)} \O(n \log 2^{2^m}) = \sum_{m=0}^{\O(\log \log h)} \O(n \cdot 2^m) = \O(n \log h),$$
+kde poslední rovnost dostaneme jako souèet prvních $\O(\log \log h)$ èlenù geometrické øady $\sum 2^m$.
+
+\bye
diff --git a/old/7-geom/7-geom1_male_obaly.eps b/old/7-geom/7-geom1_male_obaly.eps
new file mode 100644
index 0000000..94a501d
--- /dev/null
+++ b/old/7-geom/7-geom1_male_obaly.eps
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diff --git a/old/7-geom/7-geom7_naslednik_pres_konvexni_obal.eps b/old/7-geom/7-geom7_naslednik_pres_konvexni_obal.eps
new file mode 100644
index 0000000..7a14c41
--- /dev/null
+++ b/old/7-geom/7-geom7_naslednik_pres_konvexni_obal.eps
@@ -0,0 +1,106 @@
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diff --git a/old/7-geom/Makefile b/old/7-geom/Makefile
new file mode 100644
index 0000000..b06fe95
--- /dev/null
+++ b/old/7-geom/Makefile
@@ -0,0 +1,3 @@
+P=7-geom
+
+include ../Makerules
diff --git a/old/7-geom/lib.mp b/old/7-geom/lib.mp
new file mode 100644
index 0000000..2536305
--- /dev/null
+++ b/old/7-geom/lib.mp
@@ -0,0 +1,82 @@
+% implementation of figure naming and figure transparency
+string name,tag; name := ""; tag := "";
+def updatefigname =
+	if tag="": filenametemplate (name & "%c.eps");
+	else: filenametemplate (name & "%c_" & tag & ".eps");
+	fi;
+enddef;
+	
+def figname(expr n) = name := n; updatefigname; enddef;
+def figtag(expr t) = tag := t; updatefigname; enddef;
+
+picture transparent_picture;
+color transparent_color; transparent_color := 0.9white;
+def drawtransparent(expr num) =
+	transparent_picture := currentpicture;
+endfig;
+
+if tag="": filenametemplate (name & "%c_transparent.eps");
+else: filenametemplate (name & "%c_" & tag & "_transparent.eps");
+fi;
+beginfig(num);
+	draw transparent_picture withcolor transparent_color;
+endfig;
+
+updatefigname;
+enddef;
+
+def from(expr p,d,len) =
+	p+dir(d)*len
+enddef;
+
+def dirs(expr p,d,len) =
+	p--from(p,d,len)
+enddef;
+
+def drawvertices(expr s,n) =
+	for i:=s upto n:
+		draw vertex(PQ[i]);
+	endfor
+enddef;
+
+def drawfvertices(expr s,n,flags) =
+	for i:=s upto n:
+		draw vertex(PQ[i]) flags;
+	endfor
+enddef;
+
+def vertex(expr p) = p withpen pencircle scaled 4pt enddef;
+def drawemptyvertex(expr p) = unfill fullcircle scaled 4pt shifted p; draw fullcircle scaled 4pt shifted p; enddef;
+def drawendpointvertex(expr p) = draw vertex(p) withcolor red; draw fullcircle scaled 6pt shifted p; enddef;
+def createpath(expr p) = shakepath(p, 0.015cm,0.1cm) enddef;
+vardef shakepath(expr p,d,l) =
+	save r,b;
+	path r; r := point(arctime 0 of p) of p;
+	b := -1;
+	for i:=l step l until arclength(p):
+		r := r--(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*d*b;
+		b := -b;
+	endfor
+	r--point(arctime arclength(p) of p) of p
+enddef;
+	
+pen normalpen; normalpen := pencircle scaled 0.6pt;
+pen boldpen; boldpen := pencircle scaled 1.5pt;
+pen bolderpen; bolderpen := pencircle scaled 2pt;
+def dotline = withdots scaled 0.82 withpen boldpen enddef;
+
+vardef unclosedbubblec(expr p,c) =
+	bubblec((p..reverse p..cycle),c)
+enddef;	
+
+vardef bubblec(expr p,c) =
+	save r;
+	path r; r := (point(arctime 0 of p) of p)+dir(angle(direction(arctime 0 of p) of p rotated 90))*c;
+	for i:=0.01cm step 0.025cm until arclength(p):
+		r := r..(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*c;
+	endfor
+	r..(point(arctime arclength(p) of p) of p)+dir(angle(direction(arctime arclength(p) of p) of p rotated 90))*c..cycle
+enddef;
+
+vardef bubble(expr p) = bubblec(p,0.12cm) enddef;
+vardef unclosedbubble(expr p) = unclosedbubblec(p,0.12cm) enddef;
diff --git a/old/8-geom2/8-geom2.mp b/old/8-geom2/8-geom2.mp
new file mode 100644
index 0000000..5a44396
--- /dev/null
+++ b/old/8-geom2/8-geom2.mp
@@ -0,0 +1,137 @@
+input lib
+
+figname("8-geom2_");
+figtag("usecky");
+beginfig(1);
+	def drawusecka(expr p,q) = draw vertex(p); draw vertex(q); draw p--q; enddef;
+	pair A[],B[];
+	A0 := origin; B0 := (4cm,1.5cm);
+	A1 := (0.5cm,1.5cm); B1 := (1cm,0);
+	A2 := (2cm,1.3cm); B2 := (3cm, -0.9cm);
+	A3 := (1cm,-1.3cm); B3 := (3.5cm,0);
+	z0 = whatever[A0,B0]; z0 = whatever[A1,B1];
+	z1 = whatever[A0,B0]; z1 = whatever[A2,B2];
+	z2 = whatever[A2,B2]; z2 = whatever[A3,B3];
+	
+	for i:=0 upto 2: fill fullcircle scaled 7pt shifted z[i]; drawemptyvertex(z[i]); endfor
+	for i:=0 upto 3: drawusecka(A[i], B[i]); endfor
+endfig;
+
+figtag("polorovina");
+beginfig(2);
+	pair A,B;
+	A := origin;
+	B := (1.7cm,0.5cm);
+	an := angle(B-A);
+	path p; p := from(.5[A,B],an+90,3cm)--from(.5[A,B],an-90,3cm);
+	fill p--reverse(p shifted (A-0.8[A,B]))--cycle withcolor 0.8white;
+	%fill p{dir (an+180)}..-0.35[A,B]..{dir an}cycle withcolor 0.8white;
+	draw p withpen boldpen;
+
+	pair C; C := point(0.9) of p; 
+	drawarrow from(C,an+180,0.1cm)--from(C,an+180,1cm) withpen boldpen;
+	C := point(0.1) of p; 
+	drawarrow from(C,an+180,0.1cm)--from(C,an+180,1cm) withpen boldpen;
+	label.rt(btex $p$ etex, point(0.15) of p);
+	
+	draw A--B dashed evenly;
+	drawemptyvertex(A); drawemptyvertex(B);
+	label.llft(btex $a$ etex, A);
+	label.urt(btex $b$ etex, B);
+	label(btex $B_a$ etex, A+2(-0.1cm,0.5cm));
+	label(btex $B_b$ etex, B+2(-0.1cm,0.5cm));
+endfig;
+
+figtag("voroneho_diagram");
+beginfig(3);
+	u := 1.35cm;
+	pair A[],B[];
+	A0 := origin; A1 := (-u,-u); A2 := (1.3u,-u); A3 := (1.50821u,0.7u); A4 := (0.6u,0.8u); A5 := (0.2u,1.9u); A6 := (-1.3u,1u);
+	def osa(expr a, b, an,l) = from(.5[a,b], angle(b-a)+an, l) enddef;
+	vardef prusecik_os(expr p,q,r) =
+		save b; pair b;
+		b = whatever[osa(p,q, 90,1cm), osa(p,q,-90,1cm)];
+		b = whatever[osa(q,r, 90,1cm), osa(q,r,-90,1cm)];
+		b
+	enddef;
+	B[0] := prusecik_os(A0,A1,A2); B[1] := prusecik_os(A0,A1,A6); B[2] := prusecik_os(A0,A4,A6); B[3] := prusecik_os(A4,A5,A6);
+	B[4] := prusecik_os(A3,A4,A5); B[5] := prusecik_os(A0,A3,A4); B[6] := prusecik_os(A0,A2,A4);
+	draw osa(A1,A2,-90,2cm)--B[0]--B[1]--osa(A1,A6,90,1.3cm) withpen boldpen;
+	draw B[1]--B[2]--B[3]--osa(A6,A5,90,2cm) withpen boldpen;
+	draw B[3]--B[4]--osa(A3,A5,-90,2.7cm) withpen boldpen;
+	draw B[4]--B[5]--B[2] withpen boldpen;
+	draw B[5]--B[6]--osa(A2,A3,-90,1.3cm) withpen boldpen;
+	draw B[6]--B[0] withpen boldpen;
+	
+	draw A0--A1--A2--A0--A6--A1 dashed evenly;
+	draw A6--A5--A4--A0 dashed evenly; draw A3--A2 dashed evenly;
+	draw A3--A5 dashed evenly; draw A6--A4--A3 dashed evenly;
+
+	for i:=0 upto 6: draw vertex(B[i]); endfor
+	for i:=0 upto 6: drawemptyvertex(A[i]); endfor
+endfig;
+
+figtag("pasy_mnohouhelniku");
+beginfig(4);
+	u := 1.35cm;
+	pair A[],B[];
+	A0 := origin; A1 := (-u,-u); A2 := (1.3u,-u); A3 := (1.50821u,0.7u); A4 := (0.6u,0.8u); A5 := (0.2u,1.9u); A6 := (-1.3u,1u);
+	def osa(expr a, b, an,l) = from(.5[a,b], angle(b-a)+an, l) enddef;
+	vardef prusecik_os(expr p,q,r) =
+		save b; pair b;
+		b = whatever[osa(p,q, 90,1cm), osa(p,q,-90,1cm)];
+		b = whatever[osa(q,r, 90,1cm), osa(q,r,-90,1cm)];
+		b
+	enddef;
+	def drawline(expr p) = draw ((-2.3u,p)--(2.5u,p)) cutbefore (D0--D1) cutafter (D2--D3) enddef;
+	B[0] := prusecik_os(A0,A1,A2); B[1] := prusecik_os(A0,A1,A6); B[2] := prusecik_os(A0,A4,A6); B[3] := prusecik_os(A4,A5,A6);
+	B[4] := prusecik_os(A3,A4,A5); B[5] := prusecik_os(A0,A3,A4); B[6] := prusecik_os(A0,A2,A4);
+	
+	pair C; C := origin; for i:=0 upto 6: C := C + B[i]; endfor C := C/6;
+	pair D[];
+	D0 := C+(-2.25,2.25)*u; D1 := C+(-2.25,-2.25)*u; D2 := C+(2.25,-2.25)*u; D3 := C+(2.25,2.25)*u; 
+	
+	draw B[0]--B[1]--B[2]--B[3]--B[4]--B[5]--B[6]--B[0] withpen boldpen;
+	draw B[5]--B[2] withpen boldpen;
+	pair E; E := 0.6[B0,B4];
+	drawline(ypart(E));
+	drawline(ypart(E)) cutbefore (B2--B5) cutafter (B4--B5) dashed evenly withpen bolderpen;
+	drawemptyvertex(E);
+
+	for i:=0 upto 6: drawline(ypart(B[i])) dashed evenly; endfor
+	for i:=0 upto 6: draw vertex(B[i]); endfor
+endfig;
+
+figtag("upravy_stromu");
+beginfig(5);
+	u := 1cm;
+	draw (0,0.1u)--(2.05u,-2.05u)--(-2.05u,-2.05u)--cycle;
+	pair A[]; A0 := from(origin,-90,0.1u); A1 := from(A0, -70, 0.5u); A2 := from(A1, -110, 0.5u); A3 := from(A2, -80, 0.5u); A4 := from(A3, -120, 0.45u);
+	path p; p := createpath(A0--A1--A2--A3--A4);
+	path q; q := (p scaled 0.93 shifted (-0.15u,-0.15u))--(-1.85u,-1.95u)--cycle; fill q withcolor 0.8white; draw q;
+	q := (p scaled 0.93 shifted (0.15u,-0.15u))--(1.85u,-1.95u)--cycle; fill q withcolor 0.8white; draw q;
+	p := createpath(A0--from(A0,-110,0.1u)--A1--A2--A3--A4); draw p withpen boldpen;
+endfig;
+
+figtag("rychla_perzistence");
+beginfig(6);
+	u := 1cm;
+	def drawtable(expr p, lab, sa, sb) =
+		draw centersquare xscaled 2u yscaled (2u/3) shifted (p+(0,u/3));
+		draw centersquare xscaled 2u yscaled (2u/3) shifted (p-(0,u/3));
+		label(sa, p+(0,u/3));
+		label(sb, p-(0,u/3));
+		label.bot(lab, p-(0,2u/3));
+	enddef;
+
+	pair A[];
+	for i:=0 upto 2: A[i] := (0,2u) rotated 120i; endfor
+	drawtable(A1, btex $v$ etex, btex verze 2 etex, btex verze 1 etex);
+	drawtable(A2, btex $v'$ etex, btex verze 3 etex, "");
+	drawtable(A0, btex $u$ etex, btex verze 2 etex, btex verze 1 etex);
+
+	drawarrow from(A[2]+(0,u/3), 180, 7u/6)--from(A[1]+(0,u/3), 0, 7u/6);
+	drawarrow from(A[0]+(0,-u/3), 180, 7u/6)..{dir -90}from(A[1]+(-3u/4,2u/3), 90, u/6);
+	drawarrow from(A[0]+(0,u/3), 0, 7u/6)..{dir -90}from(A[2]+(3u/4,2u/3), 90, u/6);
+endfig;
+end
diff --git a/old/8-geom2/8-geom2.tex b/old/8-geom2/8-geom2.tex
new file mode 100644
index 0000000..034b395
--- /dev/null
+++ b/old/8-geom2/8-geom2.tex
@@ -0,0 +1,167 @@
+\input lecnotes.tex
+
+\prednaska{8}{Geometrie vrací úder}{(sepsal Pavel Klavík)}
+
+\>Kdy¾ s geometrickými problémy poøádnì nezametete, ony vám to vrátí! Ale kdy¾ u¾ zametat, tak urèitì ne pod koberec a místo smetáku pou¾ijte pøímku.
+V této pøedná¹ce nás spolu s dvìma geometrickými problémy samozøejmì èeká pokraèování pohádky o ledních medvìdech.
+
+{\I Medvìdi vyøe¹ili rybí problém a hlad je ji¾ netrápí. Av¹ak na severu ne¾ijí sami, za sousedy mají Eskymáky. Proto¾e je rozhodnì lep¹í se sousedy
+dobøe vycházet, jsou medvìdi a Eskymáci velcí pøátelé. Skoro ka¾dý se se svými pøáteli rád schází. Av¹ak to je musí nejprve nalézt~\dots}
+
+\h{Hledání prùseèíkù úseèek}
+
+Zkusíme nejprve Eskymákùm vyøe¹it lokalizaci ledních medvìdù.
+
+{\I Kdy¾ takový medvìd nemá co na práci, rád se prochází. Na místech, kde se trasy protínají, je zvý¹ená ¹ance, ¾e se dva medvìdi potkají a zapovídají
+-- ostatnì co byste èekali od medvìdù. To jsou ta správná místa pro Eskymáka, který chce potkat medvìda. Jenom¾e jak tato køí¾ení najít?}
+
+Pro zjednodu¹ení pøedpokládejme, ¾e medvìdi chodí po úseèkách tam a zpìt. Budeme tedy chtít nalézt v¹echny prùseèíky úseèek v rovinì.
+
+\bigskip
+\centerline{\epsfxsize=1.5in\epsfbox{8-geom2_0_bear.eps}\hskip 4em\epsfxsize2in\epsfbox{8-geom2_1_usecky.eps}}
+\smallskip
+\centerline{Problém Eskymákù: Kde v¹ude se køí¾í medvìdí trasy?}
+\bigskip
+
+Pro $n$ úseèek mù¾e existovat a¾ $\Omega(n^2)$ prùseèíkù.\foot{Zkuste takový pøíklad zkonstruovat.} Tedy optimální slo¾itosti by dosáhl i algoritmus,
+který by pro ka¾dou dvojici úseèek testoval, zda se protínají. Èasovou slo¾itost algoritmu v¹ak posuzujeme i vzhledem k velikosti výstupu $p$. Typické
+rozmístìní úseèek mívá toti¾ prùseèíkù spí¹e pomálu. Pro tento pøípad si uká¾eme podstatnì rychlej¹í algoritmus.
+
+Pro jednodu¹¹í popis pøedpokládejme, ¾e úseèky le¾í v obecné poloze. To znamená, ¾e ¾ádné tøi úseèky se neprotínají v jednom bodì a prùnikem ka¾dých
+dvou úseèek je nejvý¹e jeden bod. Navíc pøedpokládejme, ¾e krajní bod ¾ádné úseèky nele¾í na jiné úseèce a také neexistují vodorovné úseèky. Na závìr si
+uká¾eme, jak se s tìmito pøípady vypoøádat.
+
+Algoritmus funguje na principu zametání roviny, popsaném v minulé pøedná¹ce. Budeme posouvat vodorovnou pøímku odshora dolù. V¾dy, kdy¾ narazíme na
+nový prùnik, ohlásíme jeho výskyt. Samozøejmì spojité posouvání nahradíme diskrétním a pøímku v¾dy posuneme do dal¹ího zajímavého bodu.
+
+Zajímavé události jsou {\I zaèátky úseèek}, {\I konce úseèek} a {\I prùseèíky úseèek}. Po utøídìní známe pro první dva typy událostí poøadí, v jakém
+se objeví. Výskyty prùseèíkù budeme poèítat prùbì¾nì, jinak bychom celý problém nemuseli øe¹it.
+
+V ka¾dém kroku si pamatujeme {\I prùøez} $P$ -- posloupnost úseèek aktuálnì protnutých zametací pøímkou. Tyto úseèky máme utøídìné zleva doprava.  Navíc si
+udr¾ujeme kalendáø $K$ budoucích událostí. Z hlediska prùseèíkù budeme na úseèky nahlí¾et jako na polopøímky. Pro sousední dvojice úseèek si
+udr¾ujeme, zda se jejich smìry nìkde protnou. Algoritmus pro hledání prùnikù úseèek funguje následovnì:
+
+\s{Algoritmus:}
+
+\algo
+
+\:$P \leftarrow \emptyset$.
+\:Do $K$ vlo¾íme zaèátky a konce v¹ech úseèek.
+\:Dokud $K \ne \emptyset$:
+\::Odebereme nejvy¹¹í událost.
+\::Pokud je to zaèátek úseèky, zatøídíme novou úseèku do $P$.
+\::Pokud je to konec úseèky, odebereme úseèku z $P$.
+\::Pokud je to prùseèík, nahlásíme ho a prohodíme úseèky v $P$.
+\::Navíc v¾dy pøepoèítáme prùseèíkové události, v¾dy maximálnì dvì odebereme a dvì nové pøidáme.
+\endalgo
+
+Zbývá rozmyslet si, jaké datové struktury pou¾ijeme, abychom prùseèíky nalezli dostateènì rychle. Pro kalendáø pou¾ijeme napøíklad haldu. Prùøez si
+budeme udr¾ovat ve vyhledávacím stromì. Poznamenejme, ¾e nemusíme znát souøadnice úseèek, staèí znát jejich poøadí, které se mezi jednotlivými
+událostmi nemìní. Pøi pøidávání úseèek procházíme stromem a porovnáváme souøadnice v prùøezu, které prùbì¾nì dopoèítáváme.
+
+Kalendáø obsahuje v¾dy nejvý¹e $\O(n)$ událostí. Podobnì prùøez obsahuje v ka¾dém okam¾iku nejvý¹e $\O(n)$ úseèek.  Jednu událost kalendáøe doká¾eme
+o¹etøit v èase $\O(\log n)$. V¹ech událostí je $\O(n+p)$, a tedy celková slo¾itost algoritmu je $\O((n+p) \log n)$.
+
+Slíbili jsme, ¾e popí¹eme, jak se vypoøádat s vý¹e uvedenými podmínkami na vstup. Události kalendáøe se stejnou $y$-ovou souøadnicí budeme tøídit v
+poøadí zaèátky, prùseèíky a konce úseèek. Tím nahlásíme i prùseèíky krajù úseèek a ani vodorovné úseèky nebudou vadit. Podobnì se není tøeba obávat
+prùseèíkù více úseèek v jednom bodì. Úseèky jdoucí stejným smìrem, jejich¾ prùnik je úseèka, jsou komplikovanìj¹í, ale lze jejich prùseèíky o¹etøit a
+vypsat tøeba souøadnice úseèky tvoøící jejich prùnik.
+
+Na závìr poznamenejme, ¾e Balaban vymyslel efektivnìj¹í algoritmus, který funguje v èase $\O(n \log n + p)$, ale je podstatnì komplikovanìj¹í.
+
+\h{Hledání nejbli¾¹ích bodù a Voroného diagramy}
+
+Nyní se pokusíme vyøe¹it i problém druhé strany -- pomù¾eme medvìdùm nalézt Eskymáky.
+
+{\I Eskymáci tráví vìt¹inu èasu doma, ve svém iglù. Takový medvìd je na své toulce zasnì¾enou krajinou, kdy¾ tu se najednou rozhodne nav¹tívit nìjakého
+Eskymáka. Proto se podívá do své medvìdí mapy a nalezne nejbli¾¹í iglù. Má to ale jeden háèek, iglù jsou spousty a medvìd by dávno usnul, ne¾ by
+nejbli¾¹í objevil.}\foot{Zlí jazykové by øekli, ¾e medvìdi jsou moc líní a nebo v mapách ani èíst neumí!}
+
+Popí¹eme si nejprve, jak vypadá medvìdí mapa. Medvìdí mapa obsahuje celou Arktidu a jsou v ní vyznaèena v¹echna iglù. Navíc obsahuje vyznaèené
+oblasti tvoøené body, které jsou nejblí¾e k jednomu danému iglù. Takovému schématu se øíká {\I Voroného diagram}. Ten pro zadané body $x_1, \ldots, x_n$
+obsahuje rozdìlení roviny na oblasti $B_1, \ldots, B_n$, kde $B_i$ je mno¾ina bodù, které jsou blí¾e k $x_i$ ne¾ k ostatním bodùm $x_j$. Formálnì jsou
+tyto oblasti definovány následovnì:
+$$B_i = \left\{y \in {\bb R}^2\ \vert\ \forall j:\rho(x_i,y) \le \rho(x_j,y)\right\},$$
+kde $\rho(x,y)$ znaèí vzdálenost bodù $x$ a $y$.
+
+Uká¾eme si, ¾e Voroného diagram má pøekvapivì jednoduchou strukturu. Nejprve uva¾me, jak budou vypadat oblasti $B_a$ a $B_b$ pouze pro dva body
+$a$ a $b$, jak je naznaèeno na obrázku. V¹echny body stejnì vzdálené od $a$ i $b$ le¾í na pøímce $p$ -- ose úseèky $ab$. Oblasti $B_a$ a $B_b$
+jsou tedy tvoøeny polorovinami ohranièenými osou $p$. Tedy obecnì tvoøí mno¾ina v¹ech bodù bli¾¹ích k $x_i$ ne¾ k $x_j$ nìjakou polorovinu. Oblast
+$B_i$ obsahuje v¹echny body, které jsou souèasnì bli¾¹í k $x_i$ ne¾ ke v¹em ostatním bodùm $x_j$ -- tedy le¾í ve v¹ech polorovinách souèasnì.
+Ka¾dá z oblastí $B_i$ je tvoøena prùnikem $n-1$ polorovin, tedy je to (mo¾ná neomezený) mnohoúhelník.\foot{Sly¹eli jste u¾ o lineárním programování?
+Jak název vùbec nenapoví, {\I lineární programování} je teorii zabývající se øe¹ením a vlastnostmi soustav lineárních nerovnic. Lineární program je
+popsaný lineární funkcí, kterou chceme maximalizovat za podmínek popsaných soustavou lineárních nerovnic. Ka¾dá nerovnice urèuje poloprostor, ve
+kterém se pøípustná øe¹ení nachází.  Proto¾e pøípustné øe¹ení splòuje v¹echny nerovnice zároveò, je mno¾ina v¹ech pøípustných øe¹ení (mo¾ná neomezený)
+mnohostìn, obecnì ve veliké dimenzi ${\bb R}^d$, kde $d$ je poèet promìnných. Mno¾iny $B_i$ lze snadno popsat jako mno¾iny v¹ech pøípustných øe¹ení
+lineárních programù pomocí vý¹e ukázaných polorovin. Na závìr poznamenejme, ¾e dlouho otevøená otázka, zda lze nalézt optimální øe¹ení lineárního
+programu v polynomiálním èase, byla pozitivnì vyøe¹ena -- je znám polynomiální algoritmus, kterému se øíká {\I metoda vnitøního bodu}. Na druhou
+stranu, pokud chceme najít pøípustné celoèíselné øe¹ení, je úloha NP-úplná a je jednoduché na ni pøevést spoustu optimalizaèních problémù. Dokázat
+NP-tì¾kost není pøíli¹ tì¾ké. Na druhou stranu ukázat, ¾e tento problém le¾í v NP, není vùbec jednoduché.}
+Pøíklad Voroného diagramu je naznaèen na obrázku. Zadané body jsou oznaèeny prázdnými krou¾ky a hranice oblastí $B_i$ jsou vyznaèené èernými èárami.
+
+\twofigures{8-geom2_2_polorovina.eps}{Body bli¾¹í k $a$ ne¾ $b$.}{1.25in}{8-geom2_3_voroneho_diagram.eps}{Voroného diagram.}{2.5in}
+
+Není náhoda, pokud vám hranice oblastí pøipomíná rovinný graf. Jeho vrcholy jsou body, které jsou stejnì vzdálené od alespoò tøí zadaných bodù. Jeho
+stìny jsou oblasti $B_i$. Jeho hrany jsou tvoøeny èástí hranice mezi dvìma oblastmi -- body, které mají dvì oblasti spoleèné. Obecnì prùnik dvou
+oblastí mù¾e být, v závislosti na jejich sousedìní, prázdný, bod, úseèka, polopøímka nebo dokonce celá pøímka.  V dal¹ím textu si pøedstavme, ¾e celý
+Voroného diagram uzavøeme do dostateènì velkého obdélníka,\foot{Pøeci jenom i celá Arktida je omezenì velká.} èím¾ dostaneme omezený rovinný graf.
+
+Poznamenejme, ¾e pøeru¹ované èáry tvoøí hrany duálního rovinného grafu s vrcholy v zadaných bodech. Hrany spojují sousední body na kru¾nicích, které 
+obsahují alespoò tøi ze zadaných bodù. Napøíklad na obrázku dostáváme skoro samé trojúhelníky, proto¾e vìt¹ina kru¾nic obsahuje pøesnì tøi zadané
+body. Av¹ak nalezneme i jeden ètyøúhelník, jeho¾ vrcholy le¾í na jedné kru¾nici.
+
+Zkusíme nyní odhadnout, jak velký je rovinný graf popisující Voroného diagram. Podle slavné Eulerovy formule má ka¾dý rovinný graf nejvý¹e lineárnì
+mnoho vrcholù, hran a stìn -- pro $v$ vrcholù, $e$ hran a $f$ stìn je $e \le 3v-6$ a navíc $v+f = e+2$. Tedy slo¾itost diagramu je lineární vzhledem k
+poètu zadaných bodù $n=f$, $\O(n)$. Navíc Voroného diagram lze zkonstruovat v èase $\O(n \log n)$, napøíklad pomocí zametání roviny nebo metodou
+rozdìl a panuj. Tím se v¹ak zabývat nebudeme,\foot{Pro zvídavé, kteøí nemají zkou¹ku druhý den ráno: Detaily naleznete v zápiscích z pøedloòského
+ADSka.} místo toho si uká¾eme, jak v ji¾ spoèteném Voroného diagramu rychle hledat nejbli¾¹í body.
+
+\h{Lokalizace bodu uvnitø mnohoúhelníkové sítì}
+
+Problém medvìdù je najít v medvìdí mapì co nejrychleji nejbli¾¹í iglù. Máme v rovinì sí» tvoøenou mnohoúhelníky. Chceme pro jednotlivé body rychle
+rozhodovat, do kterého mnohoúhelníku patøí. Na¹e øe¹ení budeme optimalizovat pro jeden pevný rozklad a obrovské mno¾ství rùzných dotazù, které chceme
+co nejrychleji zodpovìdìt.\foot{Pøedstavujme si to tøeba tak, ¾e medvìdùm zprovozníme server. Ten jednou schroustá celou mapu a potom co nejrychleji
+odpovídá na jejich dotazy. Medvìdi tak nemusí v mapách nic hledat, staèí se pøipojit na server a poèkat na odpovìï.} Nejprve pøedzpracujeme zadané
+mnohoúhelníky a vytvoøíme strukturu, která nám umo¾ní rychlé dotazy na jednotlivé body.
+
+Uka¾me si pro zaèátek øe¹ení bez pøedzpracování. Rovinu budeme zametat pøímkou shora dolù. Podobnì jako pøi hledání prùseèíkù úseèek, udr¾ujeme si prùøez
+pøímkou. V¹imnìte si, ¾e tento prùøez se mìní jenom ve vrcholech mnohoúhelníkù. Ve chvíli, kdy narazíme na hledaný bod, podíváme se, do kterého
+intervalu v prùøezu patøí. To nám dá mnohoúhelník, který nahlásíme. Prùøez budeme uchovávat ve vyhledávacím stromì. Takové øe¹ení má slo¾itost $\O(n
+\log n)$ na dotaz, co¾ je hroznì pomalé.
+
+Pøedzpracování bude fungovat následovnì. Jak je naznaèeno na obrázku pøeru¹ovanými èárami, rozøe¾eme si celou rovinu na pásy, bìhem kterých se prùøez
+pøímkou nemìní. Pro ka¾dý z nich si pamatujeme stav stromu popisující, jak vypadal prùøez pøi procházení tímto pásem. Kdy¾ chceme lokalizovat nìjaký bod,
+nejprve pùlením nalezneme pás, ve kterém se nachází. Poté polo¾íme dotaz na pøíslu¹ný strom. Strom procházíme a po cestì si dopoèítáme souøadnice
+prùøezu, a¾ lokalizujeme správný interval v prùøezu. Dotaz doká¾eme zodpovìdìt v èase $\O(\log n)$. Hledaný bod je na obrázku naznaèen prázdným
+koleèkem a nalezený interval v prùøezu je vyta¾ený tuènì.
+
+\figure{8-geom2_4_pasy_mnohouhelniku.eps}{Mnohoúhelníky rozøezané na pásy.}{2.5in}
+
+Jenom¾e na¹e øe¹ení má jeden háèek: Jak zkonstruovat jednotlivé verze stromu dostateènì rychle? K tomu napomohou {\I èásteènì perzistentní} datové
+struktury. Pod perzistencí se myslí, ¾e struktura umo¾òuje uchovávat svoji historii. Èásteènì perzistentní struktury nemohou svoji historii
+modifikovat.
+
+Popí¹eme si, jak vytvoøit perzistentní strom s pamìtí $\O(\log n)$ na zmìnu. Pokud provádíme operaci na stromì, mìní se jenom malá èást stromu.
+Napøíklad pøi vkládání do stromu se mìní jenom prvky na jedné cestièce z koøene do listu (a pøípadnì rotací i na jejím nejbli¾¹ím okolí). Proto si
+ulo¾íme upravenou cestièku a zbytek stromu budeme sdílet s pøedchozí verzí. Na obrázku je vyznaèena cesta, její¾ vrcholy jsou upravovány.  ©edì
+oznaèené podstromy navì¹ené na tuto cestu se nemìní, a proto na nì staèí zkopírovat ukazatele. Mimochodem zmìny ka¾dé operace se slo¾itostí $\O(k)$
+lze zapsat v pamìti $\O(k)$, prostì operace nemá tolik èasu, aby mohla pozmìnit pøíli¹ velikou èást stromu. 
+
+\figure{8-geom2_5_upravy_stromu.eps}{Jedna operace mìní pouze okolí cesty -- navì¹ené podstromy se nemìní.}{2in}
+
+Celková èasová slo¾itost je tedy $\O(n \log n)$ na pøedzpracování Voroného diagramu a vytvoøení persistentního stromu. Kvùli persistenci potøebuje
+toto pøedzpracování pamì» $\O(n \log n)$. Na dotaz spotøebujeme èas $\O(\log n)$, nebo» nejprve vyhledáme pùlením pøíslu¹ný pás a poté polo¾íme dotaz
+na pøíslu¹nou verzi stromu. Rychleji to ani provést nepùjde, nebo» potøebujeme utøídit souøadnice bodù.
+
+\s{Lze to lépe?} Na závìr poznamenejme, ¾e se umí provést vý¹e popsaná persistence vyhledávacího stromu v amortizované pamìti $\O(1)$ na zmìnu. Ve
+struènosti naznaèíme my¹lenku. Pou¾ijeme stromy, které pøi insertu a deletu provádí amortizovanì jenom konstantnì mnoho úprav své struktury. To nám
+napøíklad zaruèí 2-4 stromy z pøedná¹ky a podobnou vlastnost lze dokázat i o èerveno-èerných stromech. Pøi zmìnì potom nebudeme upravovat celou cestu,
+ale upravíme jenom jednotlivé vrcholy, kterých se zmìna týká. Ka¾dý vrchol stromu si v sobì bude pamatovat a¾ dvì své verze. Pokud chceme vytvoøit
+tøetí verzi, vrchol zkopírujeme stranou. To v¹ak mù¾e vyvolat zmìny v jeho rodièích a¾ do koøene. Situace je naznaèena na obrázku. Pøi vytvoøení nové
+verze $3$ pro vrcholu $v$ vytvoøíme jeho kopii $v'$, do které ulo¾íme tuto verzi. Av¹ak musíme také zmìnit rodièe $u$, kterému vytvoøíme novou verzi
+ukazující na $v'$. Abychom dosáhli ký¾ené konstantní pamì»ové slo¾itosti, pomù¾e potenciálový argument -- zmìn se provádí amortizovanì jenom
+konstantnì mnoho. Navíc si pro ka¾dou verzi pamatujeme její koøen, ze kterého máme dotaz spustit.
+
+\figure{8-geom2_6_rychla_perzistence.eps}{Vytvoøení nové verze vrcholu.}{2in}
+
+\bye
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diff --git a/old/8-geom2/8-geom2_5_upravy_stromu.eps b/old/8-geom2/8-geom2_5_upravy_stromu.eps
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diff --git a/old/8-geom2/8-geom2_6_rychla_perzistence.eps b/old/8-geom2/8-geom2_6_rychla_perzistence.eps
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diff --git a/old/8-geom2/Makefile b/old/8-geom2/Makefile
new file mode 100644
index 0000000..087c424
--- /dev/null
+++ b/old/8-geom2/Makefile
@@ -0,0 +1,3 @@
+P=8-geom2
+
+include ../Makerules
diff --git a/old/8-geom2/lib.mp b/old/8-geom2/lib.mp
new file mode 100644
index 0000000..cbeedf5
--- /dev/null
+++ b/old/8-geom2/lib.mp
@@ -0,0 +1,84 @@
+% implementation of figure naming and figure transparency
+string name,tag; name := ""; tag := "";
+def updatefigname =
+	if tag="": filenametemplate (name & "%c.eps");
+	else: filenametemplate (name & "%c_" & tag & ".eps");
+	fi;
+enddef;
+	
+def figname(expr n) = name := n; updatefigname; enddef;
+def figtag(expr t) = tag := t; updatefigname; enddef;
+
+picture transparent_picture;
+color transparent_color; transparent_color := 0.9white;
+def drawtransparent(expr num) =
+	transparent_picture := currentpicture;
+endfig;
+
+if tag="": filenametemplate (name & "%c_transparent.eps");
+else: filenametemplate (name & "%c_" & tag & "_transparent.eps");
+fi;
+beginfig(num);
+	draw transparent_picture withcolor transparent_color;
+endfig;
+
+updatefigname;
+enddef;
+
+def from(expr p,d,len) =
+	p+dir(d)*len
+enddef;
+
+def dirs(expr p,d,len) =
+	p--from(p,d,len)
+enddef;
+
+def drawvertices(expr s,n) =
+	for i:=s upto n:
+		draw vertex(PQ[i]);
+	endfor
+enddef;
+
+def drawfvertices(expr s,n,flags) =
+	for i:=s upto n:
+		draw vertex(PQ[i]) flags;
+	endfor
+enddef;
+
+path centersquare; centersquare := (-0.5,-0.5)--(0.5,-0.5)--(0.5,0.5)--(-0.5,0.5)--cycle;
+
+def vertex(expr p) = p withpen pencircle scaled 4pt enddef;
+def drawemptyvertex(expr p) = unfill fullcircle scaled 4pt shifted p; draw fullcircle scaled 4pt shifted p; enddef;
+def drawendpointvertex(expr p) = draw vertex(p) withcolor red; draw fullcircle scaled 6pt shifted p; enddef;
+def createpath(expr p) = shakepath(p, 0.015cm,0.1cm) enddef;
+vardef shakepath(expr p,d,l) =
+	save r,b;
+	path r; r := point(arctime 0 of p) of p;
+	b := -1;
+	for i:=l step l until arclength(p):
+		r := r--(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*d*b;
+		b := -b;
+	endfor
+	r--point(arctime arclength(p) of p) of p
+enddef;
+	
+pen normalpen; normalpen := pencircle scaled 0.6pt;
+pen boldpen; boldpen := pencircle scaled 1.5pt;
+pen bolderpen; bolderpen := pencircle scaled 2pt;
+def dotline = withdots scaled 0.82 withpen boldpen enddef;
+
+vardef unclosedbubblec(expr p,c) =
+	bubblec((p..reverse p..cycle),c)
+enddef;	
+
+vardef bubblec(expr p,c) =
+	save r;
+	path r; r := (point(arctime 0 of p) of p)+dir(angle(direction(arctime 0 of p) of p rotated 90))*c;
+	for i:=0.01cm step 0.025cm until arclength(p):
+		r := r..(point(arctime i of p) of p)+dir(angle(direction(arctime i of p) of p rotated 90))*c;
+	endfor
+	r..(point(arctime arclength(p) of p) of p)+dir(angle(direction(arctime arclength(p) of p) of p rotated 90))*c..cycle
+enddef;
+
+vardef bubble(expr p) = bubblec(p,0.12cm) enddef;
+vardef unclosedbubble(expr p) = unclosedbubblec(p,0.12cm) enddef;
diff --git a/old/9-fft/9-fft.tex b/old/9-fft/9-fft.tex
new file mode 100644
index 0000000..3340759
--- /dev/null
+++ b/old/9-fft/9-fft.tex
@@ -0,0 +1,392 @@
+\def\scharfs{\char"19}
+\input lecnotes.tex
+\def\imply{\Rightarrow}
+\prednaska{9}{Fourierova transformace}{\vbox{\hbox{(K.Jakubec, M.Polák
+	a~G.Ocsovszky,}\hbox{\ V.Tùma, M.Kozák)}}}
+
+Násobení polynomù mù¾e mnohým pøipadat jako pomìrnì (algoritmicky) snadný
+problém. Asi ka¾dého hned napadne \uv{hloupý} algoritmus -- vezmeme
+koeficienty prvního polynomu a~vynásobíme ka¾dý se v¹emi koeficienty druhého
+polynomu a~pøíslu¹nì u~toho seèteme i~exponenty (stejnì jako to dìláme, kdy¾
+násobíme polynomy na~papíøe). Pokud stupeò prvního polynomu je $n$ a~druhého
+$m$, strávíme tím èas $\Omega(mn)$. Pro $m=n$ je to kvadraticky pomalé.
+Na~první pohled se mù¾e zdát, ¾e rychleji to prostì nejde (pøeci musíme
+v¾dy vynásobit \uv{ka¾dý s~ka¾dým}). Ve skuteènosti to ale rychleji fungovat
+mù¾e, ale k~tomu je potøeba znát trochu tajemný algoritmus FFT neboli {\I Fast
+Fourier Transform}.
+
+
+\ss{Trochu algebry na~zaèátek}
+Celé polynomy oznaèujeme velkými písmeny, jednotlivé èleny polynomù pøíslu¹nými
+malými písmeny (pø.: polynom $W$ stupnì $d$ má koeficienty $w_{0}, w_{1},
+w_{2},\ldots, w_{d}$).
+
+Libovolný polynom $P$ stupnì (nejvý¹e) $d$ lze reprezentovat
+jednak jeho koeficienty, tedy èísly $p_{0}, p_{1}, \ldots ,p_{d}$, druhak
+i~pomocí hodnot:
+
+\>{\bf Lemma:} Polynom stupnì nejvý¹e $d$ je jednoznaènì urèeò svými
+hodnotami v~$d+1$ rùzných bodech.
+
+\>{\it Dùkaz:}
+Polynom stupnì $d$ má maximálnì $d$ koøenù (indukcí -- je-li
+$k$ koøenem $P$, pak lze $P$ napsat jako $(x-k)Q$ kde $Q$ je polynom stupnì
+o~jedna men¹í, pøitom polynom stupnì 1 má jediný koøen); uvá¾íme-li
+dva rùzné polynomy $P$ a~$Q$ stupnì $d$ nabývající v~daných bodech stejných
+hodnot, tak $P-Q$ je polynom stupnì maximálnì $d$, ka¾dé
+z $x_{0}\ldots x_{d}$ je koøenem tohoto polynomu $\imply$ spor, polynom stupnì
+$d$ má $d+1$ koøenù $\imply$ $P-Q$ musí být nulový polynom $\imply$ $P=Q$.
+\qed
+\medskip	
+
+Pov¹imnìme si jedné skuteènosti -- máme-li dva polynomy $A$ a~$B$ stupnì $d$
+a~body $x_{0}, \ldots, x_{n}$, dále polynom $C=A \cdot B$ (stupnì $2d$), pak
+platí $C(x_{j}) = A(x_{j}) \cdot B(x_{j})$ pro $j = 0,1,2, \ldots, n$. Toto
+èiní tento druhý zpùsob reprezentace polynomu velice atraktivním pro násobení
+-- máme-li $A$ i $B$ reprezentované hodnotami v $n \geq 2d+1$ bodech, pak
+snadno (v $\Theta(n)$) spoèteme takovou reprezentaci $C$.
+Problémem je, ¾e typicky máme polynom zadaný koeficienty, a~ne hodnotami
+v~bodech. Tím pádem potøebujeme nìjaký hodnì rychlý algoritmus (tj.
+rychlej¹í ne¾ kvadratický, jinak bychom si nepomohli oproti hloupému
+algoritmu) na~pøevod polynomu z jedné reprezentace do druhé a~zase zpìt.
+
+\s{Idea, jak by mìl algoritmus pracovat:}
+\algo
+\:Vybereme $n\geq 2d+1$ bodù $x_{0}, x_{1}, \ldots , x_{n-1}$.
+\:V tìchto bodech vyhodnotíme polynomy $A$ a~$B$.
+\:Nyní ji¾ v~lineárním èase získáme hodnoty polynomu $C$ v~tìchto bodech:
+	$C(x_i) = A(x_i)\cdot B(x_i)$
+\:Pøevedeme hodnoty polynomu $C$ na~jeho koeficienty.
+\endalgo
+
+\>Je vidìt, ¾e klíèové jsou kroky 2 a~4.
+Celý trik spoèívá v~chytrém vybrání onìch bodù, ve kterých budeme polynomy
+vyhodnocovat -- zvolí-li se obecná $x_j$, tak se to rychle neumí, pro speciální
+$x_j$ ale uká¾eme, ¾e to rychle jde.
+
+\ss{Vyhodnocení polynomu metodou Rozdìl a~panuj (algoritmus FFT):}
+Mìjme polynom $P$ stupnì $\leq d$ a~chtìjme jej vyhodnotit v~$n$ bodech.
+Vybereme si body tak, aby byly spárované, èili $\pm x_{0}, \pm x_{1},
+\ldots , \pm x_{n/2-1} $. To nám výpoèet urychlí, proto¾e pak se druhé
+mocniny $x_{j}$ shodují s~druhými mocninami $-x_{j}$.
+
+Polynom $P$ rozlo¾íme na~dvì èásti, první obsahuje èleny se sudými exponenty,
+druhá s~lichými:
+$$P(x) = (p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{d-2}x^{d-2}) + (p_{1}x^{1} +
+	p_{3}x^{3} + \ldots + p_{d-1}x^{d-1})$$
+\>se zavedením znaèení:
+$$P_s(t) = p_0t^0 + p_{2}t^{1} + \ldots + p_{d - 2}t^{d - 2\over 2}$$
+$$P_l(t) = p_1t^0 + p_3t^1 + \ldots + p_{d - 1}t^{d - 2\over 2}$$
+
+\>bude $P(x) = P_s(x^{2}) + xP_l(x^{2})$ a~$P(-x) = P_s(x^{2}) -
+xP_l(x^{2})$. Jinak øeèeno, vyhodnocování polynomu $P$ v~$n$ bodech se nám
+smrskne na~vyhodnocení $P_s$ a~$P_l$ v~$n/2$ bodech -- oba jsou polynomy
+stupnì nejvý¹e $d/2$ a~vyhodnocujeme je v~$x^{2}$ (vyu¾íváme
+rovnosti $(x_{i})^{2} = (-x_{i})^{2}$).
+
+\s{Pøíklad:}
+$3 + 4x + 6x^{2} + 2x^{3} + x^{4} + 10x^{5} = (3 + 6x^{2} + x^{4}) + x(4 +
+2x^{2} + 10x^{4})$.
+
+Teï nám ov¹em vyvstane problém s~oním párováním -- druhá mocnina pøece nemù¾e
+být záporná a~tím pádem u¾ v~druhé úrovni rekurze body spárované nebudou.
+Z~tohoto dùvodu musíme pou¾ít komplexní èísla -- tam druhé mocniny záporné býti
+mohou.
+
+% komplex
+\h{Komplexní intermezzo}
+\def\i{{\rm i}}
+\def\\{\hfil\break}
+
+\s{Základní operace}
+
+\itemize\ibull
+\:Definice: ${\bb C} = \{a + b\i \mid a,b \in {\bb R}\}$
+
+\:Sèítání: $(a+b\i)\pm(p+q\i) = (a\pm p) + (b\pm q)\i$. \\
+Pro $\alpha\in{\bb R}$ je $\alpha(a+b\i) = \alpha a + \alpha b\i$.
+
+\:Komplexní sdru¾ení: $\overline{a+b\i} = a-b\i$. \\
+$\overline{\overline x} = x$, $\overline{x\pm y} = \overline{x} \pm
+\overline{y}$, $\overline{x\cdot y} = \overline x \cdot \overline y$, $x
+\cdot \overline x \in {\bb R}$.
+
+\:Absolutní hodnota: $\vert x \vert = \sqrt{x\cdot\overline{x}}$, tak¾e $\vert
+a+b\i \vert = \sqrt{a^2+b^2}$. \\
+Také $\vert \alpha x \vert = \vert \alpha\vert \cdot \vert x \vert$.
+
+\:Dìlení: $x/y = (x\cdot \overline{y}) / (y \cdot \overline{y})$.
+\endlist
+
+\s{Gau{\scharfs}ova rovina a goniometrický tvar}
+
+\itemize\ibull
+\:Komplexním èíslùm pøiøadíme body v~${\bb R}^2$: $a+b\i \leftrightarrow (a,b)$.
+
+\:$\vert x\vert$ je vzdálenost od~bodu $(0,0)$.
+
+\:$\vert x\vert = 1$ pro èísla le¾ící na~jednotkové kru¾nici ({\I komplexní jednotky}). \\
+Pak platí $x=\cos\varphi + \i\sin\varphi$ pro nìjaké $\varphi\in\left[
+0,2\pi \right)$.
+
+\:Pro libovolné $x\in{\bb C}$: $x=\vert x \vert \cdot (\cos\varphi(x) +
+\i\sin\varphi(x))$. \\
+Èíslu $\varphi(x)\in\left[ 0,2\pi \right)$ øíkáme {\I argument}
+èísla~$x$, nìkdy znaèíme $\mathop{\rm arg} x$.
+
+\:Navíc $\varphi({\overline{x}}) = -\varphi(x)$.
+\endlist
+
+\s{Exponenciální tvar}
+
+\itemize\ibull
+\:Eulerova formule: $e^{i\varphi} = \cos\varphi + \i\sin\varphi$.
+
+\:Ka¾dé $x\in{\bb C}$ lze tedy zapsat jako $\vert x\vert \cdot e^{\i\cdot
+\varphi(x)}$.
+
+\:Násobení: $xy = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right) \cdot
+	\left(\vert y\vert\cdot e^{\i\cdot\varphi(y)}\right) = \vert x\vert
+	\cdot \vert y\vert \cdot e^{\i\cdot(\varphi(x) + \varphi(y))}$. \\
+(absolutní hodnoty se násobí, argumenty sèítají)
+
+\:Umocòování: $x^\alpha = \left(\vert x\vert\cdot e^{\i\cdot\varphi(x)}\right)^
+	\alpha = {\vert x\vert}^\alpha\cdot e^{\i \alpha \varphi(x)}$.
+
+\:Odmocòování: $\root n\of x = {\vert x\vert}^{1/n} \cdot e^{\i\cdot
+\varphi(x)/n}$. \\
+Pozor -- odmocnina není jednoznaèná: $1^4=(-1)^4=\i^4=(-\i)^4=1$.
+\endlist
+
+\s{Odmocniny z~jednièky}
+
+\itemize\ibull
+\:Je-li nìjaké $x\in{\bb C}$ $n$-tou odmocninou z~jednièky, musí platit:
+$\vert x \vert = 1$, tak¾e $x=e^{\i\varphi}$ pro nìjaké~$\varphi$.
+Proto $x^n = e^{\i\varphi n} = \cos{\varphi n} + \i\sin\varphi n = 1$.
+Platí tedy $\varphi n = 2k\pi$ pro nìjaké $k\in{\bb Z}$.
+
+\:Z~toho plyne: $\varphi = 2k\pi/n$ \\
+(pro $k=0,\ldots,n-1$ dostáváme rùzné $n$-té odmocniny).
+
+\:Obecné odmocòování: $\root n \of x = {\vert x\vert}^{1/n} \cdot e^{\i\varphi
+	(x)/n} \cdot u$, kde $u=\root n\of 1$.
+
+\:Je-li $x$ odmocninou z 1, pak $\overline{x} = x^{-1}$ -- je toti¾ $1 = \vert
+x\cdot \overline{x}\vert = x\cdot \overline{x}$.
+\endlist
+
+\s{Primitivní odmocniny}
+
+\s{Definice:} $x$ je {\I primitivní} $k$-tá odmocnina z 1 $\equiv x^k=1 \land \forall j: 0<j<k \Rightarrow x^j \neq 1$.
+
+\>Tuto definici splòují napøíklad èísla $\omega = e^{2\pi \i / k}$ a $\overline\omega = e^{-2\pi\i/k}$.
+Platí toti¾, ¾e $\omega^j = e^{2\pi\i j/k}$, co¾ je rovno~1 právì tehdy,
+je-li $j$ násobkem~$k$ (jednotlivé mocniny èísla~$\omega$ postupnì obíhají
+jednotkovou kru¾nici). Analogicky pro~$\overline\omega$.
+
+\>Uka¾me si nìkolik pozorování fungujících pro libovolné èíslo~$\omega$,
+které je primitivní $k$-tou odmocninou z~jednièky (nìkdy budeme potøebovat,
+aby navíc $k$ bylo sudé):
+
+\itemize\ibull
+\:Pro $0\leq j<l<k$ je $\omega^j \neq \omega^l$, nebo» $\omega^l /
+\omega^j = \omega^{l-j} \neq 1$, proto¾e $l-j < k$ a $\omega$ je primitivní.
+\:$\omega^{k/2} = -1$, proto¾e $(\omega^{k/2})^2 = 1$, a tedy
+$\omega^{k/2}$ je druhá odmocnina z~1. Takové odmocniny jsou dvì:
+1 a $-1$, ov¹em 1 to být nemù¾e, proto¾e $\omega$ je primitivní.
+\:$\omega^j = - \omega^{k/2 + j}$ -- pøímý dùsledek pøedchozího bodu, pro
+nás ale velice zajímavý: $\omega^0,\omega^1,\ldots,\omega^{k-1}$ jsou po dvou
+spárované.
+\:$\omega^2$ je $k/2$-tá primitivní odmocnina z 1 -- dosazením.
+\endlist
+
+\h{Konec intermezza}
+% end komplex
+
+\bigskip
+
+Vra»me se nyní k algoritmu. Z poslední èásti komplexního intermezza se zdá,
+¾e by nemusel být ¹patný nápad zkusit vyhodnocovat polynom v mocninách
+$n$-té primitivní odmocniny z~jedné (tedy za $x_0,x_1,\ldots,x_{n-1}$
+z~pùvodního algoritmu zvolíme $\omega^0,\omega^1,\ldots,\omega^{n-1}$).
+Aby nám v¹e vycházelo pìknì, zvolíme $n$ jako mocninu dvojky.
+
+\ss{Celý algoritmus bude vypadat takto:}
+\>FFT($P$, $ \omega$)
+
+\>{\sl Vstup:} $p_{0}, \ldots , p_{n-1}$, koeficienty polynomu $P$ stupnì
+nejvý¹e $n-1$, a~$\omega$,
+$n$-tá primitivní odmocina z jedné.
+
+\>{\sl Výstup:} Hodnoty polynomu v~bodech $1, \omega, \omega^{2}, \ldots ,
+\omega^{n - 1}$, èili èísla $P(1), P(\omega), P(\omega^{2}),$ $\ldots ,
+P(\omega^{n - 1})$.
+
+\algo
+\:Pokud $n = 1$, vrátíme $p_{0}$ a~skonèíme.
+\:Jinak rozdìlíme $P$ na èleny se sudými a lichými exponenty (jako v pùvodní
+my¹lence) a~rekurzivnì zavoláme FFT($P_s$, $\omega^{2}$) a~FFT($P_l$,
+$\omega^{2}$) -- $P_l$ i~$P_s$ jsou stupnì max. $n/2-1$, $\omega^2$ je
+$n/2$-tá primitivní odmocnina, a mocniny $\omega^2$ jsou stále po dvou
+spárované ($n$ je mocnina dvojky, a tedy i $n/2$ je sudé; popø. $n=2$ a je to
+zøejmé).
+\:Pro $j = 0, \ldots , n/2 - 1$ spoèítáme:
+
+\:\qquad $P(\omega^{j}) = P_s(\omega^{2j}) + \omega^{j}\cdot P_l(\omega^{2j})$.
+
+\:\qquad $P(\omega^{j+n/2})=P_s(\omega^{2j})-\omega^{j}\cdot P_l(\omega^{2j})$.
+
+\endalgo
+
+
+\s{Èasová slo¾itost:}
+\>$T(n)=2T(n/2) + \Theta(n) \Rightarrow$ slo¾itost $\Theta(n \log n)$, jako
+MergeSort.
+
+
+Máme tedy algoritmus, který pøevede koeficienty polynomu na~hodnoty tohoto
+polynomu v~rùzných bodech. Potøebujeme ale také algoritmus, který doká¾e
+reprezentaci polynomu pomocí hodnot pøevést zpìt na~koeficienty polynomu.
+K~tomu nám pomù¾e podívat se na~ná¹ algoritmus trochu obecnìji.
+
+
+\s{Definice:}
+\>{\I Diskrétní Fourierova transformace} $(DFT)$
+je zobrazení $f: { {\bb C} ^n} \rightarrow { {\bb C} ^n}$, kde $$y=f(x) \equiv
+\forall j \ y_{j} = \sum \limits ^{n-1}_{k=0} x_{k} \cdot \omega ^{jk}$$
+(DFT si lze mimo jiné pøedstavit jako funkci vyhodnocující polynom
+s~koeficienty $x_k$ v~bodech $\omega^j$). Takovéto zobrazení je lineární
+a~tedy popsatelné maticí $\Omega$ s~prvky $\Omega_{jk} = \omega^{jk}$.
+Chceme-li umìt pøevádìt z~hodnot polynomu na koeficienty, zajímá nás inverze
+této matice.
+
+
+\ss{Jak najít inverzní matici?}
+Znaème $\overline{\Omega}$ matici, její¾ prvky
+jsou komplexnì sdru¾ené odpovídajícím prvkùm $\Omega$, a vyu¾ijme následující
+lemma:
+
+\ss{Lemma:}$\quad \Omega\cdot \overline{\Omega} = n\cdot E$.
+
+\proof $$ (\Omega\cdot \overline{\Omega})_{jk} = \sum_{l=0}^{n-1} \omega^{jl}
+\cdot \overline{\omega^{lk}} = \sum \omega^{jl} \cdot \overline{\omega}^{lk} =
+\sum \omega^{jl} \cdot \omega^{-lk} = \sum \omega^{l(j-k)}\hbox{.}$$
+\itemize\ibull
+\:Pokud $j=k$, pak $ \sum \limits ^{n-1}_{l=0} (\omega ^{0}) ^{l} = n$.
+
+\:Pokud $j\neq k$, pou¾ijeme vzoreèek pro souèet geometrické øady
+s kvocientem $\omega ^{(j-k) }$ a~dostaneme ${{\omega^{(j-k)n} -1} \over
+{\omega^{(j-k)} -1}} ={1-1 \over \neq 0} = 0$ (
+$\omega^{j-k} - 1$ je jistì $\neq 0$, nebo» $\omega$ je $n$-tá primitivní
+odmoncina a $j-k<n$).
+\endlist
+\qed
+
+
+\>Na¹li jsme inverzi:
+$\Omega({1 \over n} \overline{\Omega}) = {1 \over n}\Omega \cdot
+\overline{\Omega} = E$, \quad $\Omega^{-1}_{jk} = {1 \over n}\overline{\omega^
+{jk}} = {1 \over n}\omega^{-jk} = {1 \over n} {(\omega^{-1})}^{jk}$, \quad
+(pøipomínáme, $\omega^{-1}$ je $\overline{\omega}$).
+
+Vyhodnocení polynomu lze provést vynásobením $\Omega$, pøevod do pùvodní
+reprezentace vynásobením $\Omega^{-1}$. My jsme si ale v¹imli chytrého
+spárování, a vyhodnocujeme polynom rychleji ne¾ kvadraticky (proto FFT,
+jako¾e {\it fast}, ne jako {\it fuj}). Uvìdomíme-li si, ¾e $\overline \omega =
+\omega^{-1}$ je také $n$-tá primitivní odmocnina z 1 (má akorát
+úhel s opaèným znaménkem), tak mù¾eme stejným trikem vyhodnotit i~zpìtný
+pøevod -- nejprve vyhodnotíme $A$ a $B$ v $\omega^j$, poté pronásobíme
+hodnoty a~dostaneme tak hodnoty polynomu $C=A\cdot B$, a pustíme na nì
+stejný algoritmus s~$\omega^{-1}$ (hodnoty $C$
+vlastnì budou v~algoritmu \uv{koeficienty polynomu}). Nakonec jen získané
+hodnoty vydìlíme $n$ a~máme chtìné koeficienty.
+
+\s{Výsledek:} Pro $n= 2^k$ lze DFT na~${\bb C}^n$ spoèítat v~èase $\Theta(n
+\log n)$ a~DFT$^{-1}$ takté¾.
+
+\s{Dùsledek:}
+Polynomy stupnì $n$ lze násobit v~èase $\Theta(n \log n)$:
+$\Theta(n \log n)$ pro vyhodnocení, $\Theta(n)$ pro vynásobení a~$\Theta(n
+\log n)$ pro pøevedení zpìt.
+
+\s{Pou¾ití FFT:}
+
+\itemize\ibull
+
+\:Zpracování signálu -- rozklad na~siny a~cosiny o~rùzných frekvencích
+$\Rightarrow$ spektrální rozklad.
+\:Komprese dat -- napøíklad formát JPEG.
+\:Násobení dlouhých èísel v~èase $\Theta(n \log n)$.
+\endlist
+
+\s{Paralelní implementace FFT}
+
+\figure{img.eps}{Pøíklad prùbìhu algoritmu na~vstupu velikosti 8}{3in}
+
+Zkusme si prùbìh algoritmu FFT znázornit graficky (podobnì, jako jsme kreslili
+hradlové sítì). Na~levé stranì obrázku se nachází vstupní vektor $x_0,\ldots,x_{n-1}$
+(v~nìjakém poøadí), na~pravé stranì pak výstupní vektor $y_0,\ldots,y_{n-1}$.
+Sledujme chod algoritmu pozpátku: Výstup spoèítáme z~výsledkù \uv{polovièních}
+transformací vektorù $x_0,x_2,\ldots,x_{n-2}$ a $x_1,x_3,\ldots,x_{n-1}$.
+Èerné krou¾ky pøitom odpovídají výpoètu lineární kombinace $a+\omega^kb$,
+kde $a,b$ jsou vstupy krou¾ku a $k$~nìjaké pøirozené èíslo závislé na poloze
+krou¾ku. Ka¾dá z~polovièních transformací se poèítá analogicky z~výsledkù
+transformace velikosti $n/4$ atd. Celkovì výpoèet probíhá v~$\log_2 n$ vrstvách
+po~$\Theta(n)$ operacích.
+
+Jeliko¾ operace na~ka¾dé vrstvì probíhají na sobì nezávisle, mù¾eme je poèítat
+paralelnì. Ná¹ diagram tedy popisuje hradlovou sí» pro paralelní výpoèet FFT
+v~èase $\Theta(\log n\cdot T)$ a prostoru $\O(n\cdot S)$, kde $T$ a~$S$ znaèí
+èasovou a prostorovou slo¾itost výpoètu lineární kombinace dvou komplexních èísel.
+
+\s{Cvièení:} Doka¾te, ¾e permutace vektoru $x_0,\ldots,x_{n-1}$ odpovídá bitovému
+zrcadlení, tedy ¾e na pozici~$b$ shora se vyskytuje prvek $x_d$, kde~$d$ je
+èíslo~$b$ zapsané ve~dvojkové soustavì pozpátku.
+
+\s{Nerekurzivní FFT}
+
+Obvod z~pøedchozího obrázku také mù¾eme vyhodnocovat po~hladinách zleva doprava,
+èím¾ získáme elegantní nerekurzivní algoritmus pro výpoèet FFT v~èase $\Theta(n\log n)$
+a prostoru $\Theta(n)$:
+
+\algo
+\algin $x_0,\ldots,x_{n-1}$
+\:Pro $j=0,\ldots,n-1$ polo¾íme $y_k\= x_{r(k)}$, kde $r$ je funkce bitového zrcadlení.
+\:Pøedpoèítáme tabulku $\omega^0,\omega^1,\ldots,\omega^{n-1}$.
+\:$b\= 1$ \cmt{velikost bloku}
+\:Dokud $b<n$, opakujeme:
+\::Pro $j=0,\ldots,n-1$ s~krokem~$2b$ opakujeme: \cmt{zaèátek bloku}
+\:::Pro $k=0,\ldots,b-1$ opakujeme: \cmt{pozice v~bloku}
+\::::$\alpha\=\omega^{nk/2b}$
+\::::$(y_{j+k},y_{j+k+b}) \= (y_{j+k}+\alpha\cdot y_{j+k+b}, y_{j+k}-\alpha\cdot y_{j+k+b})$.
+\::$b\= 2b$
+\algout $y_0,\ldots,y_{n-1}$
+\endalgo
+
+\s{FFT v~koneèných tìlesech}
+
+Nakonec dodejme, ¾e Fourierovu transformaci lze zavést nejen nad tìlesem
+komplexních èísel, ale i v~nìkterých koneèných tìlesech, pokud zaruèíme
+existenci primitivní $n$-té odmocniny z~jednièky. Napøíklad v~tìlese ${\bb
+Z}_p$ pro prvoèíslo $p=2^k+1$ platí $2^k=-1$, tak¾e $2^{2k}=1$
+a $2^0,2^1,\ldots,2^{2k-1}$ jsou navzájem rùzná, tak¾e èíslo~2 je~primitivní
+$2k$-tá odmocnina z~jedné. To se nám ov¹em nehodí pro algoritmus FFT, jeliko¾
+$2k$ bude málokdy mocnina dvojky.
+
+Zachrání nás ov¹em algebraická vìta, která øíká, ¾e multiplikativní grupa\foot{To je
+mno¾ina v¹ech nenulových prvkù tìlesa s~operací násobení.}
+libovolného koneèného tìlesa je cyklická, tedy ¾e v¹echny nenulové prvky tìlesa lze
+zapsat jako mocniny nìjakého èísla~$g$ (generátoru). Napøíklad pro $p=2^{16}+1=65\,537$
+je jedním takovým generátorem èíslo~$3$. Jeliko¾ mezi èísly $g^1,g^2,\ldots,g^{p-1}$
+se musí ka¾dý nenulový prvek tìlesa vyskytnout právì jednou, je~$g$ primitivní
+$2^k$-tou odmocninou z~jednièky, tak¾e mù¾eme poèítat FFT pro libovolný vektor,
+jeho¾ velikost je mocnina dvojky men¹í nebo rovná $2^k$.\foot{Bli¾¹í prùzkum
+na¹ich úvah o~FFT dokonce odhalí, ¾e není ani potøeba tìleso. Postaèí libovolný
+komutativní okruh, ve~kterém existuje pøíslu¹ná primitivní odmocnina
+z~jednièky, její multiplikativní inverze (ta ov¹em existuje v¾dy, proto¾e
+$\omega^{-1} = \omega^{n-1}$) a multiplikativní inverze èísla~$n$. To nám
+poskytuje je¹tì daleko více volnosti ne¾ tìlesa, ale není snadné takové okruhy
+hledat.}
+
+\bye
diff --git a/old/9-fft/Makefile b/old/9-fft/Makefile
new file mode 100644
index 0000000..0769a79
--- /dev/null
+++ b/old/9-fft/Makefile
@@ -0,0 +1,3 @@
+P=9-fft
+
+include ../Makerules
diff --git a/old/9-fft/img.eps b/old/9-fft/img.eps
new file mode 100644
index 0000000..d7e20d4
--- /dev/null
+++ b/old/9-fft/img.eps
@@ -0,0 +1,1072 @@
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+%%BeginResource: procset XaraStudio1Dict
+% Copyright (c) 1995,1996 Xara Ltd
+/XaraStudio1Dict 300 dict def XaraStudio1Dict begin
+/bd{bind def}bind def/ld{load def}bind def/xd{exch def}bind def/sv{save}bd
+/rs{restore}bd/gs{gsave}bd/gr{grestore}bd/bg{begin}bd/en{end}bd/level2
+/languagelevel where{pop languagelevel 2 ge}{false}ifelse def/setseps{
+/v_gseps xd}bd/setplate{/v_plate xd}bd/setkgray{/v_keyg xd}bd/setmono{
+/v_mono xd}bd/rgb2gray{0.109 mul exch 0.586 mul add exch 0.305 mul
+add}bd/cmyk2rgb{3{dup 5 -1 roll add dup 1 gt{pop 1}if 1 exch sub exch}repeat
+pop}bd/rgb2cmyk{3{1.0 exch sub 3 1 roll}repeat 3 copy 2 copy gt{exch}if
+pop 2 copy gt{exch}if pop dup 0.5 gt{0.5 sub dup 3{5 1 roll dup 3 1
+roll sub}repeat 5 1 roll pop}{pop 0}ifelse}bd/cmyk2hsb{3{dup 5 -1 roll
+add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop rgb2hsb}bd/rgb2hsb{setrgbcolor
+currenthsbcolor}bd/readcurve{exch 255.0 mul 0.5 add cvi get 255.0 div}bd
+/rgb2devcmyk{3 copy dup 3 1 roll eq 3 1 roll eq v_keyg 1 eq and and{pop
+pop 1 exch sub 0 0 0 4 -1 roll}{/ucurve where{pop 3{1.0 exch sub 3
+1 roll}repeat 3 copy 2 copy gt{exch}if pop 2 copy gt{exch}if pop dup
+ucurve readcurve exch bcurve readcurve clamp01 3{5 1 roll dup 3 1 roll
+sub clamp01}repeat 5 1 roll pop 4 1 roll ycurve readcurve 4 1 roll
+mcurve readcurve 4 1 roll ccurve readcurve 4 1 roll}{rgb2cmyk}ifelse}ifelse}def
+/rgb2keyG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop}{Max3}ifelse
+1 exch sub bcurve readcurve clamp01}bd/rgb2key{Max3 1 exch sub bcurve
+readcurve clamp01}bd/rgb2cyanG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop
+pop pop 0}{rgb2cyan}ifelse}bd/rgb2cyan{3 copy Max3 1 exch sub ucurve
+readcurve 4 1 roll pop pop 1 exch sub exch sub ccurve readcurve clamp01}bd
+/rgb2magentaG{3 copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2magenta}ifelse}bd
+/rgb2magenta{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll pop
+1 exch sub 3 1 roll pop sub mcurve readcurve clamp01}bd/rgb2yellowG{3
+copy dup 3 1 roll eq 3 1 roll eq and{pop pop pop 0}{rgb2yellow}ifelse}bd
+/rgb2yellow{3 copy Max3 1 exch sub ucurve readcurve 4 1 roll 1 exch
+sub 4 1 roll pop pop sub ycurve readcurve clamp01}bd/rgb2plategray{v_keyg
+0 eq v_plate v_cpky eq{{rgb2key}{rgb2keyG}ifelse}{v_plate v_cpyl eq{{rgb2yellow}{rgb2yellowG}ifelse}{v_plate
+v_cpmg eq{{rgb2magenta}{rgb2magentaG}ifelse}{v_plate v_cpcy eq{{rgb2cyan}{rgb2cyanG}ifelse}{{rgb2key}{rgb2keyG}ifelse}ifelse}ifelse}ifelse}ifelse
+1 exch sub setgray}bd/dc{0 def}bd/aca{/v_cpnone 0 def/v_cpcy 1 def
+/v_cpyl 2 def/v_cpmg 3 def/v_cpky 4 def/v_gseps 0 def/v_keyg 0 def
+/v_plate v_cpnone def/v_mono 0 def/v_wr dc/v_fc dc/v_fm dc/v_fy dc
+/v_fk dc/v_fg dc/v_fr dc/v_fg dc/v_fb dc/v_sc dc/v_sm dc/v_sy dc/v_sk
+dc/v_sg dc/v_sr dc/v_sg dc/v_sb dc/v_sct 0 def/v_fct 0 def/v_ft 0 def
+/v_cxe 0 def/v_cxm 0 def/v_sa -1 def/v_ea -1 def/sR dc/sG dc/sB dc
+/mR dc/mG dc/mB dc/eR dc/eG dc/eB dc/sC dc/sM dc/sY dc/sK dc/eC dc
+/eM dc/eY dc/eK dc/sH dc/sS dc/sV dc/eH dc/eS dc/eV dc/sGy dc/eGy
+dc/mGy dc/ci_datasrc dc/ci_matrix dc/ci_dataleft dc/ci_buf dc/ci_dataofs
+dc/ci_y dc/rciBuf dc/cbslw dc/cmiBuf dc/cPalette dc/cpci_datasrc dc
+/cpci_matrix dc/cpci_bpp dc/cpci_y dc/cpci_sampsleft dc/cpci_nextcol
+dc/cpci_buf dc/startX dc/startY dc/endX dc/endY dc/endX2 dc/endY2 dc
+/fillX dc/urx dc/ury dc/llx dc/lly dc/incD dc/distance dc/slice dc
+/startangle dc/Steps dc/incH dc/incS dc/incV dc/incR dc/incG dc/incB
+dc/incGy dc 0.25 setlinewidth [] 0 setdash 0 setlinejoin 0 setlinecap}bd
+aca/setplategray{v_plate v_cpky eq{1 exch sub setgray pop pop pop}{v_plate
+v_cpyl eq{pop 1 exch sub setgray pop pop}{v_plate v_cpmg eq{pop pop
+1 exch sub setgray pop}{v_plate v_cpcy eq{pop pop pop 1 exch sub setgray}{1
+exch sub setgray pop pop pop}ifelse}ifelse}ifelse}ifelse}bd/setplatecolor{v_plate
+v_cpky eq{1 exch sub 0 0 0 4 -1 roll setcmykcolor pop pop pop}{v_plate
+v_cpyl eq{pop 1 exch sub 0 0 0 4 2 roll setcmykcolor pop pop}{v_plate
+v_cpmg eq{pop pop 1 exch sub 0 0 0 4 1 roll setcmykcolor pop}{v_plate
+v_cpcy eq{pop pop pop 1 exch sub 0 0 0 setcmykcolor}{1 exch sub 0 0
+0 4 -1 roll setcmykcolor pop pop pop}ifelse}ifelse}ifelse}ifelse}bd
+/setcmykcolor where{pop}{/setcmykcolor{cmyk2rgb setrgbcolor}bd}ifelse
+/setlogcmykcolor{v_gseps 1 eq{v_mono 1 eq{1 exch sub setgray pop pop
+pop}{setcmykcolor}ifelse}{v_mono 1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd
+/setlogrgbcolor{v_gseps 1 eq{v_mono 1 eq{rgbtoplategray}{rgb2devcmyk
+setplatecolor}ifelse}{v_mono 1 eq{rgb2gray setgray}{systemdict begin
+setrgbcolor end}ifelse}ifelse}bd/setfillcolor{v_fct 0 eq{v_fc v_fm
+v_fy v_fk setlogcmykcolor}{v_fr v_fg v_fb setlogrgbcolor}ifelse}bd
+/setstrokecolor{v_sct 0 eq{v_sc v_sm v_sy v_sk setlogcmykcolor}{v_sr
+v_sg v_sb setlogrgbcolor}ifelse}bd/setgfillcmyk{v_gseps 1 eq{v_mono
+1 eq{cmyk2rgb rgb2plategray}{cmyk2rgb rgb2devcmyk setplatecolor}ifelse}{v_mono
+1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd/setgfillrgb{v_gseps
+1 eq{v_mono 1 eq{rgb2plategray}{rgb2devcmyk setplatecolor}ifelse}{v_mono
+1 eq{rgb2gray setgray}{systemdict begin setrgbcolor end}ifelse}ifelse}bd
+/setgfillhsb{v_gseps 1 eq{v_mono 1 eq{systemdict begin sethsbcolor
+currentrgbcolor end rgb2plategray}{systemdict begin sethsbcolor currentrgbcolor
+end rgb2devcmyk setplatecolor}ifelse}{v_mono 1 eq{systemdict begin
+sethsbcolor currentgray end setgray}{systemdict begin sethsbcolor end}ifelse}ifelse}bd
+/Max{2 copy lt{exch}if pop}bd/Max3{2 copy lt{exch}if pop 2 copy lt{exch}if
+pop}bd/Min{2 copy gt{exch}if pop}bd/Min3{2 copy gt{exch}if pop 2 copy
+gt{exch}if pop}bd/clamp{3 1 roll Max 2 1 roll Min}bd/clamp01{0 Max
+1 Min}bd/Pythag{dup mul exch dup mul add sqrt}bd/ssc{DeviceRGB setcolorspace
+setcolor}bd/ssg{setgray}bd/p_render{}def/p_count 0 def/vis_flag true
+def/DataString 3 string def/DataSrc{currentfile DataString readhexstring
+pop}bd/DataStr1 1 string def/DataStr2 1 string def/DataStr3 1 string
+def/DataSrc1{DataStr1}bd/DataSrc2{DataStr2}bd/DataSrc3{DataStr3}bd
+/colorimage where{pop/ci{colorimage}bd}{/ci{pop pop/ci_datasrc exch
+def matrix invertmatrix/ci_matrix exch def pop/ci_dataleft 0 def/ci_buf()def
+/ci_dataofs 0 def 0 1 3 -1 roll 1 sub{/ci_y exch def dup 0 1 3 -1
+roll 1 sub{0 1 2{pop ci_dataleft 0 eq{ci_datasrc dup length/ci_dataleft
+exch def/ci_buf exch def/ci_dataofs 0 def}if ci_buf ci_dataofs get
+255 div/ci_dataofs ci_dataofs 1 add def/ci_dataleft ci_dataleft 1 sub
+def}for setrgbcolor dup ci_y 3 -1 roll 1 add ci_y 1 add 4 copy 5 1
+roll 4 2 roll 5 -1 roll 1 1 4{pop ci_matrix transform 8 2 roll}for
+m l l l closepath fill}for}for pop}bd}ifelse/rci{/rciBuf 4 index 3
+index mul 7 add 8 div floor cvi string def{currentfile rciBuf readhexstring
+pop}bind false 3 ci}bd/cbsl{2 eq/cbslL2 xd 5 index/cbslw xd translate
+scale 8 [ 3 index 0 0 5 index 0 0 ] cbslL2{/DataStr1 cbslw string def
+currentfile/ASCII85Decode filter/RunLengthDecode filter DataStr1 readstring
+pop pop/DataStr2 cbslw string def currentfile/ASCII85Decode filter
+/RunLengthDecode filter DataStr2 readstring pop pop/DataStr3 cbslw
+string def currentfile/ASCII85Decode filter/RunLengthDecode filter
+DataStr3 readstring pop pop{DataStr1}bind{DataStr2}bind{DataStr3}bind
+true}{/DataSrc load false}ifelse 3 ci}bd/gbsl{2 eq/gbslL2 xd 5 index
+/gbslw xd translate scale 8 [ 3 index 0 0 5 index 0 0 ] gbslL2{/DataStr1
+gbslw string def currentfile/ASCII85Decode filter/RunLengthDecode filter
+DataStr1 readstring pop pop{DataStr1}bind}{/DataStr1 gbslw string def
+currentfile DataSrc1 readhexstring pop pop{DataStr1}bind}ifelse image}bd
+/cmi{/cmiBuf 4 index 3 index mul 7 add 8 div floor cvi string def{currentfile
+cmiBuf readhexstring pop}bind image}bd/cpal{4 mul string/cPalette exch
+def currentfile cPalette readhexstring pop}bd/cpci{/cpci_datasrc exch
+def matrix invertmatrix/cpci_matrix exch def/cpci_bpp exch def cpci_init
+0 1 3 -1 roll 1 sub{/cpci_y exch def dup cpci_bpp 4 eq{cpci_sampsleft
+1 eq{/cpci_sampsleft 0 def}if}if 0 1 3 -1 roll 1 sub{cpci_nextcol dup
+cpci_y 3 -1 roll 1 add cpci_y 1 add 4 copy 5 1 roll 4 2 roll 5 -1 roll
+1 1 4{pop cpci_matrix transform 8 2 roll}for m l l l closepath fill}for}for
+pop}bd/cpci_init{/cpci_sampsleft 0 def}bd/cpci_buf 1 string def/cpci_nextcol{cpci_bpp
+1 eq{cpci_sampsleft 0 eq{currentfile cpci_buf readhexstring pop pop
+/cpci_sampsleft 8 def}if cpci_buf dup 0 get dup 1 and setgray -1 bitshift
+1 exch put/cpci_sampsleft cpci_sampsleft 1 sub def}{cpci_bpp 4 eq{cpci_sampsleft
+0 eq{currentfile cpci_buf readhexstring pop pop/cpci_sampsleft 2 def}if
+cpci_buf 0 get dup 15 and exch -4 bitshift cpci_buf 0 3 -1 roll put
+/cpci_sampsleft cpci_sampsleft 1 sub def}{currentfile cpci_buf readhexstring
+pop 0 get}ifelse 4 mul dup 2 add cPalette exch get 255 div exch dup
+1 add cPalette exch get 255 div exch cPalette exch get 255 div setrgbcolor}ifelse}bd
+/setup1asciiproc{[ currentfile mystring/readhexstring cvx/pop cvx
+] cvx bind}bd/setup1binaryproc{[ currentfile mystring/readstring cvx
+/pop cvx ] cvx bind}bd level2{save/dontloadlevel1 xd}if/iw 0 def/ih
+0 def/im_save 0 def/setupimageproc 0 def/polarity 0 def/smoothflag
+0 def/mystring 0 def/bpc 0 def/beginimage{/im_save save def dup 0 eq{pop
+/setup1binaryproc}{1 eq{/setup1asciiproc}{(error, can't use level2 data acquisition procs for level1)print
+flush}ifelse}ifelse/setupimageproc exch ld/polarity xd/smoothflag xd
+/imat xd/mystring exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save
+restore}bd/1bitbwcopyimage{1 setgray 0 0 moveto 0 1 rlineto 1 0 rlineto
+0 -1 rlineto closepath fill 0 setgray iw ih polarity imat setupimageproc
+imagemask}bd/1bitcopyimage{setrgbcolor 0 0 moveto 0 1 rlineto 1 0 rlineto
+0 -1 rlineto closepath fill setrgbcolor iw ih polarity imat setupimageproc
+imagemask}bd/1bitmaskimage{setrgbcolor iw ih polarity [iw 0 0 ih 0
+0] setupimageproc imagemask}bd level2{dontloadlevel1 restore}if level2
+not{save/dontloadlevel2 xd}if/setup2asciiproc{currentfile/ASCII85Decode
+filter/RunLengthDecode filter}bd/setup2binaryproc{currentfile/RunLengthDecode
+filter}bd/myimagedict 9 dict dup begin/ImageType 1 def/MultipleDataSource
+false def end def/im_save 0 def/setupimageproc 0 def/polarity 0 def
+/smoothflag 0 def/mystring 0 def/bpc 0 def/ih 0 def/iw 0 def/beginimage{
+/im_save save def dup 2 eq{pop/setup2binaryproc}{dup 3 eq{pop/setup2asciiproc}{0
+eq{/setup1binaryproc}{/setup1asciiproc}ifelse}ifelse}ifelse/setupimageproc
+exch ld{[ 1 0 ]}{[ 0 1 ]}ifelse/polarity xd/smoothflag xd/imat xd/mystring
+exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save restore}bd/1bitbwcopyimage{1
+ssg 0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill
+0 ssg myimagedict dup begin/Width iw def/Height ih def/Decode polarity
+def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
+1 def/Interpolate smoothflag def end imagemask}bd/1bitcopyimage{ssc
+0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill ssc
+myimagedict dup begin/Width iw def/Height ih def/Decode polarity def
+/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
+1 def/Interpolate smoothflag def end imagemask}bd/1bitmaskimage{ssc
+myimagedict dup begin/Width iw def/Height ih def/Decode polarity def
+/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
+1 def/Interpolate smoothflag def end imagemask}bd level2 not{dontloadlevel2
+restore}if
+level2{save/dontloadlevel1 xd}if/startnoload{{/noload save def}if}bd
+/endnoload{{noload restore}if}bd/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}bd
+/ncolors 1 def/colorimage where{pop true}{false}ifelse{/ncolors 0
+statusdict begin/processcolors where{pop pop processcolors}{/deviceinfo
+where{pop deviceinfo/Colors known{pop{deviceinfo/Colors get}}if}if}ifelse
+end def ncolors 0 ne{/colorimage testsystemdict/setcolortransfer testsystemdict
+/currentcolortransfer testsystemdict/currentcmykcolor testsystemdict
+and and and not{/ncolors 0 def}if}if}if ncolors dup 1 ne exch dup 3
+ne exch 4 ne and and{/ncolors 0 def}if ncolors 1 eq dup dup not startnoload{
+/expandbw{expandfactor mul round cvi bwclut exch get 255 div}bd/doclutimage{bwclut
+colorclut pop/bwclut xd bpc dup 8 eq{pop 255}{4 eq{15}{3}ifelse}ifelse
+/expandfactor xd [/expandbw load/exec load dup currenttransfer exch
+] cvx bind settransfer iw ih bpc imat setupimageproc image}bd}if not
+endnoload ncolors dup 3 eq exch 4 eq or dup dup not startnoload{/nullproc{{}}def
+/concatutil{/exec load 7 -1 roll/exec load}bd/defsubclut{1 add getinterval
+def}bd/spconcattransfer{/Dclut exch def/Cclut exch def/Bclut exch def
+/Aclut exch def/ncompute exch ld currentcolortransfer [{Aclut ncompute}concatutil
+] cvx [{Bclut ncompute}concatutil ] cvx [{Cclut ncompute}concatutil
+] cvx [{Dclut ncompute}concatutil ] cvx setcolortransfer}bd/setuprgbcluts{
+/bit3x rgbclut length 3 sub def/bit1x bit3x 3 idiv def/rclut rgbclut
+def/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}bd}if
+not endnoload ncolors 3 eq dup dup not startnoload{/3compute{exch bit3x
+mul round cvi get 255 div}bd/doclutimage{/rgbclut xd pop setuprgbcluts
+/3compute rclut gclut bclut dup spconcattransfer iw ih bpc imat [
+setupimageproc/exec load/dup load dup ] cvx nullproc nullproc true
+3 colorimage}bd}if not endnoload ncolors 4 eq dup dup not startnoload{
+/stuffclut{cmykindex 3 -1 roll put}bd/ftoint{1 exch sub 255 mul round
+cvi}bd/4compute{exch bit4x mul round cvi get 255 div}bd/computecmykclut{setuprgbcluts
+/bit4x rgbclut length 3 idiv 4 mul 4 sub def/cmykclut bit4x 4 add
+string def/cclut cmykclut def/mclut cclut 1 bit4x defsubclut/yclut
+cclut 2 bit4x defsubclut/kclut cclut 3 bit4x defsubclut/cmykindex 0
+def 0 1 bit1x{dup/cmykindex exch bit1x exch sub 4 mul def 3 mul dup
+rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch
+get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint
+yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut}for}bd
+/doclutimage{/rgbclut xd pop invalidcolortable?{computecmykclut}if
+/4compute cclut mclut yclut kclut spconcattransfer iw ih bpc imat
+[ setupimageproc/exec load/dup load dup dup ] cvx nullproc nullproc
+nullproc true 4 colorimage}bd}if not endnoload ncolors 0 eq dup dup
+not startnoload{/lookupandstore{3 mul 3 getinterval putinterval exch
+3 add exch 3 copy}bd/8lookup/lookupandstore ld/4lookup{/byte 1 index
+def -4 bitshift lookupandstore byte 15 and lookupandstore}bd/2lookup{
+/byte 1 index def -6 bitshift lookupandstore byte -4 bitshift 3 and
+lookupandstore byte -2 bitshift 3 and lookupandstore byte 3 and lookupandstore}bd
+/colorexpand{mystringexp 0 rgbclut 3 copy 7 -1 roll/mylookup load
+forall pop pop pop pop pop}bd/createexpandstr{/mystringexp exch mystring
+length mul string def}bd/doclutimage{/rgbclut xd pop/mylookup bpc 8
+eq{3 createexpandstr/8lookup}{bpc 4 eq{6 createexpandstr/4lookup}{12
+createexpandstr/2lookup}ifelse}ifelse ld iw ih bpc imat [ setupimageproc
+/exec load/colorexpand load/exec load] cvx false 3 colorimage}bd}if
+not endnoload/colorimage where{pop true}{false}ifelse dup{/do24image{iw
+ih 8 imat setupimageproc false 3 colorimage}bd}if dup dup startnoload
+not{/rgbtogray{/str xd/len str length def/smlen len 3 idiv def/rstr
+str def/gstr str 1 len 1 sub getinterval def/bstr str 2 len 2 sub getinterval
+def str dup 0 1 smlen 1 sub{dup 3 mul rstr 1 index get .3 mul gstr
+2 index get .59 mul add bstr 3 -1 roll get .11 mul add round cvi put
+dup}for pop 0 smlen getinterval}bd/do24image{iw ih 8 imat [ setupimageproc
+/exec load/rgbtogray load/exec load ] cvx bind image}bd}if endnoload
+/doimage{iw ih 8 imat setupimageproc image}bd level2{dontloadlevel1
+restore}if level2 not{save/dontloadlevel2 xd}if/myappcolorspace/DeviceRGB
+def/rgbclut 0 def/doclutimage{/rgbclut xd pop bpc dup 8 eq{pop 255}{4
+eq{15}{3}ifelse}ifelse/hival xd [/Indexed myappcolorspace hival rgbclut]
+setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode
+[0 hival] def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent
+bpc def/Interpolate smoothflag def end image}bd/do24image{myappcolorspace
+setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode
+[0 1 0 1 0 1] def/ImageMatrix imat def/DataSource setupimageproc def
+/BitsPerComponent 8 def/Interpolate smoothflag def end image}bd level2
+not{dontloadlevel2 restore}if
+/NumSteps{dtransform matrix defaultmatrix idtransform Pythag currentscreen
+pop pop 72 exch div div}bd/FindMinSteps{v_ft 4 eq{urx startX sub abs
+llx startX sub abs Max ury startY sub abs lly startY sub abs Max Pythag
+2 3.14159265 mul mul 0}{v_ft 2 eq{endY startY sub endX startX sub Pythag
+endY2 startY sub endX2 startX sub Pythag gt{endY startY sub endX startX
+sub}{endY2 startY sub endX2 startX sub}ifelse}{endY startY sub endX
+startX sub}ifelse}ifelse NumSteps}bd/cxe{/v_cxe exch def}bd/cxm{pop
+/v_cxm exch def}bd/cxmt{pop pop}bd/cxt{pop}bd/S_eoclip{currentflat{{eoclip}stopped{dup
+currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: eoclip]\n)print
+flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
+/S_clip{currentflat{{clip}stopped{dup currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: clip]\n)print
+flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
+/S_eofill{currentflat{{eofill}stopped{dup currentflat exch sub 20
+gt{([Error: PathTooComplex; OffendingCommand: eofill]\n)print flush
+exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd
+/gpbbx{pathbbox/ury exch def/urx exch def/lly exch def/llx exch def}bd
+/lineargfill{initgfill{false initgfx/distance endX startX sub endY
+startY sub Pythag def/incD distance Steps div def endY startY sub endX
+startX sub atan newpath llx lly urx ury Bx startX startY translate
+rotate gpbbx eGy ssg newpath llx lly urx ury Bx S_eofill sGy ssg newpath
+llx lly 0 ury Bx S_eofill/fillX 0 def 0 1 Steps 1 sub{stepgfx newpath
+fillX lly fillX incD add dup/fillX exch def ury Bx S_eofill pop}for}if}bd
+/radialgfill{initgfill{false initgfx/distance endX startX sub endY
+startY sub Pythag def/incD distance Steps div def eGy ssg newpath llx
+lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepgfx newpath
+startX startY distance 0 360 arc closepath distance incD add dup/distance
+exch def 0 rmoveto startX startY distance 0 360 arc closepath S_eofill
+pop}for}if}bd/ellipticgfill{initgfill{true initgfx sGy ssg newpath
+llx lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2
+startX sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepgfx
+/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for}if}bd/conicalgfill{initgfill{urx
+startX sub abs llx startX sub abs Max ury startY sub abs lly startY
+sub abs Max Pythag startY endY sub startX endX sub Pythag div/radius
+exch def true initgfx [ endX startX sub endY startY sub startY endY
+sub endX startX sub startX startY ] concat/slice 180 Steps div def
+/startangle 0 def Steps -1 0{pop stepgfx 0 0 moveto 0 0 radius startangle
+neg dup slice add arc fill 0 0 moveto 0 0 radius startangle dup slice
+add dup/startangle exch def arc fill}for}if}bd/initgfill{gpbbx{S_eoclip}{S_clip}ifelse
+startX endX eq startY endY eq and sGy eGy eq or dup{sGy ssg newpath
+llx lly urx ury Bx S_eofill}if not}bd/initgfx{/flag exch def flag{sGy
+eGy/sGy exch def/eGy exch def}if eGy sGy sub abs 256 mul FindMinSteps
+Min 256 Min ceiling 1 Max/Steps exch def eGy sGy sub Steps div/incGy
+exch def sGy/mGy exch def}bd/stepgfx{mGy ssg mGy incGy add/mGy exch
+def}bd/linearfill{initfill{false initfx endX startX sub endY startY
+sub Pythag/distance exch def/incD distance Steps div def endY startY
+sub endX startX sub atan newpath llx lly urx ury Bx startX startY translate
+rotate gpbbx v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse
+newpath llx lly urx ury Bx S_eofill v_cxe 0 eq{sR sG sB setgfillrgb}{sH
+sS sV setgfillhsb}ifelse newpath llx lly 0 ury Bx S_eofill/fillX 0
+def 0 1 Steps 1 sub{stepfx newpath fillX lly fillX incD add dup/fillX
+exch def ury Bx S_eofill pop}for}if}bd/radialfill{initfill{false initfx
+endX startX sub endY startY sub Pythag/distance exch def/incD distance
+Steps div def v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse
+newpath llx lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepfx
+newpath startX startY distance 0 360 arc closepath distance incD add
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