--- /dev/null
+\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
--- /dev/null
+P=0-demo
+
+include ../Makerules
+++ /dev/null
-\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
+++ /dev/null
-P=0-demo
-
-include ../Makerules
+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{1}{Hradlové sítì}{(zapsali Jirka Fajfr a Ján Èerný)}
-
-\def\land{\mathbin{\&}}
-
-Na této pøedná¹ce se budeme zabývat jednoduchým modelem paralelního poèítaèe,
-toti¾ hradlovou sítí, a uká¾eme si alespoò jeden efektivní paralelní algoritmus,
-konkrétnì sèítání dvojkových èísel v~logaritmickém èase vzhledem k~jejich délce.
-
-\h{Hradlové sítì}
-
-\s{Definice:} {\I Hradlo} je zaøízení, které poèítá nìjakou pevnì danou funkci
-s~$k$ vstupy a jedním výstupem.
-
-\s{Pøíklad:} Obvykle pracujeme s~booleovskými hradly, ta pak odpovídají funkcím
-$f: \{0,1\}^{k} \rightarrow \{0,1\} $. Z~nich nejèastìji potkáme:
-
-\itemize\ibull
-\:0-vstupové: to jsou konstanty {\sc true} a {\sc false},
-\:1-vstupové: identita (ta je vcelku k~nièemu) a negace (znaèíme~$\lnot$),
-\:2-vstupové: logický souèin ({\sc and},~$\land$) a souèet ({\sc or},~$\lor$).
-\endlist
-
-\>Hradla kreslíme tøeba následovnì:
-
-\figure{1_1_hradlo.eps}{Hradlo provádìjící logickou operaci {\sc and} se dvìma vstupy}{4cm}
-
-Z~jednotlivých hradel pak vytváøíme hradlové sítì. Pokud pou¾íváme pouze booleovská
-hradla, øíkáme takovým sítím {\I booleovské obvody,} pokud operace nad nìjakou obecnìj¹í
-(ale koneènou) mno¾inou symbolù (abecedou), nazývají se {\I kombinaèní obvody.}
-Ka¾dý vstup hradla je pøipojen buïto na~nìkterý ze~vstupù sítì nebo na~výstup nìjakého
-jiného hradla. Výstupy hradel mohou být propojeny na výstupy sítì nebo pøivedeny
-na~vstupy dal¹ích hradel, pøièem¾ je zakázáno vytváøet cykly. Ne¾ si øekneme
-formální definici, podívejme se na obrázek.
-
-\todo{OBR}
-
-\s{Definice:} {\I Hradlová sí»} je urèena:
-\itemize\ibull
-\:{\I abecedou} $\Sigma$ (to je nìjaká koneèná mno¾ina symbolù, obvykle $\Sigma=\{0,1\}$);
-\:mno¾inou {\I hradel} $H$, {\I vstupních portù} $I$ a {\I výstupních portù} $O$;
-\:acyklickým orientovaným grafem~$(V,E)$, kde~$V = H \cup I \cup O$;
-\:zobrazením~$F$, které ka¾dému hradlu $h\in H$ pøiøadí nìjakou funkci~$F(h):
- \Sigma^{a(h)} \rightarrow \Sigma$. To je funkce, kterou toto hradlo vykonává,
- a~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
-\:$\forall i \in I: \deg^{+}(i)=0$ (do~vstupù nic nevede);
-\:$\forall o \in O: \deg^{+}(o)=1 \land \deg^{-}(o)=0$ (z~výstupù nic nevede a do~ka¾dého vede právì jedna hrana);
-\:$\forall h \in H: \deg^{+}(v)=a(v)$ (do~ka¾dého hradla vede tolik hran, kolik je jeho arita);
-\:$\forall h \in H, 1\le j\le a(h)$ existuje právì jeden vrchol~$v$ takový, ¾e $z(vh)=j$
- (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, co¾ není
-ani realistické, ani pìkné. Proto pøijmìme omezení, ¾e v¹echna hradla budou mít maximálnì
-$k$ vstupù, kde~$k$ je nìjaká pevná konstanta, obvykle dvojka. Následující obrázky
-ukazují, jak hradla o~více vstupech nahradit dvouvstupovými:
-
-\twofigures{1_2_vice_vstupove_hradlo.eps}{Trojvstupové hradlo \sc and}{3cm}{1_3_vice_vstupove_hradlo.eps}{Jeho nahrazení 2-vstupovými hradly}{3cm}
-
-\s{Definice:} {\I Výpoèet sítì} probíhá v~{\I taktech.} V nultém taktu jsou definovány právì hodnoty
-vstupních portù. V~$i$-tém taktu vydají výsledek hradla, která jsou pøipojena
-na~porty nebo na~výstupy hradel, jejich¾ hodnota byla definována v~$(i-1)$-ním
-taktu. A¾ po~nìjakém koneèném poètu taktù budou definované i hodnoty výstupních
-portù, sí» se zastaví a vydá výsledek.
-
-\figure{1_7_vypocet_site.eps}{Výpoèet hradlové sítì}{6cm}
-
-\>Podle toho, jak sí» poèítá, si ji mù¾eme rozdìlit na~vrstvy:
-
-\s{Definice:} {\I $i$-tá vrstva} obsahuje v¹echny vrcholy~$v$ takové, ¾e
-nejdel¹í z~cest z~portù sítì do~$v$ má délku právì~$i$. To jsou
-pøesnì vrcholy, které vydají výsledek poprvé v~$i$-tém taktu výpoètu.
-Dává tedy smysl prohlásit za~{\I èasovou slo¾itost} sítì poèet jejích
-vrstev. Podobnì {\I prostorovou slo¾itost} definujeme jako poèet hradel
-v~síti.
-
-\s{Pøíklad:} Sestrojte sí», která zjistí, zda se mezi jejími~$n$ vstupy
-vyskytuje alespoò jedna jednièka.
-
-\>{\I První øe¹ení:} zapojíme hradla za~sebe (sériovì). Èasová a prostorová
-slo¾itost jsou~$n$. Zde vùbec nevyu¾íváme toho, ¾e by mohlo poèítat více
-hradel souèasnì.
-
-\figure{1_5_hloupy_or.eps}{Hradlová sí», která zjistí zda-li je na vstupu alespoò jedna jednièka}{7cm}
-
-\>{\I Druhé øe¹ení:} Budeme vrcholy spojovat do~dvojic, pak výsledky z~tìchto
-dvojic opìt do~dvojic a tak dále. Tak dosáhneme èasové slo¾itosti $\Theta(\log n)$,
-prostorová slo¾itost zùstane lineární.
-
-\figure{1_4_chytry_or.eps}{Chytøej¹í øe¹ení stejného problému}{8cm}
-
-\h{Sèítání binárních èísel}
-
-Pojïme se podívat na~zajímavìj¹í problém: Mìjme dvì èísla zapsané ve~dvojkové
-soustavì jako $x_{n-1}\ldots x_1x_0$ a $y_{n-1}\ldots y_1y_0$. Budeme chtít
-spoèítat jejich souèet $z_nz_{n-1}\ldots z_1z_0$.
-
-Samozøejmì mù¾eme pou¾ít algoritmus \uv{sèítání pod sebou}, který nás
-uèili na~základní ¹kole. Formálnì by se dal zapsat tøeba takto:
-$$
-z_i=x_i \oplus y_i \oplus c_{i-1},
-$$
-kde $\oplus$ znaèí operaci {\sc xor} (souèet modulo~2) a $c_{i-1}$ je {\I pøenos} z~$(i-1)$-ního
-øádu do~$i$-tého. Pøenos pøitom nastane tehdy, kdy¾ ze~tøí xorovaných èíslic
-jsou alespoò dvì jednièky:
-$$
-\eqalign{
-c_{-1} &= 0 \cr
-c_i &= (x_i \land y_i)\lor((x_i \lor y_i) \land c_{i-1}).\cr
-}
-$$
-
-\figure{1_6_hloupe_scitani.eps}{Sèítání ze~základní ¹koly}{8cm}
-
-Bohu¾el na to, abychom spoèítali $c_i$ (a~tedy~$z_i$), musíme znát hodnotu $c_{i-1}$, tedy mít
-spoèítané hodnoty pro v¹echny èísla men¹í ne¾ $i$. To dává lineární èasovou
-slo¾itost. Zamysleme se nad tím, jak by se proces sèítání mohl zrychlit.
-
-\h{Pøenosy v~blocích}
-
-Jediné, co nás pøi sèítání brzdí, jsou pøenosy. Kdybychom je dokázali spoèítat rychle
-(øeknìme v~logaritmické hloubce), souèet u¾ zvládneme dopoèítat v~konstantním èase.
-
-Podívejme se na~libovolný {\I blok} v~na¹em souètu. Tak budeme øíkat èíslùm
-$x_a\ldots x_b$ a $y_a\ldots y_b$ v~nìjakém intervalu indexù $\left<a,b\right>$.
-Pøenos $c_b$ vystupující z~tohoto bloku závisí mimo hodnot sèítancù u¾ pouze
-na~pøenosu $c_{a-1}$, který do bloku vstupuje. Záviset mù¾e pouze tøemi
-mo¾nými zpùsoby:
-
-\numlist\ndotted
-\:generuje pøenos: $c_a=1$,
-\:pohlcuje pøenos: $c_a=0$,
-\:kopíruje pøenos: $c_a=c_{b-1}$.
-\endlist
-
-\figure{1_7_blok_scitani.eps}{Blok souètu}{8cm}
-
-\s{Cvièení:} Rozmyslete si, jak pøesnì vypadají bloky s~jednotlivými typy chování.
-
-Jednobitové bloky se chovají velice jednodu¹e:
-
-\figure{1_11_tabulka_kodovani.eps}{Tabulka triviálních bitù}{3cm}
-
-Pokud máme nìjaký vìt¹í blok~$B$ slo¾ený z~men¹ích blokù $p$ a~$q$, jejich¾
-chování u¾ známe, mù¾eme z~toho odvodit, jak se chová velký blok:
-
-\figure{1_10_konvence_deleni_bloku.eps}{Skládání chování blokù}{3cm}
-
-V¹imòìme si, ¾e skládání chování blokù je asociativní operace (je to vlastnì
-úplnì obyèejné skládání funkcí), tak¾e pro libovolný blok mù¾eme jeho
-chování spoèítat v~èase $\O(\log n)$ postupným skládáním (\uv{stromeèkovým}
-zpùsobem).
-
-To nám dá nìjaký kombinaèní obvod nad trojprvkovou abecedou, ale samozøejmì
-mù¾eme chování blokù kódovat i binárnì dvojicí bitù:
-
-\itemize\ibull
-\:$(1,*) = <$,
-\:$(0,0) = 0$,
-\:$(0,1) = 1$
-\endlist
-
-\>Operaci skládání $(a,x) \odot (b,y) = (c,z)$ pak definujeme takto:
-$$
-\eqalign{
-c &= a \land b,\cr
-z &= (\neg a \land x) \lor (a \land y).\cr
-}
-$$
-
-\h{Paralelní sèítání}
-
-\>Paralelní algoritmus na~sèítání u¾ zkonstruujeme pomìrnì snadno. Bez
-újmy na~obecnosti budeme pøedpokládat, ¾e poèet bitù vstupních èísel~$n$
-je mocnina dvojky, jinak si vstup doplníme nulami.
-
-\algo
-\:Spoèteme chování blokù velikosti~1. ($\O(1)$ hladin)
-\:Postupnì poèítáme chování blokù velikosti $2^k$ na~pozicích dìlitelných $2^k$.
- ($\O(\log n)$ hladin, na~nich¾ se skládají bloky)
-\:$c_{-1} \leftarrow 0$
-\:Urèíme $c_n$ podle $c_{-1}$ a chování (jediného) bloku velikosti~$n$.
-\:Postupnì poèítáme pøenosy na~hranicích dìlitelných $2^k$ \uv{zahu¹»ováním}:
- jakmile víme $c_{2^k-1}$, mù¾eme dopoèítat $c_{2^k+2^{k-1}-1}$ podle
- chování bloku $\left< 2^k+2^{k-1}-1,2^k\right>$. ($\O(\log n)$ hladin,
- na~nich¾ se dosazuje)
-\:$\forall i: z_i = x_i \oplus y_i \oplus c_{i-1}$.
-\endalgo
-
-\figure{1_9_deleni_bloku.eps}{Výpoèet pøenosu}{8cm}
-
-Algoritmus pracuje v~èase $\O(\log n)$. Hradel je dokonce 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 se
-seète na~$\O(n)$.
-
-\bye
+++ /dev/null
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+++ /dev/null
-P=1-hradla
-
-include ../Makerules
+++ /dev/null
-\input lecnotes.tex
-
-\prednaska{10}{Pøevody problémù}{(zapsali Martin Chytil, Vladimír Kudelas)}
-
-\>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 $x\in L$.]
-
-\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, 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 pøevést) na $B$ (pí¹eme $A \rightarrow B$) $\Leftrightarrow$
-existuje funkce $f$ spoèitatelná v polynomiálním èase taková, ¾e pro $\forall
-x: A(x) = B(f(x))$.
-
-V¹imnìte si, ¾e $A\rightarrow B$ 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 $\O(a^\ell)$, tak¾e $B(f(x))$
-pobì¾í v~èase $\O(a^{k\ell})$, 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
-
-\>Podívejme se nyní na~pøevody mezi dal¹ími zajímavými problémy:
-
-\h{1. problém: SAT}
-\>Splnitelnost logických formulí, tj. dosazení \<true> èi \<false> za promìnné v logické formuli tak, aby formule dala výsledek \<true>.
-
-\>Zamìøíme se na speciální formu zadání formulí, {\I konjunktivní normální formu} (CNF).
-$$(\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$
-
-\>{\I Vstup:} Formule $\varphi$ v konjunktivní normální formì.
-
-\>{\I Výstup:} $\exists$ dosazení \<true> a \<false> za promìnné takové, ¾e hodnota formule $\varphi(\ldots) = \<true>$.
-
-\>Pro formuli platí následující podmínky:
-
-\itemize\ibull
-\:{\I formule} je zadána pomocí {\I klauzulí} oddìlených $\land$,
-\:ka¾dá {\I klauzule} je slo¾ená z {\I literálù} oddìlených $\lor$,
-\:ka¾dý {\I literál} je buïto promìnná nebo její negace.
-\endlist
-
-\>Uká¾eme, ¾e staèí vyøe¹it jednodu¹¹í problém 3-SAT.
-
-\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 (3-SAT je alespoò tak tì¾ký jako SAT)
-
-\s {Pøevod SAT na 3-SAT:}
-Musíme formuli pøevést tak, abychom neporu¹ili splnitelnost.
-
-\>Trik pro dlouhé klauzule: Ka¾dou klauzuli
-$$(\alpha \lor \beta) \hbox{, t¾. } \vert\alpha\vert + \vert\beta\vert \ge 4$$
-pøepí¹eme na: $$(\alpha \lor x) \land (\beta \lor \lnot x),$$
-kde $x$ je nová promìnná, kterou nastavíme tak, abychom neovlivnili splnitelnost formule.
-
-\>Platí-li:
-\itemize\ibull
-\:$\alpha \Rightarrow x = 0$ (zajistí splnìní druhé poloviny nové formule),
-\:$\beta \Rightarrow x = 1$ (zajistí splnìní první poloviny nové formule),
-\:$\alpha ,\beta / \lnot\alpha ,\lnot\beta \Rightarrow x = 0/1$ (je nám to jedno, celkové øe¹ení nám to neovlivní).
-\endlist
-
-\>Tento trik opakujeme tak dlouho, dokud je to tøeba.
-
-Nabízí se otázka, proè mù¾eme promìnnou $x$ nastavit, 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 následující trik:
-$$(\alpha) \rightarrow (\alpha \lor x) \land (\alpha \lor \lnot x).$$
-
-\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:} Z ka¾dé klauzule vybereme jeden literál 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 $\lnot x$.
-
-Princip je takový, ¾e z~ka¾dé klauzule si vybereme literál, který danou
-klauzuli splní, a to tak, aby literály, které si vybereme, nekolidovaly. Kolize
-o¹etøíme hranami mezi promìnnými a jejich negacemi.
-
-\figure{nezmna_graf.eps}{Ukázka pøevodu 3-SAT na nezávislou mno¾inu}{3in}
-
-Existuje nezávislá mno¾ina velikosti rovné poètu klauzulí?
-Pokud ano, tak dostaneme seznam promìnných, pomocí kterých splníme danou formuli.
-
-\s{Pøevod NzMna na SAT:}
-Máme promìnné $v_1, \ldots , v_n$ pro vrcholy.
-
-\>Nyní uká¾eme, jak pøevést problém hledání nezávislé mno¾iny, 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.
-\: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 klauzuje, které nám øeknou, ¾e vybrané do nezávislé mno¾iny jsou právì
- ty vrcholy, které jsou touto maticí oèíslované: $\forall i,j$, $x_{ij} \Rightarrow v_j$.
-\: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
-
-\s{Pøíklad matice:} Jako pøíklad pou¾ijeme nezávislou mno¾inu z ukázky nezávislé mno¾iny.
-Nech» jsou vrcholy grafu oèíslované zleva a ze zhora. 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 ($k$-tý) vrchol mno¾iny bude vybrán vrchol $v_4$, proto na druhém ($k$-tém) 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.
-
-\s{Dùvod:} Pokud existuje úplný graf na $k$ vrcholech, tak v~\uv{invertovaném} grafu tyto vrcholy nejsou spojeny hranou, tj. tvoøí nezávislou mno¾inu.
-
-\figure{doplnek_nm.eps}{Prohození hran a nehran}{2in}
-
-\h{5. 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 tento problém pøevést na 3,3-SAT (ov¹em to a¾ na dal¹í pøedná¹ce).
-
-\figure{3d_parovani.eps}{Ukázka 3D párování}{3in}
-
-
-\h{6. 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 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.
-
-\s{Závìr:} Obrázek ukazuje problémy, jimi¾ jsme se dnes zabývali, a vztahy mezi tìmito problémy.
-\figure{prevody.eps}{Pøevody mezi problémy}{3in}
-
-\bye
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-\input lecnotes.tex
-
-\prednaska{11}{NP-úplné problémy}{(zapsali F. Kaèmarik, R. Krivák, D. Remi¹)}
-
-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~pøípade 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 zadal nìjaký objekt, umìli jsme efektivnì øíci, zda je to objekt, který hledáme. Napøíklad pokud dostaneme ohodnocení promìnných logické formule, staèí jen dosadit a spoèítat, ¾e formule je \<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. Definujeme 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 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 odpovídá tomu, o èem jsme se shodli, ¾e umíme efektivnì øe¹it. Nadefinujme tedy 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, tedy 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.
-
-Nasbírali jsme problémy, které jsou v~NP, ale nevíme, jestli jsou v~P. 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)
-
-Také platí, ¾e pokud umíme øe¹it nìjaký NP-tì¾ký problém v~polynomiálním èase,
-pak umíme vyøe¹it v~polynomiálním èase v¹e v~NP, a tedy ${\rm P}={\rm NP}$.
-(To u¾ také víme z~minulé pøedná¹ky.)
-
-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 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 $M\in{\rm NP}$ ($L \rightarrow M$), 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 zhora nadolu podle obtí¾nosti problémù, 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.
-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.
-
-\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 (mno¾ina alespoò~$k$ vrcholù taková, ¾e ¾ádné dva nejsou propojeny hranou)
- \:Klika (úplný podgraf na~$k$ vrcholech)
- \:3D párování (tøi mno¾iny se zadanými trojicemi, najít takovou mno¾inu disjunktních trojic, ve~které jsou v¹echny prvky)
- \:Barvení grafu (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)
- \:Loupe¾níci (rozdìlit mno¾inu 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
-
-\h { Pøevoditelnost 3,3-SAT na 3D-párování }
-
-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.
-Jenom pro pøipomenutí, máme mno¾inu klukù, dìvèat, zvíøátek a nìjaké trojice, kdo se s~kým snese, a chceme vybrat trojice tak, aby se v~nich ka¾dý kluk, holka, zvíøátko vyskytovalo právì jednou.
-Najdeme si takovouto konfiguraci:
-
-\fig{3d.eps}{4cm}
-
-\>4 zvíøátka, 2 kluci, 2 dívky a~takové 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.
-
-Takovýto obrázek 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~$D$ odpovídá $x=0$. 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 tyto nespárovaná zvíøátka mù¾eme pøedávat informaci, jestli promìnná $x$ má hodnotu \<true> nebo \<false> do dal¹ích èástí grafu.
-
-Zbývá vymyslet, jak reprezentovat klauzule. Klauzule budou vypadat jako trojice literálù:
-$\kappa = (x \lor y \lor \lnot r) $
-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. 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.
-
-Je¹tì nám ale urèitì zbude $2p-k$ zvíøátek, kde $p$ je poèet promìnných, tak pøidáme je¹tì $2p-k$ párù kluk-dìvèe, kteøí milují v¹echna zvíøátka, a~ti vytvoøí zbývající páry.
-
-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 také 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 to 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. 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.
-
-\s{Vìta:} Obvodový SAT je NP-úplný.
-
-\proof
-Náznakem. Na základì zku¹eností z Principù poèítaèù intuitivnì chápeme poèítaèe
-jako nìjaké slo¾ité booleovské obvody, jejich¾ stav se mìní v~èase. Uva¾me nìjaký
-problém $L \in {\rm P}$ a polynomiální algoritmus, který ho øe¹í. Pro vstup velikosti~$n$
-tedy 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~$n$.
-
-Je¹tì si dovolíme drobnou úpravu v~definici tøídy NP. Budeme chtít, aby nápovìda byla
-mìla pevnou velikost, závislou pouze na~velikosti vstupu (tedy: $\vert y \vert
-= g(\vert x \vert)$). 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.)
-
-Máme tedy nìjaký problém $L$ z~NP a~chceme dokázat, ¾e $L$ se dá pøevést na obvodový
-SAT. Kdy¾ nám nìkdo pøedlo¾í nìjaký vstup $x$ (chápeme jako vektor $(x_1, x_2, \ldots, x_n)$),
-spoèítáme velikost nápovìdy $g(\vert x\vert)$. Víme, ¾e kontrolní
-algoritmus~$K$ (který kontroluje, zda nápovìda je správnì) je v~P. Vyu¾ijeme
-intuice o~obvodech, 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 $\vert x\vert + g(\vert x\vert)$, co¾ je 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.
-\qed
-
-\s{Lemma:} Obvodový SAT se dá pøevést na 3-SAT.
-
-\proof
-Budeme postupnì budovat formuli v~konjunktivní normální formì. Pro ka¾dé hradlo v~obvodu zavedeme novou promìnnou popisující jeho výstup. Pøidáme klauzule, které nám kontrolují, ¾e toto hradlo máme ohodnocené konzistentnì. Ka¾dý booleovský obvod se dá pøevést na ekvivalentní obvod, ve~kterém se vyskytují jen hradla {\sc and} a {\sc not}, tak¾e staèí najít klauzule odpovídající tìmto hradlùm.
-
-\>{\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
-
+++ /dev/null
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-\input lecnotes.tex
-\prednaska{12}{Aproximaèné algoritmy}{(F. Ha¹ko, J. Menda, M. Mare¹)}
-
-\>Na~minulých predná¹kach sme sa zaoberali rôzne »a¾kými rozhodovacími problémami. Táto sa zaoberá postupmi ako sa v~praxi vysporiada» s~rie¹ením týchto problémov.
-
-\h{Prvý spôsob: ©peciálny prípad}
-
-\>Èasto si vystaèíme s~vyrie¹ením ¹peciálneho prípadu NP-úplného problému, ktorý le¾í v~P. Napríklad, ak rie¹ime grafovú úlohu, tak nám mô¾e staèi» rie¹enie pre~¹peciálny druh grafov (strom, bipartitný graf, \dots). Farbenie grafu je µahké pre~nejaký malý poèet farieb. 2SAT, ako ¹peciálny prípad SAT-u, sa dá rie¹i» v~lineárnom èase.
-
-\s{Problém: Maximálna nezávislá mno¾ina v strome (nie rozhodovacia)}
-
-\>{\I Vstup:} zakorenený strom~$T$
-
-\>{\I Výstup:} Maximálna (èo do poètu vrcholov) nezávislá mno¾ina vrcholov~$M$
-
-\>BUNV mô¾eme predpoklada», ¾e v~$M$ sú v¹etky listy~$T$. Ak by nejaký list $l$ nebol v~$M$, tak sa pozrieme na jeho otca:
-\itemize\ibull
-\:Ak otec nie je v~$M$, tak vytvoríme novú nezávislú mno¾inu~$M'$ obsahujúcu aj~$l$ (veµkos» nezávislej mno¾iny stúpla o~1).
-\:Ak tam otec je, tak ho z~$M$ vyjmeme a~namiesto neho vlo¾íme~$l$ (veµkos» nezávislej mno¾iny sa nezmen¹ila).
-\endlist
-\>Tieto listy aj ich otcov z~$T$ odstránime a~postup opakujeme. $T$~sa mô¾e rozpadnú» na~les; potom tento postup aplikujeme na~v¹etky stromy v~lese.
-
-\s{Algoritmus:}
-\algo
-\:Polo¾íme $M_1$:=$\{$listy stromu $T\}$.
-\:Polo¾íme $M_2$:=$\{$otcovia vrcholov z~$M_1\}$.
-\:Vrátime $M_1 \cup$ MaxNz$(T\setminus(M_1 \cup M_2)$.
-\endalgo
-\>{\I Poznámka:} Toto doká¾eme naprogramova» v $\O(n)$ (vrcholy máme vo fronte a prechádzame).
-
-\s{Problém: Batoh}
-
-\>Je daná mno¾ina $n$~predmetov s~hmotnos»ami $h_1,\ldots,h_n$
-a cenami $c_1,\ldots,c_n$ a~batoh, ktorý unesie hmotnos»~$H$. Nájdite takú
-podmno¾inu predmetov, ktorých celková hmotnos» je najviac~$H$ a~celková cena je
-maximálna 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 jednu mno¾inu, v~èase $\O(nC)$ pak v¹echny.
-
-\>Podle $A_n$ snadno nalezneme maximální cenu mno¾iny, která se vejde do batohu. To bude
-nejvìt¹í~$c^*$, pro nì¾ je $A_n(c^*) < \infty$. 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 ( $C$~mu¾e být a¾ exponenciálnì
-velké vzhledem k~velikosti vstupu), ale pouze ve~velikosti èísel na~vstupu.
-Takovým algoritmùm se øíká {\I pseudopolynomiální.}
-
-\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~$Z_{k-1}$ a ze~$Z_n$ vyèteme výsledek. V¹echny tyto mno¾iny
-mají nejvý¹e $H$ prvkù, tak¾e celková èasová slo¾itost algoritmu je~$\O(nH)$.
-
-\h{Druhý spôsob: Aproximácia}
-
-\>V predchádzajúcich problémoch sme sa zamerali na ¹peciálne prípady. Obèas v¹ak také ¹tastie nemáme a~musíme vyrie¹i» celý NP-úplný problém. Mo¾eme si v¹ak pomôc» tým, ¾e sa nebudeme sna¾i» vyrie¹i» ho optimálne -- iba v nejakom pomere k~optimálnosti ({\I aproximácia}), t.j. budeme vedie», o~koµko maximálne je na¹e rie¹enie hor¹ie ako optimálne.
-
-\s{Problém: Obchodný cestujúci}
-
-\>{\I Vstup:} neorientovaný graf $G$, popisujúci nejaku krajinu a~ka¾dá hrana je ohodnotená funkciou $w: E(G)\rightarrow {\bb R}^+_0$
-
-\>{\I Vystup:} Hamiltonovská kru¾nica (v¹etky vrcholy grafu), a~to tá najkrat¹ia (podµa ohodnotenia).
-
-\>Tento problém je hneï na~prvý pohµad nároèný -- u¾ problém, èi existuje Hamiltonovská kru¾nica, je NP-úplný. BUNV nech graf~$G$ je úplný (doplnime zvy¹né hrany ohodnotené $max(w)+1$ alebo viac, nie v¹ak nekoneènom, lebo by neplatila trojuholníková nerovnos», ktorú neskôr budeme potrebova»). Vyrie¹me tento problém najprv za~predpokladu, ¾e vrcholy grafu spåòajú trojuholníkovú nerovnos», potom bez nej.
-
-\>{\I a) trojuholníková nerovnos»:} $\forall x,y,z \in V: w(xz)\le w(xy)+w(yz)$
-
-\>Existuje pekný algoritmus, ktory nájde Hamiltonovsku kru¾nicu, èo je
-maximálne dvakrát tak veµká ako optimálna.
-
-\>Nájdeme najmen¹iu kostru a~obchodnému cestujúcemu poradíme, nech ide po~nej (staèí zakoreni» a~prejs» do~håbky). Problémom v¹ak je, ¾e daný sled obsahuje ka¾dý vrchol viackrát a~preto musíme nahradi» nepovolené vracania sa, t.j.~pre ka¾dý vrchol nájs» e¹te nenav¹tívený vrchol v~na¹om slede a~ís» priamo naò. Keï¾e platí trojuholníková nerovnos», tak si týmito skratkami neu¹kodíme. Nech minimálna kostra má váhu~$T$. Váha obídeného sledu tak bude~$2T$. Skrátenia urèite nezväè¹ujú, tak¾e váha nájdene Hamiltonovskej kru¾nice bude nanajvý¹~$2T$.
-
-\>Ak máme Hamiltonovskú kru¾nicu~$C$ a~z~nej vy¹krtneme hranu, tak máme kostru grafu~$G$ s~váhou najviac~$w(C)$, teda to aspoò takú, aká je váha minimálnej kostry --~$T$. To je optimálny prípad Hamiltonovskej kru¾nice. Ak to teda zlo¾íme dohromady, algoritmus nám vráti Hamiltonovskú kru¾nicu s~váhou najviac dvojnásobnou od~optimálnej Hamiltonovskej kru¾nice. Takéto algoritmy sa nazývajú {\I 2-aproximaèné}, keï¾e rie¹enie je maximálne dvojnásobné od~optimálneho.
-
-\>{\I b) bez~trojuholníkovej nerovnosti:}
-
-\>Tu sa budeme naopak sna¾i» ukáza», ¾e ¾iaden polynomiálny aproximaèný algoritmus neexistuje.
-
-\s{Veta:} Ak existuje polynomiálny $(1+\varepsilon)$-aproximaèný algoritmus pre~algoritmus obchodného cestujúceho bez~trojuholníkovej nerovnosti pre~µubovoµné $\varepsilon>0$, tak potom $P = NP$.
-
-\proof Uká¾eme, ¾e v~tom prípade doká¾eme v~polynomiálnom èase nájs» Hamiltonovskú kru¾nicu.
-
-\>Dostali sme graf $G$, v~ktorom hµadáme Hamiltonovskú kru¾nicu. Doplníme $G$ na~uplný graf~$G'$ a~váhy hrán~$G'$ nastavíme takto:
-\itemize\ibull
-\: $w(e) = 1$, ak $e \in E(G)$
-\: $w(e) = c \ll 1$, ak $e \not\in E(G)$
-\endlist
-\>Ak existuje Hamiltonovská kru¾nica v~$G'$ zlo¾ená iba z~hrán, ktoré boli pôvodne v~$G$, tak optimálné rie¹enie bude ma» váhu $n$, inak bude urèite minimálne $n-1+c$. Ak máme aproximaèný algoritmus s~pomerom $1+\varepsilon$, musí by»
-$$
-\eqalign{
-(1+\varepsilon).n &< n-1+c \cr
-\varepsilon n+1 &< c
-}
-$$
-\>Ak by taký algoritmus existoval, tak na~neho máme polynomiálny algoritmus
-na~Hamiltonovsku kru¾nicu. Inak neexistuje ani pseudo-polynomialny algoritmus.
-\qed
-
-\h{Aproximaèní schéma pro problém batohu}
-
-\s{POZOR:} Verze algoritmu, kterou jsem øíkal na~pøedná¹ce, obsahovala jednu
-pomìrnì zásadní chybu, které jsem si nev¹iml: Verze se zaokrouhlováním dolù
-mohla produkovat nepøípustná (pøíli¹ tì¾ká) øe¹ení, verze se zaokrouhlováním nahoru pro zmìnu
-nìkdy spoèítala øe¹ení pøíli¹ daleká od~optima. Algoritmus lze opravit (budeme-li
-zvlá¹» zpracovávat lehké a tì¾ké pøedmìty), ale radìji budeme místo hmotností
-kvantovat ceny. Tak dojdeme k~následujícímu aproximaènímu algoritmu. --M.M.
-
-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~$M$
-a zobrazíme interval cen $[0, c_{max}]$ na $[0,M]$.
-Jak jsme tím zkreslili výsledek? V¹imnìme si, ¾e efekt je stejný, jako kdybychom jednotlivé
-ceny zaokrouhlili na~násobky èísla $c_{max}/M$. Ka¾dé $c_i$ jsme tím
-zmìnili o~nejvý¹e $c_{max}/M$, celkovou cenu libovolné podmno¾iny pøedmìtù tedy
-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 $M\ge n/\varepsilon$.
-
-\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: $\hat{c}_i = \lfloor c_i \cdot M/c_{max} \rfloor$.
-\: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 $\hat{C}\le nM \le n^2/\varepsilon$, 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, jak dopadne optimální øe¹ení $OPT$ pùvodního zadání,
-kdy¾ ceny v~nìm pou¾itých pøedmìtù 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 naopak spoèítejme, jak dopadne 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}).
-
-\bye
+++ /dev/null
-P=12-apx
-R=../..
-
-include ../../Makerules
+++ /dev/null
-\input lecnotes.tex
-
-\def\\{\setminus}
-\def\gcd{{\rm gcd}}
-
-\prednaska{13}{Z teorie èísel}{(zapsali L. Banáková, O. Hoferek, J. Bøeèka)}
-
-Na této pøedná¹ce se budeme zabývat rùznými problémy okolo teorie èísel. Zopakujme si
-nìkteré ze základních pojmù:
-
-\itemize\ibull
-\:$a \\ b$ ($a$ dìlí $b$) $\Leftrightarrow \exists c: b = a \cdot c$.
-\:$\gcd(a,b)$ je oznaèení nejvìt¹ího spoleèného dìlitele èísel $a$ a $b$.
-\:$a \equiv_n b \Leftrightarrow n \perp (a-b)$ (nebo také $a \bmod n = b \bmod n$).
-\:$a \perp b$ ($a$ a $b$ jsou nesoudìlná) $\Leftrightarrow \gcd(a,b) = 1$.
-\endlist
-
-Dále si zopakujeme, jakou èasovou slo¾itost mají základní operace s~èísly,
-které budeme potøebovat. Pro $N$-bitová èísla:
-
-\itemize\ibull
-\:$a+b$, $a-b$ \dots $\O(N)$
-\:$a*b$, $a/b$, $a \bmod b$ \dots $\O(N^2)$
-\:$\gcd(a,b)$ \dots $\O(N^3)$
-\endlist
-
-\s{Definice:} {\I Komutativní (Abelovská) grupa} je ètveøice $(G,\cdot,1,^{-1})$,
-kde $G$ je nosná mno¾ina prvkù, $\cdot$ je binární operace $G^2 \rightarrow G$,
-$1$ je prvkem $G$, $^{-1}$ je unární operace $G \rightarrow G$ a platí následující
-axiomy:
-\numlist\ndotted
-\:$\cdot$ je komutativní a asociativní
-\:$\forall a \in G: a \cdot 1 = a$
-\:$\forall a \in G: a \cdot a^{-1} = 1$
-\endlist
-
-Zkusme si pro následujících nìkolik kandidátù urèit, zda jsou komutativními grupami:
-
-\itemize\ibull
-\:$({\bb Z},+,0,-x)$ (sèítání na mno¾inì celých èísel, nula a zmìna znaménka) je komutativní grupa.
-\:$({\bb Z}_n,+ \bmod n,0,-x)$ (${\bb Z}_n = \{0,\dots,n-1\}$, sèítání modulo $n$) je komutativní grupa, navíc koneèná.
-\:$({\bb Q}-\{0\},*,1,1/x)$ (násobení nad racionálními èísly) je komutativní grupa.
-\:$({\bb Z}_n - \{0\},* \bmod n,1,?)$ (násobení modulo $n$) nemusí být v¾dy grupa, proto¾e se nám nepovede v¾dy najít inverzní prvek.
-(napø. pro $n=4$ neexistuje inverzní prvek pro dvojku).
-\:$($permutace na $\{1,\dots,n\},\circ,{\rm id},^{-1})$ (permutace se skládáním a inverzní permutací) je grupa, ale není komutativní.
-\endlist
-
-\s{Definice:} $(H,\cdot,1,^{-1})$ je {\I podgrupa} grupy $(G,\cdot,1,^{-1})$ právì tehdy,
-pokud $H \subseteq G$ a $H$ je grupa.
-
-Pøíkladem podgrupy mù¾e být $(2 * {\bb Z}_n,+ \bmod n,0,-x) \subseteq ({\bb Z}_n,+ \bmod n,0,-x)$
-(grupa tvoøená jen sudými èísly).
-
-\s{Vìta (Lagrangeova):} Pokud $H$ je podgrupa $G$, pak poèet prvkù $G$ je dìlitelný poètem prvkù $H$.
-
-\s{Definice:} Umocòování prvkù grupy definujeme takto: $x^0 = 1$, $x^{n+1} = x^n \cdot x$.
-
-\s{Definice:} Grupa $G$ je {\I cyklická} právì tehdy, pokud $\exists g \in G: {g^0, g^1, \dots} = G$.
-Pak $g$ je {\I generátor} grupy G.
-
-\itemize\ibull
-\:$({\bb Z},+,0,-x)$ není cyklická.
-\:$({\bb Z}_n,+ \bmod n,0,-x)$ je cyklická, $g=1$.
-\:$({\bb Q},\dots)$ není cyklická.
-\endlist
-
-\s{Definice:} {\I Multiplikativní grupa modulo $n$} je grupa
-${\bb Z}^*_n = (\{ x~\vert~1 \le x < n $ a zároveò $ \exists y: xy \equiv_n 1\}, * \bmod n, 1, ^{-1})$
-
-Multiplikativní grupa ${\bb Z}^*_n$ obsahuje pouze invertibilní prvky. Snadno lze nahlédnout,
-¾e toto je opravdu grupa, pokud ovìøíme, ¾e mno¾ina prvkù je uzavøená na násobení:
-
-$x_1, x_2 \in {\bb Z}^*_n \Rightarrow \exists y_1, y_2 \in {\bb Z}^*_n: x_{1}y_1 \equiv_n 1, x_{2}y_2 \equiv_n 1$
-
-$x \equiv_n x_{1}x_2, y \equiv_n y_{1}y_2 \dots xy \equiv_n x_{1}x_{2}y_{1}y_{2} \equiv_n 1 \cdot 1 = 1 \Rightarrow $ mno¾ina
-je uzavøená vzhledem k operaci $* \bmod n$.
-
-Otázkou zùstavá, jak najít tyto invertibilní prvky v~${\bb Z}_n$. Pøedpokládejme nejprve,
-¾e $n$ je prvoèíslo. Pomù¾eme si vìtou, kterou doká¾eme pozdìji.
-
-\s{Vìta (Malá Fermatova):} Pro ka¾dé prvoèíslo $n$ a ka¾dé èíslo $a$, které je nesoudìlné s~$n$, platí: $a^{n-1} \equiv_n 1$.
-
-Pokud je tedy $n$ prvoèíslo, tak z~{\I Malé Fermatovy vìty} vyplývá, ¾e invertibilní jsou v¹echny
-prvky ${1, \dots, n-1}$. Snadno toti¾ nahlédneme, ¾e $\forall a \in {\bb Z}_n$ platí $a^{-1} \equiv_n a^{n-2}$,
-proto¾e $a \cdot a^{-1} \equiv_n 1 \equiv_n a^{n-1} \equiv_n a \cdot a^{n-2}$. Jak to v¹ak bude
-s~$n$, která nejsou prvoèísla? Invertibilní jsou prvky $a \in {\bb Z}_n$, pro které existuje
-$x \in {\bb Z}_n$ tak, ¾e $a \cdot x \equiv_n 1$. Jejich zkoumání zaèneme následující vìtou:
-
-\s{Vìta:} Rovnice $a \cdot x \equiv_n b$ má øe¹ení pro $a,b,n \in {\bb Z} \Longleftrightarrow
-g = \gcd(a,n) \perp b$. Navíc existuje algoritmus s èasovou slo¾itostí $\O(N^3)$, který øe¹ení najde.
-
-\proof
-Rovnice $a \cdot x \equiv_n b$ je ekvivalentní s~rovnicí $a \cdot x - n \cdot y = b$.
-
-\noindent
-\uv{$\Rightarrow$}: Doká¾eme sporem. $g \\ a \cdot x$ a $g \\ n \cdot y \Rightarrow
-g \\ a \cdot x - n \cdot y$. Zároveò v¹ak platí $g$ nedìlí $b$, co¾ vede ke sporu.
-
-\noindent
-\uv{$\Leftarrow$}: Pokud $b = g$, pak lze úlohu hledání $x, y$ øe¹ení rovnice $a \cdot x - n \cdot y = g$
-pøevést na Eukleidùv algoritmus. V Eukleidovì algoritmu se vstupem $(x,y)$ jsou toti¾ v¹echny
-mezivýsledky i koneèný výsledek lineární kombinace typu $\alpha \cdot x - \beta \cdot y$. Pokud
-$b = k \cdot g$, pak najdeme $x_0, y_0$ taková, ¾e $a \cdot x_0 - n \cdot y_0 = g$. Pro $x= k \cdot x_0$
-a $y = k \cdot y_0$ pak platí $a \cdot x - n \cdot y = b$.
-\qed
-
-Z pøedchozí vìty vyplývá, ¾e invertibilní jsou právì taková $a \in {\bb Z}_n$, která
-jsou nesoudìlná s $n$. Zbývá u¾ jenom urèit, jak vypadají prvky k nim inverzní.
-
-\s{Definice:} {\I Eulerova funkce} $\varphi(n) = \vert\{ x~\vert~1 \le x < n$ a zároveò $x \perp n\}\vert$.
-
-\s{Pozorování:} {\I(vlastnosti Eulerovy funkce)}
-
-\itemize\ibull
-\:Je-li $n$ prvoèíslo, pak $\varphi(n) = n - 1$.
-\:$\varphi(n^k) = (n - 1) \cdot n^{k-1}$.
-\:Pro $a \perp b$ platí $\varphi(a \cdot b) = \varphi(a) \cdot \varphi(b)$.
-\:$\vert{\bb Z}^*_n\vert = \varphi(n)$.
-\endlist
-
-\s{Vìta (Eulerova):} Pro ka¾dá dvì pøirozená èísla $n$ a $a$ nesoudìlná platí: $a^{\varphi(n)} \equiv_n 1$.
-
-\proof
-Uva¾me posloupnost $a^0, a^1, a^2, \dots$. V této posloupnosti se urèitì vyskytuje jednièka, proto¾e
-mo¾ných hodnot $a^m$ je nejvý¹e $n$. Proto existují $i$ a $j$ $(i < j)$ taková, ¾e $a^i \equiv_n a^j$,
-tedy $a^{j - i} \equiv_n 1$.
-Zvolme $m > 0$ nejmen¹í takové, ¾e $a^m \equiv_n 1$. Èísla $a^0, a^1, \dots, a^{m-1}$ tvoøí podgrupu
-multiplikativní grupy ${\bb Z}^*_n$. Proto podle Lagrangeovy vìty platí $m$ dìlí $\varphi(n)$, tedy
-$\varphi(n) = k \cdot m$ pro nìjaké $k$. Snadno nahlédneme, ¾e
-$a^{\varphi(n)} \equiv_n a^{k \cdot m} \equiv_n 1^k = 1$.
-\qed
-
-Tato vìta nám dokázala i døíve zmínìnou Malou Fermatovu vìtu a snadno díky ní nahlédneme, ¾e
-pro $a \perp n$ platí $a^{-1} \equiv_n a^{\varphi(n)-1}$.
-
-\h{Testování prvoèíselnosti}
-
-Nyní vyu¾ijeme na¹e novì získané poznatky k~ovìøování prvoèíselnosti. Tzv. {\I faktorizace},
-neboli rozklad èísla na souèet prvoèísel, je urèitì v~{\I NP}, av¹ak zatím se neví, zda to je
-nebo není {\I NP}-úplný problém. Naproti tomu je ji¾ znám
-polynomiální algoritmus, který pøesnì ovìøí, zda je zadané èíslo prvoèíslem (s èasovou
-slo¾itostí $\O(N^{6,5})$). Tento algoritmus je v¹ak velmi komplikovaný a stejnì tak odhad
-jeho èasové slo¾itosti. My se proto spokojíme s~tzv. Monte Carlo testováním prvoèísel.
-Pokud takto oznaèený test prvoèíselnosti odpoví, ¾e èíslo $n$ zadané na vstupu je slo¾ené,
-je to pravda. Pokud odpoví, ¾e se jedná o prvoèíslo, mýlí se s~pravdìpodobností men¹í,
-ne¾ $1 \over 2$, èili celková pravdìpodobnost, ¾e se mýlí je men¹í ne¾ $1 \over 4$.
-Pokud takový test opakujeme $k$-krát za sebou, je pravdìpodobnost, ¾e se test zmýlil
-$({1 \over 4})^k$, co¾ je u¾ pro $k = 100$ dostaèující. Uka¾me si tedy první z takových testù.
-
-\s{Algoritmus:} {\I(Fermatùv test)}
-
-\algo
-\:Zvolme náhodnì $a \in \{2, \dots, n-1\}$.
-\:Pokud $\gcd(a,n) \not= 1$, je $n$ slo¾ené.
-\:Pokud $a^{n-1} \not\equiv_n 1$, je $n$ slo¾ené (a $a$ je {\I Fermatùv svìdìk}).
-\:Jinak je $n$ prvoèíslo.
-\endalgo
-
-\s{Pozorování:} Pokud odpoví Fermatùv test, ¾e zadané èíslo je slo¾ené, nemýlí se.
-
-\s{Poznámka:} V¹imnìme si, ¾e druhý krok algoritmu je zbyteèný. Pokud toti¾ není $a \perp n$,
-neboli existuje $g > 1$ spoleèný dìlitel $n$ a $a$, pak $g$ dìlí i $a^{n -1} \bmod n$ nebo
-$a^{n -1} \bmod n = 0$ a tøetí krok algoritmu prohlásí $n$ za slo¾ené èíslo.
-
-©patná zpráva pro tento algoritmus je, ¾e existují tzv. {\I Carmichaelova èísla},
-co¾ jsou slo¾ená èísla, která Fermatova svìdka nemají. Je jich sice \uv{øídko},
-ale zato nekoneènì mnoho. Pokud se ale zrovna do Carmichaelova èísla nestrefíme, mají
-slo¾ená èísla Fermatových svìdkù dostatek.
-
-\s{Vìta:} Pokud $n$ není Carmichaelovo èíslo ani prvoèíslo, pak existuje alespoò
-${\varphi(n)} \over 2$ Fermatových svìdkù.
-
-\proof
-Zvolme $H = \{ a \in \{1, \dots, n-1\}~\vert~a^{n-1} \equiv_n 1, a \perp n\}$, tedy mno¾inu
-v¹ech èísel, která nejsou Fermatovými svìdky. V¹echna èísla v $H$ jsou invertibilní
-a jejich souèin je opìt v $H$. $H$ je tedy podgrupa multiplikativní grupy ${\bb Z}^*_n$.
-Podle Lagrangeovy vìty tak existuje $k$ tak, ¾e $\vert{\bb Z}^*_n\vert = \varphi(n) = k \cdot \vert H\vert$.
-Navíc $n$ není Carmichaelovo, tak¾e platí $\vert{\bb Z}^*_n\vert \not= \vert H\vert$.
-Proto $k \ge 2$ a èísel, která nemají Fermatova svìdka, je $\vert H\vert \le {\varphi(n) \over 2}$.
-\qed
-
-Vidíme tedy, ¾e pokud $n$ není Carmichaelovo èíslo, pak je Fermatùv test Monte Carlo
-test prvoèíselnosti. Na závìr pøedná¹ky si uvedeme Monte Carlo test, který
-øe¹í problém Carmichaelových èísel (co¾ není zrovna jednoduché):
-
-\s{Algoritmus:} {\I(Rabin-Millerùv test)}
-
-\algo
-\:Zvolme náhodnì $a \in \{2, \dots, n-1\}$.
-\:Pokud $a^{n-1} \not\equiv_n 1$, je $n$ slo¾ené (a $a$ je {\I Fermatùv svìdìk}).
-\:Pro $i = 1, 2, \dots$ dokud $2^i$ dìlí $n - 1$:
- \::Spoèteme $t_i\equiv_n a^{(n-1)/ 2^i}$.
- \::Pokud je $t_i\equiv_n -1$, je $n$ prvoèíslo.
- \::Pokud je $t_i\not\equiv_n 1$, je $n$ slo¾ené (a $a$ je {\I Riemannùv svìdìk}).
-\:Jinak je $n$ prvoèíslo.
-\endalgo
-
-\bye
+++ /dev/null
-P=13-cisla
-
-include ../Makerules
+++ /dev/null
-\input lecnotes.tex
-
-\prednaska{2}{Toky v sítích}{(pøedná¹el T. Valla, zapsali J. Machálek a K. Vandas)}
-
-\s{Motivaèní úlohy:}
-\itemize\ibull
-\:Mìjme orientovaný graf se speciálními vrcholy ®elivka a Kanál pøedstavující pra¾ské vodovody. V tomto grafu budou vrcholy vodovodními stanicemi a hrany trubkami mezi nimi. Kolik vody proteèe ze ®elivky do Kanálu?
-\:Mìjme orientovaný graf pøedstavující ¾eleznièní sí»; graf má význaèné vrcholy Moskva a Fronta, ka¾dá hrana grafu má kapacitu, kterou mù¾e uvézt. Kolik vojákù je schopna sí» pøevézt z~Moskvy a spotøebovat na Frontì?
-\endlist
-
-\s{Definice:} {\I Sí»} je uspoøádaná ètveøice $(G,z,s,c)$, kde $G$ je
-orientovaný graf, $z$~a~$s$~jsou nìjaké dva jeho vrcholy ({\I zdroj} a {\I stok}) a $c$ je
-{\I kapacita hran,} kterou pøedstavuje funkce $c:E(G)\to{\bb
-R}_{0}^{+}$.
-
-\figure{sit.eps}{Pøíklad sítì. Èísla pøedstavují kapacity jednotlivých hran.}{3in}
-
-\par\noindent {\sl Intuice:} Toky v sítích pøedstavují rozvr¾ení, jakým suroviny sítí poteèou.
-
-\s{Definice:} {\I Tok} je funkce $f:E(G)\to{\bb R}_{0}^{+}$ taková, ¾e platí:
-\numlist{\ndotted}
-\:Tok po~ka¾dé hranì je omezen její kapacitou: $0\le f(e)\le c(e)$.
-\:Kirchhoffùv zákon -- \uv{sí» tìsní}: $$\sum_{xu \in E}{f(xu)}=\sum_{ux \in E}{f(ux)}\quad\hbox{pro ka¾dé }u\in V(G) \setminus \{z,s\}.$$
-\endlist
-
-\s{Poznámka:} Pokud bychom se chtìli v definici toku u bodu 2 vyhnout podmínkám pro $z$ a $s$, mù¾eme zdroj a stok vzájemnì propojit (pak jde o tzv. cirkulaci).
-
-\s{Poznámka:} V angliètinì se obvykle zdroj znaèí \uv{$s$} a stok \uv{$t$} (jako source a~target).
-
-\figure{tok.eps}{Pøíklad toku. Èísla pøedstavují toky po~hranách, v~závorkách jsou kapacity.}{4in}
-
-\s{Definice:} {\I Velikost toku} $f$ je: $$\vert f\vert:=\sum_{zx \in E}{f(zx)}-\sum_{xz \in E}{f(xz)}.$$
-
-\>Budeme tedy chtít najít v~zadané síti tok, jeho¾ velikost je maximální. Musí v¾dycky existovat?
-
-\s{Vìta:} Pro ka¾dou sí» existuje maximální tok.
-
-\par\noindent {\sl Idea dùkazu:} Doká¾e se pomocí metod matematické analýzy s~tím, ¾e mno¾ina tokù je kompaktní a funkce velikosti toku je spojitá (dokonce lineární).
-
-Dobrá, maximální tok v¾dy existuje. Ale kdy¾ nám ho nìkdo uká¾e, umíme poznat,
-¾e je skuteènì maximální? K~tomu se budou hodit øezy:
-
-\par\noindent {\sl Intuice:} Øez v~grafu je mno¾ina hran oddìlující zdroj a stok.
-
-\s{Definice:} {\I Øez} $R$ v síti $(G,z,s,c)$ je mno¾ina hran $R$ taková, ¾e
-neexistuje orientovaná cesta ze $z$~do $s$~v~grafu $(V(G),E(G)\setminus R)$.
-
-\s{Definice:} {\I Kapacita øezu} $c(R)=\sum_{uv \in R}{c(uv)}$.
-
-Nìkdy je lep¹í se na~øezy dívat takto:
-
-\s{Definice:} Pro rozklad mno¾iny vrcholù na dvì disjunktní mno¾iny $A,B$, kde $z\in A$ a $s\in B$ zavedeme
-{\I separátor} $S(A,B)$, co¾ bude mno¾ina hran vedoucích z~$A$ do~$B$. Pokud je $g$ funkce
-definovaná na~hranách (tøeba tok nebo kapacita), definujeme $g(A,B):=\sum_{uv \in E,u\in A,v\in B}{g(uv)}.$
-
-\s{Pozorování:} Ka¾dý separátor je øezem (libovolná orientovaná cesta ze~$z$ do~$s$ musí
-nìkdy opustit mno¾inu~$A$, a to jde pouze po~hranì patøící do~separátoru). Opaènì to sice
-platit nemusí, ale platí, ¾e ke~ka¾dému øezu existuje separátor takový, ¾e souèet kapacit
-jeho hran je nejvý¹e kapacita daného øezu. Staèí si zvolit mno¾inu $A$ jako vrcholy dosa¾itelné
-ze~zdroje po~hranách nele¾icích v~øezu a do~$B$ dát v¹echny ostatní vrcholy. Separátor
-$S(A,B)$ pak bude tvoøen výhradnì hranami øezu (ne~nutnì v¹emi) a sám bude øezem.
-
-\s{Definice:} Pro libovolnou mno¾inu vrcholù $A$ zavedeme její {\I doplnìk} $\overline A:=V(G)\setminus A$.
-
-Nyní si v¹imneme, ¾e velikost toku mù¾eme mìøit pøes libovolný separátor: je to mno¾ství
-tekutiny, které teèe pøes separátor z~$A$ do~$B$ minus to, které se vrací zpátky
-(zatím jsme velikost mìøili u~zdroje, vlastnì na~triviálním separátoru $A=\{z\}$, $B=\overline A).$
-
-\s{Lemma:} Nech» $A\subseteq V(G),z\in A,s\not\in A$ a $f$ je libovolný tok. Potom platí,
-¾e: $$\vert f\vert=f(A,\overline A)-f(\overline A,A).$$
-
-\proof
-provedeme pomocí Kirchhoffova zákona a definice velikosti toku:
-$$\hbox{Pro ka¾dý vrchol~$u\ne z,s$ platí:~~} \sum_{ux \in E}{f(ux)}-\sum_{xu \in E}{f(xu)}=0,$$
-$$\hbox{pro zdroj pak:~~} \sum_{zx \in E}{f(zx)}-\sum_{xz \in E}{f(xz)=\vert f\vert}.$$
-Rovnice seèteme:
-$$\sum_{u\in A}{\left(\sum_{ux \in E}{f(ux)}-\sum_{xu \in E}{f(xu)}\right)}=\vert f\vert.$$
-V¹imneme si, ¾e hrany, jeji¾ oba koncové vrcholy le¾í v~mno¾inì~$A$, pøispívají
-k~této sumì jednou kladnì a jednou zápornì, hrany, její¾ ani jeden vrchol nele¾í v~$A$,
-nepøispívají vùbec, a koneènì hrany vedoucí z~$A$ do~$B$, resp. opaènì pøispìjí jen jednou
-(kladnì, resp. zápornì). Tedy:
-$$f(A,\overline A)-f(\overline A,A)=\sum_{u\in A,v\not\in A}{f(uv)}-\sum_{u\not\in A,v\in A}{f(uv)}=\vert f \vert.$$
-\qed
-
-\s{Dùsledek:} Pokud $f$ je tok, $R$ je øez, pak platí: $\vert f \vert\le c(R)$.
-
-\proof
-Doká¾eme pro separátory (u¾ víme, ¾e ke~ka¾dému øezu najdeme stejnì velký nebo men¹í separátor):
-$$\vert f \vert=f(A,\overline A)-f(\overline A,A)\le f(A,\overline A)\le c(A,\overline A)\le c(R).$$
-\qed
-
-Nyní víme, ¾e velikost ka¾dého toku je shora omezena velikostí ka¾dého øezu. Kdybychom tedy
-k~na¹emu toku umìli najít stejnì velký øez, hned víme, ¾e tok je maximální a øez minimální.
-Pøekvapivì platí, ¾e se to povede v¾dy. K~tomu se nám budou hodit zlep¹ující cesty:
-
-\s{Definice:} {\I Zlep¹ující cesta} budeme øíkat takové cestì mezi danými dvìma vrcholy,
-která pou¾ívá buïto hrany sítì (hrany orientované po~smìru cesty) nebo hrany k~nim opaèné (v~síti jsou
-tedy proti smìru cesty).
-
-\figure{cesta.eps}{Pøíklad zlep¹ující cesty.}{3in}
-
-\s{Definice:} {\I Zlep¹ující cesta} $P$ ze $z$ do $s$ je {\I nasycená}, pokud:
-$$\exists e \in P:\cases{
- f(e)=c(e) & \hbox{je-li $e$ orientovaná po smìru} \cr
- f(e)=0 & \hbox{je-li $e$ orientovaná proti smìru} \cr
-}$$
-\>Jinak je zlep¹ující cesta nenasycená.
-
-\s{Definice:} {\I Tok je nasycený,} pokud je ka¾dá zlep¹ující cesta $P$ ze $z$ do $s$ nasycená.
-
-\s{Vìta:} Tok $f$ je nasycený $\Leftrightarrow$ $f$ je maximální. Navíc pro ka¾dý maximální tok $f$ existuje øez $R$ takový, ¾e $\vert f \vert=c(R)$.
-
-\proof
-
-\>\uv{$\Leftarrow$} doká¾eme nepøímo -- uká¾eme, ¾e pokud nìjaký tok není nasycený, tak ho
-je¹tì lze zlep¹it, proèe¾ nemù¾e být maximální. Mìjme nìjaký nenasycený tok~$f$. Existuje tedy
-nenasycená zlep¹ující cesta $P$. Podél této cesty budeme tok vylep¹ovat.
-Zvolíme:
-$$
-\eqalign{
-\varepsilon_1&:=\min_{e\in P\hbox{\sevenrm~po smìru}}{\{c(e)-f(e)\}}, \cr
-\varepsilon_2&:=\min_{e\in P\hbox{\sevenrm~proti smìru}}{\{f(e)\}}, \cr
-\varepsilon&:=\min{\{\varepsilon_1,\varepsilon_2\}}. \cr
-}
-$$
-Jeliko¾ cesta byla nenasycená, musí být $\varepsilon>0$. Nyní z~toku~$f$ vytvoøíme
-tok~$f'$ takto:
-$$f'(e):=\cases{
- f(e)+\varepsilon & \hbox{je-li $e$ po smìru cesty,} \cr
- f(e)-\varepsilon & \hbox{je-li $e$ proti smìru,} \cr
- f(e) & \hbox{pokud $e\not\in P$.} \cr
-}
-$$
-\>Nyní je potøeba ovìøit, ¾e $f^{'}$ je skuteènì tok:
-$$0\le f^{'}(e)\le c(e)\dots\hbox{platí stále díky volbì }\varepsilon.$$
-
-\>Platnost Kirchhoffova zákona ovìøíme rozborem pøípadù:
-\figure{kirch.eps}{Rozbor pøípadù.}{4in}
-
-\>Funkce $f'$ je tedy tok. Jeho velikost se ov¹em oproti~$f$ zvý¹ila o~$\varepsilon$, tak¾e~$f$ nebyl maximální.
-
-\>\uv{$\Rightarrow$}: Uvá¾íme mno¾inu vrcholù $A\subseteq V(G)$ definovanou tak, ¾e $v\in A$
-právì tehdy, kdy¾ existuje nenasycená cesta ze $z$ do $v$. V¹imnìme si, ¾e $z \in A$ a $s \not\in A$.
-Potom $S(A,\overline A)$ je øez, který je stejnì velký jako tok~$f$. Jak u¾ víme, pokud k~toku
-najdeme stejnì velký øez, je tok maximální.
-\qed
-
-\figure{nenasyc.eps}{Rozdìlení $V(G)$ na mno¾inu $A$ a $\overline A$ v dùkazu hlavní vìty o tocích.}{2.5in}
-
-\>To, co jsme o~tocích zjistili, mù¾eme shrnout do~následující \uv{minimaxové} vìty:
-
-\s{Vìta (Hlavní vìta o tocích, Ford-Fulkerson):} Pro ka¾dou sí» platí: $$\max_{f\hbox{\sevenrm~tok}}{\vert f \vert=\min_{R\hbox{\sevenrm~øez}}{c(R)}}.$$
-
-\proof
-Nerovnost \uv{$\le$} je ná¹ Dùsledek lemmatu o~tocích a øezech, rovnost
-platí proto, ¾e k~maximálnímu toku existuje podle pøedcházející vìty stejnì
-velký øez.
-\qed
-
-\>Zlep¹ování tokù pomocí zlep¹ujících cest není jen pìkný dùkazový prostøedek,
-dá se pomocí nìj také formulovat elegantní algoritmus na~hledání maximálního toku:
-
-\s{Algoritmus (hledání maximálního toku v síti, Ford-Fulkerson)}
-
-\algo
-\:$f \leftarrow \hbox{libovolný tok}$, tøeba v¹ude nulový ($\forall e \in E: f(e) \leftarrow 0 $).
-\:Dokud $\exists$ zlep¹ující cesta $P$: vylep¹íme $f$ podél $P$ jako v~dùkazu vìty.
-\:Prohlásíme $f$ za~maximální tok.
-\endalgo
-
-\h{Cvièení:}
-\itemize\ibull
-\:Je pro pøirozené kapacity F-F algoritmus koneèný? Ano -- v ka¾dém zlep¹ujícím
-kroku algoritmu se celkový tok zvìt¹í aspoò o jedna. Proto¾e máme horní odhad
-na velikost maximálního toku (napø. souèet kapacit v¹ech hran), máme i~horní odhad na dobu
-bìhu algoritmu.
-\:Je F-F algoritmus koneèný pro racionální kapacity hran? Ano -- v¹echny
-kapacity vynásobíme spoleèným jmenovatelem a pøevedeme na pøedchozí pøípad.
-Algoritmus se pøitom chová stejnì jako s~pùvodními kapacitami.
-\:A~pro reálné kapacity? Obecnì ne -- zkuste najít sí» s~nìkterými kapacitami
-iracionální, kde se algoritmus pøi ne¹ikovné volbì zlep¹ujících cest nikdy
-nezastaví a dokonce ani nekonverguje k~maximálnímu toku.
-\:Kolik krokù bude muset algoritmus na následující síti maximálnì udìlat, aby
-úspì¹nì dobìhl? ($2M$ krokù)
-\figure{2M.eps}{Pøíklad sítì. Kolik krokù musí maximálnì udìlat F-F algoritmus?}{2in}
-\:Pokud bychom v¾dy hledali {\I nejkrat¹í} zlep¹ující cestu, co¾ je snad
-nejpøímoèaøej¹í mo¾ná implementace (prohledávání do ¹íøky), algoritmus by
-se zastavil po~$\O(M^2N)$ krocích (tomu se øíká Edmondsùv-Karpùv algoritmus).
-To si nebudeme dokazovat a místo toho si na~pøí¹tí pøedná¹ce rovnou odvodíme
-efektivnìj¹í Dinicùv algoritmus.
-\endlist
-
-\bye
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-ipeMakeFont
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-%!PS-AdobeFont-1.1: CMSY10 1.0
-%%CreationDate: 1991 Aug 15 07:20:57
-% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
-11 dict begin
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-\input lecnotes.tex
-
-\prednaska{3}{Dinicùv algoritmus}{(zapsali Jakub Melka, Petr Musil)\foot{\rm s~díky Bernardovi Lidickému za obrázky}}
-
-Na~minulé pøedná¹ce jsme si ukázali {\I Fordùv-Fulkersonùv} algoritmus. Víme o~nìm, ¾e kdy¾ se zastaví, tak vydá maximální tok. Jen¾e zastavit se nemusí (napøíklad pro~sítì s~reálnými kapacitami), nebo trvá pøíli¹ dlouho.
-Uká¾eme si lep¹í algoritmus, {\I Dinicùv}, který má výraznì men¹í slo¾itost a zastaví se v¾dy.
-
-
-Idea je následující: v~algoritmu budeme pou¾ívat {\I sí» rezerv}, která bude obsahovat rezervy -- kolik je¹tì po~dané hranì mù¾eme pustit, aby to nepøekroèilo její kapacitu.
-Sí» rezerv pak budeme vyu¾ívat k~vylep¹ování toku.
-
-\s{Definice:} {\I Sí» rezerv }$R$ k síti $S=(V,E,z,s,c)$ a toku $f$ v $S$ je sí» $R=(V,E\cup\overleftarrow{E}, z,s, r)$, pro $\forall e\in E$:
-\itemize\ibull
-\:$r(e)=c(e)-f(e)$,
-\:$r(\overleftarrow{e})=f(e)$,
-\endlist
-\>kde hrana $\overleftarrow{e}$ vznikne z~hrany $e$ tak, ¾e se zorientuje opaèným smìrem. V pøípadì, ¾e v~síti rezerv u¾ opaènì orientovaná hrana\foot{vznikne tak, ¾e v pùvodní síti jsou dvì opaènì orientované hrany mezi stejnými vrcholy.} je, pak vznikne multigraf\foot{graf, který mù¾e mít mezi dvìma vrcholy více stejnì orientovaných hran.} s~multiplicitou maximálnì dva.
-
-
-\>Sí» rezerv budeme pou¾ívat v~algoritmu k~hledání vylep¹ujících tokù. K~tomu nám bude slou¾it následující vìta:
-
-\s{Vìta:} Je-li $f$ tok v~síti $S$ a $g$ tok v~pøíslu¹né síti rezerv, pak $\exists$ tok $f'$ v $S$ takový, ¾e $\vert f'\vert = \vert f\vert + \vert g\vert$, co¾ znamená $\forall e \in E : f'(e) = f(e) + g(e) - g(\overleftarrow{e})$.
-
-\proof
-Rozebereme si jednotlivé pøípady pro~$\forall e\in E$:
-\numlist\nalpha
-\:$g(e)=g(\overleftarrow{e})=0 \Rightarrow f'(e)=f(e) $.
-\:$g(e)>0$ a zároveò $g(\overleftarrow{e})=0\Rightarrow f'(e) = f(e) + g(e)$.
-\:$g(e)=0$ a zároveò $g(\overleftarrow{e})>0\Rightarrow f'(e) = f(e) - g(\overleftarrow{e})$.
-\:Nastává cirkulace, tu snadno odstraníme: odeèteme $\varepsilon$ od obou hran $e$ a $\overleftarrow{e}$,
-kde $\varepsilon=\min( g(e), g(\overleftarrow{e}) )$. Pøevedeme tím tento pøípad na jeden ze tøí uvedených vý¹e.
-\endlist
-
-Je v¹ak $f'$ tok? Víme, ¾e $f'$ urèitì nemù¾e klesnout pod nulu, proto¾e se odeèítá jen v pøípadì c), a tam je z~definice vidìt, ¾e $f'$ pod nulu klesnout nemù¾e. Kapacita také nemù¾e
-být pøekroèena, pøièítá se jen v~pøípadì b) a z~definice se nepokazí, proto¾e $g(e)=c(e)-f(e)$, tedy v~nejhor¹ím pøípadì $f'(e) = c(e)$.
-
-Dále doká¾eme, ¾e $f'$ dodr¾uje Kirchhoffùv zákon. V~následujících sumách pøedpokládejme, ¾e v¹echny vrcholy jsou rozdílné od~zdroje a stoku. Musí platit, ¾e:
-$$\sum\limits_{ab \in E} f'(ab) = \sum\limits_{ba \in E} f'(ba).$$
-Rozepí¹eme si tuto rovnici dle definice:
-$$\sum\limits_{ab \in E} f'(ab) - \sum\limits_{ba \in E} f'(ba) = 0,$$
-$$\sum\limits_{uv\in E}(f(uv)+g(uv)-g(\overleftarrow{uv})) - \sum\limits_{vu\in E}(f(vu)+g(vu)-g(\overleftarrow{vu})) = 0.$$
-Roztrhneme si to na ètyøi sumy a dostaneme:
-$$\underbrace{\sum\limits_{uv\in E}f(uv)-\sum\limits_{vu\in E}f(vu)}\limits_0+$$
-$$+\underbrace{\sum\limits_{uv\in E}g(uv)-g(\overleftarrow{uv})-\sum\limits_{vu\in E}g(vu)-g(\overleftarrow{vu})}\limits_0 = 0,$$
-nebo» $f$ i $g$ jsou toky a musí splòovat Kirchhoffùv zákon.
-\qed
-
-Tato vìta nám øíká, ¾e pokud existuje nenulový tok v~síti rezerv, pak lze tok v~pùvodní síti je¹tì zvìt¹it. Naopak pokud takový tok neexistuje, je tok v~pùvodní síti maximální.
-
-
-\s{Definice:} $f$ je {\I blokující tok}, pokud na~ka¾dé orientované cestì $P$ ze~zdroje do~spotøebièe $\exists e\in P : f(e)=c(e)$.
-
-\s{Definice:} $C$ je {\I proèi¹tìná sí»}, pokud obsahuje pouze vrcholy a hrany na~nejkrat¹ích $z\rightarrow s$ cestách. Proèi¹tìná sí» nemá slepé ulièky, ani hrany vedoucí ze~stoku nìkam do~dal¹ího vrcholu.
-
-\figure{dinic-cistasit.eps}{Pøíklad proèi¹tìné sítì}{0.5\hsize}
-
-\s{Algoritmus (hledání maximálního toku v síti, Dinicùv)}
-
-\algo
-\:$f\leftarrow$ nulový tok.
-\:Sestrojíme sí» rezerv $R$, vynecháme hrany s nulovou rezervou.
-\:$l\leftarrow$ délka nejkrat¹í cesty $z\rightarrow s$ v~$R$.
-\:Kdy¾ $l=\infty$, tak skonèíme.
-\:Sestrojíme proèi¹tìnou sí» $C$, a to následujícím zpùsobem:%\foot{Ponecháme vrcholy a hrany z $R$, které le¾í na nejkrat¹ích $z\rightarrow s$ cestách}
-\::Spustíme BFS\foot{Breadth-First Search, standardní prohledávání do ¹íøky.} algoritmus ze zdroje.
-\::BFS nám rozdìlí uzly do vrstev, vyhodíme hrany za spotøebièem a slepé ulièky.
-\:$g\leftarrow$ blokující tok v $C$.
-\:Zlep¹íme tok $f$ podle $g$ a jdeme na bod 3.
-\endalgo
-
-\s{Postup tvorby proèi¹tìné sítì podrobnìji:} Prohledáním do~¹íøky vytvoøíme vrstvy $C_i$, zahodíme ty za~spotøebièem, ponecháme
-pouze hrany mezi $C_i$ a $C_{i+1}$. Je¹tì musíme odstranit slepé ulièky -- cesty, které konèí v~$C_m : m < l$, proto¾e ty urèitì nejsou souèástí nejkrat¹í $z\rightarrow s$ cesty.
-
-Proèi¹tìní zvládneme v~lineárním èase $\O(N+M)$, v~pøípadì souvislého grafu pouze $\O(M)$.
-
-
-\figure{dinic-neprocistenasit.eps}{Pøíklad neproèi¹tìné sítì}{0.5\hsize}
-
-Na obrázku neproèi¹tìné sítì vidíme, co se má smazat. Èerné hrany ponecháme, ty tam jsou správnì. Èervené hrany jsou zpìtné, ty sma¾eme. Modré hrany jsou hrany ve~stejné vrstvì, ty rovnì¾ sma¾eme. Èervené teèkované hrany nevedou vùbec do stoku, tak¾e ty rovnì¾ sma¾eme.
-
-\s{Definice:} {\I Fází} algoritmu oznaèíme jeden bìh cyklu -- kroky 3 a¾ 9.
-
-\>Provedeme podrobnou analýzu algoritmu z~hlediska slo¾itosti a uvidíme, ¾e má slo¾itost $\O(N^2M)$. Nejprve analyzujeme hledání
-blokujícího toku, pak se podíváme, kolik fází maximálnì mù¾e Dinicùv algoritmus mít.
-
-
-\s{Algoritmus hledání blokujícího toku}
-\algo
-\:$g\leftarrow$ nulový tok.
-\:Dokud $\exists z\rightarrow s$ cesta $P$ v proèi¹tìné síti $C$:
-\::$\varepsilon \leftarrow \min\limits_{e\in P} (c(e)-f(e)) $.
-\::$\forall e \in P :g(e)\leftarrow g(e)+\varepsilon$, pokud $g(e)$ vzroste na $r(e)$, tak sma¾eme hranu $e$.
-\::Doèistíme sí» tím, ¾e odstraníme slepé ulièky, které mohly vzniknout smazáním hrany $e$.
-\endalgo
-
-Pøi ka¾dém prùchodu se sma¾e v¾dy alespoò jedna hrana, tedy maximálnì $M$-krát provádíme $\O(N)$ -- právì tolik trvá nalezení cesty $P$, proto¾e délka cesty bude krat¹í nebo rovna $N$. Èi¹tìní pak maximálnì sma¾e celý graf, jedno mazání nás stojí konstantní èas, tedy celková slo¾itost tohoto algoritmu bude $\O(MN)$.
-
-Doká¾eme si, ¾e poèet fází je men¹í nebo roven $N$. Algoritmus se ukonèí, pokud $l>N$, proto¾e pak u¾ neexistuje nejkrat¹í $z\rightarrow s$ cesta, pro¹li jsme v¹echny vrcholy.
-
-\s{Lemma:} Pøi ka¾dé fázi vzroste $l$ alespoò o~jedna.
-
-\proof
-Uva¾me sí» $R$, rozdìlenou na~vrstvy, je¹tì pøed~proèi¹tìním. Po~proèi¹tìní nìkteré hrany zmizí. Pøibýt\foot{Pøibudou tak, ¾e po~hranì s~nulovým tokem po¹leme nìjaký tok, v~opaèném smìru v~síti rezerv vytvoøíme z~nulové hrany nenulovou.} mohou jen zrcadlové
-protìj¹ky ji¾ existujících hran.
-
-Uva¾me cestu $P$ délky $l$ nebo men¹í ze $z\rightarrow s$ a novou hranu $e$ vzniklou pøi~poslání toku po~hranì s~nulovým tokem:
-\numlist\nalpha
-\:Hrana $e \not\in P\Rightarrow$ zablokování, taková cesta neexistuje.
-\:Hrana $e \in P\Rightarrow$ délka $ > l$, proto¾e hrana $e$ vede z nìjakého vrcholu ve vrstvì $C_i$ do vrcholu ve vrstvì $C_{i-1}$.
-\qeditem
-\endlist
-
-\s{Vìta:} Dinicùv algoritmus najde maximální tok v~èase $\O(MN^2)$.
-
-\proof
-Slo¾itost plyne pøímo z~pøedchozího lemmatu a slo¾itosti algoritmu hledání blokujícího toku.
-
-Korektnost (najde v¾dy maximální tok) doká¾eme takto: Nech» algoritmus probìhne. Skonèit mù¾e jedinì tehdy, kdy¾ nenajde ¾ádnou cestu ze~zdroje do~spotøebièe v~síti rezerv, kterou by
-mohl být tok vylep¹en. Tedy tok musí být nutnì maximální.
-\qed
-
-
-Takto napsaný algoritmus je je¹tì pøíli¹ pomalý, ale jde výraznì zrychlit. Napøíklad namísto pøepoèítávání tokù na~rezervy a naopak mù¾eme
-minimálnì vnitøní cyklus poèítat jenom v~rezervách. Proèi¹tìní se dá dìlat jednou namísto dvakrát, mù¾eme prohledávat do~hloubky a mazat pøi vracení se z~vrcholù. Dokonce i~hledání blokujícího toku lze øe¹it v~rámci prohledávání do~hloubky.
-Cesta si pamatuje minimum rezerv a pøi vracení se rezerva sni¾uje.
-
-
-\>Problematika tokù v~sítích má velké uplatnìní v~kombinatorice a teorii grafù. Zde uvedeme jeden pøíklad:
-
-
-\>\s{Hledání maximálního párování v bipartitních grafech:} Zorientujeme v¹echny hrany zleva doprava a pøidáme zdroj, z~nìho¾
-vedou hrany do první partity, a stok, do~nìho¾ vedou hrany z~druhé partity. Hrany mezi partitami mají jednotkovou kapacitu.
-
-\bye
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-showpage
+++ /dev/null
-%version 1.8
-
-\input lecnotes.tex
-
-\prednaska{4}{Goldbergùv algoritmus}{(zapsali R. Tupec,
-J. Volec,
-J. Záloha)}
-
-\noindent
-Pøedstavíme si nový algoritmus pro~hledání maximálního toku v~síti, který se uká¾e stejnì dobrý jako {\I Dinicùv algoritmus} ($\O(MN^{2})$) a po~nìkolika vylep¹eních bude i lep¹í.
-
-\noindent
-Tento algoritmus narozdíl od~Dinicova algoritmu zaèíná s~pøebytky v~sousedních vrcholech zdroje a sna¾í se jich zbavit pomocí pøevádìní. Pokud bychom toto pøevádìní dìlali \uv{tupým zpùsobem}, mohl by se algoritmus zacyklit. Proto pro~ka¾dý vrchol budeme definovat vý¹ku, a jak uvidíme, s~její pomocí se vyhneme zacyklení.
-
-\s{Definice:} Funkce $f:E \rightarrow {\bb R}_{0}^{+}$
-je {\I vlna} v~síti~$(V, E, z, s, c)$ tehdy, kdy¾ $ \forall uv \in E : f(uv) \leq c(uv) $, kde $c(uv)$ je kapacita hrany~$uv$, a $ \forall v \ne z, s : f^{\Delta}(v) \geq 0 $. Funkcí $f^{\Delta}(v)$ pro libovolný vrchol~$v$ rozumíme {\I pøebytek} v~tomto vrcholu, co¾ je souèet v¹eho, co do~vrcholu~$v$ pøiteèe, minus souèet v¹eho, co z~$v$ odteèe. To mù¾eme zapsat jako:
- $$f^{\Delta}(v):=\sum_{uv \in E}{f(uv)} - \sum_{vu \in E}{f(vu)}.$$
-Ka¾dý tok je vlna, kde $\forall v \ne z,s: f^{\Delta}(v) = 0$.
-
-\noindent
-Algoritmus pou¾ívá sít rezerv, kterou jsme nadefinovali ji¾ v~pøedchozí kapitole vìnované Dinicovi.
-
-\noindent
-Dále budeme provádìt následující dvì operace na~vrcholech sítì. K~tomu budeme potøebovat pøiøadit v¹em vrcholùm vý¹ku pomocí funkce $h : V \rightarrow \bb{N}$.
-
-\s{Operace:} Pro~hranu~$uv \in E$ definujme {\I pøevedení pøebytku}:
-
-\noindent
-Pokud platí, ¾e:
-\numlist \ndotted
- \:ve~vrcholu~$u$ je nenulový pøebytek, tj. $f^{\Delta}(u) > 0$,
- \:vrchol~$u$ je vý¹ ne¾ vrchol~$v$, tj. $h(u) > h(v)$, a
- \:hrana $uv$ má nenulovou rezervu, tj. $r(uv)>0$,
-\endlist
-\noindent pøevedeme tok o~velikosti $\delta:=\min(f^{\Delta}(u),r(uv))$ z~$u$ do~$v$ tímto zpùsobem:
-\numlist \ndotted
- \:$f^{\Delta}(u) \leftarrow f^{\Delta}(u)-\delta$ a $f^{\Delta}(v) \leftarrow f^{\Delta}(v)+\delta$,
- \:$r(uv) \leftarrow r(uv)-\delta$ a $r(vu) \leftarrow r(vu)+\delta$.
-\endlist
-
-\s{Definice:} Øekneme, ¾e pøevedení je {\I nasycené}, pokud je po~pøevodu rezerva na~hranì $uv$ nulová, tedy $r(uv)=0$.
-Naopak pøevedení je {\I nenasycené}, pokud po~pøevodu $f^{\Delta}(u) = 0$. Pokud $r(uv)=0$ a $f^{\Delta}(u) = 0$,
-budeme pøevedení pova¾ovat za~{\I nasycené}.
-
-\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 (hledání maximálního toku v síti, Goldberg)}
-
-\algo
-\:$\forall v \in V: h(v)\leftarrow 0$ (v¹em vrcholùm nastavíme poèáteèní vý¹ku nula).
-\:$h(z)\leftarrow N$ (zdroj zvedneme do~vý¹ky $N$).
-\:$\forall e \in E: f(e)\leftarrow 0$ (po~hranách na~poèátku nenecháme protékat nic).
-\:$\forall zu \in E : f(zu)\leftarrow c(zu)$ (ze~zdroje pustíme maximální mo¾nou vlnu).
-\:Dokud $\exists u \in V \setminus \{z,s\}, f^{\Delta}(u)>0$:
-\::Pokud $\exists uv \in E, r(uv)>0$ a $h(u)>h(v)$: pøevedeme pøebytek po~hranì $uv$.
-\::V~opaèném pøípadì zvedneme $u$.
-\:Vrátíme tok $f$ jako výsledek.
-\endalgo
-
-\noindent
-Nyní bude následovat nìkolik lemmat a invariantù, jimi¾ doká¾eme správnost a èasovou slo¾itost vý¹e popsaného algoritmu.
-
-\s{Invariant A:} Funkce $f:E \rightarrow \bb{R}$ je v~ka¾dém kroku algoritmu vlna, $h(v)$ nikdy neklesá, $h(z)=N$ a $h(s)=0$.
-
-\proof
-Pro~první èást invariantu si staèí rozmyslet, ¾e v~¾ádném kroku algoritmu nepøekroèíme kapacity hran a nevytvoøíme záporný pøebytek. Pro~$v \in V \setminus \{z,s\}$ skuteènì vý¹ku pouze zvy¹ujeme a z~podmínky v~pátém kroku algoritmu vyplývá, ¾e nás pøebytky v~$z$ a $s$ v~podstatì nezajímají, tudí¾ se $h(z)$ a $h(s)$ nemìní.
-\qed
-
-\s{Invariant S (o~Spádu):} Neexistuje hrana se~spádem vìt¹ím ne¾ jedna a nenulovou rezervou, neboli $\forall uv \in E, r(uv)>0 : h(u) \leq h(v)+1$.
-
-\proof %todo pøeformulovat:BEGIN OK
-Podívejme se, kdy by mohla vzniknout nenasycená hrana se~spádem vìt¹ím ne¾ jedna. Bìhem inicializace k~tomu evidentnì nedojde, proto¾e v¹echny hrany jsou nenasycené nebo mají kapacitu nula, proto je mù¾eme vypustit. Bìhem práce algoritmu k~tomu v¹ak také nedojde, jak uvidíme z~rozebrání následujících pøípadù. Pokud ji¾ existuje vrchol~$v$ s~kladným pøebytkem, dále existuje nenasycená hrana $vu$ a $h(v)=h(u)+1$, vrchol~$v$ algoritmus nezvedne, ale pøebytek po¹le po~této hranì. Uva¾me tedy je¹tì druhý pøípad, kdy existuje nasycená hrana $uv$ se~spádem vìt¹ím ne¾ jedna a tuto hranu se pokusíme odsytit. Jen¾e pokud bychom chtìli nìco poslat v protismìru, sna¾ili bychom se o pøelití proti smìru funkce $h$.
-\qed %todo pøeformulovat:END
-
-\s{Lemma K (o~Korektnosti):} Kdy¾ se algoritmus zastaví, je $f$ maximální tok.
-
-\proof
-Nejprve uká¾eme, ¾e $f$ je tok, a pak jeho maximalitu. Vyjdìme z~toho, ¾e $f$ je vlna a algoritmus se mù¾e zastavit, jen pokud nastanou oba následující pøípady souèasnì:
-\itemize\ibull
-\:Ve~vrcholech grafu nejsou ¾ádné pøebytky (mimo $z$ a~$s$), proto¾e jinak by se algoritmus nezastavil a pokraèoval dále ve~výpoètu. Tudí¾ $f$ je tok.
-\:Neexistuje nenasycená cesta ze~zdroje do~stoku, èím¾ z~{\I Ford-Fulkersonovy vìty} okam¾itì vyplývá, ¾e $f$ je tok maximální. A jak tuto neexistenci nahlédneme? Pro~spor pøedpokládejme, ¾e nìjaká nenasycená cesta~$P$ ze~$z$ do~$s$ existuje. Tato cesta mù¾e mít maximálnì $N-1$ hran. O~nich víme, ¾e v¹echny mají kladnou rezervu, a dále víme, ¾e po~celou dobu výpoètu je vý¹ka zdroje $N$ a vý¹ka stoku $0$. Tak¾e celkový spád cesty $P$ je $N$, co¾ ale znamená, ¾e na cestì $P$ existuje hrana s~kladnou rezervou, která má spád alespoò $2$. To je v¹ak v~rozporu s~invariantem~S.
-\qeditem
-\endlist
-
-\s{Invariant C (Cesta domù, do~zdroje):} Je-li $v \in V \setminus \{z,s\}$ a $f^{\Delta}(v) > 0$, pak existuje nenasycená cesta z~$v$ do~$z$.
-
-\proof
-Mìjme nìjaký vrchol~$v \in V$ takový, ¾e $f^{\Delta}(v) > 0$.
-Potom definujme mno¾inu $A := \{ u \in V : \exists$ nenasycená cesta z~$v$ do~$u \}$.
-Mìjme vrcholy $a \in A$ a $b \in V \setminus A$ takové, ¾e $ba\in E$. O~nich víme, ¾e $f(ba)=0$, proto¾e pokud by tomu tak nebylo, muselo by platit $r(ab)>0$, a tudí¾ by $b$ patøilo do~mno¾iny $A$.
-
-Seètìme pøebytky ve~v¹ech vrcholech mno¾iny $A$. Proto¾e pøebytek ka¾dého vrcholu se spoèítá jako souèet tokù do~nìj vstupujících minus souèet tokù z~nìj vystupujících a v¹echny hrany, jejich¾ oba vrcholy le¾í v~$A$, se jednou pøiètou a jednou odeètou, platí:
- $$\sum_{u \in A}f^{\Delta}(u)=\sum_{\scriptstyle{ab \in E \cap {\bb A}} \atop \scriptstyle{{\bb A} = \bar{A}\times A}} f(a,b)-\sum_{{\scriptstyle ba \in E \cap {\bb A}} \atop {\scriptstyle {\bb A} = A\times \bar{A}}} f(b,a).$$
-My v¹ak víme, ¾e do~$A$ nic neteèe, a proto $\sum_{v \in A}{f^\Delta(v) \le 0}$. Zároveò v¹ak v~$A$ je vrchol s~kladným pøebytkem, toti¾ $v$, proto v~$A$ musí být také vrchol se záporným pøebytkem a jediný takový je $z$.
-\qed
-\s{Invariant V (Vý¹ka):} $\forall v \in V$ platí $h(v)\le 2N$.
-
-\proof
-Víme, ¾e poèet hran na~cestì ze~$z$ do~$\forall v \in V$ je maximálnì $N-1$. Pokud by existoval vrchol~$v$ s~vý¹kou $h(v)>2N$, museli jsme tento vrchol zvednout alespoò $(2N+1)$-krát. Snadno si uvìdomíme, ¾e $z$ nikdy nezvedáme, a tudí¾ by na cestì ze $z$ do $v$ musela být hrana se spádem vìt¹ím ne¾ jedna, co¾ je spor s~invariantem~S.
-\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
-
-\s{Lemma S (naSycená pøevedení):} Poèet v¹ech nasycených pøevedení je nejvý¹ $NM$.
-
-\proof
-Mìjme hranu~$uv \in E$, kterou jsme právì nasytili. Tedy platí $h(v)<h(u)$ a zároveò $r(uv)=0$. Aby se rezerva této hrany zmìnila, musel by ji nìkdo odsytit. Pro~odsycení hrany se musí otoèit nerovnost mezi~vý¹kami koncových vrcholù. Tedy $h(v)>h(u)$. Proto, abychom tuto hranu opìt nasytili, musíme opìt zmìnit nerovnost vý¹ek na~$h(v)<h(u)$. Mezi dvìma nasyceními hranami $uv$ proto probìhla minimálnì dvì zvednutí vrcholu~$u$. Algoritmus nikdy vý¹ku vrcholu nesni¾uje, a tedy poèet v¹ech nasycených pøevedení je skuteènì $NM$.
-\qed
-
-\s{Lemma N (Nenasycená pøevedení):} Poèet v¹ech nenasycených pøevedení je $\O(N^2M)$.
-
-\proof
-Dùkaz provedeme pomocí potenciálu -- nadefinujme si následující funkci jako potenciál:
- $$ \psi := \sum_{\scriptstyle{v: f^{\Delta}(v) > 0} \atop \scriptstyle{v \ne z,s}} h(v). $$
-Nyní se podívejme, jak se ná¹ potenciál bìhem algoritmu vyvíjí a jaké má vlastnosti:
-\itemize\ibull
-\:Bìhem celého algoritmu je $ \psi \ge 0 $, nebo» je souètem nezáporných èlenù.
-\:Na poèátku je $ \psi = 0 $.
-\:Zvednutí vrcholu zvý¹í $\psi$ o~jednièku. 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 $\psi$ nejvý¹e o~$2N$, proto¾e buï po~pøevodu hranou $uv$ v~$u$ zùstal nìjaký pøebytek, tak¾e se mohl potenciál zvý¹it a¾ o~$2N$, 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~$u$ 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 $\psi$ 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 $v \ne z,s$ takových, ¾e $f^{\Delta}(v) > 0$. Kdy¾ mìníme pøebytek nìjakého vrcholu, mù¾eme ná¹ seznam v~konstantním èase aktualizovat (napø. tak, ¾e si ka¾dý vrchol pamatuje pozici, na~které v~seznamu je). A v~konstantním èase také umíme odpovìdìt, zda existuje nìjaký vrchol s~pøebytkem. Dále si $\forall u \in V$ budeme pamatovat $L(u) := $ seznam $uv \in E$ takových, ¾e $r(uv) > 0$ a $h(v) < h(u)$. Díky tomu mù¾eme pøistupovat k~patøièným sousedùm $u$ v~èase $\O(1)$, stejnì jako provádìt operace pøidání do~$L(u)$, resp. smazání v~nìm. Ka¾dé pøevedení po~hranì $uv$ nás stojí konstantní èas na~aktualizaci rezerv hran $uv$ a $vu$, stejnì tak i na aktualizaci pøebytkù ve~vrcholech $u$ a $v$. V~pøípadì, ¾e se jedná o~nasycené pøevedení, musíme je¹tì odstranit hranu~$uv$ z~$L(u)$, co¾ také stihneme v~èase $\O(1)$. A koneènì zvedání vrcholu~$v$ nám zabere èas $\O(N)$, proto¾e musíme obejít v¹echny hrany~$uv$, kterých je $\O(N)$, porovnat vý¹ky a pøípadnì odebrat $uv$ z~seznamu $L(u)$ resp. pøidat do $L(v)$. Abychom pro odebrání hrany~$uv$ ze~seznamu $L(u)$ nemuseli procházet celý seznam, budeme si $\forall v \in V$ pamatovat je¹tì $L^{-1}(v) := $ seznam ukazatelù na~hrany~$uv$ v~seznamech $L(u)$.
-
-\s{Vìta:} Goldbergùv algoritmus najde maximální tok v~èase $\O(N^2M)$.
-
-\proof
-Z~lemmatu~Z vyplývá, ¾e celkový poèet zvednutí je maximálnì $2N^2$, pøièem¾ ka¾dé zvednutí jsme schopni provést v~èase $\O(N)$. Tak¾e dohromady pro~zvedání spotøebujeme èas $\O(N^3)$, co¾ je pro souvislé sítì urèitì $\O(N^2M)$. Z~lemmatu~S pro~zmìnu vyplývá, ¾e nasycená pøevedení nás stojí $\O(NM)$, a na~závìr z~lemmatu~N dostáváme èasovou slo¾itost $\O(N^2M)$ pro~pøevedení nenasycená. Proto celková slo¾itost algoritmu je $\O(N^2M)$.
-\qed %todo ? pro zmìnu vs. prozmenu ?
-
-Dokázali jsme, ¾e algoritmus má èasovou slo¾itost $\O(N^2M)$ pro libovolnou posloupnost zvedání a pøevádìní. Nabízí se otázka, zda není mo¾né vhodným výbìrem tìchto operací výpoèet zrychlit. Uká¾eme, ¾e pokud v~$5.$ kroku algoritmu budeme v¾dy brát vrchol~$u$ takový, ¾e $h(u)$ je maximální, poèet nenasycených pøevedení se sní¾í.
-
-\s{Lemma N':} Poèet nenasycených pøevedení v~upravené verzi Goldbergova algoritmu je $\O(N^2\sqrt{M})$, co¾ je maximálnì $\O(N^3)$. Díky tomu je i slo¾itost celého algoritmu $\O(N^3)$.
-
-\proof
-Viz pøí¹tí pøedná¹ku.
-\bye
+++ /dev/null
-P=4-goldberg
-
-include ../Makerules
+++ /dev/null
-\input lecnotes.tex
-\input epsf
-\def\itm{\item{$\bullet$}}
-
-\prednaska{5}{Tøídicí sítì}{(zapsaly T.~Klimo¹ová
-a~K.~B\"ohmová)}
-
-\h{Goldbergùv algoritmus -- pokraèování}
-
-Algoritmus upøesníme tak, ¾e místo libovolného vrcholu s~pøebytkem
-budeme v¾dy pracovat s~takovým vrcholem~$v$, jeho¾ vý¹ka~$h(v)$ je
-nejvìt¹í. Jak doká¾eme v~následujícím lemmatu, sní¾í se tím poèet
-potøebných nenasycených pøevedení. Takto modifikovaný algoritmus
-budeme znaèit~$G'$.
-
-\s{Lemma $N'$:} V~algoritmu $G'$ je poèet nenasycených pøevedení
-$\O(N^3)$.
-
-\proof
-Definujme $H$~jako maximální vý¹ku vrcholù s~pøebytkem:
- $$H:=\max\{h(v) : v \not= z,s, f^\triangle (v) > 0\}.$$
-Bìh algoritmu $G'$ rozdìlíme na fáze tak, ¾e fáze skonèí po ka¾dé zmìnì
-$H$. Odhadneme poèet nenasycených pøevedení v~jedné fázi: bude jich
-nanejvý¹ stejnì jako vrcholù, které se na zaèátku fáze nacházely na
-nejvy¹¹í hladinì -- z~jiných vrcholù v~prùbìhu fáze nic nepøevádíme
-a~nenasycené pøevedení mù¾eme provést z~ka¾dého vrcholu nejvý¹e
-jednou. Poèet nenasycených pøevedení za fázi je proto nejvý¹e~$N$.
-
-Odhadneme poèet fází: rozdìlíme fáze podle toho, jestli konèí sní¾ením
-nebo zvý¹ením~$H$ a~odhadneme jejich poèty. Maximálnì $2N^2$ fází mù¾e
-konèit zvý¹ením~$H$, proto¾e dle lemmatu Z~provedeme nanejvý¹ $2N^2$
-zvednutí. Sní¾ením~$H$ mù¾e konèit nanejvý¹ $2N^2$ fází, proto¾e~$H$
-klesne v¾dy alespoò o~jedna, pùvodnì bylo nula a~nikdy nemù¾e být záporné --
-klesnout tedy nemù¾e víckrát ne¾ vzrùst.
-
-Máme maximálnì $4N^2$ fází o~nejvý¹e~$N$ nenasycených pøevedení,
-celkem tedy algoritmus $G'$ provede $\O(N^3)$ nenasycených pøevedení.
-\qed
-
-\medskip
-\>Ve skuteènosti je algoritmus je¹tì lep¹í (alespoò pro øídké
-grafy):
-
-\s{Lemma $N''$:} V~algoritmu $G'$ je poèet nenasycených pøevedení $\O(N^2
-\sqrt M)$.
-
-\proof
-(Nezkou¹í se.) Algoritmus rozdìlíme na fáze stejnì jako v~dùkazu
-pøedchozího lemmatu a~zvolíme pøirozené èíslo~$K$ (jak velké ho
-zvolíme, se rozhodneme a¾ na konci dùkazu).
-Fáze tentokrát budeme dìlit na drahé, v~nich¾ se provede více ne¾~$K$
-nenasycených pøevedení, a~laciné, v~nich¾ se nenasycených pøevedení
-provede maximálnì~$K$.
-
-Nyní budeme zkoumat poèty pøevedení v~obou typech fází.
-Jak víme z~minulého lemmatu, v¹ech fází je nanejvý¹ $4N^2$, tak¾e
-v~laciných se provede nanejvý¹ $4N^2K$ nenasycených pøevedení.
-Za úèelem zkoumání drahých fází definujeme potenciál:
-$$\psi:=\sum_{v\not=z,s; f^\triangle(x)>0}{ p(v)\over K},$$ kde
-$p(v):= \vert\{u : h(u) \leq h(v)\}\vert$, èili poèet vrcholù ve stejné
-nebo men¹í vý¹ce ne¾~$v$. Víme, ¾e na zaèátku bude $\psi \leq
-{N^2/K}$ a~po celou dobu bude $\psi \geq 0$.
-
-Zvednutím vrcholu~$v$ se hodnota $p(v)$ zvý¹í maximálnì o~$N$, u~libovolného
-jiného vrcholu~$w$ jeho $p(w)$ klesne nebo se nezmìní, potenciál tedy
-vzroste maximálnì o~$N/K$.
-Pøi sytém pøevedení po hranì z~$u$ do~$v$ (z~minula víme, ¾e se v¾dy
-provádí po hranì spádu nejvý¹e jedna) se mohl potenciál zmen¹it o~$p(u)/K$
-a~mohl se zvìt¹it o~$p(v)/K$. Zvìt¹í se tedy nanejvý¹ o~$N/K$
-Pøi nenasyceném pøevedení z~potenciálu urèitì ubude $p(u)/K$
-a~mo¾ná pøibude $p(v)/K$. Celkovì se tedy sní¾í nanejvý¹
-o~${p(u)/K} - {p(v)/K}$, co¾ je $1/K$~poètu prvkù na nejvy¹¹í
-hladinì. Z~minulého lemmatu víme, ¾e poèet nenasycených pøevedení
-v~jedné fázi je men¹í nebo roven poètu vrcholù na nejvy¹¹í hladinì na
-zaèátku fáze. Vzhledem k~tomu, ¾e zkoumáme drahé fáze, v~nich¾ probìhne
-více ne¾~$K$ nenasycených pøevedení, vrcholù na nejvy¹¹í hladinì musí
-být na zaèátku fáze také více ne¾~$K$, potenciál se tedy sní¾í o~více
-ne¾ ${K/K}=1$.
-
-Potenciál $\psi$ tedy pøijme na zaèátku nanejvý¹ ${N^2/K}$, pøi
-zvedání nevzroste o~víc ne¾ $(N/K)2N^2$, pøi sytém pøevádìní
-nevzroste o~víc ne¾ $(N/K)NM$, tak¾e za celý prùbìh algoritmu potenciál vzroste maximálnì o
-$(N/2)(N+2N^2+NM)=\O({N^2}M/2)$, v~drahých fázích tak mù¾e probìhnout
-nanejvý¹ $\O({N^2}M/2)$ nenasycených pøevedení.
-Celkem algoritmus vykoná $\O(N^2K+{N^2}M/K)$ nenasycených
-pøevedení. Kdy¾ za~$K$ dosadíme $\sqrt M$, dostaneme slíbený poèet
-$\O(N^2\sqrt M)$.
-\qed
-
-\s{Implementace:}
-Narozdíl od pùvodní verze algoritmu si ve verzi se zvedáním nejvy¹¹ího
-vrcholu nebudeme pamatovat seznam vrcholù s~kladným pøebytkem, ale
-setøídìný seznam pøihrádek. V~ka¾dé pøihrádce budou jen vrcholy
-s~pøebytkem s~urèitou vý¹kou. Vyhledání nejvy¹¹ího vrcholu tedy
-zvládneme v konstantním èase, stejnì pro zvý¹ení vrcholu nám staèí
-$\O(1)$ (buï vrchol pøesuneme do vedlej¹í pøihrádky, nebo pro nìj
-zalo¾íme novou). Pøevádíme-li pøebytek do vrcholu, kde pøedtím nebyl,
-pak musí mít vý¹ku o~$1$ ni¾¹í, ne¾ vrchol, ze kterého pøebytek
-pøevádíme (jinak by existovala nenasycená hrana se spádem dva, co¾
-nejde). Najít (pøípadnì vytvoøit) pøihrádku novì vzniklému vrcholu
-s~pøebytkem tak také stihneme v~konstantním èase.
-Pro zvednutí nám tedy stále staèí èas $\O(N)$ a libovolné
-pøevedení pøebytku zvládneme v~$\O(1)$.
-
-\medskip
-\h{Tøídìní}
-
-\s{Definice:} {\I Komparátorová sí»} je kombinaèní obvod, jeho¾ hradla jsou
-komparátory:
-
-\centerline{\epsfbox{sortnet.0}} Komparátor dostane na
-vstupu dvì èísla, porovná je a~na levý výstup vrátí men¹í z~nich, na
-pravý výstup naopak vrací vìt¹í èíslo ze zadané dvojice.
-
-\>Výstupy komparátorù se nevìtví.
-
-\medskip
-\s{Pøíklad:} {\sl Bubble sort}
-
-Obrázek Bubble.1 ilustruje pou¾ití komparátorù pro tøídìní bubble
-sortem. ©ipky pøedstavují jednotlivé komparátory.
-
-\twofigures{sortnet.1}{Bubble.1}{143pt}{sortnet.2}{Bubble.2}{143pt}
-
-Sna¾íme se výpoèet co nejvíce paralelizovat (viz obrázek Bubble.2).
-Takto se nám podaøilo výpoèet provést pomocí $\O(n^2)$ komparátorù
-rozmístìných na $\O(n)$ hladinách. Tøídíme v~èase~$n$ a~prostoru
-$n^2$.
-
-%\medskip
-%\s{Pøíklad:} {\>\sl Merge sort}
-%\centerline{\epsfbox{sortnet.4}}
-
-\medskip
-\s{Definice:} Øekneme, ¾e posloupnost $x_0,\dots,x_{n-1} $ je {\I èistì bitonická},
-pokud pro nìjaké $x_j\in\{1, \dots, n-1\} $ platí:
-$$x_0\leq x_1\leq \dots \leq x_j \geq x_{j+1}\geq\dots \geq x_{n-1}.$$
-Posloupnost je {\I bitonická}, pokud existuje $k\in \{1,\dots ,n-1\}$, pro
-které je rotace pùvodní posloupnosti o $k$ prvkù, tedy posloupnost
-$x_k,x_{(k+1) \bmod n},\dots, x_{(k+n-1) \bmod n}$, 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 bude~$i$-tým,
-maximum $(i+{n/2})$-ním prvkem výstupu.
-\figure{sortnet.3}
-{$(y_i, y_{i+{n/2}}) = CMP(x_i, x_{i+{n/2}})$} {300pt}
-
-\s{Lemma:} Pokud vstup $S_n$ obvodu je bitonická posloupnost, pak výstup
-$y_0,\dots, y_{n-1}$ je posloupnost, která 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 $y_i < y_{j + {n/2}}$.
-
-\proof
-(i) Nejprve nahlédneme, ¾e lemma platí, je-li vstupem èistì bitonická
-posloupnost. Tehdy najdeme nejmen¹í~$k$ takové, ¾e $x_k$ a~$x_{k+{n/2}}$
-se prohodí. (Pokud takové~$k$ neexistuje, separátor pouze zkopíruje
-vstup na výstup a~obì tvrzení lemmatu zøejmì platí.)
-Øeknìme, ¾e $x_m$ je maximum
-vstupní posloupnosti. 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.
-Uvìdomíme si, ¾e 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. Úsek $k+{n/2}$ a~$n-1$ nahradíme èistì
-bitonickou posloupností, která bude neklesající, je-li $m\geq {n/2}$,
-v~opaèném pøípadì je úsek ${n/2}-1$ a¾ $k+{n/2}$
-nerostoucí. V~obou pøípadech budou v~posloupnosti maximálnì dva zlomy
-a~mezi $x_{n/2}$ a~$x_{n-1}$ bude správná nerovnost na to, aby
-posloupnost byla bitonická.
-
-Dostaneme-li na vstupu obecnou bitonickou posloupnost, pøedstavíme si,
-¾e je to èistì bitonická posloupnost zrotovaná o~$r$ prvkù (BÚNO
-doprava), a~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 $y_0\dots y_{n/2-1}$ 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}}
-
-\medskip
-\s{Definice:} {\I Bitonická tøídièka $B_n$} je obvod sestavený ze separátorù, který dostane-li na vstupu bitonickou posloupnost délky $n$ (konstruujeme tøídièku pro $n=2^k$), vydá setøídìnou posloupnost délky $n$.
-
-\centerline{\epsfbox{sortnet.5}}
-
-Separátor má jednu hladinu s~${\O}(n)$ hradly, tøídièka tedy bude
-mít
-$\log n$ hladin s~${\O}(n\log n)$ hradly.
-
-\s{Pøíklad:} {\sl Merge sort}
-
-Bitonická tøídièka se dá pou¾ít ke slévání setøídìných posloupností.
-S~její pomocí sestavíme souèástky mergesortové sítì.
-Setøídìné posloupnosti
-$x_0,\dots, x_{n-1}$ a~$y_0,\dots, y_{n-1}$ spojíme do jedné bitonické
-$x_0,\dots, x_{n-1},y_{n-1},\dots, y_0$. Z~takové posloupnosti pomocí
-$B_{2n}$ vytvoøíme setøídìnou posloupnost.
-Blok $M_{2n}$ sestává z~bloku $B_{2n}$, jeho¾ druhá polovina vstupù je
-zapojena v~obráceném poøadí.
-
-\medskip
-\centerline{\epsfbox{sortnet.6}}
-
-Z~bitonických tøídièek tedy mù¾eme postavit mergesortovou sí», která
-bude mít
-${\O}(\log^2 n)$ hladin a~${\O}(n\log^2 n)$ hradel.
-
-Existuje tøídicí algoritmus, kterému staèí ${\O}(\log n)$ hladin,
-ale jeho multiplikativní konstanta je pøíli¹ veliká, tak¾e je v~praxi
-nepou¾itelný.
-
-\bye
+++ /dev/null
-P=5-sortnet
-
-include ../Makerules
+++ /dev/null
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-
-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;
+++ /dev/null
-\input lecnotes.tex
-
-\prednaska{6}{Vyhledávání v textu}{(zapsal K. Ka¹èák, M. Klauèo, M. Vachna)}
-
-\s{Úkol:} V~textu s~délkou $S$ najít v¹echny výskyty hledaného slova s~délkou $J$.
-
-\h{Hloupý algoritmus}
-
-Algoritmus prochází sekvenènì textem a hledaným vzorovým slovem. Pøi neshodì se ve vzorovem slovì vrací na zaèátek a v~textu pokraèuje znakem, ve~kterém nastala neshoda. Èasová slo¾itost je $\O(S)$. Tento algoritmus funguje pouze pro vzorová slova, ve kterých se neopakuje první znak.
-
-\s{Pøíklad:} Hledání vzorového slova |jehla| v~textu |vkupcejejehla|. Ve chvíli kdy máme prefix |je| a na vstupu dostaneme |j|, dochází k~neshodì a pokraèujeme v~hledání od tohoto znaku. Pro tenhle pøípad algoritmus najde vzorové slovo. To ale ji¾ neplatí
-pro vzorové slovo |kokos| v~textu |clanekokokosu|. Ve chvíli kdy máme prefix |koko| a na vstupu dostaneme |k|, dochází k~neshodì a pokraèujeme v~hledání od tohoto znaku, tím ale zahodíme potøebnou èást a algoritmus sel¾e.
-
-\h{Neefektivní algoritmus}
-
-Algoritmus prochází text od zaèátku a¾ do konce a pro ka¾dou pozici v~textu zkontroluje, zda na této pozici nezaèíná hledané slovo. Tak pro ka¾dou pozici provede a¾ $S$ porovnání znakù, èili celkem a¾ $SJ$ porovnání. Proto je èasová slo¾itost $\O(SJ)$.
-
-\h{Chytrý algoritmus}
-
-Algoritmus je vylep¹ením Neefektivního algoritmu, konkrétnì zpùsobu, jakým sa vrací v textu pøi neshodì mezi znakem textu a
-znakem vzorového slova.
-
-\s{Pøíklad:} Pro vzorové slovo |ajaajak| jsme na¹li v~textu prefix |ajaaja|. Oèekávame |k|.
-\itemize\ibull
-\:Kdy¾ ale dostaneme |a| a budeme mít prefix |ajaajaa|, vracíme se v~textu za první |aja|, tedy prefix zkrátíme na |ajaa| a pokraèujeme v~hledání.
-\:Kdy¾ je nasledující znak |j| a budeme mít prefix |ajaajaj|, vracíme se v~textu za |ajaaj|, tedy prefix zkrátíme na |aj| a pokraèujeme v~hledání.
-\:V~pøípadì, ¾e dostaneme jiný znak, se v~textu nevracíme a pokraèujeme dal¹ím znakem textu.
-\endlist
-
-\s{Definice a znaèení pro øetìzce (slova):}
-
-\s{Definice:}
-\itemize\ibull
-\:{\I Abeceda $\Sigma$} je koneèná mno¾ina znakù, ze~kterých tvoøíme text, øetìzce, slova jako koneèné posloupnosti znakù z $\Sigma$. Pøíkladem extrémních abeced je binární abeceda slo¾ená z~nul a jednièek. Pøíklad z~druhého konce je abeceda, která má jako znaky slova èeského jazyka. V algoritmech nebudeme uva¾ovat velikost abecedy (poèet znakù), budeme pøedpokládat, ¾e je to konstanta.
-\:{\I $\Sigma^*$} je mno¾ina v¹ech slov nad abecedou $\Sigma$.
-\endlist
-\s{Znaèení:}
-\itemize\ibull
-\:{\I Slova} budeme znaèit malými písmeny øecké abecedy $\alpha$,$\beta$... a {\I znaky} malými písmeny latinky $a$,$b$... .
-\:{\I Prázdné slovo} znaèíme písmenem $\varepsilon$.
-\:{\I Délka slova} $\vert \alpha \vert$ pro $\alpha \in \Sigma^*$ je poèet jeho znakù.
-\:{\I Zøetìzení} $\alpha\beta$ vznikne zapsáním slov $\alpha$ a $\beta$ za sebe. Platí $\alpha\varepsilon=\varepsilon\alpha=\alpha$, $\vert \alpha\beta \vert=\vert \alpha \vert+\vert \beta \vert$.
-\:$\alpha[i]$ je $i$-té písmeno slova $\alpha$, indexuje se od $0$.
-\:$\alpha[i:j]$ je podslovo tvoøené písmeny $\alpha[i]$,...,$\alpha[j-1]$. Pøíklady: $\alpha[i:i+1]=\alpha[i]$, $\alpha[i:i]=\varepsilon$. Vynechaním první meze získame prefix ($\alpha[:j]$), druhé meze suffix ($\alpha[i:]$), obou mezí dostaneme celé slovo ($\alpha[:]$=$\alpha$).
-\:$\alpha[:j]$ je {\I prefix} obsahující prvních $j$ znakù slova $\alpha$.
-\:$\alpha[i:]$ je {\I suffix} obsahující znaky slova $\alpha$ poèínaje $i$-tým znakem.
-\:Ka¾dé slovo je prefixem i suffixem sebe sama, takovému prefixu/suffixu øíkáme {\I nevlastní}. V¹em ostatním {\I vlastní}.
-\:Prázdné slovo je podslovem, prefixem i suffixem ka¾dého slova vèetnì prázdného slova.
-\endlist
-
-
-\s{Problém:}
-
-Vstupem je $\iota$ hledané slovo (jehla) délky $J=\vert \iota \vert$ a $\sigma$ text (seno) délky $S=\vert \sigma \vert$.
-
-Výstupem jsou v¹echny vyskyty hledaného slova $\iota$ v textu $\sigma$: $\left\{ i\vert \sigma[i:i+J]=\iota \right\}$
-
-\h{Vyhledávací automat (Knuth, Morris, Pratt)}
-Vyhledávací automat bude graf, jeho¾ vrcholùm øíkame stavy automatu. Jména stavù budou v¹echny prefixy slova $\iota$. 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 $d(\alpha , X)$), tedy ka¾dá taková hrana bude oznaèena písmenem $X$ a bude popisovat dané zvìt¹ení délky jména stavu, tedy $\alpha \rightarrow \alpha X$. Zpìtné hrany grafu budú popisovat pøechod (zpìtná funkce $z(\alpha)$) mezi stavem $\alpha$ a nejdel¹ím vlastním suffixem $\alpha$, který je prefixem $\iota$, kdy¾ nastane neshoda.
-
-\figure{vautomat.eps}{Vyhledávací automat}{5.5in}
-
-\s{Vyhledávání:}
-\algo
-\:$\alpha \leftarrow \varepsilon$.
-\:Pro $c\in\Sigma$ postupnì:
-\:$\indent$Dokud $\neg \exists d(\alpha , c) \wedge \alpha\neq\varepsilon : \alpha \leftarrow z(\alpha)$.
-\:$\indent$Kdy¾ $\exists d(\alpha , c)\Rightarrow \alpha \leftarrow d(\alpha , c)$.
-\:$\indent$Kdy¾ $\alpha = \iota \Rightarrow$ hledané slovo je v~textu.
-\endalgo
-
-\s{Alternatíva:} Automat mù¾e být reprezentovaný i polem. Pøi této reprezentaci odpadá starost o dopøední hrany (staèí zvìt¹it hodnotu, kterou v poli indexujeme). Hodnota na dané pozici v poli urèuje kam smìruje zpìtná hrana (index v poli).
-
-\s{Alternatívní vyhledávání:}
-\algo
-\:$k \leftarrow 0$.
-\:pro $c\in\Sigma$ postupnì:
-\:$\indent$Dokud $c\neq \iota[k] \wedge k>0: k \leftarrow z[k]$
-\:$\indent$Je-li $c=\iota[k] \Rightarrow k \leftarrow k+1$
-\:$\indent$Kdy¾ $k = J \Rightarrow$ hledané slovo je v~textu
-\endalgo
-
-\s{Invariant:} Nejdel¹í suffix $\beta$, který je prefixem $\iota$ $=$ $\alpha(\beta)$. Kde $\beta$ je pøeètení vstup.
-Z~invariantu vyplýva korektnost vyhledávací èásti algoritmu KMP.
-
-\proof
-Indukcí podle $\vert \beta \vert$. Na zaèátku pro prázdný naètený vstup platí invariant, tedy prázdny suffix $\beta$ je prefixem $\iota$. V~kroku $n$ máme naètený vstup $\beta$ a k~nìmu naèteme znak $c$. Jestli si odmyslíme $c$, tedy kdy¾ si od jména stavu odmyslíme poslední písmenko, dostaneme znovu jméno stavu. Tak stav, který pasuje na konec vstupu bez toho $c$ je stav, který pasuje na konec pùvodního vstupu, toho o~jeden znak krat¹ího. Tím pádem to musí být nìco, co je maximálnì tak dlouhé jako pùvodní stav, u~kterého jsme byli, proto¾e to byl nejdel¹í, který pasoval. Staèí procházet postupnì v¹echny stavy, které pasují na konec toho vstupu od nejdel¹ího k~nejkrat¹ímu a vzít první, který se dá roz¹íøit o $c$. To je pøesnì to, co algoritmus dìlá. Proto¾e zpìtná funkce øekne nejbli¾¹í krat¹í jméno stavu. Tak¾e algoritmus iteruje pøes stavy, které tam pasují, a¾ najde jeden, který se dá roz¹íøit o~$c$ a jeliko¾ iteroval od toho nejdel¹ího, tak to je logicky ten nejdel¹í, který tam pasuje.
-\qed
-
-\s{Lemma:} Vyhledávání dobìhne v~èase $\O(S)$.
-
-\proof
-Pro ka¾dý znak vstupního textu mohou nastat dva pøípady. Znak roz¹iruje aktuální prefix, nebo musíme pou¾ít zpìtnou funkci (zpìtnou hranu). Roz¹irování trvá konstantnì mnoho èasu, zatímco zpìtná funkce mu¾e být pro jeden znak volána a¾ $J$-krát. Pøi ka¾dém volání klesne délka aktuálního stavu minimálnì o~jedna a zároveò platí, ¾e kdykoliv stav prodlu¾ujeme, roste právì o~jeden znak. Proto v¹ech zkrácení dohromady mu¾e být nejvý¹e tolik, kolik bylo v¹ech prodlou¾ení, t.j. kolik jsme pøeèetli znaku textu. Celkem je tedy poèet krokù lineární vzhledem k~délce textu.
-\qed
-
-\s{Konstrukce zpìtné funkce:}
-\algo
-\:Sestrojíme dopøedné hrany
-\:$z( \varepsilon ) \leftarrow 0$, $z( \iota [0]) \leftarrow \varepsilon $
-\:$\indent$ $\alpha \leftarrow \varepsilon$
-\:pro $i = 1$ do $J$
-\:$\indent$$\alpha \leftarrow krok( \alpha , \iota [i])$
-\:$\indent$$z( \iota [0:i+1]) \leftarrow \alpha$
-\endalgo
-
-\s{Vysvìtlení:} V¹imnìte si, ¾e $z(i)$ je pøesnì stav, do nej¾ se dostaneme pøi spu¹tìní na¹eho vyhledávacího algoritmu na øetìzec $\iota [2:i]$, èili na $i$-tý prefix bez prvního písmenka. Proè to tak je? Zpìtná funkce øíká, jaký je nejdel¹í vlastní suffix daného stavu, který je také stavem, zatímco $\alpha$ oznaèuje nejdel¹í suffix textu, který je stavem. Tyto dvì vìci se pøeci li¹í jen v~tom, ¾e ta druhá pøipou¹tí i nevlastní suffixy, a právì tomu zabráníme odstranìním prvního znaku. Tak¾e $z()$ získáme tak, ¾e spustíme vyhledávání na èást samotného slova $\iota$. Jen¾e k~vyhledávání zase potøebujeme zpìtnou funkci $z$. Proto budeme zpìtnou funkci vytváøet postupne od nejkrat¹ích prefixù. Zøejmì $z(1) = \varepsilon$. Pokud ji¾ máme $z(i)$, pak výpoèet $z(i+1)$ odpovídá spu¹tení automatu na slovo délky $i$ a pritom budeme zpìtnou funkci potøebovat jen pro stavy délky $i$ nebo men¹í, pro které ji ji¾ máme hotovou.
-
-Navíc nemusíme pro jednotlivé prefixy spou¹tìt výpoèet v¾dy znovu od zaèátku, proto¾e $(i+1)$-ní prefix
-je prodlou¾ením $i$-tého prefixu o~jeden znak. Staèí tedy spustit algoritmus na celý øetìzec $\iota$ a sledovat, jakými stavy bude procházet. To budou pøesnì hodnoty zpìtné funkce. Vytvoøení zpìtné funkce se tak nakonec zredukovalo na jediné vyhledávání v~textu o~délce $J-1$, a proto pobì¾í v èase $\O(J)$. Èasová slo¾itost celého algoritmu tedy bude $\O(S+J)$.
-
-\h{Algoritmus Rabin \& Karp}
-Tento algoritmus funguje tak, ¾e porovnává hash hledaného øetìzce s~hashem aktuálního podøetìzce (\uv{posuvné okénko} stejné délky jako hledaný øetìzec) v~textu a aktuální podøetìzec porovná se vzorkem pouze v~pøípadì, kdy¾ mají shodný hash. Kdy¾ si zvolíme tu správnou hashovací funkci, budeme moci vypoèítat hash následujíciho podøetìzce na základe hashe toho aktuálního. Jako hashovací funkci $h: \Sigma^J \rightarrow \bb Z$ pou¾ijeme následující: $h(x_{0},...,x_{J-1}) = ( \sum_{i=0}^{J-1} x_{i}.p^{J-1-i}) \bmod N$, kde $N$ je velikost prostoru, do kterého hashujeme. Jak zjistíme hash následujícího podøetìzce?
-\itemize\ibull
-\:$h = x_{0}.p^{J} + x_{1}.p^{J-1} + ... + x_{J-1}.p^{1}$
-\:$h^{'} = x_{1}.p^{J} + x_{2}.p^{J-1} + ... + x_{J}.p^{1}$
-\:$h^{'} = (h - x_{0}.p^{J}).p + x_{J}.p^{1}$
-\endlist
-Tady mù¾eme vidìt, ¾e hash následujícího øetìzce lze pøepoèítat na základì toho pøedchozího v konstantním èase.
-Èasová slo¾itost je v nejlep¹ím pøípadì lineární vzhledem k~délce textu, zatímco nejhor¹í pøípad mú¾e trvat a¾ $\Theta(JS)$.
-
-\bye
+++ /dev/null
-P=6-kmp
-
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+++ /dev/null
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+++ /dev/null
-\input ../lecnotes.tex
-
-\prednaska{7}{Vyhledávání v textu}{(zapsali J. Kunèar, M. Demin a J. Chludil)}
-
-Na minulých predná¹kách jsme si ukázali, jak se v~textu (senì) vyhledává slovo (jehlu). Teï si ov¹em úlohu zobecníme a uká¾eme si, jak v kupce sena hledat souèasnì více ne¾ jednu jehlu.
-
-\h{Hledání výskytu v¹ech slov}
-
-\itemize\ibull
-\:$\iota_1, \ldots, \iota_k$ -- vyhledávaná slova (jehly) délek $ J_i= \vert\iota_i\vert $
-\:$\sigma$ -- text (seno) délky $ S= \vert\sigma\vert $
-\endlist
-Nejprve si øekneme, jak chceme, aby vypadal výstup. Výstupem pro nás budou v¹echny uspoøádané dvojice $(i,j)$ ($i$ je index jehly, kterou jsme nalezli, a $j$ je poèáteèní pozice v senì, kde se jehla nachází) takové, ¾e $$\iota_i=\sigma[j:j+J_i].$$ Postavme si proto vyhledávací automat podobný tomu, který jsme vidìli na minulé pøedná¹ce. Tento automat nám v¹echny takové uspoøádané dvojice najde.
-
-
-\h{Vyhledávací automat}
-Vyhledávací automat je strom\foot{http://en.wikipedia.org/wiki/Trie}, kde ka¾dý vrchol mù¾e mít stupeò a¾ do velikosti abecedy a kde jednotlivé hrany odpovídají písmenùm této abecedy. Vrcholy, ve kterých konèí slovo, jsou oznaèené (na obrazcích èernì). Dále si èasem do tohoto vyhledávacího stromu pøidáme zpìtné hrany a \uv{zkratky}.
-
-\s{\I{Stavy}} automatu jsou urèeny vrcholy stromu, pro které platí rovnì¾ stejný {\I{invariant}} z pøedchozí pøedná¹ky.\par
-\s{\I{Zpìtná hrana}} $z$($\alpha$) := nejdel¹í vlastní suffix\foot{definováno na minulé pøedná¹ce} slova $\alpha$, který je stavem.\par
-
-\figure{vyhl_automat_dopr.eps}{Vyhledávací automat}{1in}
-
-\h{Výstup z automatu}
-Pøi vypisování výsledkù mu¾eme narazit na urèité problémy, které jsou dobøe vidìt na následujícím obrázku. První problém urèitì nastane, proto¾e v automatu není pøesnì øeèeno, které slovo konèí v jakém vrcholu.
-Napøíklad ve stavu, kde konèí slovo BARBARA, konèí také slovo ARA, ale o tom nevíme.
-Druhý problém nastává, kdy¾ v automatu není informace o~konci slova. Pøíkladem je seno BARAB (jednoduché k~nahlédnutí, viz obrázek).
-Teï nám nezbývá nic jiného, ne¾ najít øe¹ení tìchto záludných problémù. Øe¹e¹í se nám naskýtá hned nìkolik:
-\itemize\ibull
-\:Projdeme v¹echy zpìtné hrany a vypí¹eme slova, je¾ v daných stavech konèí. Toto øe¹ení funguje, ale je pomalé, proto¾e poka¾dé procházíme v¹echny zpìtné hrany.
-%\:Pøedpoèítání mno¾in slov. Najdeme mno¾inu slov tak, aby celková velikost slov byla vìt¹í ne¾ lineární. Funkèní, ale konstrukce je pomalá.
-\:Pro následující øe¹ení, jen¾ spoèívá v nalezení zkratek ve stromì, si zavedeme toto znaèení:\par
-\s{\<slovo>($s$)} = index slova $\iota$, které konèí ve stavu $s$, nebo $\emptyset$ \par
-\s{\<out>($s$)} = nejbli¾¹í vrchol, do kterého se lze z $s$ dostat po zpìtných hranách a \<slovo(v)> $\ne 0$ (konèí tam slovo)
-\figure{Graphic2.eps}{Vyhledávací automat se zpìtnými hranami}{1.3in}
-\endlist
-
-\>Podle posledního bodu vytvoøíme algoritmus na vyhledávání \uv{jehel v senì}.
-\algo
-\:$s \leftarrow$ \<koøen> ($s$ bude aktuální stav vyhledávacího automatu).
-\:Procházíme v¹echny písmena $c$ v senì $\sigma$:
-\::$s \leftarrow krok(s,c)$.
-\::Je-li $\<slovo>(s) \ne 0$, vypí¹eme $\<slovo>(s)$.
-\::$v \leftarrow \<out>(s)$.
-\::Dokud $v \ne 0$:
-\:::Vypí¹eme $\<slovo>(v)$.
-\:::$v \leftarrow \<out>(v)$.
-\endalgo
-
-\s{\<krok(s,c)>}:= jeden \<krok> vyhledávacího automatu:
-\algo
-\:Dokud $\not\exists f(s,c) \wedge s \ne$ \<koøen>: $s \leftarrow z(s)$.
-\:Pokud $\exists f(s,c)$: $s \leftarrow f(s,c)$.
-\:Vrátíme $s$.
-\endalgo
-
-\h{Reprezentace v pamìti}
-První mo¾nost, jak reprezentovat vyhledávací automat, je jednorozmìrné pole vrcholù stromu, v nìm¾ ukládáme seznam synù pro ka¾dý vrchol. Je to jednoduchá varianta, ale má nevýhodu pro velké abecedy, proto¾e procházení seznamu synù mù¾e trvat neúmìrnì dlouho. Proto se nabízí druhá mo¾nost a to hashovací tabulka $(\<stav>,\<znak>) \rightarrow f(\<stav>,\<znak>)$, kde se \uv{ztratí} pou¾ívání hashovací funkce.
-
-\h{Slo¾itost}
-\itemize\ibull
-\:Kroky 2.--5. mají èasovou slo¾itost $\O(\vert \sigma \vert)$, kterou jednodu¹e doká¾eme pomocí potenciálu -- poèet krokù nahoru je men¹í nebo roven poètu krokù dolù. A to je maximálnì $S$.
-\:Kroky 6.--8. mají èasovou slo¾itost $\O(\<poèet výskytù>)$, proto¾e rychleji doopravdy nelze v¹echny výskyty vypsat.
-\endlist
-
-\s{Konstrukce automatu} (Aho, Corasicková)
-\algo
-\:Postavíme strom dopøedných hran, $r \leftarrow$ koøen stromu.
-\:Spoèteme $\<slovo>(\ast)$ -- oznaèíme si stavy, kde konèí slova.
-\:Spoèteme $z(\ast)$: $z(\beta)=\alpha(\beta[1:])$:
- {\parindent=6em \itemize\ibull
- \:\>\>\>$z(\beta) = \alpha(\beta[1:])$ -- v¹echny zpìtné hrany vedou do vy¹¹ích hladin
- \:$z(v) = \<krok>(z(u),c)$
- \endlist}
-\figure{Graphic100.eps}{$z(v) = \<krok>(z(u),c)$}{0.7in}
-\:$z(r) \leftarrow 0$, do fronty $Q$ pøiøadíme v¹echny syny $r$, pro v¹echny $v$ prvky $Q: z(v) \leftarrow r$.
-\:Dokud fronta $Q$ není prázdná:
-\::$u\leftarrow$ vybereme z~$Q$.
-\::Pro syny $v$ vrcholu $u$:
-\:::$R \leftarrow \<krok>(z(u))$ [znak na hranì \<uv>].
-\:::$z(v)\leftarrow R$.
-
-\figure{Graphic101.eps}{$z(v) = R$}{0.7in}
-\:::Je-li $slovo(R) \not= 0 \Rightarrow \<out>(v) \leftarrow R$, jinak $\<out>(v) \leftarrow \<out>(R)$.
-\figure{Graphic102.eps}{Nastavení $\<out>(v)$}{0.7in}
-\endalgo
-\figure{vyhl_automat_full.eps}{Vyhledávací automat -- kompletní}{1in}
-
-\s{Vìta:}
-Algoritmus Aho-Corasicková najde v¹echny výskyty slov $\iota_1, \ldots, \iota_k$ ve~slovì $\sigma$ v~èase $\O(\sum_i \vert \iota_i \vert + \vert \sigma \vert + \<poèet výskytù>)$.
-
-\h{Polynomy a násobení}
-\>Mìjme dva polynomy definované jako:
-$$P(x) = \sum_{j=0}^{n-1} p_j x^j, \quad Q(x) = \sum_{j=0}^{n-1} q_j x^j.$$
-Násobení dvou polynomù $R=P \cdot Q$ je ekvivalentní s operací $R = \sum_{j,k} p_j q_k x^{j+k}$. Pøièem¾ na vypoèítání èlenu $r_l = \sum_{j=0}^l p_j q_{l-j}$ pou¾ijeme $\Theta(n)$ operací, tedy na spoèítaní celého polynomu $R$ potøebujeme $\Theta(n^2)$ operací.
-
-Podíváme se na jinou mo¾nost, jak tento problém øe¹it. Poslou¾í nám k~tomu následující vìta o jednoznaèné existenci polynomu nejvý¹e $k$-tého stupnì, pokud známe hodnoty
-ve~více ne¾ $k$ bodech.
-
-\s{Vìta:} Jsou-li $x_0, \ldots, x_k \in \bb{R} $ navzájem ruzná a $y_0, \ldots, y_k \in \bb{R}$, pak $\exists !$ polynom $P$ stupnì $\leq k : \forall j: P(x_j) = y_j$.
-
-\figure{polynom.eps}{Polynom}{2in}
-
-\ss{Plán:}
-
-\>Nech» $k=2^{n-1}$. Zvolíme èísla $x_0, \ldots, x_k$ libovolná, ale rùzná, a spoèteme $P(x_0)$, \dots, $P(x_k)$ a $Q(x_0), \ldots, Q(x_k)$.
-Poté $\forall j: y_j=P(x_j)Q(x_j)$
-musíme najít polynom $R$ stupnì $\leq k: \forall j: R(x_j)=y_j$.
-
-\s{Vyhodnocování polynomù} (metodou Rozdìl a panuj)
-
-\>BÚNO $n=2^m$. Uva¾me polynom:
-$$P(x) = p_0 x^0 + p_1 x^1 + \ldots + p_{n-1} x^{n-1}.$$
-Tento polynom si mu¾eme rozdìlit na dvì èasti. V levé budeme mít èleny se sudými exponenty a v~pravé budou èleny s~exponenty lichými:
-$$P(x) = (p_0 + p_2 x^2 + \ldots + p_{n-2}x^{n-2}) + (p_1 x^1 + p_3 x^3 + \ldots + p_{n-1} x^{n-1}).$$
-Z pravé strany mù¾eme vytknout $x$ a dostaneme:
-$$P(x) = (p_0 + p_2 x^2 + \ldots + p_{n-2}x^{n-2}) + x(p_1 + p_3 x^2 + \ldots + p_{n-1} x^{n-2}),$$
-$$ \vdots $$
-$$P(x) = L(x^2) + xN(x^2),$$
-$$P(-x) = L(x^2) - xN(x^2),$$
-kde $L(x)$ a $N(x)$ jsou polynomy stupnì $n/2$. Umocnìním $x^2$ se nám poru¹í párování $x$ a $-x$, proto musíme poèítat v~$\bb{C}$ místo~$\bb{R}$.
-V~tomto pøípadì jsme z~polynomu s~$n$ koeficienty v~$n$ bodech dostali $2$ polynomy s~$n/2$ koeficienty v~$n/2$ bodech. Z~toho vyplývá èasová slo¾itost definována vztahem:
-$$T(n) = 2T(n/2) + \O(n).$$
-Ten mù¾eme vyøe¹it s pou¾itím Master Theoremu z~ADS~I a dostaneme:
-$$T(n) = \O(n \log n).$$
-
-\bye
+++ /dev/null
-P=7-ac
-
-include ../Makerules
+++ /dev/null
-\input lecnotes.tex
-\prednaska{8}{Fourierova transformace}{(K.Jakubec, M.Polák a G.Ocsovszky)}
-
-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:}
-\>Libovolný polynom $P$ stupnì $n$ mù¾e být reprezentován dvìma rùznými zpùsoby:
-
-\itemize\ibull
-\:svými koeficienty, èili èísly $p_{0}, p_{1}, \ldots ,p_{n}$, nebo
-\:svými hodnotami v $n$ rùzných bodech $x_{0}, x_{1}, \ldots , x_{n}$, èili èísly $P(x_{0}),$ $P(x_{1}),$ $\ldots , P(x_{n})$.
-\endlist
-
-\ss{Konvence:}
-\>Celé polynomy oznaèujeme velkými písmeny, jednotlivé èleny polynomù pak pøíslu¹nými malými písmeny. (Pø.: Polynom $W$ stupnì $n$ má koeficienty $w_{0}, w_{1}, w_{2},\ldots, w_{n}$.)
-
-Pov¹imnìme si jedné skuteènosti -- máme-li dva polynomy $A$ a $B$ stupnì $n$ a body $x_{0}, \ldots, x_{k}$, dále polynom $C=A \cdot B$ (stupnì $2n$), pak platí $C(x_{k}) = A(x_{k}) \cdot B(x_{k}), k = 0,1,2, \ldots, n.$ Toto èiní tento druhý zpùsob reprezentace polynomu velice atraktivním pro násobení. Problémem je, ¾e typicky máme polynom zadaný koeficienty a ne hodnotami v bodech. Tím pádem potøebujeme nìjaký hodnì rychlý algorimtus (tj. rychlej¹í ne¾ kvadratický, jinak bychom si nepomohli oproti hloupému algoritmu) na pøevod polynomu z jedné reprezentace do druhé a zase zpìt.
-
-Dále bychom si mìli uvìdomit, ¾e stupeò na¹eho výsledného polynomu $C$ bude $\leq 2n$ (kde $n$ je stupeò výchozích polynomù). Pokud chceme polynom $C$ reprezentovat pomocí jeho hodnot v bodech, musíme tedy vzít alespoò $2n$ bodù. Tímto konèí malá algebraická vsuvka.
-
-\s{Idea, jak by mìl algoritmus pracovat:}
-\algo
-\:Vybereme $2n$ bodù $x_{0}, x_{1}, \ldots , x_{2n-1}$.
-\:V tìchto bodech vyhodnotíme polynomy $A$ a $B$.
-\:Nyní ji¾ v lineárním èase získáme hodnoty polynomu $C$ v tìchto bodech (viz vý¹e).
-\:Pøevedeme hodnoty polynomu $C$ na jeho koeficienty.
-\endalgo
-
-\>Je asi vidìt, ¾e klíèové jsou kroky 2 a 4. Vybrání bodù jistì stihneme pohodlnì v lineárním èase a vynásobení samotných hodnot té¾ (máme $2n$ bodù a $C(x_{k}) = A(x_{k}) \cdot B(x_{k}), k = 0,1,2, \ldots , 2n-1$, tak¾e na to nepotøebujeme více ne¾ $2n$ násobení).
-
-Celý trik spoèívá v chytrém vybrání onìch bodù, ve kterých budeme polynomy vyhodnocovat. Je na to potøeba vìdìt pár zajímavostí o komplexních èíslech, na webové stránce pøedná¹ky jsou k dispozici slajdy, zde to bude zapsáno o trochu struènìji.
-
-\ss{ Vyhodnocení polynomu metodou Rozdìl a panuj (algoritmus FFT):}
-Mìjme polynom $P$ øádu $n$ 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:
-$P(x) = (p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{n-2}x^{n-2}) + (p_{1}x^{1} + p_{3}x^{3} + \ldots + p_{n-1}x^{n-1})$.
-
-$S(x^{2}) = p_{0}x^{0} + p_{2}x^{2} + \ldots + p_{n - 2}x^{n - 2}$,
-$L(x^{2}) = p_{1}x^{1} + p_{3}x^{3} + \ldots + p_{n - 1}x^{n - 1}$
-
-\>Tak¾e obecnì $P(x) = S(x^{2}) + xL(x^{2})$ a $P(-x) = S(x^{2}) - xL(x^{2})$.
-Jinak øeèeno, vyhodnocování $P$ v $n$ bodech se nám smrskne na vyhodnocení $S(x)$ a $L(x)$ (oba jsou polynomy stupnì $n/2$ a vyhodnocujeme je nyní v $x^{2}$) v $n/2$ bodech (proto¾e $(x_{i})^{2} = (-x_{i})^{2}$).
-
-\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 mocina 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. Jako $x_{0}, \ldots , x_{n-1} $ si zvolíme mocniny $n$-té primitvní odmocniny z jedné (oznaèíme si ji jako $\omega$). Máme $n$ $n$-tých primitivních odmocnin z jednièky, rovnomìrnì rozesetých po jednotkové kru¾nici, BÚNO $n=2^{k}, k \in N$ (jinak viz slajdy Martina Mare¹e). Jednotlivé mocniny vypadají takto: $1, \omega, \omega^{2}, \ldots , \omega^{n - 1} $, kde $\omega = e^{2 \pi i/ n}$.
-
-\s{Dvì poznámky:}
-\itemize\ibull
-\:primitivní $n$-té odmocniny z jednièky jsou spárované, èili $\omega^{j} = -\omega^{n/2 + j}$,
-\:umocníme-li v¹echny na druhou, vznikne nám $n/2$ $n/2$-tých odmocnin z jedné, které jsou i nadále spárované.
-\endlist
-
-\ss{Celý algoritmus bude vypadat takto:}
-\>FFT($P$, $ \omega$)
-
-\>{\sl Vstup:} $p_{0}, \ldots , p_{n-1}$, koeficienty polynomu $P$, a $\omega$, $n-$tá 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 sudé a liché koeficienty a rekurzivnì zavoláme FFT($S$, $\omega^{2}$) a FFT($L$, $\omega^{2}$).
-\:Pro $j = 0, \ldots , n/2 - 1$ spoèítáme:
-
-$P(\omega^{j}) = S(\omega^{2j}) + \omega^{j} \cdot L(\omega^{2j})$.
-
-$P(\omega^{j+n/2}) = S(\omega^{2j}) - \omega^{j} \cdot L(\omega^{2j})$.
-
-$j$ je z intervalu $[0, {n \over 2}-1]$
-
-\endalgo
-
-
-\s{Èasová slo¾itost:}
-\>$T(n)=2T(n/2) + \O(n) \Rightarrow$ slo¾itost $\O(n \log n)$, stejnì 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 Diskretní Fourierova transformace} $(DFT)$
-je funkce $f: { {\bb C} ^n} \rightarrow { {\bb C} ^n}$, kde $y=f(x) \equiv \forall j \ y_{j} = \sum \limits ^{n-1}_{k=0} x_{k} \cdot \omega ^{k}$.
-
-
-\s{Jak najít inverzní matici?} Víme, ¾e $\Omega =\Omega ^{T}$ proto¾e $\omega ^{jk} = \omega ^{kj}$.
-
-Vyu¾ijeme následující lemma:
-
-\ss{Lemma:}
-
-\quad $\Omega _{j} \cdot \Omega _{k} = \left\{
-{\displaystyle 0 \ldots j\neq k}\atop
-{\displaystyle 1 \ldots j=k}
-\right.$.
-
-\s{Poznámka:}
-$\Omega _{j} \cdot \Omega _{k}$ myslíme skalární souèin.
-Jsou-li $x = (x _{0}, \ldots, x _{n})$ a $y = (y _{0}, \ldots, y _{n})$ dva komplexní vektory,
-pak jejich skalární souèin definujeme jako:
-$x^{*}y = \sum \limits ^{n}_{i=0} \overline{x _{i}}y_{i}$.
-
-
-\>V této definici $\overline{x}$ oznaèuje èíslo komplexnì sdru¾ené k èíslu $x$.
-
-\>
-
-\proof Souèin
-$$\Omega _{j} \Omega _{k} = \sum \limits ^{n-1}_{l=0} \Omega _{jl} \overline{\Omega _{kl}} = \sum \limits _{l} \omega ^{jl} \overline{\omega ^{kl}} = \sum \limits _{l} \omega ^{jl} \omega ^{-kl} = \sum \limits _{l} \omega ^{(j-k)l } = \sum \limits ^{n-1}_{l=0} (\omega^{j-k}) ^{l}, $$
-
-proto¾e $ \overline{\omega^{kl}} = \overline{\omega} ^{kl} = {({1 \over \omega} )}^{kl} = \omega ^{-kl}$.
-
-\itemize\ibull
-\:Pokud $j\neq k$, pou¾ijeme vzoreèek pro souèet geometrické posloupnosti, kde $a_{1}=1$ a $q=\omega ^{(j-k) }$ a dostaneme ${{\omega^{(j-k)n} -1} \over {\omega^{(j-k)} -1}} ={1-1 \over r- 1} = {0 \over \neq 0} = 0$. Kde $r$ je èíslo rùzné od jednièky.
-
-\:Pokud $j=k$, pak $ \sum \limits ^{n-1}_{l=0} (\omega ^{0}) ^{l} = n$.
-\endlist
-\qed
-
-
-
-\s{Dùsledek:} \quad $\Omega \cdot \overline{\Omega} = nE$.
-
-\>Jedná se o násobení matic, èili prvek na pozici $ij$ je $0$ nebo $n$. $\Rightarrow\Omega^{-1} = {1 \over n} \overline{\Omega}$.
-
-
-\>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
-kde $\omega^{-1}$ je $\overline{\omega}$ a $\omega _{n}$ je $n$-tá primitivní odmocnina z jednièky.
-
-\>Ná¹ algoritmus poèítá tedy i inverzní transformaci, pouze místo $\omega_n$ pou¾ijeme komplexnì zdru¾ené
- $\overline{\omega_n}$ a matici vynásobíme $(1/n)$. Co¾ je skvìlé --
- staèí znát pouze jeden algoritmus u~kterého staèí v~jednom pøípadì pou¾ít transformovanou matici a vydìlit $n$.
-
-\s{Výsledek:} Pro $n= 2^k$ lze DFT na ${\bb C}^n$ spoèítat v~èase $\O(n \log n)$ a DFT$^{-1}$ takté¾.
-
-\s{Dùsledek:}
-
-\>Polynomy stupnì $n$ lze násobit v èase $\O(n \log n)$:
-$\O(n \log n)$ pro vyhodnocení, $\O(n)$ pro vynásobení a $\O(n \log n)$ pro pøevedení zpìt.
-
-\s{Pou¾ití:}
-
-\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 $\O(n \log n)$.
-\endlist
-
-\s{Hardwarová implementace FFT}
-
-\figure{img.eps}{Pøíklad prùbìhu algoritmu na vstupu velikosti 8}{3in}
-
-
-\>Obrázek ukazuje zapojení kombinaèního obvodu pro DFT pro vstup velikosti~8. Hladin bude v¾dy $\log_2 n$, tj. v~na¹em pøípadì $\log_2 8 = 3$ hladiny.
-
-\>Podívejme se na pravou èást obrázku, tedy výstup celého obvodu. Èerná koleèka pøedstavují podobvody, rovnice vedle nich operaci, kterou provádìjí. Hodnoty $y_j$ znaèí hodnotu polynomu $P$ v bodì $\omega^j$ kde $\omega^j$ je $j-tá$ mocnina primitivní $n$-té odmocniny z jednièky. K jejímu spoètení ale potøebujeme znát hodnoty $s_k$ a $l_k$ kde $k$ je z intervalu $[0, {n/2} -1]$ a $s_k$ a $l_k$ jsou hodnoty polynomu stupnì ${n/2}$ v bodì $\omega^{2k}$. V polynomu $s$ jsou sudé koeficienty a v polynomu $l$ liché koeficienty polynomu $P$. Vidíme ze se jedná pøesnì o ná¹ rekurzivní algoritmus pro poèítání FFT a tímto zpùsobem postavíme celou sí».
-\>Tímto obvodem jsme tedy získali nerekurzivní algoritmus pro poèítání FFT. V¹imìme si poøadí vstupních hodnot (koeficientù). Èísla jsou v binárním tvaru 0--7 pøeètená pozpátku. Pro pøedstavu jaké koeficienty polynomu $P$ se objevují v rùzných hladinách, na obrázku jsou naznaèena jejich èísla spolu s pøíslu¹nými mocninami primitivní $n$-té odmocniny z jednièky.
-
-\s{Z toho:}
-
-\itemize\ibull
-\:Kombinaèní obvod pro DFT
-s~$\O(\log n)$ hladinami
-a $\O(n)$ hradly na hladinì.
-\:Nerekurzivní algoritmus (postupujeme zleva) v~èase $\O(n \log n)$.
-
-\endlist
-
-\bye
+++ /dev/null
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-def/DataSrc1{DataStr1}bd/DataSrc2{DataStr2}bd/DataSrc3{DataStr3}bd\r
-/colorimage where{pop/ci{colorimage}bd}{/ci{pop pop/ci_datasrc exch\r
-def matrix invertmatrix/ci_matrix exch def pop/ci_dataleft 0 def/ci_buf()def\r
-/ci_dataofs 0 def 0 1 3 -1 roll 1 sub{/ci_y exch def dup 0 1 3 -1\r
-roll 1 sub{0 1 2{pop ci_dataleft 0 eq{ci_datasrc dup length/ci_dataleft\r
-exch def/ci_buf exch def/ci_dataofs 0 def}if ci_buf ci_dataofs get\r
-255 div/ci_dataofs ci_dataofs 1 add def/ci_dataleft ci_dataleft 1 sub\r
-def}for setrgbcolor dup ci_y 3 -1 roll 1 add ci_y 1 add 4 copy 5 1\r
-roll 4 2 roll 5 -1 roll 1 1 4{pop ci_matrix transform 8 2 roll}for\r
-m l l l closepath fill}for}for pop}bd}ifelse/rci{/rciBuf 4 index 3\r
-index mul 7 add 8 div floor cvi string def{currentfile rciBuf readhexstring\r
-pop}bind false 3 ci}bd/cbsl{2 eq/cbslL2 xd 5 index/cbslw xd translate\r
-scale 8 [ 3 index 0 0 5 index 0 0 ] cbslL2{/DataStr1 cbslw string def\r
-currentfile/ASCII85Decode filter/RunLengthDecode filter DataStr1 readstring\r
-pop pop/DataStr2 cbslw string def currentfile/ASCII85Decode filter\r
-/RunLengthDecode filter DataStr2 readstring pop pop/DataStr3 cbslw\r
-string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
-DataStr3 readstring pop pop{DataStr1}bind{DataStr2}bind{DataStr3}bind\r
-true}{/DataSrc load false}ifelse 3 ci}bd/gbsl{2 eq/gbslL2 xd 5 index\r
-/gbslw xd translate scale 8 [ 3 index 0 0 5 index 0 0 ] gbslL2{/DataStr1\r
-gbslw string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
-DataStr1 readstring pop pop{DataStr1}bind}{/DataStr1 gbslw string def\r
-currentfile DataSrc1 readhexstring pop pop{DataStr1}bind}ifelse image}bd\r
-/cmi{/cmiBuf 4 index 3 index mul 7 add 8 div floor cvi string def{currentfile\r
-cmiBuf readhexstring pop}bind image}bd/cpal{4 mul string/cPalette exch\r
-def currentfile cPalette readhexstring pop}bd/cpci{/cpci_datasrc exch\r
-def matrix invertmatrix/cpci_matrix exch def/cpci_bpp exch def cpci_init\r
-0 1 3 -1 roll 1 sub{/cpci_y exch def dup cpci_bpp 4 eq{cpci_sampsleft\r
-1 eq{/cpci_sampsleft 0 def}if}if 0 1 3 -1 roll 1 sub{cpci_nextcol dup\r
-cpci_y 3 -1 roll 1 add cpci_y 1 add 4 copy 5 1 roll 4 2 roll 5 -1 roll\r
-1 1 4{pop cpci_matrix transform 8 2 roll}for m l l l closepath fill}for}for\r
-pop}bd/cpci_init{/cpci_sampsleft 0 def}bd/cpci_buf 1 string def/cpci_nextcol{cpci_bpp\r
-1 eq{cpci_sampsleft 0 eq{currentfile cpci_buf readhexstring pop pop\r
-/cpci_sampsleft 8 def}if cpci_buf dup 0 get dup 1 and setgray -1 bitshift\r
-1 exch put/cpci_sampsleft cpci_sampsleft 1 sub def}{cpci_bpp 4 eq{cpci_sampsleft\r
-0 eq{currentfile cpci_buf readhexstring pop pop/cpci_sampsleft 2 def}if\r
-cpci_buf 0 get dup 15 and exch -4 bitshift cpci_buf 0 3 -1 roll put\r
-/cpci_sampsleft cpci_sampsleft 1 sub def}{currentfile cpci_buf readhexstring\r
-pop 0 get}ifelse 4 mul dup 2 add cPalette exch get 255 div exch dup\r
-1 add cPalette exch get 255 div exch cPalette exch get 255 div setrgbcolor}ifelse}bd\r
-/setup1asciiproc{[ currentfile mystring/readhexstring cvx/pop cvx\r
-] cvx bind}bd/setup1binaryproc{[ currentfile mystring/readstring cvx\r
-/pop cvx ] cvx bind}bd level2{save/dontloadlevel1 xd}if/iw 0 def/ih\r
-0 def/im_save 0 def/setupimageproc 0 def/polarity 0 def/smoothflag\r
-0 def/mystring 0 def/bpc 0 def/beginimage{/im_save save def dup 0 eq{pop\r
-/setup1binaryproc}{1 eq{/setup1asciiproc}{(error, can't use level2 data acquisition procs for level1)print\r
-flush}ifelse}ifelse/setupimageproc exch ld/polarity xd/smoothflag xd\r
-/imat xd/mystring exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save\r
-restore}bd/1bitbwcopyimage{1 setgray 0 0 moveto 0 1 rlineto 1 0 rlineto\r
-0 -1 rlineto closepath fill 0 setgray iw ih polarity imat setupimageproc\r
-imagemask}bd/1bitcopyimage{setrgbcolor 0 0 moveto 0 1 rlineto 1 0 rlineto\r
-0 -1 rlineto closepath fill setrgbcolor iw ih polarity imat setupimageproc\r
-imagemask}bd/1bitmaskimage{setrgbcolor iw ih polarity [iw 0 0 ih 0\r
-0] setupimageproc imagemask}bd level2{dontloadlevel1 restore}if level2\r
-not{save/dontloadlevel2 xd}if/setup2asciiproc{currentfile/ASCII85Decode\r
-filter/RunLengthDecode filter}bd/setup2binaryproc{currentfile/RunLengthDecode\r
-filter}bd/myimagedict 9 dict dup begin/ImageType 1 def/MultipleDataSource\r
-false def end def/im_save 0 def/setupimageproc 0 def/polarity 0 def\r
-/smoothflag 0 def/mystring 0 def/bpc 0 def/ih 0 def/iw 0 def/beginimage{\r
-/im_save save def dup 2 eq{pop/setup2binaryproc}{dup 3 eq{pop/setup2asciiproc}{0\r
-eq{/setup1binaryproc}{/setup1asciiproc}ifelse}ifelse}ifelse/setupimageproc\r
-exch ld{[ 1 0 ]}{[ 0 1 ]}ifelse/polarity xd/smoothflag xd/imat xd/mystring\r
-exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save restore}bd/1bitbwcopyimage{1\r
-ssg 0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill\r
-0 ssg myimagedict dup begin/Width iw def/Height ih def/Decode polarity\r
-def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
-1 def/Interpolate smoothflag def end imagemask}bd/1bitcopyimage{ssc\r
-0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill ssc\r
-myimagedict dup begin/Width iw def/Height ih def/Decode polarity def\r
-/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
-1 def/Interpolate smoothflag def end imagemask}bd/1bitmaskimage{ssc\r
-myimagedict dup begin/Width iw def/Height ih def/Decode polarity def\r
-/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
-1 def/Interpolate smoothflag def end imagemask}bd level2 not{dontloadlevel2\r
-restore}if\r
-level2{save/dontloadlevel1 xd}if/startnoload{{/noload save def}if}bd\r
-/endnoload{{noload restore}if}bd/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}bd\r
-/ncolors 1 def/colorimage where{pop true}{false}ifelse{/ncolors 0\r
-statusdict begin/processcolors where{pop pop processcolors}{/deviceinfo\r
-where{pop deviceinfo/Colors known{pop{deviceinfo/Colors get}}if}if}ifelse\r
-end def ncolors 0 ne{/colorimage testsystemdict/setcolortransfer testsystemdict\r
-/currentcolortransfer testsystemdict/currentcmykcolor testsystemdict\r
-and and and not{/ncolors 0 def}if}if}if ncolors dup 1 ne exch dup 3\r
-ne exch 4 ne and and{/ncolors 0 def}if ncolors 1 eq dup dup not startnoload{\r
-/expandbw{expandfactor mul round cvi bwclut exch get 255 div}bd/doclutimage{bwclut\r
-colorclut pop/bwclut xd bpc dup 8 eq{pop 255}{4 eq{15}{3}ifelse}ifelse\r
-/expandfactor xd [/expandbw load/exec load dup currenttransfer exch\r
-] cvx bind settransfer iw ih bpc imat setupimageproc image}bd}if not\r
-endnoload ncolors dup 3 eq exch 4 eq or dup dup not startnoload{/nullproc{{}}def\r
-/concatutil{/exec load 7 -1 roll/exec load}bd/defsubclut{1 add getinterval\r
-def}bd/spconcattransfer{/Dclut exch def/Cclut exch def/Bclut exch def\r
-/Aclut exch def/ncompute exch ld currentcolortransfer [{Aclut ncompute}concatutil\r
-] cvx [{Bclut ncompute}concatutil ] cvx [{Cclut ncompute}concatutil\r
-] cvx [{Dclut ncompute}concatutil ] cvx setcolortransfer}bd/setuprgbcluts{\r
-/bit3x rgbclut length 3 sub def/bit1x bit3x 3 idiv def/rclut rgbclut\r
-def/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}bd}if\r
-not endnoload ncolors 3 eq dup dup not startnoload{/3compute{exch bit3x\r
-mul round cvi get 255 div}bd/doclutimage{/rgbclut xd pop setuprgbcluts\r
-/3compute rclut gclut bclut dup spconcattransfer iw ih bpc imat [\r
-setupimageproc/exec load/dup load dup ] cvx nullproc nullproc true\r
-3 colorimage}bd}if not endnoload ncolors 4 eq dup dup not startnoload{\r
-/stuffclut{cmykindex 3 -1 roll put}bd/ftoint{1 exch sub 255 mul round\r
-cvi}bd/4compute{exch bit4x mul round cvi get 255 div}bd/computecmykclut{setuprgbcluts\r
-/bit4x rgbclut length 3 idiv 4 mul 4 sub def/cmykclut bit4x 4 add\r
-string def/cclut cmykclut def/mclut cclut 1 bit4x defsubclut/yclut\r
-cclut 2 bit4x defsubclut/kclut cclut 3 bit4x defsubclut/cmykindex 0\r
-def 0 1 bit1x{dup/cmykindex exch bit1x exch sub 4 mul def 3 mul dup\r
-rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch\r
-get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint\r
-yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut}for}bd\r
-/doclutimage{/rgbclut xd pop invalidcolortable?{computecmykclut}if\r
-/4compute cclut mclut yclut kclut spconcattransfer iw ih bpc imat\r
-[ setupimageproc/exec load/dup load dup dup ] cvx nullproc nullproc\r
-nullproc true 4 colorimage}bd}if not endnoload ncolors 0 eq dup dup\r
-not startnoload{/lookupandstore{3 mul 3 getinterval putinterval exch\r
-3 add exch 3 copy}bd/8lookup/lookupandstore ld/4lookup{/byte 1 index\r
-def -4 bitshift lookupandstore byte 15 and lookupandstore}bd/2lookup{\r
-/byte 1 index def -6 bitshift lookupandstore byte -4 bitshift 3 and\r
-lookupandstore byte -2 bitshift 3 and lookupandstore byte 3 and lookupandstore}bd\r
-/colorexpand{mystringexp 0 rgbclut 3 copy 7 -1 roll/mylookup load\r
-forall pop pop pop pop pop}bd/createexpandstr{/mystringexp exch mystring\r
-length mul string def}bd/doclutimage{/rgbclut xd pop/mylookup bpc 8\r
-eq{3 createexpandstr/8lookup}{bpc 4 eq{6 createexpandstr/4lookup}{12\r
-createexpandstr/2lookup}ifelse}ifelse ld iw ih bpc imat [ setupimageproc\r
-/exec load/colorexpand load/exec load] cvx false 3 colorimage}bd}if\r
-not endnoload/colorimage where{pop true}{false}ifelse dup{/do24image{iw\r
-ih 8 imat setupimageproc false 3 colorimage}bd}if dup dup startnoload\r
-not{/rgbtogray{/str xd/len str length def/smlen len 3 idiv def/rstr\r
-str def/gstr str 1 len 1 sub getinterval def/bstr str 2 len 2 sub getinterval\r
-def str dup 0 1 smlen 1 sub{dup 3 mul rstr 1 index get .3 mul gstr\r
-2 index get .59 mul add bstr 3 -1 roll get .11 mul add round cvi put\r
-dup}for pop 0 smlen getinterval}bd/do24image{iw ih 8 imat [ setupimageproc\r
-/exec load/rgbtogray load/exec load ] cvx bind image}bd}if endnoload\r
-/doimage{iw ih 8 imat setupimageproc image}bd level2{dontloadlevel1\r
-restore}if level2 not{save/dontloadlevel2 xd}if/myappcolorspace/DeviceRGB\r
-def/rgbclut 0 def/doclutimage{/rgbclut xd pop bpc dup 8 eq{pop 255}{4\r
-eq{15}{3}ifelse}ifelse/hival xd [/Indexed myappcolorspace hival rgbclut]\r
-setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode\r
-[0 hival] def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
-bpc def/Interpolate smoothflag def end image}bd/do24image{myappcolorspace\r
-setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode\r
-[0 1 0 1 0 1] def/ImageMatrix imat def/DataSource setupimageproc def\r
-/BitsPerComponent 8 def/Interpolate smoothflag def end image}bd level2\r
-not{dontloadlevel2 restore}if\r
-/NumSteps{dtransform matrix defaultmatrix idtransform Pythag currentscreen\r
-pop pop 72 exch div div}bd/FindMinSteps{v_ft 4 eq{urx startX sub abs\r
-llx startX sub abs Max ury startY sub abs lly startY sub abs Max Pythag\r
-2 3.14159265 mul mul 0}{v_ft 2 eq{endY startY sub endX startX sub Pythag\r
-endY2 startY sub endX2 startX sub Pythag gt{endY startY sub endX startX\r
-sub}{endY2 startY sub endX2 startX sub}ifelse}{endY startY sub endX\r
-startX sub}ifelse}ifelse NumSteps}bd/cxe{/v_cxe exch def}bd/cxm{pop\r
-/v_cxm exch def}bd/cxmt{pop pop}bd/cxt{pop}bd/S_eoclip{currentflat{{eoclip}stopped{dup\r
-currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: eoclip]\n)print\r
-flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
-/S_clip{currentflat{{clip}stopped{dup currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: clip]\n)print\r
-flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
-/S_eofill{currentflat{{eofill}stopped{dup currentflat exch sub 20\r
-gt{([Error: PathTooComplex; OffendingCommand: eofill]\n)print flush\r
-exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
-/gpbbx{pathbbox/ury exch def/urx exch def/lly exch def/llx exch def}bd\r
-/lineargfill{initgfill{false initgfx/distance endX startX sub endY\r
-startY sub Pythag def/incD distance Steps div def endY startY sub endX\r
-startX sub atan newpath llx lly urx ury Bx startX startY translate\r
-rotate gpbbx eGy ssg newpath llx lly urx ury Bx S_eofill sGy ssg newpath\r
-llx lly 0 ury Bx S_eofill/fillX 0 def 0 1 Steps 1 sub{stepgfx newpath\r
-fillX lly fillX incD add dup/fillX exch def ury Bx S_eofill pop}for}if}bd\r
-/radialgfill{initgfill{false initgfx/distance endX startX sub endY\r
-startY sub Pythag def/incD distance Steps div def eGy ssg newpath llx\r
-lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepgfx newpath\r
-startX startY distance 0 360 arc closepath distance incD add dup/distance\r
-exch def 0 rmoveto startX startY distance 0 360 arc closepath S_eofill\r
-pop}for}if}bd/ellipticgfill{initgfill{true initgfx sGy ssg newpath\r
-llx lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2\r
-startX sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepgfx\r
-/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for}if}bd/conicalgfill{initgfill{urx\r
-startX sub abs llx startX sub abs Max ury startY sub abs lly startY\r
-sub abs Max Pythag startY endY sub startX endX sub Pythag div/radius\r
-exch def true initgfx [ endX startX sub endY startY sub startY endY\r
-sub endX startX sub startX startY ] concat/slice 180 Steps div def\r
-/startangle 0 def Steps -1 0{pop stepgfx 0 0 moveto 0 0 radius startangle\r
-neg dup slice add arc fill 0 0 moveto 0 0 radius startangle dup slice\r
-add dup/startangle exch def arc fill}for}if}bd/initgfill{gpbbx{S_eoclip}{S_clip}ifelse\r
-startX endX eq startY endY eq and sGy eGy eq or dup{sGy ssg newpath\r
-llx lly urx ury Bx S_eofill}if not}bd/initgfx{/flag exch def flag{sGy\r
-eGy/sGy exch def/eGy exch def}if eGy sGy sub abs 256 mul FindMinSteps\r
-Min 256 Min ceiling 1 Max/Steps exch def eGy sGy sub Steps div/incGy\r
-exch def sGy/mGy exch def}bd/stepgfx{mGy ssg mGy incGy add/mGy exch\r
-def}bd/linearfill{initfill{false initfx endX startX sub endY startY\r
-sub Pythag/distance exch def/incD distance Steps div def endY startY\r
-sub endX startX sub atan newpath llx lly urx ury Bx startX startY translate\r
-rotate gpbbx v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse\r
-newpath llx lly urx ury Bx S_eofill v_cxe 0 eq{sR sG sB setgfillrgb}{sH\r
-sS sV setgfillhsb}ifelse newpath llx lly 0 ury Bx S_eofill/fillX 0\r
-def 0 1 Steps 1 sub{stepfx newpath fillX lly fillX incD add dup/fillX\r
-exch def ury Bx S_eofill pop}for}if}bd/radialfill{initfill{false initfx\r
-endX startX sub endY startY sub Pythag/distance exch def/incD distance\r
-Steps div def v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse\r
-newpath llx lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepfx\r
-newpath startX startY distance 0 360 arc closepath distance incD add\r
-dup/distance exch def 0 rmoveto startX startY distance 0 360 arc closepath\r
-S_eofill pop}for}if}bind def/ellipticalfill{initfill{true initfx v_cxe\r
-0 eq{sR sG sB setgfillrgb}{sH sS sV setgfillhsb}ifelse newpath llx\r
-lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2 startX\r
-sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepfx\r
-/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for v_cxe 0\r
-eq{sR sG sB eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def\r
-/eG exch def/eR exch def}if}if}bd/conicalfill{initfill{urx startX\r
-sub abs llx startX sub abs Max ury startY sub abs lly startY sub abs\r
-Max Pythag startY endY sub startX endX sub Pythag div/radius exch def\r
-true initfx [ endX startX sub endY startY sub startY endY sub endX\r
-startX sub startX startY ] concat/slice 180 Steps div def/startangle\r
-0 def Steps -1 0{pop stepfx 0 0 moveto 0 0 radius startangle neg dup\r
-slice add arc fill 0 0 moveto 0 0 radius startangle dup slice add dup\r
-/startangle exch def arc fill}for}if}bd/initfill{gpbbx{S_eoclip}{S_clip}ifelse\r
-startX endX eq startY endY eq and v_cxe 2 ne sR eR eq sG eG eq and\r
-sB eB eq and and or dup{sR sG sB setgfillrgb newpath llx lly urx ury\r
-Bx S_eofill}if not}bd/initfx{/flag exch def v_cxe 0 eq{flag{sR sG sB\r
-eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def/eG exch def\r
-/eR exch def}if eR sR sub abs 256 mul eG sG sub abs 256 mul Max eB\r
-sB sub abs 256 mul Max FindMinSteps Min 256 Min ceiling 1 Max/Steps\r
-exch def/incR eR sR sub Steps div def/incG eG sG sub Steps div def\r
-/incB eB sB sub Steps div def sR/mR exch def sG/mG exch def sB/mB\r
-exch def}{sR sG sB rgb2hsb/sV exch def/sS exch def/sH exch def eR eG\r
-eB rgb2hsb/eV exch def/eS exch def/eH exch def eH sH sub abs v_cxe\r
-1 eq{dup 0.5 gt{1 exch sub}if}{dup 0.5 lt{1 exch sub}if}ifelse 256\r
-mul eS sS sub abs 256 mul Max eV sV sub abs 256 mul Max FindMinSteps\r
-Min 256 Min ceiling 1 Max/Steps exch def v_cxe 1 eq{/incH eH sH sub\r
-dup abs 0.5 gt{dup 0 ge{1 sub}{1 add}ifelse}if Steps div def/incS eS\r
-sS sub Steps div def/incV eV sV sub Steps div def}{/incH eH sH sub\r
-dup abs 0.5 le{dup 0 ge{1 sub}{1 add}ifelse}if Steps div def/incS eS\r
-sS sub Steps div def/incV eV sV sub Steps div def}ifelse flag{/sH eH\r
-/eH sH def def/sS eS/eS sS def def/sV eV/eV sV def def/incH incH neg\r
-def/incS incS neg def/incV incV neg def}if}ifelse}bd/stepfx{v_cxe 0\r
-eq{mR mG mB setgfillrgb mR incR add/mR exch def mG incG add/mG exch\r
-def mB incB add/mB exch def}{sH sS sV setgfillhsb sH incH add dup 0\r
-le{1 add}{dup 1 ge{1 sub}if}ifelse/sH exch def sS incS add/sS exch\r
-def sV incV add/sV exch def}ifelse}bd\r
-/ar{}bd/arr{pop pop pop pop pop pop pop}bd/ae{pop pop pop pop pop\r
-pop}bd/aoa{pop}bd/apl{pop}bd/apc{}bd/aof{pop pop}bd/aafs{pop pop pop}bd\r
-/O{pop}bd/R{pop}bd/axop{pop pop pop pop}bd/g{/v_ft 0 def/v_fc 0 def\r
-/v_fm 0 def/v_fy 0 def 1 exch sub/v_fk exch def/v_fct 0 def}bd/G{\r
-/v_sc 0 def/v_sm 0 def/v_sy 0 def 1 exch sub/v_sk exch def/v_sct 0\r
-def}bd/k{/v_fk exch def/v_fy exch def/v_fm exch def/v_fc exch def/v_ft\r
-0 def/v_fct 0 def}bd/K{/v_sk exch def/v_sy exch def/v_sm exch def/v_sc\r
-exch def/v_sct 0 def}bd/Xa{/v_fb exch def/v_fg exch def/v_fr exch def\r
-/v_ft 0 def/v_fct 1 def}bd/XA{/v_sb exch def/v_sg exch def/v_sr exch\r
-def/v_sct 1 def}bd/a_tc{exch pop 1 exch sub dup 3 -1 roll exch div\r
-exch dup 4 -1 roll exch div 3 1 roll dup 5 -1 roll exch div 4 1 roll\r
-5 -1 roll exch div 4 1 roll}bd/x{a_tc k}bd/X{a_tc K}bd/Xx{0 eq{x}{pop\r
-pop Xa}ifelse}bd/XX{0 eq{X}{pop pop XA}ifelse}bd/awr{/v_wr exch def}bd\r
-/w{setlinewidth}bd/j{setlinejoin}bd/J{setlinecap}bd/d{setdash}bd/asc{pop\r
-pop J}bd/aec{pop pop J}bd/csah{pop pop pop}bd/ceah{pop pop pop}bd/cst{pop\r
-pop}bd/cdp{pop}bd/m{moveto}bd/l{lineto}bd/c{curveto}bd/Bx{4 copy 5\r
-1 roll 4 2 roll 5 -1 roll m l l l closepath}bd/Cp{Bx clip newpath}bd\r
-/a_fp{gsave v_wr 0 ne v_ft 0 eq{setfillcolor{eofill}{fill}ifelse}{v_ft\r
-1 eq{linearfill}{v_ft 2 eq{ellipticalfill}{v_ft 3 eq{radialfill}{v_ft\r
-4 eq{conicalfill}{v_ft 8 eq{lineargfill}{v_ft 9 eq{ellipticgfill}{v_ft\r
-10 eq{radialgfill}{conicalgfill}ifelse}ifelse}ifelse}ifelse}ifelse}ifelse}ifelse}ifelse\r
-grestore}bd/*u{/p_count p_count 1 add def}bd/*U{/p_count p_count 1\r
-sub def p_count 0 eq{vis_flag{p_render}{newpath}ifelse}if}bd/B{/p_render{a_fp\r
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-% Copyright (c) 1995,1996 Xara Ltd\r
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-v_sg v_sb setlogrgbcolor}ifelse}bd/setgfillcmyk{v_gseps 1 eq{v_mono\r
-1 eq{cmyk2rgb rgb2plategray}{cmyk2rgb rgb2devcmyk setplatecolor}ifelse}{v_mono\r
-1 eq{cmyk2rgb rgb2gray setgray}{setcmykcolor}ifelse}ifelse}bd/setgfillrgb{v_gseps\r
-1 eq{v_mono 1 eq{rgb2plategray}{rgb2devcmyk setplatecolor}ifelse}{v_mono\r
-1 eq{rgb2gray setgray}{systemdict begin setrgbcolor end}ifelse}ifelse}bd\r
-/setgfillhsb{v_gseps 1 eq{v_mono 1 eq{systemdict begin sethsbcolor\r
-currentrgbcolor end rgb2plategray}{systemdict begin sethsbcolor currentrgbcolor\r
-end rgb2devcmyk setplatecolor}ifelse}{v_mono 1 eq{systemdict begin\r
-sethsbcolor currentgray end setgray}{systemdict begin sethsbcolor end}ifelse}ifelse}bd\r
-/Max{2 copy lt{exch}if pop}bd/Max3{2 copy lt{exch}if pop 2 copy lt{exch}if\r
-pop}bd/Min{2 copy gt{exch}if pop}bd/Min3{2 copy gt{exch}if pop 2 copy\r
-gt{exch}if pop}bd/clamp{3 1 roll Max 2 1 roll Min}bd/clamp01{0 Max\r
-1 Min}bd/Pythag{dup mul exch dup mul add sqrt}bd/ssc{DeviceRGB setcolorspace\r
-setcolor}bd/ssg{setgray}bd/p_render{}def/p_count 0 def/vis_flag true\r
-def/DataString 3 string def/DataSrc{currentfile DataString readhexstring\r
-pop}bd/DataStr1 1 string def/DataStr2 1 string def/DataStr3 1 string\r
-def/DataSrc1{DataStr1}bd/DataSrc2{DataStr2}bd/DataSrc3{DataStr3}bd\r
-/colorimage where{pop/ci{colorimage}bd}{/ci{pop pop/ci_datasrc exch\r
-def matrix invertmatrix/ci_matrix exch def pop/ci_dataleft 0 def/ci_buf()def\r
-/ci_dataofs 0 def 0 1 3 -1 roll 1 sub{/ci_y exch def dup 0 1 3 -1\r
-roll 1 sub{0 1 2{pop ci_dataleft 0 eq{ci_datasrc dup length/ci_dataleft\r
-exch def/ci_buf exch def/ci_dataofs 0 def}if ci_buf ci_dataofs get\r
-255 div/ci_dataofs ci_dataofs 1 add def/ci_dataleft ci_dataleft 1 sub\r
-def}for setrgbcolor dup ci_y 3 -1 roll 1 add ci_y 1 add 4 copy 5 1\r
-roll 4 2 roll 5 -1 roll 1 1 4{pop ci_matrix transform 8 2 roll}for\r
-m l l l closepath fill}for}for pop}bd}ifelse/rci{/rciBuf 4 index 3\r
-index mul 7 add 8 div floor cvi string def{currentfile rciBuf readhexstring\r
-pop}bind false 3 ci}bd/cbsl{2 eq/cbslL2 xd 5 index/cbslw xd translate\r
-scale 8 [ 3 index 0 0 5 index 0 0 ] cbslL2{/DataStr1 cbslw string def\r
-currentfile/ASCII85Decode filter/RunLengthDecode filter DataStr1 readstring\r
-pop pop/DataStr2 cbslw string def currentfile/ASCII85Decode filter\r
-/RunLengthDecode filter DataStr2 readstring pop pop/DataStr3 cbslw\r
-string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
-DataStr3 readstring pop pop{DataStr1}bind{DataStr2}bind{DataStr3}bind\r
-true}{/DataSrc load false}ifelse 3 ci}bd/gbsl{2 eq/gbslL2 xd 5 index\r
-/gbslw xd translate scale 8 [ 3 index 0 0 5 index 0 0 ] gbslL2{/DataStr1\r
-gbslw string def currentfile/ASCII85Decode filter/RunLengthDecode filter\r
-DataStr1 readstring pop pop{DataStr1}bind}{/DataStr1 gbslw string def\r
-currentfile DataSrc1 readhexstring pop pop{DataStr1}bind}ifelse image}bd\r
-/cmi{/cmiBuf 4 index 3 index mul 7 add 8 div floor cvi string def{currentfile\r
-cmiBuf readhexstring pop}bind image}bd/cpal{4 mul string/cPalette exch\r
-def currentfile cPalette readhexstring pop}bd/cpci{/cpci_datasrc exch\r
-def matrix invertmatrix/cpci_matrix exch def/cpci_bpp exch def cpci_init\r
-0 1 3 -1 roll 1 sub{/cpci_y exch def dup cpci_bpp 4 eq{cpci_sampsleft\r
-1 eq{/cpci_sampsleft 0 def}if}if 0 1 3 -1 roll 1 sub{cpci_nextcol dup\r
-cpci_y 3 -1 roll 1 add cpci_y 1 add 4 copy 5 1 roll 4 2 roll 5 -1 roll\r
-1 1 4{pop cpci_matrix transform 8 2 roll}for m l l l closepath fill}for}for\r
-pop}bd/cpci_init{/cpci_sampsleft 0 def}bd/cpci_buf 1 string def/cpci_nextcol{cpci_bpp\r
-1 eq{cpci_sampsleft 0 eq{currentfile cpci_buf readhexstring pop pop\r
-/cpci_sampsleft 8 def}if cpci_buf dup 0 get dup 1 and setgray -1 bitshift\r
-1 exch put/cpci_sampsleft cpci_sampsleft 1 sub def}{cpci_bpp 4 eq{cpci_sampsleft\r
-0 eq{currentfile cpci_buf readhexstring pop pop/cpci_sampsleft 2 def}if\r
-cpci_buf 0 get dup 15 and exch -4 bitshift cpci_buf 0 3 -1 roll put\r
-/cpci_sampsleft cpci_sampsleft 1 sub def}{currentfile cpci_buf readhexstring\r
-pop 0 get}ifelse 4 mul dup 2 add cPalette exch get 255 div exch dup\r
-1 add cPalette exch get 255 div exch cPalette exch get 255 div setrgbcolor}ifelse}bd\r
-/setup1asciiproc{[ currentfile mystring/readhexstring cvx/pop cvx\r
-] cvx bind}bd/setup1binaryproc{[ currentfile mystring/readstring cvx\r
-/pop cvx ] cvx bind}bd level2{save/dontloadlevel1 xd}if/iw 0 def/ih\r
-0 def/im_save 0 def/setupimageproc 0 def/polarity 0 def/smoothflag\r
-0 def/mystring 0 def/bpc 0 def/beginimage{/im_save save def dup 0 eq{pop\r
-/setup1binaryproc}{1 eq{/setup1asciiproc}{(error, can't use level2 data acquisition procs for level1)print\r
-flush}ifelse}ifelse/setupimageproc exch ld/polarity xd/smoothflag xd\r
-/imat xd/mystring exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save\r
-restore}bd/1bitbwcopyimage{1 setgray 0 0 moveto 0 1 rlineto 1 0 rlineto\r
-0 -1 rlineto closepath fill 0 setgray iw ih polarity imat setupimageproc\r
-imagemask}bd/1bitcopyimage{setrgbcolor 0 0 moveto 0 1 rlineto 1 0 rlineto\r
-0 -1 rlineto closepath fill setrgbcolor iw ih polarity imat setupimageproc\r
-imagemask}bd/1bitmaskimage{setrgbcolor iw ih polarity [iw 0 0 ih 0\r
-0] setupimageproc imagemask}bd level2{dontloadlevel1 restore}if level2\r
-not{save/dontloadlevel2 xd}if/setup2asciiproc{currentfile/ASCII85Decode\r
-filter/RunLengthDecode filter}bd/setup2binaryproc{currentfile/RunLengthDecode\r
-filter}bd/myimagedict 9 dict dup begin/ImageType 1 def/MultipleDataSource\r
-false def end def/im_save 0 def/setupimageproc 0 def/polarity 0 def\r
-/smoothflag 0 def/mystring 0 def/bpc 0 def/ih 0 def/iw 0 def/beginimage{\r
-/im_save save def dup 2 eq{pop/setup2binaryproc}{dup 3 eq{pop/setup2asciiproc}{0\r
-eq{/setup1binaryproc}{/setup1asciiproc}ifelse}ifelse}ifelse/setupimageproc\r
-exch ld{[ 1 0 ]}{[ 0 1 ]}ifelse/polarity xd/smoothflag xd/imat xd/mystring\r
-exch string def/bpc xd/ih xd/iw xd}bd/endimage{im_save restore}bd/1bitbwcopyimage{1\r
-ssg 0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill\r
-0 ssg myimagedict dup begin/Width iw def/Height ih def/Decode polarity\r
-def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
-1 def/Interpolate smoothflag def end imagemask}bd/1bitcopyimage{ssc\r
-0 0 moveto 0 1 rlineto 1 0 rlineto 0 -1 rlineto closepath fill ssc\r
-myimagedict dup begin/Width iw def/Height ih def/Decode polarity def\r
-/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
-1 def/Interpolate smoothflag def end imagemask}bd/1bitmaskimage{ssc\r
-myimagedict dup begin/Width iw def/Height ih def/Decode polarity def\r
-/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
-1 def/Interpolate smoothflag def end imagemask}bd level2 not{dontloadlevel2\r
-restore}if\r
-level2{save/dontloadlevel1 xd}if/startnoload{{/noload save def}if}bd\r
-/endnoload{{noload restore}if}bd/testsystemdict{where{systemdict eq{true}{false}ifelse}{false}ifelse}bd\r
-/ncolors 1 def/colorimage where{pop true}{false}ifelse{/ncolors 0\r
-statusdict begin/processcolors where{pop pop processcolors}{/deviceinfo\r
-where{pop deviceinfo/Colors known{pop{deviceinfo/Colors get}}if}if}ifelse\r
-end def ncolors 0 ne{/colorimage testsystemdict/setcolortransfer testsystemdict\r
-/currentcolortransfer testsystemdict/currentcmykcolor testsystemdict\r
-and and and not{/ncolors 0 def}if}if}if ncolors dup 1 ne exch dup 3\r
-ne exch 4 ne and and{/ncolors 0 def}if ncolors 1 eq dup dup not startnoload{\r
-/expandbw{expandfactor mul round cvi bwclut exch get 255 div}bd/doclutimage{bwclut\r
-colorclut pop/bwclut xd bpc dup 8 eq{pop 255}{4 eq{15}{3}ifelse}ifelse\r
-/expandfactor xd [/expandbw load/exec load dup currenttransfer exch\r
-] cvx bind settransfer iw ih bpc imat setupimageproc image}bd}if not\r
-endnoload ncolors dup 3 eq exch 4 eq or dup dup not startnoload{/nullproc{{}}def\r
-/concatutil{/exec load 7 -1 roll/exec load}bd/defsubclut{1 add getinterval\r
-def}bd/spconcattransfer{/Dclut exch def/Cclut exch def/Bclut exch def\r
-/Aclut exch def/ncompute exch ld currentcolortransfer [{Aclut ncompute}concatutil\r
-] cvx [{Bclut ncompute}concatutil ] cvx [{Cclut ncompute}concatutil\r
-] cvx [{Dclut ncompute}concatutil ] cvx setcolortransfer}bd/setuprgbcluts{\r
-/bit3x rgbclut length 3 sub def/bit1x bit3x 3 idiv def/rclut rgbclut\r
-def/gclut rclut 1 bit3x defsubclut/bclut rclut 2 bit3x defsubclut}bd}if\r
-not endnoload ncolors 3 eq dup dup not startnoload{/3compute{exch bit3x\r
-mul round cvi get 255 div}bd/doclutimage{/rgbclut xd pop setuprgbcluts\r
-/3compute rclut gclut bclut dup spconcattransfer iw ih bpc imat [\r
-setupimageproc/exec load/dup load dup ] cvx nullproc nullproc true\r
-3 colorimage}bd}if not endnoload ncolors 4 eq dup dup not startnoload{\r
-/stuffclut{cmykindex 3 -1 roll put}bd/ftoint{1 exch sub 255 mul round\r
-cvi}bd/4compute{exch bit4x mul round cvi get 255 div}bd/computecmykclut{setuprgbcluts\r
-/bit4x rgbclut length 3 idiv 4 mul 4 sub def/cmykclut bit4x 4 add\r
-string def/cclut cmykclut def/mclut cclut 1 bit4x defsubclut/yclut\r
-cclut 2 bit4x defsubclut/kclut cclut 3 bit4x defsubclut/cmykindex 0\r
-def 0 1 bit1x{dup/cmykindex exch bit1x exch sub 4 mul def 3 mul dup\r
-rclut exch get 255 div exch dup gclut exch get 255 div exch bclut exch\r
-get 255 div setrgbcolor currentcmykcolor ftoint kclut stuffclut ftoint\r
-yclut stuffclut ftoint mclut stuffclut ftoint cclut stuffclut}for}bd\r
-/doclutimage{/rgbclut xd pop invalidcolortable?{computecmykclut}if\r
-/4compute cclut mclut yclut kclut spconcattransfer iw ih bpc imat\r
-[ setupimageproc/exec load/dup load dup dup ] cvx nullproc nullproc\r
-nullproc true 4 colorimage}bd}if not endnoload ncolors 0 eq dup dup\r
-not startnoload{/lookupandstore{3 mul 3 getinterval putinterval exch\r
-3 add exch 3 copy}bd/8lookup/lookupandstore ld/4lookup{/byte 1 index\r
-def -4 bitshift lookupandstore byte 15 and lookupandstore}bd/2lookup{\r
-/byte 1 index def -6 bitshift lookupandstore byte -4 bitshift 3 and\r
-lookupandstore byte -2 bitshift 3 and lookupandstore byte 3 and lookupandstore}bd\r
-/colorexpand{mystringexp 0 rgbclut 3 copy 7 -1 roll/mylookup load\r
-forall pop pop pop pop pop}bd/createexpandstr{/mystringexp exch mystring\r
-length mul string def}bd/doclutimage{/rgbclut xd pop/mylookup bpc 8\r
-eq{3 createexpandstr/8lookup}{bpc 4 eq{6 createexpandstr/4lookup}{12\r
-createexpandstr/2lookup}ifelse}ifelse ld iw ih bpc imat [ setupimageproc\r
-/exec load/colorexpand load/exec load] cvx false 3 colorimage}bd}if\r
-not endnoload/colorimage where{pop true}{false}ifelse dup{/do24image{iw\r
-ih 8 imat setupimageproc false 3 colorimage}bd}if dup dup startnoload\r
-not{/rgbtogray{/str xd/len str length def/smlen len 3 idiv def/rstr\r
-str def/gstr str 1 len 1 sub getinterval def/bstr str 2 len 2 sub getinterval\r
-def str dup 0 1 smlen 1 sub{dup 3 mul rstr 1 index get .3 mul gstr\r
-2 index get .59 mul add bstr 3 -1 roll get .11 mul add round cvi put\r
-dup}for pop 0 smlen getinterval}bd/do24image{iw ih 8 imat [ setupimageproc\r
-/exec load/rgbtogray load/exec load ] cvx bind image}bd}if endnoload\r
-/doimage{iw ih 8 imat setupimageproc image}bd level2{dontloadlevel1\r
-restore}if level2 not{save/dontloadlevel2 xd}if/myappcolorspace/DeviceRGB\r
-def/rgbclut 0 def/doclutimage{/rgbclut xd pop bpc dup 8 eq{pop 255}{4\r
-eq{15}{3}ifelse}ifelse/hival xd [/Indexed myappcolorspace hival rgbclut]\r
-setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode\r
-[0 hival] def/ImageMatrix imat def/DataSource setupimageproc def/BitsPerComponent\r
-bpc def/Interpolate smoothflag def end image}bd/do24image{myappcolorspace\r
-setcolorspace myimagedict dup begin/Width iw def/Height ih def/Decode\r
-[0 1 0 1 0 1] def/ImageMatrix imat def/DataSource setupimageproc def\r
-/BitsPerComponent 8 def/Interpolate smoothflag def end image}bd level2\r
-not{dontloadlevel2 restore}if\r
-/NumSteps{dtransform matrix defaultmatrix idtransform Pythag currentscreen\r
-pop pop 72 exch div div}bd/FindMinSteps{v_ft 4 eq{urx startX sub abs\r
-llx startX sub abs Max ury startY sub abs lly startY sub abs Max Pythag\r
-2 3.14159265 mul mul 0}{v_ft 2 eq{endY startY sub endX startX sub Pythag\r
-endY2 startY sub endX2 startX sub Pythag gt{endY startY sub endX startX\r
-sub}{endY2 startY sub endX2 startX sub}ifelse}{endY startY sub endX\r
-startX sub}ifelse}ifelse NumSteps}bd/cxe{/v_cxe exch def}bd/cxm{pop\r
-/v_cxm exch def}bd/cxmt{pop pop}bd/cxt{pop}bd/S_eoclip{currentflat{{eoclip}stopped{dup\r
-currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: eoclip]\n)print\r
-flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
-/S_clip{currentflat{{clip}stopped{dup currentflat exch sub 20 gt{([Error: PathTooComplex; OffendingCommand: clip]\n)print\r
-flush exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
-/S_eofill{currentflat{{eofill}stopped{dup currentflat exch sub 20\r
-gt{([Error: PathTooComplex; OffendingCommand: eofill]\n)print flush\r
-exit}{currentflat 2 add setflat}ifelse}{exit}ifelse}loop setflat}bd\r
-/gpbbx{pathbbox/ury exch def/urx exch def/lly exch def/llx exch def}bd\r
-/lineargfill{initgfill{false initgfx/distance endX startX sub endY\r
-startY sub Pythag def/incD distance Steps div def endY startY sub endX\r
-startX sub atan newpath llx lly urx ury Bx startX startY translate\r
-rotate gpbbx eGy ssg newpath llx lly urx ury Bx S_eofill sGy ssg newpath\r
-llx lly 0 ury Bx S_eofill/fillX 0 def 0 1 Steps 1 sub{stepgfx newpath\r
-fillX lly fillX incD add dup/fillX exch def ury Bx S_eofill pop}for}if}bd\r
-/radialgfill{initgfill{false initgfx/distance endX startX sub endY\r
-startY sub Pythag def/incD distance Steps div def eGy ssg newpath llx\r
-lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepgfx newpath\r
-startX startY distance 0 360 arc closepath distance incD add dup/distance\r
-exch def 0 rmoveto startX startY distance 0 360 arc closepath S_eofill\r
-pop}for}if}bd/ellipticgfill{initgfill{true initgfx sGy ssg newpath\r
-llx lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2\r
-startX sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepgfx\r
-/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for}if}bd/conicalgfill{initgfill{urx\r
-startX sub abs llx startX sub abs Max ury startY sub abs lly startY\r
-sub abs Max Pythag startY endY sub startX endX sub Pythag div/radius\r
-exch def true initgfx [ endX startX sub endY startY sub startY endY\r
-sub endX startX sub startX startY ] concat/slice 180 Steps div def\r
-/startangle 0 def Steps -1 0{pop stepgfx 0 0 moveto 0 0 radius startangle\r
-neg dup slice add arc fill 0 0 moveto 0 0 radius startangle dup slice\r
-add dup/startangle exch def arc fill}for}if}bd/initgfill{gpbbx{S_eoclip}{S_clip}ifelse\r
-startX endX eq startY endY eq and sGy eGy eq or dup{sGy ssg newpath\r
-llx lly urx ury Bx S_eofill}if not}bd/initgfx{/flag exch def flag{sGy\r
-eGy/sGy exch def/eGy exch def}if eGy sGy sub abs 256 mul FindMinSteps\r
-Min 256 Min ceiling 1 Max/Steps exch def eGy sGy sub Steps div/incGy\r
-exch def sGy/mGy exch def}bd/stepgfx{mGy ssg mGy incGy add/mGy exch\r
-def}bd/linearfill{initfill{false initfx endX startX sub endY startY\r
-sub Pythag/distance exch def/incD distance Steps div def endY startY\r
-sub endX startX sub atan newpath llx lly urx ury Bx startX startY translate\r
-rotate gpbbx v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse\r
-newpath llx lly urx ury Bx S_eofill v_cxe 0 eq{sR sG sB setgfillrgb}{sH\r
-sS sV setgfillhsb}ifelse newpath llx lly 0 ury Bx S_eofill/fillX 0\r
-def 0 1 Steps 1 sub{stepfx newpath fillX lly fillX incD add dup/fillX\r
-exch def ury Bx S_eofill pop}for}if}bd/radialfill{initfill{false initfx\r
-endX startX sub endY startY sub Pythag/distance exch def/incD distance\r
-Steps div def v_cxe 0 eq{eR eG eB setgfillrgb}{eH eS eV setgfillhsb}ifelse\r
-newpath llx lly urx ury Bx S_eofill/distance 0 def 0 1 Steps 1 sub{stepfx\r
-newpath startX startY distance 0 360 arc closepath distance incD add\r
-dup/distance exch def 0 rmoveto startX startY distance 0 360 arc closepath\r
-S_eofill pop}for}if}bind def/ellipticalfill{initfill{true initfx v_cxe\r
-0 eq{sR sG sB setgfillrgb}{sH sS sV setgfillhsb}ifelse newpath llx\r
-lly urx ury Bx S_eofill [ endX startX sub endY startY sub endX2 startX\r
-sub endY2 startY sub startX startY ] concat Steps 1 sub -1 0{stepfx\r
-/i exch def 0 0 moveto 0 0 i Steps div 0 360 arc fill}for v_cxe 0\r
-eq{sR sG sB eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def\r
-/eG exch def/eR exch def}if}if}bd/conicalfill{initfill{urx startX\r
-sub abs llx startX sub abs Max ury startY sub abs lly startY sub abs\r
-Max Pythag startY endY sub startX endX sub Pythag div/radius exch def\r
-true initfx [ endX startX sub endY startY sub startY endY sub endX\r
-startX sub startX startY ] concat/slice 180 Steps div def/startangle\r
-0 def Steps -1 0{pop stepfx 0 0 moveto 0 0 radius startangle neg dup\r
-slice add arc fill 0 0 moveto 0 0 radius startangle dup slice add dup\r
-/startangle exch def arc fill}for}if}bd/initfill{gpbbx{S_eoclip}{S_clip}ifelse\r
-startX endX eq startY endY eq and v_cxe 2 ne sR eR eq sG eG eq and\r
-sB eB eq and and or dup{sR sG sB setgfillrgb newpath llx lly urx ury\r
-Bx S_eofill}if not}bd/initfx{/flag exch def v_cxe 0 eq{flag{sR sG sB\r
-eR eG eB/sB exch def/sG exch def/sR exch def/eB exch def/eG exch def\r
-/eR exch def}if eR sR sub abs 256 mul eG sG sub abs 256 mul Max eB\r
-sB sub abs 256 mul Max FindMinSteps Min 256 Min ceiling 1 Max/Steps\r
-exch def/incR eR sR sub Steps div def/incG eG sG sub Steps div def\r
-/incB eB sB sub Steps div def sR/mR exch def sG/mG exch def sB/mB\r
-exch def}{sR sG sB rgb2hsb/sV exch def/sS exch def/sH exch def eR eG\r
-eB rgb2hsb/eV exch def/eS exch def/eH exch def eH sH sub abs v_cxe\r
-1 eq{dup 0.5 gt{1 exch sub}if}{dup 0.5 lt{1 exch sub}if}ifelse 256\r
-mul eS sS sub abs 256 mul Max eV sV sub abs 256 mul Max FindMinSteps\r
-Min 256 Min ceiling 1 Max/Steps exch def v_cxe 1 eq{/incH eH sH sub\r
-dup abs 0.5 gt{dup 0 ge{1 sub}{1 add}ifelse}if Steps div def/incS eS\r
-sS sub Steps div def/incV eV sV sub Steps div def}{/incH eH sH sub\r
-dup abs 0.5 le{dup 0 ge{1 sub}{1 add}ifelse}if Steps div def/incS eS\r
-sS sub Steps div def/incV eV sV sub Steps div def}ifelse flag{/sH eH\r
-/eH sH def def/sS eS/eS sS def def/sV eV/eV sV def def/incH incH neg\r
-def/incS incS neg def/incV incV neg def}if}ifelse}bd/stepfx{v_cxe 0\r
-eq{mR mG mB setgfillrgb mR incR add/mR exch def mG incG add/mG exch\r
-def mB incB add/mB exch def}{sH sS sV setgfillhsb sH incH add dup 0\r
-le{1 add}{dup 1 ge{1 sub}if}ifelse/sH exch def sS incS add/sS exch\r
-def sV incV add/sV exch def}ifelse}bd\r
-/ar{}bd/arr{pop pop pop pop pop pop pop}bd/ae{pop pop pop pop pop\r
-pop}bd/aoa{pop}bd/apl{pop}bd/apc{}bd/aof{pop pop}bd/aafs{pop pop pop}bd\r
-/O{pop}bd/R{pop}bd/axop{pop pop pop pop}bd/g{/v_ft 0 def/v_fc 0 def\r
-/v_fm 0 def/v_fy 0 def 1 exch sub/v_fk exch def/v_fct 0 def}bd/G{\r
-/v_sc 0 def/v_sm 0 def/v_sy 0 def 1 exch sub/v_sk exch def/v_sct 0\r
-def}bd/k{/v_fk exch def/v_fy exch def/v_fm exch def/v_fc exch def/v_ft\r
-0 def/v_fct 0 def}bd/K{/v_sk exch def/v_sy exch def/v_sm exch def/v_sc\r
-exch def/v_sct 0 def}bd/Xa{/v_fb exch def/v_fg exch def/v_fr exch def\r
-/v_ft 0 def/v_fct 1 def}bd/XA{/v_sb exch def/v_sg exch def/v_sr exch\r
-def/v_sct 1 def}bd/a_tc{exch pop 1 exch sub dup 3 -1 roll exch div\r
-exch dup 4 -1 roll exch div 3 1 roll dup 5 -1 roll exch div 4 1 roll\r
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-\input lecnotes.tex
-
-\prednaska{9}{Geometrické algoritmy}{(B. Maslowski, J. Návrat, M. Sta¹a)}
-
-\>Budeme se teï bavit o geometrických problémech v~rovinì. Vìt¹ina algoritmù,
-které zde uvedeme, má sice své obdoby i pro prostory vy¹¹í nebo ni¾¹í dimenze,
-ale jednorozmìrné pøípady bývají triviální a vícerozmìrné jsou zase vìt¹inou
-moc slo¾ité.
-
-Budeme se tedy zabývat tím, jak tyto problémy øe¹it v~dimenzi dva
-(v~Euklidovské rovinì).
-
-\h{Hledání konvexního obalu}
-
-Ptáte se o co pùjde? Zkusme si to pøiblí¾it na problému ledních medvìdù :) {\I
-Lední medvìdi si po dlouhé dobì zmapovali vody severního moøe a zjistili
-pøesnì místa, kde se nacházejí jejich oblíbené ryby. No a proto¾e to jsou
-medvìdi chytøí, rozhodli se v¹echny tyto rybky pochytat najednou do jedné
-velké sítì. A problém, který tady mají, je následující: jaký nejmen¹í obvod
-mù¾e mít taková sí», aby se dovnitø ve¹ly v¹echny rybky?!}
-
-Neboli budeme øe¹it, jak nìjakou zadanou mno¾inu bodù v~rovinì obalit co
-nejkrat¹í uzavøenou køivkou, do které se je¹tì v¹echny body vejdou.
-
-Intuice nám napovídá ¾e výsledek bude nìjaký konvexní\foot{Mno¾ina bodù
-v~rovinì je konvexní, pokud platí, ¾e pro ka¾dé dva body této mno¾iny le¾í
-úseèka spojující tyto dva body také celá v~této mno¾inì.} mnohoúhelník, který
-bude mít vrcholy v~nìkterých uvedených bodech. Ostatní vrcholy pak budou buï
-nìkde na hranách mnohoúhelníku, nebo uvnitø. Tomuto mnohoulehníku se øíká {\I
-konvexní obal} dané mno¾iny.
-
-\>Mo¾ná by se teï hodilo pøedvést názornì, jak vypadají nejmen¹í konvexní
-obaly:
-
-\figure{ZakladniObaly.eps}{Základní obaly.}{3in}
-
-\itemize\ibull
-
-\:Konvexní obal prázdné mno¾iny je prázdná mno¾ina.
-
-\:Konvexní obal 1 bodu je bod samotný.
-
-\:Konvexní obal 2 bodù je úseèka spojující tyto body.
-
-\:Konvexní obal 3 bodù je trojúhleník s vrcholy v~tìchto bodech.
-
-\:Konvexní obal 4 bodù \dots to u¾ je slo¾itìj¹í\dots
-
-\endlist
-
-Konvexní obaly 4 a více bodù, jak si mù¾eme v¹imnout, u¾ nejsou jednoznaèné.
-Pro $N$-prvkovou mno¾inu bude konvexní obal mnohoúhelník se tøemi a¾ $N$
-vrcholy.
-
-Jeden dobrý zpùsob, jak tento konvexní obal sestrojit se nazývá {\I Zametání
-roviny.}
-
-Algoritmus funguje tak, ¾e si v~rovinì zvolíme nìjaký smìr, a v~tomto smìru
-zaèneme posouvat pøímku. Budeme takto potkávat body le¾ící v~na¹í mno¾inì.
-V~ka¾dém okam¾iku budeme chtít, aby body v~èásti, kterou jsme ji¾ zametli, u¾
-mìli spoèítaný konvexní obal. V¾dy kdy¾ pak zametací pøímkou narazíme na nový
-bod, u¾ si jen rozmyslíme, jak ho do konvexního obalu pøidat.
-
-BÚNO pøedpokládáme body v~obecné poloze, tedy takové, ¾e ¾ádné tøi nele¾í
-na~jedné pøímce. Dále také budeme pøedpokládat, ¾e budeme zametat ve smìru
-$x$-ové osy a ¾e v¹echny body mají rùznou $x$-ovou souøadnici.
-
-Je také vidìt, ¾e bod s nejmen¹í a nejvìt¹í $x$-ovou souøadnicí bude le¾et
-na~konvexním obalu.
-
-\s{My¹lenka algoritmu:}
-
-\algo
-
-\:Setøídíme body podle jejich $x$-ové souøadnice.
-
-\:Vezmeme první tøi body a sestrojíme jejich konvexní obal.
-
-\:Opakuj: Najdeme dal¹í bod a podíváme se, jestli ho mù¾eme do konvexního
-obalu rovnou pøidat: \::Pokud jej mù¾eme rovnou pøidat, tak jej pøidáme.
-
-\::Pokud jej pøidat rovnou nemù¾eme, pak je potøeba nejdøíve nìjaké body
-odzadu odebrat a pak teprve pøipojit ná¹ nový bod. \endalgo
-
-Ka¾dá iterace tedy bude probíhat tak, ¾e nìjaké body z pùvodního konvexního
-obalu pozapomínáme a pøidáme nový bod. Aby se to lépe popisovalo, tak celý
-konvexní obal rozdìlíme na {\I horní obálku} a {\I dolní obálku.}
-
-\figure{HD-obalka.eps}{Obrázek obálek.}{3in}
-
-Vidíme teï, ¾e dolní obálka je nìjaká lomená èára, která zatáèí doleva. Horní
-obálka zatáèí doprava. V~na¹em algoritmu si budeme obálky pamatovat jako dva
-seznamy vrcholù. Kdy¾ pak v~algoritmu narazíme na nový bod, budeme zvlá¹»
-øe¹it jak to ovlivní horní obálku a jak ovlivní dolní. Je vidìt ¾e bod nejvíce
-vlevo a bod nejvíce vpravo le¾í v obou obálkách. Ostatní body buï le¾í jen
-v~jedné z obálek, nebo nele¾í v~¾ádné z nich (tedy nejsou souèástí konvexního
-obalu).
-
-\s{Algoritmus:}
-
-\algo
-
-\:Setøídíme body podle souøadnice $x$, dostaneme mno¾inu bodù $b_1~-~b_n$.
-\:Spoèítáme konvexní obal ${b_1, b_2, b_3}$, z toho získáme horní a dolní
-obálku\foot{Body $b_1$ a $b_3$ budou v obou obálkách. Bod $b_2$ bude v~horní
-obálce pokud le¾í nad pøímkou spojující $b_1$ a $b_3$, v~dolní obálce bude
-pokud le¾í pod pøímkou.}. \:Pro $b$ postupnì zpracováváme $b_3 - b_n$:
-
-\::Pøepoèítáme Horní obálku:
-
-\:::Dokud $(\vert H\vert \geq 2)$ a úhel $(H[-2], H[-1], b)$ je orientovaný
-doleva: \::::Odebereme poslední prvek z obálky.
-
-\:::Pøidáme do obálky nový vrchol.
-
-\::Pøepoèítáme dolní obálku:
-
-\:::Dokud $(\vert D\vert \geq 2)$ a úhel $(D[-2], D[-1], b)$ je orientovaný
-doprava: \::::Odebereme poslední prvek z obálky.
-
-\:::Pøidáme do obálky nový vrchol.
-
-\endalgo
-
-Setøídit body podle $x$-ové souøadnice a sestrojit konvexní obal prvních tøech
-bodù stihneme v~èase $\O(n \log n)$. Zbytek pak u¾ udìláme dokonce v~èase
-lineárním $\O(n)$\foot{V této èásti u¾ jen do obálek pøidáváme a odebíráme
-body.Pøidáváme jich $N$. A odebrat jich mù¾eme maximálnì tolik kolik jsme jich
-pøidali. Tedy zase maximálnì~$N$.}. Platí tedy:
-
-\s{Vìta:} Konvexní obal doká¾eme sestrojit v~èase $\O(n \log n)$.
-
-\>Na¹i lední medvìdi se tedy ji¾ nauèili, jak si efektivnì obstarat potravu a
-mohly se pustit do øe¹ení dal¹ího velmi dùle¾itého problému. Pojïme se na nìj
-podívat s nimi. \>A o co ¾e to pùjde?
-
-{\I Lední medvìdi nejsou na antarktidì sami, kromì nich tam taky bydlí
-kamarádi eskymáci ve svých iglù. Medvìdi by si teï rádi udìlali mapu, podle
-které by hned poznali, ke kterému ekymákovi to mají nejblí¾e na náv¹»evu.}
-
-My tuhle medvìdí mapu od teï budeme nazývat {\I Voroneho diagramem}.
-
-\h{Voroného diagramy}
-
-Pøed tím, ne¾ vás vystra¹ím nìjakou definicí, si øekneme, co jsi pod tímto, na
-první pohled ne zøejmým pojmem, pøedstavit. Mìjme mno¾inu teèek $T$
-rozmístìných náhodnì po papíru. Ke ka¾dému bodu nakreslíme okraje tak, aby
-vniklá plo¹ka obsahovala body, které jsou nejblí¾e právì té na¹í vybrané
-teèce. Samozøejmì \uv{sousední} teèky budou mít tyto hranice spoleèné.
-Výsledkem na¹eho dlouhého sna¾ení pak bude právì Voroného diagram. V dal¹ích
-odstavcích se budeme zajímat o to, jak takový útvar správnì popsat, jak ho
-sestrojit a jaké datové struktury k tomu pou¾ít.
-
-\s{Definice:} {\I Voroného diagram} pro koneènou mno¾inu $M = \{m_1, \dots,
-m_n\} \in {\bb R}^2$ míst je systém mno¾in $O_1,\dots,O_n$ takových, ¾e pro
-v¹echna $i$ a $j$ a pro v¹echna $x \in M_i$ je vzdálenost $x$ od $m_i$ men¹í
-nebo rovna vzdálenosti $x$ od $m_j$ a zároveò sjednocení $O_i$ pøes v¹echna
-$i$ je celý prostor ${\bb R}^2$, neboli:
-
-$$d(x,m_i) \leq d(x,m_j) \wedge {\bigcup}_i O_i = {\bb R}^2.$$
-
-Jednoduchý Voroného diagram:
-
-\figure{voroneho2.eps}{Voroneho diagramu pro dvì místa.}{3in}
-
-Voroného diagram se tedy skládá z nìjakých míst, oblastí a hran, které ty
-oblasti oddìlují.
-
-\figure{voronoi.eps}{Èásti Voroneho diagramu.}{2in}
-
-\s{Definice:} Øekneme, ¾e {\I hrana} $H$ je taková mno¾ina bodù, ¾e pro ka¾dý
-bod $x \in H$ platí, ¾e existují dvì místa $m_i$ a $m_j$, od kterých má bod
-$x$ stejnou vzdálenost. Tyto dvì místa jsou pro v¹echny body $x$ stejná a
-platí, ¾e v¹echny ostatní místa mají od ka¾dého bodu $x$ del¹í vzdálenost.
-
-\s{Definice:} Øekneme, ¾e {\I vrchol} je takový bod, kde se potkávají alespoò
-dvì hrany.
-
-\s{Pozorování:}
-
-\itemize\ibull
-
-\:Voroneho diagramem pro dvì místa jsou dvì poloroviny odìlené takovou pøímkou,
- ¾e ka¾dý bod pøímky je stejnì vzdálený od obou míst. \:Ka¾dá mno¾ina $M_i$ je
-ohranièena konvexní lomenou èarou, tak¾e oblasti mají tvar konvexních
-mnohoúhelníkù, ale je mo¾né, ¾e jsou oteveøené do nekneèna. \:Pro ka¾dou hranu
-$h$ ve Voroného diagramu existuje $i$ a $j$ takové, ¾e kdy¾ $x \in H$, pak
-vzdálenost $d(x,m_i) = d(x,m_j)$. \:Pro ka¾dý vrchol $v$ Voroného diagramu
-existují alespoò tøi místa le¾ící na kru¾nici se støedem $v$.
-
-\figure{body.eps}{Body na kru¾nici.}{3in}
-
-\:Poèet vrcholù a hran je lineární k poètu míst\foot{Voroneho diagram si lze
-pøedstavit jako graf, kde místa Voroneho diagramu odpovídají stìnám, vrcholy
-diagramu vrcholùm a hrany odpovídají hranám grafu. Pokud si teï nìkam mimo
-graf pøidáme je¹tì jeden vrchol a v¹echny pøímky vedoucí do nekoneèna navedeme
-do toho bodu, vidíme, ¾e ná¹ graf je roviný. Pro roviný graf platí Eulerova
-formule a z ní u¾ plyne ¾e na¹e linearita.}. \:Poèet krajních oblastí je tak
-velký, jak velký je konvexní obal té zadané mno¾iny. (Je to dobré vìdìt, ale
-asi to nebudeme potøebovat.)
-
-\endlist
-
-Pojïme teï vymyslet, jak takový diagram vyrobit. Mohli bychom zkonstruovat
-v¹echny dìlící pøímky a poslepovat je, ale vznikl by nám kvadratický
-algoritmus a to nám nemù¾e staèit.
-
-Mluvili jsme o zametání roviny, a tak bychom tento trik mohli vyu¾ít právì pøi
-øe¹ení na¹eho problému. Ov¹em tentokrát má zametání jeden podstatný háèek.
-Kdy¾ si vezmeme nìjakou zametací pøímku a pojedeme s ní shora dolù, tak nad ní
-máme nìjakou u¾ zkonstruovanou èást diagramu a kdy¾ narazíme na dal¹í bod, tak
-se nám mù¾e právì tato èást diagramu pomìrnì slo¾itì zmìnit. Pomù¾eme si malým
-trikem. Nebudeme pova¾ovat za hotovou celou oblast nad zametací pøímkou, ale
-jen takové body, které mají blí¾e k místùm ($m_i$) ne¾ k zametací pøímce. Tak
-dostaneme nìjakou posloupnost (mno¾inu) parabol. V¹echno, co jsme spoèítali
-uvnitø této oblasti nám u¾ nikdo nepokazí (ani nevylep¹í), je tam bezpeèno.
-Vezmeme si tedy dolní obálku tìchto parabol, budeme jí øíkat {\I pobøe¾í}.
-
-\figure{pobrezi.eps}{Pobøe¾ní linie.}{3in}
-
-Pobøe¾í je tedy nìjaká posloupnost parabolických obloukù s tím, ¾e nejlevìj¹í
-a nejpravìj¹í jdou do nekoneèna. Prùseèíky tìchto obloukù vykreslují hrany
-diagramu. Proè? Odpovìï na tuto otázku není tì¾ká, staèí vyjít z definice
-paraboly tak, jak jí zde pou¾íváme. Nebo-li je to mno¾ina bodù, která je od
-ohniska (pro nás místa) stejnì vzdálená jako od pøímky (øídící). A tedy prùnik
-dvou parabol je místo, které je stejnì vzdálené od obou ohnisek, co¾ je
-vlastnì definice bodu le¾ícího na nìjaké hranì.
-
-Kdykoliv v prùbìhu zametání narazíme na nìjaký bod, mù¾e nám ovlivnit u¾ jen
-èást diagramu pod pobøe¾ím. Dostáváme se tedy k tomu, co se dìje, kdy¾ hýbeme
-zametací pøímkou. Pakli¾e nenará¾íme na ¾ádné body, tak se v podstatì nedìje
-nic zajímavého. Zajímavá situace nastává a¾ tehdy, narazíme-li na dal¹í bod.
-V~tom okam¾iku vzniká nová parabola. V tuto chvíli je znaènì degenerovaná. Je
-to toti¾ zatím jen polopøímka kolmá na zametací pøímku. S dal¹ím pohybem se
-zaène parabola rozevírat. V¹imnìme si, ¾e prùnik oné pøímky a pobøe¾í je
-vlastnì vrchol Voroného diagramu.
-
-Mù¾e nastat je¹tì jeden problém. Nìjaká parabola se mù¾e natolik rozevøít, ¾e
-pohltí jiné a ty zmizí z pobøe¾ní linie. V takovém pøípadì, se nám ale
-netriviálnì zmìní vzhled pobøe¾í, a proto se této situaci budeme muset více
-vìnovat.
-
-Shrneme-li na¹e úvahy, mohou se dít celkem tøi vìci. Jedna z nich je posun
-pøímky. To se vlastnì dìje poøád. Pobøe¾í se témìø nemìní a prùseèíky parabol
-nám kreslí hrany. To mù¾eme poèítat najednou. Navíc nejen ¾e bychom mohli s
-pøímkou skoèit o nìjaké epsilon, ale my dokonce mù¾eme skoèit o poøádný kus a
-prostì pouze dopoèítat, jak se pobøe¾í zmìnilo a co se vykreslilo. Dùle¾itým
-místùm, kde se budeme zastavovat, budeme øíkat {\I události}.
-
-\>{\I Místní událost}
-
-Pokud narazíme na bod, musíme najít místo, kde pobøe¾í rozetnou a kam vklínit
-dal¹í výbì¾ek (parabolu). Takovéto události budeme øíkat místní událost.
-
-\figure{mistni.eps}{\vbox{\hsize=0.6\hsize\leftskip=0pt plus
-0.3\hsize\rightskip=\leftskip\parfillskip=0pt \>Místní událost -- èervená
-kolmice je novì vznikající parabola, pøi postupu zametací pøímky dále se bude
-rozevírat a vytvoøí dal¹í parabolu.}}{3in}
-
-\>{\I Kru¾nicová událost}
-
-Poslední situace, která mù¾e nastat, je, ¾e se nìjaká parabola schová za jiné.
-Kouknìme se na první obrázek ní¾e, fialový bod je støed kru¾nice opsané
-trojùhelníku tvoøenému tøemi místy. Jak víme, ten le¾í na osách stran takového
-trojùhelníku. Po tìchto osách se v¹ak budou i pohybovat prùseèíky parabol a s
-posunováním øídící pøímky se pak dostanou v¹echny tøi do støedu této kru¾nice.
-Stane se to pravì tehdy, kdy¾ se zametací pøímka dotkne kru¾nice zespodu. Je
-mo¾né nahlédnout, ¾e pøi postupu dále se pak dvì krajní paraboly roz¹íøí
-natolik, ¾e prostøední pohltí a ta zanikne. Takto vzniklé události budeme
-øíkat kru¾nicová.
-
-Pojïme si to ukázat lépe na následujících dvou obrázcích. První pøedstavuje
-situaci pøed kru¾nicovou událostí a druhý právì kru¾nicovou událost. Mimo jiné
-by tedy z obrázkù mìlo být patrné, ¾e kru¾nicové události jsou urèeny
-trojicemi sousedních obloukù v pobøe¾í.
-
-\figure{kruznicova.eps}{Pøed kru¾nicovou událostí.}{3in}
-
-\figure{kruznicovakonec.eps}{Kru¾nicová událost.}{3in}
-
-\s{Datové struktury}
-
-Budeme potøebovat haldu událostí (místních i kru¾nicových dohromady).
-
-Dále bude zapotøebí udr¾ovat si pobøe¾ní linii, neboli posloupnost míst
-v~ohniscích parabolických obloukù. Zde je potøeba si definovat operace {\I
-Insert, Delete} a {\I FindX}, jinak øeèeno pøidat a odebrat oblouk a najít
-oblouk podle x-sové souøadnice nebo-li oblouk do kterého jsme se trefili pøi
-místní události. Navíc budeme potøebovat vyhledávací strom nad prùseèíky s
-implicitní reprezentací, co¾ znamená, ¾e si ve vrcholech nebudeme pamatovat
-pøímo souøadnice prùseèíkù, ale jen instrukci, jak je spoèítat. Tak¾e jakkoli
-se mìní poloha prùseèíkù, tak struktura stromu zùstává stejná.
-
-S haldou událostí lze pracovat s logaritmickou èasovou slo¾itostí na operaci
-a ve stejném èase doká¾eme pracovat s pobøe¾ní linií. Kdy¾ si pobøe¾í je¹tì
-reprezentujeme zvlá¹» jako seznam tak Insert a Delete budou v konstatním èase
-a operace se stromem pak $\O(\log n)$.
-
-Poslední datovou strukturou bude samotný diagram, reprezentovaný grafem se
-souøadnicemi a vazbami hran na prùseèík.
-
-\s{Fortunùv~algoritmus}
-
-\algo
-
-\:Pøipravíme si haldu $H$ a vlo¾íme do ní v¹echny místní události.
-
-\:Pøipravíme pobøe¾ní linii $P \leftarrow 0$.
-
-\:Dokud existuje $h \leftarrow DeleteMin(H)$:
-
-\::Je-li na øadì místní událost:
-
-\:::Najdeme prùseèík s $P(FindX(P))$.
-
-\:::Vlo¾íme do $P$ novou parabolu.
-
-\:::Poznamenáme do $D$ vznik hran.
-
-\:::Pøepoèítáme kru¾nicové události.
-
-\::Je-li na øadì kru¾nicová událost:
-
-\:::Sma¾eme oblouk z $P$.
-
-\:::Poznamenáme vznik a zánik do $D$.
-
-\:::Pøepoèítáme kru¾nicové události.
-
-\endalgo
-
-\s{Slo¾itost:}
-
-Poèet místních událostí je roven $n$ (na ka¾dé místo narazíme právì jednou).
-Poèet kru¾nicových událostí není vìt¹í ne¾ $n$, proto¾e kru¾nicová událost sma¾e
-parabolu a ty vznikají jen pøi místních událostech, tak¾e kru¾nicových událostí
-není více ne¾ místních. Speciálnì z toho plyne, ¾e velikost pobøe¾ní linie je
-v¾dy lineární, proto¾e s ka¾dou místní událostí pøibudou dva úseky do pobøe¾ní
-linie, tak¾e velikost pobøe¾ní linie je maximálnì $2n$. Velikost haldy je pak
-také $2n$, tak¾e pak operace urèitì zvládneme v èase $\O(\log n)$. Jeliko¾
-diagram je lineárnì velký tak i jeho struktura je lineárnì velká. Operace se
-strukturami nás stojí nejvíce $\O(\log n)$. Tak¾e místní i kru¾nicové události
-zvládneme v èase $\O(\log n)$ na jednu na konstantním poètu struktur. Halda má
-velikost $2n$, tak¾e maximálnì provedeme $\O(2n\log n)$ operací.
-Celý algoritmus potøebuje na~inicializaci maximálnì $\O(n \log n)$ (i kdybychom
-ji dìlali neefektivnì) a $\O(2n\log n)$ výpoèet.
-
-Pokud tedy shrneme v¹echny na¹e odhady, pak èasová slo¾itost algoritmu je
-$\O(n \log n)$ a prostorová $\O(n)$.
-
-\bye
-
--------------------
-
-1) Pøipravíme si haldu $H$ a vlo¾íme do ní v¹echny místní události.
-
-2) pøipravíme pobøe¾ní linii P <- 0 } O(n\log n)
-
-3) pøipraváme reprezentaci diagramu D /
-
-4) dokud existuje h <- DeleteMin(H)
-
-5) je-li na øadì místní událost: ----
-
-a) najdeme prùseèík s P(FindX(P)) \
-
-b) vlo¾íme do P novou parabolu \
-
-c) poznamenáme do D vznik hran } O(\log n)
-
-d) pøepoèítáme kru¾nicové události / } <= 2n
-
-6) je-li na øadì kru¾nicová událost:
-
-a) sma¾eme oblouk z P \
-
-b) poznamenáme vznik a zánik do D } O(\log n)
-
-c) pøepoèítáme kru¾nicové události / ----
-
-\bye
+++ /dev/null
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projects / ads2.git / commitdiff
