--- /dev/null
+%%%%%%%%%%%%%%%%%%%%%%%\r
+% Zápisek prvého semináøe z grafových algoritmù - ze dne ? 7.6.2006 ?\r
+% Téma Toky v sítích a zejména Ford-Fulkersonùv algoritmus.\r
+% Zapsal Radovan ©esták - radovan.sestak@gmail.com\r
+%%%%%%%%%%%%%%%%%%%%%%\r
+\r
+\input ../sgr.tex\r
+\r
+\prednaska{1}{Toky v sítích a FF algoritmus}{zapsal Radovan ©esták}\r
+\r
+\h{Toky v sítích}\r
+\r
+V následujícím odstavci uká¾eme nìkolik tvrzení o sítích, které je\r
+mo¾né si pøedstavit jako sí» potrubí køí¾ící se v uzlech. Vrchol\r
+z kterého se kapalina ¹íøí nazveme zdroj a vrchol kde odtéká stok. Pro\r
+ostatní vrcholy platí, ¾e objem kapaliny pøitékající se rovná objemu\r
+kapaliny odtékající. Toto pravidlo, které pochází z teorie o elektrických obvodech,\r
+se zvykne nazývat Kirchhoffùv zákon. Hledání maximálního toku je duální úlohou k hledání \r
+minimálního øezu. Ford-Fulkersonùv(FF) algoritmus, který bude popsán ní¾e je základním \r
+algoritmem pro hledání tokù. Na úlohu hledání maximálního toku se dají pøevést mnohé\r
+grafové problémy a i proto existuje mnoho algoritmù a nìkteré z nich jsou pouze modifikace \r
+základního FF algoritmu. V dal¹í pøedná¹ce najdete Dinitzùv algoritmus pro tenhle problém.\r
+\r
+\r
+\s{Definice:}\r
+\r
+\itemize\ibull\r
+\:sí»: $N=(V,E,s,t,w)$ kde $V$ je mno¾ina vrcholù, $E\subseteq V\times V$,\r
+$s\in V$ je zdroj, $t\in V$ je stok, $w:\, E\rightarrow{R}^{+}$ jsou kapacity hran\r
+\:tok: $f:\, E\rightarrow{R}^{+}$kde $\forall e\in E\, f(e)\leq w(e)$ a\r
+$\forall v\in V,\, v\neq s,\, v\neq t\,\,\sum_{uv\in E}f(uv)=\sum_{vw\in E}f(vw)$\r
+\:velikost toku: $\|f\|=\sum_{sv\in E}f(sv)-\sum_{vs\in E}f(vs)$ kde $s$ je zdroj\r
+\:øez: $st$-øez je mno¾ina hran $C\subseteq E$ taková, ¾e v grafu $(V,\, E\setminus C)$ neexistuje\r
+cesta z s do t\r
+\:separátor: $st$-separátor je mno¾ina vrcholù $S\subseteq V$ taková, ¾e v grafu\r
+$(V\setminus S,\, E\cap(V\setminus S)\times(V\backslash S))$ neexistuje\r
+cesta z s do t\r
+\:velikost øezu: $\|c\|=\sum_{e\in c}w(e)$ je velikost øezu\r
+\endlist\r
+\r
+\figure{graf.eps}{Pøíklad orientovaného grafu ze zvoleným zdrojem a stokem.}{0.4\hsize}\r
+\r
+\s{Vìta(Ford-Fulkerson) o souvislosti tokù a øezù}\r
+Nech» $N=(V,E,s,t,w)$ je sí», $F$ je mno¾ina pøípustných tokù pro tuto sí» a $C$ je\r
+mno¾ina øezù oddìlující zdroj od stoku. Pak \r
+$$max_{f\in F}|f|=min_{c\in C}\|c\|$$. \r
+Vìta platí pro v¹echny výbìry zdrojù a stokù $\forall s,t\in V, s\neq t$ nech» $st-F$ je mno¾ina\r
+v¹ech tokù ze zdroje $s$ do stoku $t$ a $st-C$ je mno¾ina v¹ech øezù oddìlujících $s$ od $t$. Pak\r
+$$max_{f\in st-F}\|f\|=min_{c\in st-C}\|c\|$$\r
+pro graf $G=(V,E,w)$.\r
+\r
+\s{Dùkaz vìty:}\r
+\r
+\s{$max_{f}\|f\|\leq min_{c}\|c\|$:} Vìznìme libovolný øez oddìlující zdroj od stoku $c$ a uva¾me\r
+graf $G'=(V,E\setminus c)$, který vznikl odebráním hran nacházejících se v øezu. Buï $S$ mno¾ina \r
+vrcholù dosa¾itelných ze zdroje v $G'$.\r
+Pro libovolnej tok $f$ platí: \r
+\r
+$$\|f\| = \sum_{uv\in E, u\in S, v\notin S}f(uv) - \sum_{uv\in E, u\in S, v\notin S}f(vu)$$\r
+velikost toku je rovna rozdílu velikostí tokù pøes hrany opou¹tìjící $S$ a pøicházející do $S$.\r
+$$\|f\| = \sum_{u\in V}f(su) - \sum_{u\in V}f(us)$$\r
+proto¾e se tok zachovává ve v¹ech vrcholech partity $S$ kromì $s$ dostáváme\r
+$$= \sum_{v\in S}(\sum_{u\in V}f(uv)-\sum_{u\in V}f(vu))$$\r
+$f(uv)$ pro $u,v\in S$ jsme jednou pøièítali a jednou odèítali\r
+$$= \sum_{uv\in E, u\in S, v\notin S}f(uv) - \sum_{uv\in E, u\in S, v\notin S}f(vu)$$\r
+\r
+Teï si u¾ staèí jenom uvìdomit, ¾e \r
+$$\|f\| = \sum_{uv\in E, u\in S, v\notin S}f(uv)- \sum_{uv\in E, u\in S, v\notin S}f(vu) \leq \sum_{uv\in E, u\in S, v\notin S}f(uv)\r
+\leq \sum_{uv\in E, u\in S, v\notin S}w(uv) \leq \|c\|$$\r
+Poslední nerovnost je dùsledkem toho, ¾e øez obsahuje v¹echny hrany opou¹tìjící $S$.\r
+\r
+$max_{f}|f|\geq min_{c}|c|$: dùkaz plyne z korektnosti\r
+Ford-Fulkersonova algoritmu viz. algoritmus dále\r
+\r
+\s{Dùsledek:}Pro ka¾dou sí» s celoèíselnými kapacitami je její maximální tok celoèíselný.\r
+\r
+\h{Maximální párování v bipartitním grafu}\r
+Maximální párování v bipartitním grafu $(A\cup B,E)$ se dá najít\r
+pomocí maximálního toku v síti $(A\cup B\cup s\cup t,E',s,t,w)$ kde\r
+$w(e)=1$ a $E'=\{ uv\| u\in A, uv\in E\} \cup \{su\| u\in A\} \cup\{ut\| u\in B\}$ a\r
+obdobnì mù¾eme nalézt minimální vrcholové pokrytí.\r
+\r
+\figure{bip-graf.eps}{Bipartitní graf pro který hledáme maximální párování.}{0.2\hsize}\r
+\figure{bip-tok.eps}{Sí» v které najdeme maximální tok.}{0.3\hsize}\r
+\r
+\s{Definice}\r
+\itemize\ibull\r
+\:neorientovaný graf G je vrcholovì k-souvislý $\Leftrightarrow$ G má alespoò\r
+k+1 vrcholù a neexistuje separátor s ménì ne¾ k vrcholy\r
+\:neorientovaný graf G je hranovì k-souvislý $\Leftrightarrow$ G má alespoò k+1 vrcholù\r
+a neexistuje øez s ménì ne¾ k hranami\r
+\:orientovaný graf je silnì souvislý $\Leftrightarrow$ existuje orientovaná cesta mezi\r
+v¹emi vrcholy v obou smìrech\r
+\:cirkulace je nulový tok t.j. $\forall v\in V, \sum f(uv)=\sum f(vu)$\r
+\endlist\r
+\r
+\s{Vìta:}(Menger) o souvislosti existencí disjunktních cest a souvislostí\r
+grafù\r
+\r
+buï $G$ neorientovaný graf, pak:\r
+\itemize\ibull\r
+\:$G$ je vrcholovì k-souvislý $\Leftrightarrow$$\forall v,w\in V\,\exists$\r
+k vrcholovì disjunktních cest z $v$ do $w$\r
+\:$G$ je hranovì k-souvislý $\Leftrightarrow$$\forall v,w\in V\,\exists$\r
+k hranovì disjunktních cest z $v$ do $w$\r
+\:kdy¾ $u$ a $v$ jsou nesousední vrcholy v $G$ pak maximální poèet vrcholovì disjunktních\r
+cest mezi $u$ a $v$ se rovná minimálnímu poètu vrcholù z $G-{u,v}$ kterých odebrání oddìlí\r
+$u$ od $v$ \r
+\:kdy¾ $u$ a $v$ jsou vrcholy v $G$ pak maximální poèet hranovì disjunktních\r
+cest mezi $u$ a $v$ se rovná minimálnímu poètu hran, kterých odebrání oddìlí $u$ od $v$\r
+\endlist\r
+buï $G$ orientovaný graf, pak:\r
+\itemize\ibull\r
+\:kdy¾ $u$ a $v$ jsou vrcholy $G$, $uv\notin E(G)$ pak maximální poèet vrcholovì disjunktních\r
+cest z $u$ do $v$ je rovný minimálnímu poètu vrcholù z $G-{u,v}$ kterých odebrání oddìlí $u$ od $v$\r
+\:kdy¾ $u$ a $v$ jsou vrcholy $G$ pak maximální poèet hranovì disjunktních cest z $u$ do $v$ je\r
+rovný minimálnímu poètu hran, kterých odebrání oddìlí $u$ od $v$\r
+\endlist\r
+\r
+Vrcholová i hranová souvislost grafu se dá zjistit pomocí maximálního\r
+toku. Pro hranovou souvislost pøímo zjistíme maximální tok pro ka¾dou\r
+dvojici vrcholù (volba zdroje a stoku). V pøípadì neorientovaného grafu vyrobíme orientovaný graf tak, ¾e ka¾dou\r
+neorientovanou hranu nahradíme orientovanými v obou smìrech. Pro vrcholovou souvislost\r
+navíc musíme ka¾dý vrchol nahradit dvìma novými a spojit je hranou\r
+v obou smìrech (pod-rozdìlení vrcholu). Kapacity hran volíme 1.\r
+Pro nalezení tìchto cest staèí v¾dy v síti odebírat postupnì cestu\r
+z $s$ do $t$. Odebrání jedné cesty sní¾í tok o jedna a tedy cest\r
+musíme nalézt $k$.\r
+\figure{vrchol.eps}{Vrchol který chceme pod-rozdìlit.}{0.1\hsize}\r
+\figure{podrozdeleni.eps}{Výsledek pod-rozdìlení vrcholu.}{0.15\hsize}\r
+\r
+\r
+\s{Ford-Fulkersonùv algoritmus}\r
+Algoritmus nepracuje pøímo s kapacitami a s toky pøes hrany, ale s rezervami\r
+tìchto hran. Funkce rezerv definujeme jako $r(uv)\equiv w(uv)-f(uv)+f(vu)$.\r
+Jestli $uv\notin E(G)$ pak $w(uv)=0$.\r
+Kdy¾ zmen¹ujeme rezervu hrany tak o stejnou hodnotu zvý¹íme hodnotu\r
+rezervy opaèné hrany. Dále pak FF cesta, je cesta pro kterou má ka¾dá hrana kladnou rezervu.\r
+\r
+\algo\r
+\:nastav rezervy pro nulový tok na v¹ech hranách (rezervy se rovnají kapacitám)\r
+\:while $\exists$FF cesta z $s$ do $t$\r
+\::buï $p$ FF cesta z $s$ do $t$\r
+\::zvìt¹i tok zmen¹ením rezerv, o $m=min_{uv\in p}r(uv)$ \r
+\:end while\r
+\:spoèti tok z rezerv\r
+\endalgo\r
+\r
+Pro dùkaz korektnosti algoritmu uva¾ mno¾inu $X=\{ v,\,\exists$FF\r
+cesta z $s$ do $v\}$ po dobìhnutí algoritmu. Proto¾e neexistuje FF cesta z $s$ do $t$, $t\notin X$.\r
+Pak mno¾iny $X$ a $V\setminus X$ urèují øez tvoøený hranami spojujícími\r
+vrcholy z rùzných mno¾in $c=\{uv: uv\in E, u\in X, v\in V\setminus X\}$. V¹echny hrany $uv\in c$\r
+mají nulovou rezervu, nebo» jinak by se dala prodlou¾it FF cesta do vrcholu z $V\setminus X$. Tak¾e velikost\r
+toku se rovná souètu kapacit hran z $X$ do $V\setminus X$ (tohle tvrzení jsme dokázali pøi dùkazu opaèné implikace\r
+FF vìty). Souèasnì tyhle hrany tvoøí $st$-øez, kterého kapacita je rovna velikosti toku. Na¹li jsme tedy øez, \r
+kterého velikost se rovná velikosti pøípustného toku a tím je dùkaz FF vìty ukonèen. \r
+\r
+Kdy¾ máme celoèíselné kapacity hran tak v ka¾dém kroku se tok zvìt¹í\r
+alespoò o jedna a maximální tok je z hora omezen, napøíklad souètem\r
+v¹ech kapacit hran. Pro racionální kapacity staèí vynásobit kapacity\r
+a dostaneme celoèíselné. Z toho plyne koneènost FF algoritmu pro hrany s racionálními kapacitami.\r
+Táto varianta FF algoritmu není obecnì koneèná pøi reálných kapacitách. Volíme-li ov¹em zlep¹ující cestu, která\r
+má maximální minimum pøes rezervy tak je FF algoritmus koneèný i pro reálné kapacity hran.\r
+\r
+Pro FF algoritmus s jednotkovými kapacitami a vyu¾itím BFS (procházení\r
+do ¹íøky) na hledání FF cest je èasová slo¾itost algoritmu $O(n.m)$.\r
+Edmunds s Karpem dokázali, ¾e pøi volbì nejkrat¹í cesty je èasová\r
+slo¾itost v obecném pøípadì $O(n.m^{2})$.\r
+\r
+\bye\r
--- /dev/null
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+%%%%%%%%%%%%
+% Zápisek druhého semináøe z grafových algoritmù - ze dne 13.3.2006
+% Téma Dinicùv algoritmus a v¹emo¾né jeho modifikace.
+% Zapsal Bernard Lidický - bernard@matfyz.cz
+%
+% Verze z 29. dubna 2006
+%
+%%%%%%%%%%%
+
+\input ../sgr.tex
+
+\prednaska{2}{Dinicùv algoritmus a jeho analýza}{zapsal Bernard Lidický}
+
+\h{Dinicùv algoritmus}
+
+Dinicùv algoritmus je zalo¾en na my¹lence, ¾e ve Ford-Fulkersonovì algoritmu
+není potøeba pøièítat jen zlep¹ující cesty, ale je mo¾né pøièítat rovnou zlep¹ující
+toky. Funguje takto:
+
+\algo
+\:Zaèni s~libovolným tokem~$f$, napøíklad prázdným (v¹ude nulovým).
+\:Iterativnì vylep¹uj tok podle zlep¹ujících tokù v síti rezerv: {\I (vnìj¹í cyklus)}
+\::Sestroj sí» rezerv.
+\::Najdi nejkrat¹í $st$-cestu. Kdy¾ ¾ádná neexistuje, skonèi.
+\::Proèisti sí», tj. nech v ní pouze vrcholy a hrany na nejkrat¹ích $st$-cestách.
+\::Najdi blokující tok $f_B$:
+\:::$f_B \leftarrow \<prázdný tok>$
+\:::Postupnì pøidáváme $st$-cesty: {\I (vnitøní cyklus)}
+\::::Najdi $st$-cestu. Napø. hladovì metodou rovnou za nosem.
+\::::Po¹li co nejvíce po cestì.
+\::::Sma¾ nasycené hrany. (Pozor, smazáním hran mohou vzniknout slepé ulièky,
+èím¾ se zneèistí sí» a nebude fungovat metoda rovnou za nosem.)
+\::::Doèisti sí».
+\::Zlep¹i $f$ podle $f_B$
+\endalgo
+
+\figure{dinic-cistasit.eps}{Proèi¹tìná sí» rozdìlená do vrstev}{0.3\hsize}
+
+\s{Slo¾itost algoritmu:}
+Oznaèíme $l$ délku nejkrat¹í $st$-cesty, $n$ poèet vrcholù sítì a $m$ poèet hran sítì.
+
+\itemize\ibull
+\:Jeden prùchod vnitøním cyklem trvá $\O(l)$, co¾ mù¾eme odhadnout jako $\O(n)$, proto¾e v¾dy platí $l \leq n$.
+\:Vnitøní cyklus se provede maximálnì $m$-krát, proto¾e se v¾dy alespoò jedna hrana nasytí
+ a ze sítì vypadne, tak¾e krok~6 mimo podkroku~12 bude trvat $\O(mn)$.
+\:Èi¹tìní a doèi¹»ování sítì dohromady provedeme takto:
+ \itemize\ibull
+ \:Rozvrstvíme vrcholy do vrstev podle vzdálenosti od $s$.
+ \:Zaøízneme dlouhé cesty (del¹í, ne¾ do vrstvy obsahující $t$).
+ \:Dr¾íme si frontu vrcholù, které mají $\deg^+ = 0$ èi $\deg^- = 0$.
+ \:Vrcholy z~fronty vybíráme a zahazujeme vèetnì hran, které vedou do/z nich.
+ A pøípadnì pøidáváme do~fronty vrcholy, kterým klesl jeden ze stupòù na 0
+ pøi vyhazování hran. Vyma¾ou se tím slepé ulièky, které by vadily v podkroku~9.
+ \endlist
+ Tedy kroky 5 a 12 dohromady spotøebují èas $\O(m)$\foot{Pøesnìji $\O(m+n)$, pokud by v síti byly izolované
+ vrcholy, ale ty mù¾eme zahodit úplnì u¾ na zaèátku.}.
+ \:Jeden prùchod vnìj¹ím cyklem tedy trvá $\O(mn)$.
+\endlist
+
+\s{Vìta:}
+V~ka¾dém prùchodu Dinicova algoritmu vzroste $l$ alespoò~o~1.
+
+\s{Dùsledek:}
+Vnìj¹ím cyklem projdeme $\O(n)$ krát, proto¾e nejdel¹í cesta bude mít
+délku $\leq n$. Celková slo¾itost algoritmu tedy bude $\O(n^2m)$
+
+\s{Korektnost algoritmu:}
+Korektnost Dinicova algoritmu plyne z korektnosti FF algoritmu. Kdy¾ se Dinicùv
+algoritmus zastaví, musí vydat maximální tok. Pokud ne, tak podle FF algoritmu
+existuje zlep¹ující cesta, ale to je ve sporu s~krokem~4, který po¾aduje, ¾e ¾ádná
+zlep¹ující cesta neexistuje.
+
+\s{Dùkaz vìty:}
+Podíváme se na výpoèet jednoho prùchodu vnìj¹ím cyklem.
+Délku aktuálnì nejkrat¹í $st$-cesty oznaème~$l$.
+Vypozorujeme, ¾e délka cest, které pøípadnì vzniknou pøi hledání a sycení
+v¹ech cest délky $l$, musí být vìt¹í ne¾ $l$. Jinak øeèeno, nestane se,
+¾e by vznikaly nové krátké cesty.
+
+Vnìj¹í cyklus zaèíná s neproèi¹tìnou sítí. Pøíklad takové sítì je na obrázku nìkde
+okolo. Po proèi¹tìní zùstanou v~síti jen èerné hrany. Èervené a modré\foot{Modré jsou
+ty, které vedou v rámci jedné vrstvy, èervené vedou zpìt èi za~spotøebiè èi do~slepých
+ulièek. Pøi vyti¹tìní na papír není snadné je barevnì odli¹it od èerných.} se zahodí.
+\figure{dinic-neprocistenasit.eps}{Neproèi¹tìná sí». Obsahuje zpìtné hrany, hrany uvnitø vrstvy a slepé ulièky.}{0.3\hsize}
+Nové hrany tak mohou vznikat jen díky èerným hranám, proto¾e s hranami zahozenými pøi èi¹tìní algoritmus
+dále nepoèítá.
+
+Nová hrana mù¾e vzniknout zlep¹ení toku po nìjaké zlep¹ující cestì, proto¾e
+kdy¾ po hranì po¹leme nìjaký tok, zvedne se rezerva hrany opaèné, která mohla být
+nulová\foot{Tedy po zpìtné hranì ne¹lo nic poslat a nepoèítalo se s ní.}.
+Tím vznikne nová zpìtná hrana. ®ádné novì dopøedné hrany pøeskakující více vrstev nevznikají.
+\figure{dinic-zpetnahrana.eps}{Vznik nové zpìtné hrany}{0.3\hsize}
+
+Poka¾dé, kdy¾ se zlep¹í tok podél nìjaké cesty, budou v grafu ke v¹em hranám cesty
+existovat hrany opaèného smìru s~nenulovou rezervou. Nìkteré
+tyto opaèné hrany u¾ v~grafu mohly být, ale tím lépe pro nás\foot{Tento odstavec
+je spí¹e informativní a slou¾í malinko pro osvícení ètenáøe. V dùkazu nemá ¾ádný smysl.}.
+
+Vznikem nových hran by mohly vzniknout nové $st$-cesty, které pou¾ívají
+zpìtné hrany. Ale $st$-cesta, která pou¾ívá zpìtnou hranu, musí alespoò jednou skoèit zpìt.
+Tedy její délka je alespoò $l+2$.
+
+\figure{dinic-cestashranouzpet.eps}{Cesta u¾ívající novou zpìtnou hranu}{0.3\hsize}
+
+Tedy ve vnitøním cyklu zaniknou v¹echny $st$-cesty délky $l$ a pøitom nevzniknou ¾ádné nové krat¹í $st$-cesty.
+Proto délka nejkrat¹ích cest vzroste alespoò o 1.
+\qed
+
+\h{Implementaèní poznámky}
+
+\itemize\ibull
+\:Není potøeba tak brutální èi¹tìní. Vrcholy se vstupním stupnìm 0 nám
+ nevadí -- stejnì se do nich nedostaneme. Vadí jen vrcholy s výstupním stupnìm 0, kde by mohl
+ havarovat postup v podkroku 9.
+\:Je mo¾né dìlat prohledávání a èi¹tìní souèasnì. Jednodu¹e metodou HRRR na nì a kdy¾
+ to nevyjde (dostaneme se do slepé ulièky), kus ustoupíme a pøi ústupu èistíme sí»
+ odstraòováním slepé ulièky.
+\:U¾ pøi prohledáváni si rovnou udr¾ujeme minimum z rezerv a pøi zpáteèní cestì
+ opravujeme kapacity.\foot{Jak se asi zkombinuje s pøedchozím, kde musíme ustupovat?
+ V ka¾dé úrovni rekurze si zapamatujeme minimum, jaké bylo, kdy¾ jsme tam pøi¹li. Pokud
+ se vracíme, recyklujeme toto minimum.}
+\:V prùbìhu výpoètu udr¾ujeme jen sí» rezerv a tok vypoèteme a¾ nakonec z rezerv a kapacit.
+\:Kdy¾ budeme chtít hledat minimální øez, spustíme Dinicùv algoritmus, pak jednou iterací FF
+ algoritmu dostaneme jeden z minimálních øezù\foot{Samozøejmì získáme i maximální tok.}.
+\endlist
+
+\s{Na pøemý¹lení:} Máme minimální øez. Jak zjistíme maximální tok?
+
+\h{Speciální sítì (ubíráme na obecnosti)}
+
+Dále se budeme vìnovat modifikacím algoritmu na speciální druhy sítí,
+kde je mo¾né dostat lep¹í èasovou slo¾itost ne¾ v~obecném pøípadì.
+
+\s{Jednotkové kapacity:}
+V¹echny rezervy jsou jen 0 nebo~1. Na $st$-cestì má v¹echno kapacitu/rezervu~1.
+Mù¾eme poslat~1~a rovnou celou cestu zahodit -- není potøeba si pamatovat minimum z~rezerv.
+Kdy¾ máme $m$ hran, zlep¹eních po cestách délky $l$ bude maximálnì $m/l$.
+Proto slo¾itost podkrokù 9, 10 a 11 bude $m/l \cdot \O(l) = \O(m)$.%
+\foot{Nebo by ¹lo argumentovat, ¾e ka¾dou hranu pou¾ijeme jen $1\times$.}
+Tedy pro jednotkové kapacity dostáváme slo¾itost $\O(nm)$.
+
+\s{Jednotkové kapacity znovu a lépe:}
+Vnitøní cyklus lépe udìlat nepùjde. Je potøeba alespoò $\O(m)$ pro èi¹tìní.
+Mù¾eme se ale pokusit lépe odhadnout poèet iterací vnìj¹ího cyklu.
+
+Sledujeme stav sítì po $k$ iteracích vnìj¹ího cyklu a pokusíme se odhadnout, kolik iterací
+je¹tì algoritmus udìlá. Oznaème délku nejkrat¹í $st$-cesty $l$.
+Víme, ¾e $l \geq k$, proto¾e v ka¾dé iteraci vzroste $l$ alespoò o 1.
+
+Máme tok $f_k$ a chceme dostat maximální tok $f$. $f - f_k$ je tok v~síti rezerv.
+Tedy $\exists f_R$ tok v síti rezerv takový, ¾e
+zlep¹ení $f_k$ podle $f_R$ je $f$.
+Ka¾dá iterace velkého cyklu zlep¹í $f_k$ alespoò o 1. Tedy nám zbývá je¹tì
+nejvý¹e $\vert f_R \vert$ iterací.
+Proto bychom chtìli omezit velikost toku $f_R$. Napøíklad øezem.
+
+Najdeme v síti rezerv øez $C$. Kde ho vzít?\foot{Pøeci v øeznictví. Kdepak, spí¹e v cukrárnì.
+Myslíte, ¾e v cukrárnì mají Dinicovy øezy? Myslím, ¾e v cukrárnì je vìt¹ina øezù minimální.}
+Poèítejme jen hrany zleva doprava. Tìch je jistì nejvý¹e $m$ a tvoøí $k-1$ rozhraní mezi
+$k$ vrstvami. Tedy existuje rozhraní vrstev
+s~nejvý¹e $m/k$ hranami\foot{Princip holubníku a nìjaká ta $\pm1$.}.
+Toto rozhraní je øez. Tedy existuje øez $C$, ¾e $\vert C\vert \leq m/k$
+Algoritmu tedy zbývá maximálnì $m/k$ krokù.
+Celkový poèet krokù je nejvý¹ $k + m/k$. Staèí zvolit $k = \sqrt{m}$. Poèet krokù
+pak bude $\O(\sqrt{m})$.
+
+Celkovì slo¾itost Dinicova algoritmu pro jednotkové kapacity znovu a lépe
+je $\O(\sqrt{m} m) = \O(m^{3/2})$. Pomohli jsme si pro øídké grafy.
+
+\s{Jednotkové kapacity a jeden ze stupòù roven 1:}
+Úlohu hledání maximálního párování v bipartitním grafu lze
+hezky pøevést na hledání maximálního toku. Z grafu se vytvoøí
+sí» tak, ¾e v¹echny vrcholy jedné partity se napojí na $s$ a v¹echny
+vrcholy druhé partity se napojí na $t$.\foot{$s,t$ jsou novì pøidané vrcholy.}
+Na v¹echny hrany se dají jednotkové kapacity a orientace ka¾dé bude smìrem\foot{%
+V¹echny hrany v bipartitním grafu se zorientují smìrem ke komponentì, kde je napojený
+vrchol $t$. Hrany obsahující vrchol $t$ se zorientují smìrem do $t$. Hrany obsahující vrchol
+$s$ se zorientují smìrem od $s$. Snad je to intuitivnì vidìt. Pro jistotu je nìkde okolo
+obrázek popisované sítì.} k~$t$.
+Najde se maximální tok a hrany v~toku tvoøí maximální párování. Proto mù¾e být
+u¾iteèné zkoumat tento speciální pøípad sítí.
+
+\figure{dinic-bipartitni.eps}{Sí» vzniklá z bipartitního grafu}{0.4\hsize}
+
+Sí» se vyznaèuje tím, ¾e ka¾dý vrchol krom $s$ a $t$ má vstupní nebo výstupní stupeò 1.
+Pro takovou sí» mù¾eme odhad je¹tì tro¹ku upravit. My¹lenka je stejná jako minule. Pokusíme
+se nalézt v síti po $k$ krocích nìjaký malý øez. Místo rozhraní budeme hledat jednu malou
+vrstvu a z malé vrstvy vytvoøíme malý øez. Pro ka¾dý vrchol z vrstvy vezmeme tu hranu,
+která je ve svém smìru sama, a tím získáme øez.
+
+\figure{dinic-vrcholrez.eps}{Øez podle vrcholù ve vrstvì}{0.2\hsize}
+
+Tedy $\forall v \neq s,t: \min(\deg^-(v), \deg^+(v)) \leq 1$.
+Po $k$ krocích máme $k$ vrstev. Tedy existuje vrstva $\delta$ s nejvý¹e $n/k$ vrcholy\foot{Holubníkový princip.}.
+Tedy existuje øez $C$, ¾e $\vert C\vert \leq n/k$, který získáme z vrstvy $\delta$ vý¹e popsaným postupem.
+Algoritmu zbývá do konce $\leq n/k$ iterací vnìj¹ího cyklu. Algoritmus celkem udìlá $k + n/k$ krokù%
+\foot{Mo¾ná chybí nìjaké $\pm1$, ale to nehraje roli.}. Staèí zvolit $k = \sqrt{n}$ a slo¾itost
+celého algoritmu vyjde $\O(\sqrt{n}m)$\foot{Poèet iterací vnìj¹ího cyklu je $\O(\sqrt{n})$ a
+vnitøní cyklus má $\O(m)$.}.
+
+\s{Tøetí pokus pro jednotkové kapacity bez omezení na stupnì vrcholù v síti:}
+Hlavní my¹lenkou je opìt po $k$ krocích najít nìjaký malý øez. Najdeme dvì malé\foot{Co do souètu
+poètu vrcholù v obou vrstvách.} sousední vrstvy a v¹echny hrany mezi nimi budou tvoøit námi hledaný malý øez.
+
+Oznaème $s_i$ poèet vrcholù v $i$-té vrstvì. Souèet poètu vrcholù ve dvou
+sousedních vrstvách oznaèíme $t_i = s_i + s_{i+1}$. Bude tedy platit nerovnost
+$$\sum_i t_i \leq 2\sum_i s_i \leq 2n {\rm .}$$
+
+Tedy existuje $i$, ¾e $t_i \leq 2n/k$\foot{Holubníkový princip. Zase.}.
+Tedy $\exists i: s_i + s_{i+1} \leq 2n/k$. Poèet hran mezi $s_i$ a $s_{i+1}$ je velikost øezu
+$C$ a to je men¹í ne¾ $s_i \cdot s_{i+1}$.
+Nejhor¹í pøípad je, kdy¾ $s_i = s_{i+1} = n/k$. Tedy $\vert C\vert \leq {(n/k)}^2$.
+Pro jistotu je¹tì jednou v celku jako dlouhá nerovnost:
+$$ \<\# hran mezi>\ s_i, s_{i+1} = \vert C\vert \leq s_i\cdot s_{i+1} \leq {(n/k)}^2 {\rm .}$$
+Proto poèet iterací velkého cyklu je $\leq \<konstanta> + k + {(n/k)}^2$.
+Chytøe zvolíme $k = n^{2/3}$. Slo¾itost celého algoritmu pak bude $\O(n^{2/3}m)$.
+
+\s{Poslední odhad pro celoèíselné kapacity:}
+Poslední\foot{Samozøejmì následuje je¹tì jeden algoritmus i s odhadem pro celoèíselné kapacity.
+Ale to u¾ není pøímo Dinicùv algoritmus. Jen Dinice pou¾ívá jako podprogram.} odhad je zalo¾en na velikosti
+maximálního toku $f$ a pøedpokladu celoèíselných kapacit.
+Za jednu iteraci velkého cyklu projdeme malým cyklem maximálnì tolikrát,
+o kolik se v nìm zvedl tok, proto¾e ka¾dá zlep¹ující cesta ho zvedne alespoò o $1$.
+Zlep¹ující cesta se tedy hledá maximálnì $\vert f\vert$ krát za celou dobu bìhu algoritmu.
+Cestu najedeme v èase $\O(n)$. Celkem na hledání
+cest spotøebujeme $\O(\vert f\vert n)$ za celou dobu bìhu algoritmu.
+
+Nesmíme ale zapomenout na cleanupy. V jedné iteraci velkého cyklu stojí cleanupy $\O(m)$ a velkých
+iterací je $\leq n$. Proto celková slo¾itost algoritmu je $\O(\vert f\vert n + nm)$
+
+Pokud navíc budeme pøedpokládat omezení na maximální kapacitu hran $ \leq C$,
+lze získat je¹tì jeden odhad tím, ¾e odhadneme maximální velikost toku jako $ \leq Cn$.
+Co¾ lze odùvodnit napøíklad tak, ¾e z $s$ jde maximálnì $n$~hran\foot{Nepøedpokládáme multigraf.}
+a kapacita ka¾dé je $ \leq C$. Proto celkový tok musí být $ \leq Cn$.
+Dostaneme odhad $\O(Cn^2 + nm)$.
+
+\h{Scaling kapacit}
+
+Základní my¹lenka je podobná, jako u algoritmu pro tøídìní dlouhých èísel postupnì po øádech pomocí
+radix-sortu. Pro jistotu si ho pøipomeòme. Algoritmus nejprve setøídí èísla podle poslední%
+\foot{Poslední cifrou myslíme nejménì významnou cifru.}
+cifry, poté podle pøedposlední, pøedpøedposlední a tak dále.
+
+\figure{dinic-sort.eps}{Kroky postupného tøídìní podle øádù}{0.2\hsize}
+
+V na¹em pøípadì budeme postupnì budovat sítì a v nich poèítat toky, a¾ nakonec získáme tok pro celou sí».
+
+Pøedpokládejme celoèíselné kapacity. Maximální tok v síti $G$ budeme hledat tak, ¾e hranám postupnì
+budeme zvìt¹ovat kapacity bit po bitu v binárním zápisu, a¾ k jejich skuteèné kapacitì.
+Pøitom po~ka¾dém posunu zavoláme Dinicùv algoritmus. Pomocí pøedchozího odhadu uká¾eme, ¾e jedno
+volání Dinice nebude pøíli¹ drahé.
+
+\figure{dinic-scaling-original.eps}{Pùvodní sí», na hranách jsou jejich kapacity v binárním zápisu}{0.2\hsize}
+
+Oznaème $k$ poèet bitù v zápisu nejvìt¹í kapacity z celé sítì. $k = \lceil \log_2C \rceil$.
+Postupnì budeme budovat sítì $G_i$ s kapacitami $C_i(e) = \lfloor {C(e) / 2^{k-i}} \rfloor$.
+$G_0$ je nejoøezanìj¹í sí», kde ka¾dá hrana má kapacitu jako je nejvy¹¹í bit v binárním zápisu
+její skuteèné kapacity. $G_k$ je pùvodní sí» $G$.
+
+ $$ C_{i+1}(e) = \left\{
+\eqalign{
+ 2C_i(e), &\hbox{\quad pokud $(k-i-1)$-tý bitík je 0} \cr
+ 2C_i(e)+1, &\hbox{\quad pokud $(k-i-1)$-tý bitík je 1} \cr}
+\right.
+ $$
+
+\figure{dinic-scaling-g.eps}{Sítì $G_0$, $G_1$ a $G_2$, jak vyjdou pro sí» z~pøedchozího obrázku}{0.8\hsize}
+
+Na spoètení maximálního toku $f_i$ v síti $G_i$ zavoláme Dinicùv algoritmus,
+ov¹em do zaèátku nepou¾ijeme nulový tok, nýbr¾ tok $2f_{i-1}$. Rozdíl toku z inicializace
+a výsledného bude malý.
+
+Tvrdíme, ¾e:
+$$ \vert f_i\vert - \vert 2f_{i-1}\vert \leq m.$$
+
+\s{Dùkaz:}
+Vezmeme minimální øez $R$ v $G_{i-1}$. Platí $\vert f_{i-1}\vert = \vert R\vert$\foot{Minimaxová vìta.}.
+Øez $R$ obsahuje $\leq m$ hran. V $G_{i}$ má øez $R$ kapacitu maximálnì $2\vert R\vert+m$.
+Maximální tok je omezen ka¾dým øezem. Tedy i øezem $R$. Proto tok vzroste o $\leq m$.
+
+Proto podle pøedchozího odhadu výpoèet toku $f_i$ trvá $\O(mn)$. Tok se bude poèítat $k$-krát.
+Celková slo¾itost bude $\O(mn\log C)$.
+\qed
+
+\h{Dinicova tabulka}
+
+$$\vbox{\halign{# \hfil \quad &# \hfil \cr
+\it verze &\it èas \cr\noalign{\smallskip\hrule\smallskip}
+standardní &$\O(n^2m)$ \cr
+jednotkové kapacity &$\O(nm)$ \cr
+jednotkové kapacity podruhé &$\O(\sqrt{m}m) = \O(m^{3/2})$ \cr
+jednotkové kapacity, 1 stupeò $\leq 1$ &$\O(\sqrt{n}m)$ \cr
+jednotkové kapacity veseleji &$\O(n^{2/3}m)$ \cr
+celoèíselné kapacity &$\O(\vert f\vert n + nm)$ \cr
+celoèíselné kapacity $ \leq C$ &$\O(Cn^2 + mn)$ \cr
+celoèíselné kapacity $ \leq C$ &$\O(mn\log C)$ \cr
+}}$$
+
+\bye
--- /dev/null
+P=2-dinic
+
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+%%%%%%%%%%%%
+% Zápisek tøetího semináre z grafových agoritmù - ze dne 20.3.2006
+% Zapsal Jiøí Peinlich - peinlich@seznam.cz a Michal Kùrka - michal.kurka@gmail.com
+%%%%%%%%%%%
+
+\input ../sgr.tex
+
+\prednaska{3}{Bipartitní párování a globální k-souvislost}{zapsali Jiøí Peinlich, Michal Kùrka}
+
+\h{Maximální párování v $k$-regulárním bipartitním grafu}
+
+\s{Operace Degree Split} provádí rozdìlení sudì-regulárního grafu na dva
+podgrafy s polovièní regularitou. Operaci Degree Split definujme na
+sudì-regulárním grafu takto: V grafu najdeme eulerovský tah. Sestrojíme dva
+grafy, které budou mít oba stejnou mno¾inu vrcholù jako graf pùvodní a ka¾dý
+bude mít polovinu jeho hran. Mno¾inu hran prvního grafu budou tvoøit sudé hrany
+z nalezeného eulerovského tahu, mno¾inu hran druhého grafu pak hrany liché. Tuto
+operaci lze jistì provést v lineárním èase (v lineárním èase najdeme eulerovský
+tah i rozklad na sudé a liché hrany).
+
+Dále budeme pracovat s $2^d$-regulárními grafy. Operací Degree Split tedy získáme dva grafy, které budou $2^{d-1}$-regulární.
+
+Provedeme-li operaci Degree Split $\log k = d$ krát (polovinu hran v¾dy
+zahodíme), získáme 1-regulární podgraf a tedy párování pro zadaný graf (ve
+skuteènosti mù¾eme graf rozkládat na párování a vyrobit si 1-faktorizaci
+grafu). Slo¾itost nalezení párování bude tedy pro $2^d$ regulární grafy $\O(2^d n
+d)$.
+
+Pokud zadaný graf nebude $2^d$-regulární budeme muset pøidat hrany tak, aby nový
+graf tuto vlastnost mìl. Operaci Degree Split pak budeme provádìt tak, abychom
+se k párování blí¾ili.
+
+Místo toho, abychom do grafu hrany jen pøidávali, budeme v pøípadech, kdy je to
+mo¾né, pouze zvìt¹ovat násobnost hrany. U ka¾dé hrany si tedy budeme pamatovat
+její násobnost.
+
+\s{Degree Split s~násobnostmi:} Pro sudì-regulární grafy s násobnostmi zavedeme operaci Degree Split takto:
+Graf $G=(V,E)$ rozdìlíme na dva grafy $G_1$ a $G_2$, bude platit $V(G_1) = V(G_2) = V$. Hrany nyní pøidìlíme následujícím zpùsobem:
+\algo
+\:Pokud $e\in E$ v $G$ má sudou násobnost (znaèíme $n(e)$), umístíme ji do $E_1$ i do $E_2$ s násobností ${n(e) \over 2}$, v opaèném pøípadì pøidáme do obou grafù hranu s násobností $\lfloor {n(e) \over 2} \rfloor$.
+\:Graf se zbylých hran má v¹echny stupnì sudé a je bez multiplicit. Provedeme na nìj pùvodní operaci Degree Split a rozdìlené mno¾iny hran pøidáme do $G_1$ a $G_2$.
+\endalgo
+
+Celý proces lze stihnout v èase $\O(m)$ ($m$ je poèet hran $G$), nebo» v první èásti u ka¾dé hrany pouze zjistíme, zda má sudou násobnost, pøidáme nové hrany v konstantním èase (upravíme násobnosti), a v druhé èásti se provede Split, který má té¾ lineární slo¾itost. Operace Degree Split má tedy slo¾itost $\O(m)$ i v grafu s násobnostmi.
+
+Mìjme nyní $k$-regulární bipartitní graf. Zvolme $t$ tak aby $2^t\geq kn$.
+Zvolme dále
+$\alpha := \lfloor {2^t \over k} \rfloor$ a
+$\beta := 2^t \bmod k$.
+Do grafu pøidáme hrany a upravíme násobnosti hran tak, aby byl $2^t$ regulární. Dále pøidáme triviální párování ($i$-tý vrchol vlevo se spojí s $i$-tým vrcholem vpravo) s násobností $\beta$. Tuto mno¾inu hran oznaème $F$. Platí $\beta < k \Rightarrow \vert F \vert < 2^t$.
+
+Takto získáme $2^t$-regulární graf. Na tento graf budeme aplikovat operaci Degree Split a vybereme si v¾dy tu polovinu, kde bude ménì hran z $F$. Tímto zpùsobem budem graf dìlit dokud budou stupnì vrcholù vìt¹í ne¾ jedna. Tedy $t$-krát. Poslední takto získaný graf bude 1-regulární (párování). V ka¾dém kroku se zbavíme alespoò poloviny hran z $F$. Provedeme to ceklem $t$-krát a tedy výsledné párování bude perfektní párování zadaného grafu.
+
+Slo¾itost algoritmu je $\O(kn \log n)$, proto¾e inicializace algoritmu se dá provést v lineárním èase, provede se $\log (kn)$ iterací po $\O(m)$.
+
+\h{Algoritmy na hledání globální k-souvislosti}
+\s{Hranová k-souvislost}
+Problém zji¹tìní stupnì souvislosti grafu lze pøevést na problém hledání minimálního øezu v~grafu. Algoritmus pro hledání minimálního øezu lze zkonstruovat napøíklad tak, ¾e pro ka¾dé dva vrcholy $s$, $t$ zjistíme minimální $st$-øez. Algoritmus mù¾eme zrychlit, pokud si uvìdomíme, ¾e jeden z vrcholù $s$ nebo $t$ lze zvolit pevnì. Pokud pracujeme s orientovanými grafy, musíme projít jak $s \rightarrow t$ øezy tak i $t \rightarrow s$ øezy. Algoritmus bude mít slo¾itost $\O(n^{{5/3}}m)$.
+
+U vrcholové $k$-souvislosti to ov¹em tak snadno nepùjde. Pokud by toti¾ fixovaný vrchol byl souèástí nìjakého minimálního separátoru, algoritmus mù¾e selhat. Pøesto ale nemusíme procházet v¹echny dvojice vrcholù. Staèí si pamatovat, kolik vrcholù $s$ jsme u¾ pro v¹echny $t$ zkontrolovali a nejmen¹í zatím nalezený separátor. Kdy¾ bude poèet vrcholù vìt¹í ne¾ nejmen¹í separátor, tak u¾ jsme jistì na¹li jeden z minimálních øezù. Slo¾itost takového algoritmu pak bude $\O(\kappa (G) n^{3/2} m)$, kde $\kappa(G)$ je stupeò souvislosti $G$, který hledáme.
+
+Bez tokù se dá ov¹em najít hranová k-souvislost efektivnìji (i v ohodnocených grafech). Uka¾me si, jak na to.
+
+\h{Algoritmus pro nalezení minimálního globálního øezu v neorientovaných grafech (Namagochi, Ibaraki)}
+
+\noindent Zavedeme znaèení:
+
+\itemize\ibull
+\:{$r(u,v)$ buï kapacita minimálního $uv$-øezu}
+\:{$d(v)$ buï celková kapacita hran vedoucích z $v$}
+\:{$d(U)$ buï kapacita hran vedoucích z mno¾iny $U \subseteq V$, speciálnì $d(U)=d(\overline{U})$}
+\:{$d(P,Q)$ buï kapacita hran vedoucích mezi mno¾inami $P,Q$}
+\endlist
+
+\s{Definice}
+{\it Legálním uspoøádáním vrcholù} (LU) budeme nazývat lineární uspoøádání vrcholù $v_1, v_2,... ,v_n$ takové, ¾e platí
+$d(\{v_1 \ldots v_{i-1}\},v_i) \geq d(\{v_1 \ldots v_{i-1}\},v_j)$ pro $1 \leq i<j\leq n$.
+
+\s{Lemma} Je-li $v_1 ... v_n$ LU na $G$, pak $r(v_{n-1},v_n)=d(v_n)$.
+
+\s{Dùkaz} Buï $C$ nìjaký øez oddìlující $v_{n-1}$ a $v_n$. Utvoøme posloupnost vrcholù $u_i$ takto:
+
+\algo
+\:$u_0 := v_1$
+\:$u_i := v_j$ tak, ¾e $j>i$, $v_i$ a $v_j$ jsou oddìleny øezem $C$ a $j$ je minimální takové.
+\endalgo
+
+Ka¾dé $u_{i-1}$ je tedy buï $u_i$, pokud jsou $v_i$ a $v_{i-1}$ na stejné stranì øezu, nebo $u_{i-1}$ je $v_i$ pokud jsou $v_i$ a $v_{i-1}$ na opaèné stranì øezu.
+Dostáváme tedy, ¾e $d(\{v_1...v_{i-1}\},u_i)\leq d(\{v_1...v_{i-1}\},u_{i-1})$, proto¾e buïto $u_i=u_{i-1}$ a pak je nerovnost splnìna jako rovnost, nebo je $u_i=v_j$, $j>i$ a nerovnost plyne z LU vrcholù $v_i$.
+
+Chceme ukázat, ¾e velikost libovolného øezu je alespoò taková jako velikost øezu kolem vrcholu $v_n$.
+Platí, ¾e $ \vert C \vert \geq \sum_{i=1}^{n-1} d(v_i,u_i)$. Uká¾eme, ¾e pravá strana je alespoò $d(v_n)$.
+
+$$\sum_{i=1}^{n-1} d(v_i,u_i) = \sum_{i=1}^{n-1} d(\{v_1..v_i\},u_i) - d(\{v_1 \ldots v_{i-1}\},u_i) \geq \sum_{i=1}^{n-1} d(\{v_1 \ldots v_i\},u_i) - d(\{v_1 .. v_{i-1}\},u_{i-1}) = $$
+
+$$ = d(\{v_1 \ldots v_{n-1}\},u_{n-1}) - d(\{v_1 \ldots v_0\},u_0) = d(\{v_1 \ldots v_{n-1}\},v_n) - 0.$$
+\qed
+
+Dokázali jsme, ¾e libovolný øez separující $v_{n-1}$ a $v_n$ je vìt¹í ne¾ jednoduchý øez skládající se jen z hran kolem~$v_n$. Kdy¾ tedy sestavíme nìjakou LU posloupnost vrcholù, budeme mít k dispozici jednoduchý minimální øez $v_{n-1}$ a~$v_n$. Následnì vytvoøíme graf $G'$, v nìm¾ $v_{n-1}$ a $v_n$ skontrahujeme. Rekurzivnì najdeme minimální øez v $G'$. Hledaný minimální øez poté buïto oddìluje vrcholy $v_n$ a $v_{n-1}$ a potom je øez kolem vrcholu $v_n$ minimální, nebo vrcholy $v_n$ a $v_{n-1}$ neoddìluje, a v takovém pøípadì jej najdeme rekurzivnì. Hledaný øez je tedy men¹í z rekurzivnì nalezeného øezu a øezu kolem $v_n$.
+
+Zbývá ukázat, jak konstruovat LU. Postaèí hladovì: Pamatujeme si $\forall v\neq v_1 \ldots v_{i-1}$ hodnotu $d(\{v_1 \ldots v_{i-1},v)$, oznaème ji $z_v$. V ka¾dém kroku vybereme vrchol $v$ s maximální hodnotou $z_v$ a prohlásíme ho za $v_i$ a pøepoèítáme~$z_v$.
+
+Zde se hodí datová struktura, která doká¾e rychle hledat maxima a zvy¹ovat hodnoty prvkù,
+napøíklad Fibonacciho halda. Ta zvládne \<DeleteMax> v~èase $(\log n)$ a \<Increase> v~$\O(1)$
+amortizovanì. Celkem pak ná¹ algoritmus bude mít slo¾itost $\O(n(m+n\log n))$ pro obecné kapacity.
+
+Pokud jsou kapacity malá celá èísla, mù¾eme vyu¾ít balíèkové struktury. Budeme
+si udr¾ovat obousmìrný seznam zatím pou¾itých hodnot $z_v$, ka¾dý prvek takového
+seznamu bude obsahovat v¹echny vrcholy se spoleènou hodnotou $z_v$. Kdy¾ budeme
+mít seznam seøazený, vybrání minimálního prvku znamená pouze podívat se na
+první prvek seznamu a z nìj odebrat jeden vrchol, pøípadnì celý prvek ze seznamu
+odstranit. Operace \<Increase> poté bude reprezentovat pouze pøesunutí vrcholu o
+malý poèet pøihrádek, pøípadnì zalo¾ení nové pøihrádky na správném místì.
+\<DeleteMax> i \<Increase> pak budou mít slo¾itost $\O(1)$ amortizovanì a celý algoritmus $\O(mn)$.
+
+\bye
--- /dev/null
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+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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+ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef ]
+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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+end readonly def
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+/UniqueID 5000820 def
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--- /dev/null
+% Written by Milan Straka, 2006
+
+\def\li{\discretionary{-}{-}{-}li}
+\def\d{\delta}
+\def\st{$st$}
+\def\rr{$r_1r_2$}
+\def\GHT{GHT}
+\def\PGHT{ÈGHT}
+\def\th#1{\s{#1}}
+\def\proof{\noindent{\it Dùkaz:} }
+
+\input ../sgr.tex
+
+\prednaska{4}{Gomory-Hu Trees}{zapsal Milan Straka}
+
+\h{Gomory-Hu Trees}
+
+Cílem této pøedná¹ky bylo vytvoøit datovou strukturu, která po urèitém
+pøedzpracování doká¾e rychle konstruovat minimální øezy pro libovolnou
+dvojici vrcholù v~grafu.
+
+Zatím umíme nalézt minimální \st-øez pro zadanou dvojici vrcholù v~èase
+$\tau=\O(n^{2/3}m)$. Nalézt minimální \st-øez pro ka¾dou dvojici vrcholù
+bychom tedy dokázali v~èase $\O(n^2\tau)$. Tento výsledek budeme
+chtít zlep¹it.
+
+\s{Znaèení:} Máme\li{} graf $(V,E)$ a $U\subseteq V$, $\d(U)$ znaèí hrany vedoucí
+mezi $U$ a $V\setminus U$. Formálnì tedy $\d(U)=E \cap (U \times (V \setminus U))$.
+Kapacitu øezu $W$ budeme znaèit $c(W)$ a $r(s,t)$ budeme znaèit kapacitu nejmen¹ího \st-øezu.
+
+\s{Pozorování:} Minimální øez rozdìluje graf jen na~dvì komponenty (v¹imnìte si, ¾e pro
+separátory nic takového neplatí) a ka¾dý minimální øez je tím pádem v¾dy mo¾né zapsat jako $\d(W)$
+pro nìjakou mno¾inu $W\subset V$.
+
+\s{Definice:} {\I Gomory-Hu Tree} (dále jen \GHT) pro neorientovaný ohodnocený graf $G=(V,E)$
+je strom $T=(V,F)$ takový,\nobreak{}\ \nobreak{}¾e $$\forall st \in F: \d(K_1)=\d(K_2)\hbox{ je minimální \st-øez, kde
+$K_1$ a $K_2$ jsou komponenty $T\setminus st$}.$$
+Pozor, $F$ nemusí být podmno¾ina pùvodních hran $E$.
+
+\s{Dal¹í znaèení:} Pro $e\in F$ budeme øezem $\d(e)$ oznaèovat øez $\d(K_1)=\d(K_2)$ a $r(e)$ bude jeho kapacita.
+
+K~èemu takový \GHT{} je (existuje-li)? To nám poví následující
+
+\th{Vìta (o~vyu¾ití \GHT):} Buï $T=(V,F)$ \GHT{} pro graf $G=(V,E)$ a mìjme dva vrcholy $s$ a $t$. Dále
+nech» $P$ je cesta v~$T$ mezi vrcholy $s$ a $t$ a $e$ je hrana na cestì $P$ s~minimálním $r(e)$.
+Pak $\d(e)$ je minimální \st-øez.
+
+\proof Nejprve si doká¾eme jedno drobné
+
+{\advance\leftskip by 2em
+\th{Lemmátko:} Pro ka¾dou trojici vrcholù $x,y,z$ platí, ¾e $r(x,z) \ge \min(r(x,y),r(y,z))$.
+
+\proof Buï $W$ minimální $xz$-øez.
+
+\centerline{\epsfysize=1.5cm\epsfbox{4-ght-rez.eps}}
+
+\noindent Vrchol $y$ musí být v~jedné z~komponent, BÚNO v~komponentì s~$x$. Pak ale $r(y,z) \le c(W)$,
+proto¾e $W$ je také $yz$-øez. Tedy $\min(r(x,y),r(y,z)) \le r(x,z)$.\qed
+}
+
+\noindent Zpìt k~dùkazu vìty:
+Chceme dokázat, ¾e $\d(e)$ je minimální \st-øez. To, ¾e je to nìjaký øez, plyne z~definice \GHT.
+Minimalitu doká¾eme indukcí podle délky cesty $P$.\itemize\ibull
+\:$\vert\,P\,\vert = 1$: Hrana $e$ je v~tomto pøípadì pøímo $st$, tak¾e i minimalita plyne z~definice \GHT.
+\:$\vert\,P\,\vert > 1$: Cesta $P$ spojuje vrcholy $s$ a $t$, její první hrana je $sx$.
+Na¹e právì dokázané lemmátko øíká, ¾e $r(s,t) \ge \min (r(s,x),r(x,t))$.
+Urèitì je pravda, ¾e $r(s,x) \ge r(e)$, proto¾e $e$ byla hrana cesty $P$ s~nejmen¹ím $r(e)$.
+To, ¾e $r(x,t) \ge r(e)$, plyne z~indukèního pøedpokladu, proto¾e cesta mezi $x$ a $t$
+je krat¹í ne¾ cesta $P$. Dostáváme tak, ¾e $r(s,t) \ge \min(r(s,x),r(x,t)) \ge r(e)$.
+\endlist
+\qed
+Pokud doká¾eme \GHT{} sestrojit, nalézt minimální \st-øez pro libovolnou dvojici vrcholù
+doká¾eme stejnì rychle jako nalézt hranu s~nejmen¹í kapacitou na cestì mezi $s$ a $t$ v~\GHT.
+K~tomu mù¾eme pou¾ít napøíklad Sleator-Tarjanovy stromy, které tuto operaci
+doká¾ou provést v~amortizovaném èase $\O(\log n)$, nebo mù¾eme vyu¾ít toho,
+¾e máme spoustu èasu na~pøedvýpoèet, a minimální hrany si pro ka¾dou dvojici
+prostì pøichystat pøedem. Také lze vymyslet redukci na problém nalezení spoleèného
+pøedchùdce vrcholù ve stromì (nebude to \GHT) a pou¾ít jedno z~øe¹ení tohoto problému.
+
+\h{Konstrukce GHT}
+
+Nyní se nauèíme \GHT{} konstruovat, èím¾ také rozptýlíme obavy o~jejich existenci.
+Nejprve v¹ak budeme potøebovat jedno u¾iteèné lemma s~hnusnì technickým dùkazem:
+
+\vbox to 0pt{\vskip-\baselineskip\rightline{\epsfysize=2cm\epsfbox{4-ght-htl.eps}}\vss}\vskip-\baselineskip
+{
+\advance\rightskip by 17em
+\th{Hnusnì technické lemma (HTL):} Buïte¾ $s,t$ vrcholy grafu $(V,E)$, $\d(U)$ minimální \st-øez a $u\ne v$ dva rùzné
+vrcholy z~$U$. Pak existuje mno¾ina vrcholù $W \subseteq U$ taková, ¾e $\d(W)$ je minimální $uv$-øez.
+[To dùle¾ité a netriviální je, ¾e celá $W$ le¾í v~$U$.]
+
+\proof Nech» je $\d(X)$ minimální $uv$-øez.
+BÚNO mù¾eme pøedpokládat, ¾e $s$ je v~$U$ a $t$ není v~$U$, $u$ je v~$X$ a $v$ není v~$X$ a $s$ je v~$X$.
+Pokud by tomu tak nebylo, mù¾eme vrcholy pøeznaèit.
+
+}
+
+Nyní mohou nastat dva pøípady:\numlist\nalpha
+\vbox to 0pt{\vskip -1cm\rightline{\epsfysize=2.5cm\epsfbox{4-ght-htl-a.eps}}\vss}\vskip-\baselineskip
+{\advance\hsize by -14em
+\:$t$ není prvek $X$.\par
+Doká¾eme dvì nerovnosti. Nerovnost $$\eqalignno{c(\d(U \cup X)) &\ge c(\d(U))&(1)}$$
+platí proto, ¾e $U \cup X$ je nìjaký \st-øez, zatímco $U$ je minimální \st-øez. Dal¹í
+$$\eqalignno{c(\d(U \cap X)) + c(\d(U \cup X)) &\le c(\d(U)) + c(\d(X))&(2)}$$
+doká¾eme \uv{rozborem pøípadù}.
+
+}
+
+Mno¾inu vrcholù si disjunktnì rozdìlíme na $X\setminus U$, $X \cap U$, $U \setminus X$ a $\<ostatní>$.
+Ka¾dý z~øezù v~nerovnosti $(2)$ se skládá z~hran mezi tìmito skupinami vrcholù.
+Vytvoøíme tedy tabulku hran mezi ètyømi oznaèenými skupinami vrcholù a ka¾dému
+øezu z~$(2)$ oznaèíme jemu odpovídající hrany. Proto¾e je graf neorientovaný,
+staèí nám jen horní trojúhelník tabulky.
+Pro pøehlednosti si oznaèíme $L_1=\d(U \cap X), L_2=\d(U \cup X), P_1=\d(U)$ a $P_2=\d(X)$.
+$$\matrix{&X\setminus U&X \cap U&U \setminus X&\<ostatní>\cr\noalign{\smallskip}
+X\setminus U&\hbox{---}&L_1,P_1&P_1,P_2&L_2,P_2\cr
+X \cap U&&\hbox{---}&L_1,P_2&L_1,L_2,P_1,P_2\cr
+U \setminus X&&&\hbox{---}&L_2,P_1\cr
+\<ostatní>&&&&\hbox{---}\cr
+}$$
+
+Vidíme, ¾e ke ka¾dé hranì øezu na levé stranì nerovnosti máme vpravo její protìj¹ek
+a navíc hrany mezi $U\setminus X$ a $X \setminus U$ poèítáme jenom vpravo. Nerovnost
+$(2)$ tedy platí.
+
+Nyní staèí nerovnosti $(2)$ a $(1)$ odeèíst, èím¾ získáme $$c(\d(U \cap X)) \le c(\d(X)),$$
+co¾ spolu s~obrázkem dokazuje, ¾e $\d(U \cap X)$ je také minimální $uv$-øez.
+
+\vbox to 0pt{\rightline{\epsfysize=2.5cm\epsfbox{4-ght-htl-b.eps}}\vss}\vskip-\baselineskip
+{\advance\hsize by -14em\itemcount=1
+\:$t$ je prvek $X$.\par
+Postupovat budeme obdobnì jako v~pøedchozím pøípadì. Nerovnost $$\eqalignno{c(\d(X \setminus U)) &\ge c(\d(U))&(3)}$$
+platí proto, ¾e $X \setminus U$ je nìjaký \st-øez, zatímco $U$ je minimální \st-øez. Dal¹í
+$$\eqalignno{c(\d(U \setminus X)) + c(\d(X \setminus U)) &\le c(\d(U)) + c(\d(X))&(4)}$$
+doká¾eme opìt \uv{rozborem pøípadù}.
+
+}
+
+Oznaème $L_1=\d(U \setminus X), L_2=\d(X \setminus U), P_1=\d(U)$ a $P_2=\d(X)$ a vytvoøme tabulku:
+$$\matrix{&X\setminus U&X \cap U&U \setminus X&\<ostatní>\cr\noalign{\smallskip}
+X\setminus U&\hbox{---}&L_2,P_1&L_1,L_2,P_1,P_2&L_2,P_2\cr
+X \cap U&&\hbox{---}&L_1,P_2&P_1,P_2\cr
+U \setminus X&&&\hbox{---}&L_1,P_1\cr
+\<ostatní>&&&&\hbox{---}\cr
+}$$
+
+Stejnì jako v~pøedchozím pøípadì nerovnost $(4)$ platí. Odeètením $(4)$ a $(3)$ získáme
+$$c(\d(U \setminus X)) \le c(\d(X)),$$
+z~èeho¾ opìt dostaneme, ¾e $\d(U \setminus X)$ je také minimální $uv$-øez.
+\endlist
+\qed
+
+\bigskip
+Nyní se koneènì dostáváme ke konstrukci \GHT{}. Abychom mohli pou¾ívat
+indukci, zavedeme si trochu obecnìj¹í \GHT{}.
+
+\s{Definice:} Mìjme neorientovaný graf $(V,E)$. {\I Èásteèný Gomory-Hu Tree} (alias \PGHT{}) pro $R \subseteq V$ je $((R,F),C)$,
+kde $(R,F)$ je strom a mno¾ina $C=\{C(r) \;\vert\; r\in R\}$ je rozklad vrcholù $V$. Tento rozklad
+nám øíká, k~jakým vrcholùm \PGHT{} máme pøilepit které vrcholy pùvodního grafu.
+Navíc musí platit, ¾e:\numlist\ndotted
+\:$\forall r: r\in C(r)$, neboli ka¾dý vrchol \PGHT{} je pøilepen sám k~sobì,
+\:$\forall st \in F: \d\left(\bigcup_{r\in K_1} C(r)\right)=\d\left(\bigcup_{r\in K_2} C(r)\right)
+\hbox{ je minimální \st-øez, kde $K_1$ a $K_2$ jsou komponenty $(R,F) \setminus st$}.$
+\endlist
+
+
+\th{Vìta (o~existenci \PGHT{}):} Buï $(V,E)$ neorientovaný ohodnocený graf. Pro ka¾dou podmno¾inu vrcholù $R$
+existuje \PGHT{}.
+
+\proof Doká¾eme indukcí podle velikosti mno¾iny $R$.\itemize\ibull
+\:$\vert R \vert = 1$: \PGHT{} má jediný vrchol $r\in R$ a $C(r)=V$.
+\:$\vert R \vert > 1$: Najdeme dvojici vrcholù $s,t\in R$ takové, ¾e jejich minimální \st-øez $\d(W)$
+je ze v¹ech mo¾ných minimálních \st-øezù nejmen¹í. Nyní vytvoøíme graf $G_1$ z~grafu $G$ zkontrahováním
+v¹ech vrcholù $w\in W$ do jednoho vrcholu, který oznaèíme $v_1$, a vytvoøíme graf $G_2$ z~$G$ zkontrahováním
+v¹ech vrcholù $w\in V\setminus W$ do jednoho vrcholu $v_2$.
+
+\medskip
+\centerline{\epsfysize=2cm\epsfbox{4-ght-g1g2.eps}}
+
+[Proè to dìláme \uv{tak slo¾itì} a pøidáváme do $G_1$ vrchol $v_1$? Mì osobnì to dlouho pøi¹lo zbyteèné.
+Problém je v~tom, ¾e i kdy¾ dle HTL le¾í v¹echny minimální øezy oddìlující vrcholy z~$W$ v~mno¾inì vrcholù
+$W$, \<hrany> tìchto øezù celé ve $W$ le¾et nemusí. K~tìmto øezùm toti¾ patøí i hrany, které
+mají ve $W$ jenom jeden konec. Proto jsme do $G_1$ pøidali $v_1$, vedou do nìj v¹echny zajímavé
+hrany, které mají ve $W$ jeden konec. Tím {\I zajímavé} myslíme to, ¾e z~ka¾dého vrcholu $w\in W$ vede
+do $v_1$ \<nejlevnìj¹í> hrana, která z~nìj vedla do mno¾iny $V\setminus W$, pøípadnì ¾ádná, pokud
+do této mno¾iny ¾ádná hrana nevedla.]
+
+Dále vytvoøíme mno¾iny vrcholù $R_1=R \cap (V\setminus W)$ a $R_2=R \cap W$. Dle indukèního
+pøedpokladu ($R_1$ i $R_2$ jsou men¹í ne¾ $R$) existuje \PGHT{} $T_1=((R_1,F_1),C_1)$
+pro $R_1$ na $G_1$ a $T_2=((R_2,F_2),C_2)$ pro $R_2$ na $G_2$.
+
+Nyní vytvoøíme \PGHT{} pro pùvodní graf. Oznaème $r_1$ vrchol $R_1$, pro který je $v_1 \in C(r_1)$,
+obdobnì $r_2$. Oba \PGHT{} $T_1$ a $T_2$ spojíme hranou $r_1r_2$, tak¾e \PGHT{} pro $G$
+je $T=((R_1 \cup R_2,F_1 \cup F_2 \cup {r_1r_2}),C)$, pøièem¾ pro $r\in R_1$ je $C(r)=C_1(r)\setminus\{v_1\}$
+a pro $r\in R_2$ je $C(r)=C_2(r)\setminus\{v_2\}$ [odebrali jsme vrcholy $v_1$ a $v_2$ z~rozkladu $C$].
+
+Chceme ukázat, ¾e tento $T$ je opravdu \PGHT. $C$ je urèitì rozklad v¹ech vrcholù a ka¾dé
+$r\in C(r)$ z~indukèního pøedpokladu, tak¾e podmínka $1.$ je splnìna. Co se týèe podmínky $2.$, tak\itemize\ibull
+\:pro hranu $r1r2$ je $\d(W)$ urèitì minimální $r_2r_2$-øez, proto¾e øez mezi \rr\ to
+je a byl ze v¹ech mo¾ných minimálních øezù na $R$ nejmen¹í,
+\:pro hranu $e\ne r_1r_2$ je $\d(e)$ z~indukce minimální øez na jednom z~grafù $G_1$, $G_2$.
+Tento øez také pøesnì odpovídá øezu v~grafu $G$, proto¾e v~$G_1$ i v~$G_2$ jsme poèítali
+s~hranami vedoucími do $v_1$, $v_2$ a proto¾e jsme \PGHT{} napojili pøes vrcholy,
+k~nim¾ byly $v_1$ a $v_2$ pøilepeny.
+
+HTL nám navíc øíká, ¾e existuje minimální øez, který ¾ije pouze v~pøíslu¹ném z~grafù $G_1$, $G_2$,
+tak¾e nalezený øez je minimální pro celý graf $G$.
+\endlist
+\endlist
+\qed
+
+Nyní víme, ¾e \GHT{} existují, a také víme, jak by se daly konstruovat. Nicménì nalezení
+vrcholù $s,t$ tak, aby byl minimální \st-øez nejmen¹í mo¾ný, je èasovì nároèné.
+Proto si poslední vìtu je¹tì o~nìco vylep¹íme.
+
+\th{Vylep¹ení vìty o~existenci \PGHT{}:} Na zaèátku dùkazu není nutné hledat vrcholy $s$ a $t$
+takové, aby byl minimální \st-øez nejmen¹í mo¾ný. Staèí zvolit \<libovolné> vrcholy $s,t\in R$
+a nalézt minimální øez $\d(W)$.
+
+\proof Nejprve si uvìdomme, proè jsme v~pøedchozím dùkazu potøebovali, aby byl $W$ nejmen¹í ze v¹ech
+mo¾ných \st-øezù. Bylo to jenom proto, ¾e jsme jím v~\PGHT{} nakonec separovali vrcholy $r_1$ a $r_2$
+a potøebovali jsme záruku, aby byl $\d(W)$ opravdu minimální \rr-øez. Nyní musíme ukázat,
+¾e námi nalezený \st-øez $\d(W)$ je také minimálním \rr-øezem.
+
+Pro spor tedy pøedpokládejme, ¾e minimální \rr-øez $\d(X)$ má men¹í kapacitu ne¾ $\d(W)$.
+Navíc vezmìme ten pøípad, kdy se to stalo \uv{poprvé}, tak¾e pro ka¾dé men¹í $R$ je
+v¹echno v~poøádku (to mù¾eme, proto¾e pro $\vert R \vert=1$ v¹echno v~poøádku bylo).
+
+Proto¾e byl $\d(W)$ minimální \st-øez a $\d(X)$ má men¹í kapacitu, $\d(X)$ nemù¾e separovat
+$s$ a $t$. Pøitom ale separuje $r_1$ a $r_2$, tak¾e musí separovat buï $s$ a $r_1$, nebo $t$ a $r_2$.
+BÚNO nech» $X$ separuje $s$ a $r_1$.
+
+\medskip
+\centerline{\epsfysize=2.2cm\epsfbox{4-ght-rezx.eps}}
+\smallskip
+
+Podívejme se nyní na \PGHT{} $T_1$ a naleznìme v~nìm nejlevnìj¹í hranu $e$ na cestì spojující $s$ a $r_1$.
+Tato hrana definuje øez $\d(U)$, co¾ je minimální $sr_1$-øez. Proto¾e $\d(X)$ je $sr_1$-øez,
+$c(\d(U)) \le c(\d(X)) < c(\d(W))$. Teï si staèí uvìdomit, ¾e $v_1\in C(r_1)$, tak¾e $\d(U)$
+separuje nejenom $s$ a $r_1$, ale také $s$ a $v_1$. Tím pádem ale separuje také $s$ a $t$.
+To je spor, proto¾e $c(\d(U)) < c(\d(W))$ a pøitom $\d(W)$ mìl být minimální.
+\qed
+
+Teï u¾ doká¾eme \GHT{} konstruovat efektivnì -- v~ka¾dém kroku vybereme dva vrcholy $s$ a $t$,
+nalezneme v~èase $\O(\tau)$ minimální \st-øez, a výsledné komponenty s~pøidanými $v_1,v_2$
+zpracujeme rekurzivnì. Celou výstavbu tedy zvládneme v~èase $\O(n\tau)=\O(n^{5/3}m)$.
+Nalézt minimální \st-øezy pro v¹echny dvojice pak doká¾eme v~èase $\O(n^2)$, pøípadnì
+v~èase $\O(n^2\log n)$ (zále¾í na zpùsobu hledání nejlevnìj¹í hrany v~\GHT).
+
+\bye
--- /dev/null
+P=4-ght
+
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+%%%%%%%%%%%%%%%%%%%%%%
+% lecture 04-04-2006 %
+%%%%%%%%%%%%%%%%%%%%%%
+
+\input ../sgr.tex
+\prednaska{5}{Minimální kostry}{zapsali Martin Kruli¹ \& Petr Su¹il }
+
+\h{Minimální kostry a základní vìty okolo nich}
+
+\s{Definice:}
+Kostru pro nesouvislý graf definujeme jako sjednocení koster komponent grafu.
+
+\s{Definice:}
+Nech» $G$ je neorientovaný graf. Váhy (ohodnocení) hran budi¾ zobrazení $w : E \to R$. Pro podgraf $H \subseteq G$ definujeme váhu $w(H)=\sum_{e \in E} {w(e)}$.
+
+\s{Poznámka:}
+V dal¹ím textu nebudeme rozli¹ovat mezi podgrafy a podmno¾inami hran. Pro na¹e úèely jde o toté¾. Kostry jednovrcholových komponent nám slo¾itost nepokazí.
+
+\s{Definice:}
+Minimální kostra (alias MST) je taková kostra $T$, která má $w(T)$ minimální.
+
+\s{Poznámka:}
+Taková definice MST je ne¹ikovná. Vy¾aduje toti¾ existenci sèítání, pøesto¾e postaèuje pouze lineární uspoøádání.
+
+\s{Definice:}
+Buï $T \subseteq G$ nìjaká kostra grafu~$G$. Pak
+\itemize\ibull
+\:$T[x,y] := $ cesta mezi $x,y$, její¾ hrany le¾í v $T$
+\:Hrana $e=(x,y) \in E \setminus T$ je {\I lehká} vzhledem ke~kostøe $ \equiv \exists e' \in T[x,y]:w(e)<w(e')$.
+
+\centerline{\epsfysize=3cm\epsfbox{01.eps}}
+
+\:Hrana je {\I tì¾ká} $\equiv$ není lehká.
+\endlist
+
+\s{Definice:} operace {\sl swap(e,e')}
+Nech» $T$ je kostra grafu, $e'=(x,y) \in E \setminus T$ a $T[x,y]$ je cesta mezi $x$ a $y$.
+Pro libovolnou hranu $e = (a,b) \in T[x,y]$ dostaneme provedením operace $swap(e,e')$ na $T$ kostru $T' = T \cup \{e'\} \setminus \{e\}$,
+pro kterou bude zároveò platit, ¾e $e' = (x,y) \in T'[a,b]$.
+
+\s{Pozorování:}
+Mno¾ina hran $T'$, kterou jsme dostali z $T$ operací $swap(e,e')$, je opravdu kostra. $T'$ dostaneme tak, ¾e z $T$ sebereme hranu $e = (a,b)$
+a následnì pøidáme hranu $e' = (x,y)$. Sebráním $(a,b)$ se nám kostra rozpadne právì na dvì komponenty. Proto¾e ale hrana $(a,b)$ le¾ela na cestì
+mezi $x$ a $y$, budou tyto vrcholy le¾et v rùzných komponentách. Hrana $(x,y)$ pak tyto komponenty opìt spojí a výsledkem bude opìt kostra.
+
+\s{Lemma:}
+Máme-li libovolné kostry $T$ a $T'$, pak lze z $T$ dostat $T'$ koneèným poètem operací $swap$.
+
+\s{Dùkaz:}
+Oznaème si $D$ mno¾inu hran, které jsou v $T$ a nejsou v $T'$ ($D = T \setminus T'$) a $D'$ mno¾inu hran,
+které naopak jsou v $T'$ a nejsou v $T$ ($D' = T' \setminus T$). Lze jednodu¹e nahlédnout, ¾e $\vert D \vert = \vert D' \vert$,
+proto¾e i $\vert T \vert = \vert T' \vert$.
+Postupnì budeme generovat kostry $T_1$, $T_2$, ... a¾ vygenerujeme $T_n = T'$
+V ka¾ém kroku (swapu) odebereme z kostry $T_i$ nìjakou hranu z mno¾iny $D$ a nahradíme ji vhodnou hranou z $D'$ tak, aby $T_{i+1}$ byla opìt kostra.
+Pou¾ité hrany z $D$ a $D'$ samozøejmì vylouèíme. Mno¾iny $D$ a $D'$ jsou koneèné, tak¾e budeme potøebovat koneèný poèet swapù.
+Zbývá ukázat, ¾e pro libovolnou hranu z $D$ umíme najít hranu v $D'$.
+Vezmìme libovolnou hranu $e = (a,b) \in D$. $e$ nele¾í v $T'$, tak¾e $T'[a,b] \cup {e}$ tvoøí cyklus. Celý cyklus nemù¾e le¾et v kostøe a $e \in T$,
+tak¾e musí $\exists e' \in T[a,b]$, která zároveò $e' \notin T$ a tedy i $e' \in D'$. Tím jsem na¹el korespondující pár hran $e,e'$ z mno¾in $D,D'$,
+které zároveò splòují potøebné pøedpoklady pro swap ($e' \in T[a,b]$).
+\qed
+
+\s{Vìta:}
+Kostra $T$ je minimální
+$\Leftrightarrow$ $\forall e \in E \setminus T$ je $e$ tì¾ká vzhledem k $T$.
+
+\s{Dùkaz:}
+%%% proof start
+\itemize\ibull
+\:$\Rightarrow$ Sna¾íme se dokázat, ¾e $\exists$ lehká hrana $\Rightarrow$ není minimální.
+
+Nech» $\exists e = (x,y)$ lehká. Najdeme $e' \in T[x,y] : w(e) < w(e')$ (ta musí existovat z def. lehké hrany).
+Na kostru $T$ provedeme operaci $swap(e,e')$ a dostanu novou kostru $T'$. Proto¾e $w(e) < w(e')$,
+musí být i $w(T') < w(T)$ a kostra $T$ tedy nebyla minimální.
+
+\bigskip
+\:$\Leftarrow$
+
+Mìjme kostru $T$ bez lehkých hran a minimální kostru $T_{min}$. Doká¾eme, ¾e lze z $T$ dostat $T_{min}$ opakovaným pou¾itím operace $swap$.
+Z dùsledku definice operace $swap$ víme, ¾e kostry lze mezi sebou libovolnì pøeswapovat. Zbývá nám dokázat, ¾e $w(T) = w(T_{min})$.
+Postupným swapováním budu dostávat vzájemnì rùzné kostry $T$, $T_1$, $T_2$, ... $T_{min}$ tak,
+¾e ka¾dá následující kostra je výsledkem operace $swap$ na pøedchozí. Podíváme se nejprve na $T$ a $T_1$.
+
+Nech» $T_1 = swap(e,e') T$ tj. $e = (a,b) \in T$ a $e' = (x,y) \in T_1$. Hrana $e'$ musí z definice swapu le¾et na $T[a,b]$.
+Ke kostøe $T$ neexistují ¾adné lehké hrany, tak¾e $\forall f \in T[a,b]$ je $w(f) \ge w(e)$ tedy i $w(e') \ge w(e)$ a $w(T) \le w(T_1)$.
+
+Analogicky lze postupovat, a¾ dostaneme nerovnost $w(T) \le w(T_1) \le w(T_2) \le ... \le w(T_{min})$.
+$T_{min}$ je minimální kostra, tak¾e $w(T) = w(T_{min})$ a $T$ je tedy také minimální.
+
+\endlist
+\qed
+
+\s{Poznámka:}
+Ji¾ víme, ¾e pro definici minimální kostry nám staèí lineární uspoøádání.
+Pokud bude $\forall e,e':e \neq e' \Rightarrow w(e) \neq w(e')$, pak pro $T_1, T_2$ minimální kostry, $T_1$ mù¾u pøeswapovat na $T_2$, proto¾e jsou ale váhy rùzné $\#swapù = 0$.
+
+\s{Poznámka:}
+Pokud dostaneme kvaziuspoøádání, mù¾eme douspoøádat ekvivalentní váhy napø. lexikograficky.
+
+\s{Dùsledek:}
+Jsou-li v¹echny váhy rùzné, pak je MST urèen jednoznaènì. Tj.
+$\forall e_1,e_2 \in E(G)$ $w(e_1) \neq w(e_2)$ $\Rightarrow$ $\exists ! MST(G)$
+\vfill\eject
+\s{Dùkaz:}
+mám MST $T_1$ a $T_2$ a zkusím mezi nimi pøeswapovat. Pøi pøeswapovávání se ale zmìní váhy, nebo» v¹echny váhy jsou rùzné. Tedy jedna z koster nemù¾e být minimální.
+
+
+\h{Èervenomodrý meta-algoritmus}
+
+\s{Meta-algoritmus:} (pøedpokládáme, ¾e v¹echny váhy jsou rùzné)
+
+\algo
+\:na poèátku v¹echny hrany bezbarvé
+\:dokud lze opakujeme: zvol modré nebo èervené pravidlo, proveï pravidlo
+\endalgo
+\itemize\ibull %%% pravidla
+\:Modré pravidlo
+
+vezmìme øez v G takový, ¾e jeho nejlehèí hrana není modrá (prevence zacyklení) a obarvíme ji modøe
+
+\:Èervené pravidlo
+
+vezmìme kru¾nici v G takovou, ¾e její nejtì¾¹í hrana není èervená (prevence zacyklení) a obarvíme ji èervenì
+
+\endlist %%%pravidla
+
+\s{Vìta:}
+Pro Èervenomodrý meta-algoritmus platí:
+\algo
+\:zastaví se
+\:po zastavení jsou v¹echny hrany obarvené
+\:modøe obarvené hrany tvoøí kostru
+\endalgo
+
+
+\s{Lemma 1:} Modrá hrana $\in$ MST.
+
+\s{Dùkaz:} Minimální kostra $T_{min}$ je urèena jednoznaènì (váhy jsou rùzné).
+Mìjme øez $C$ a minimální kostru $T_{min}$. Podle modrého pravidla je hrana $e = (x,y)$ nejlehèí v øezu $C$.
+Pokud $\exists e' \ne e \in C \cap T_{min}[x,y]$ (tj. jiná hrana z øezu $C$ patøí do min. kostry) mù¾eme provést $swap(e',e)$.
+Hrana $e$ byla nejlehèí, tak¾e $w(e) < w(e')$. Tím jsme si sní¾ili váhu kostry a $T_{min}$ nemohla být minimální $\Rightarrow$ spor.
+\qed
+
+\centerline{\epsfysize=3cm\epsfbox{02.eps}}
+
+
+\s{Lemma 2:} Èervená hrana $\notin$ MST.
+
+\s{Dùkaz:} Sporem: Pøedpokládejme, ¾e máme kru¾nici a na ní èervenou hranu $e = (x,y) \in T_{min}$.
+Odebráním èervené hrany se nam $T_min$ rozpadne na dvì komponenty $T_x$ a $T_y$.
+Nìkteré vrcholy z kru¾nice pøipadnou do komponenty $T_x$ a ostatní do $T_y$.
+Lze jednodu¹e nahlédnout, ¾e musí existovat hrana $e' = (x',y')$ taková,
+¾e $x' \in T_x$ a $y' \in T_y$. Hrana $e$ byla nejtì¾¹í na kruønici, tak¾e $w(e') < w(e)$.
+Z toho vyplývá, ¾e provedením $swap(e,e')$ na $T_{min}$ dostaneme lehèí kostru a $T_{min}$ tedy není minimální $\Rightarrow$ spor.
+\qed
+
+\centerline{\epsfysize=4cm\epsfbox{03.eps}}
+
+\s{Lemma 3:} Po zastavení jsou v¹echny hrany obarvené.
+
+\s{Dùkaz:} Opìt sporem: Nech» existuje hrana $e = (x,y)$, která je stále bezbarvá. Oznaèíme si mno¾inu vrcholù
+$M_x := \{v \in V \vert \exists$ modrá cesta x $\to$ v $\}$. Nyní mohou nastat dvì mo¾nosti:
+
+\itemize\ibull
+\:$y \in M_x$ (tj. modrá cesta vede z $x$ a¾ do $y$)
+
+Modrá cesta je v minimální kostøe a minimální kostra nemá ¾ádné lehké hrany $\Rightarrow$ hrana $e$ je tì¾ká a mù¾u na ní pou¾ít èervené pravidlo.
+
+\centerline{\epsfysize=3cm\epsfbox{04.eps}}
+
+\:$y \notin M_x$
+
+V tom pøípadì existuje øez $\delta(M_x)$, takový, ¾e hrana $e$ patøí do øezu a zároveò $\delta(M_x)$ neobsahuje modré hrany
+$\Rightarrow$ mù¾u pou¾ít modré pravidlo.
+
+\centerline{\epsfysize=3cm\epsfbox{05.eps}}
+
+\endlist
+
+\algo
+
+\s{Dùkaz vìty:}
+\:zastaví se
+
+V algoritmu nikdy nepøebarvujeme a v ka¾dém kroku obarvíme právì jednu hranu (a» u¾ èerveným, nebo modrým pravidlem).
+Máme koneèný poèet hran $\Rightarrow$ koneèný poèet krokù.
+
+\:po zastavení jsou v¹echny hrany obarvené
+viz. lemma 3
+
+\:modøe obarvené hrany tvoøí kostru
+
+Lemma 1 a 2 øíká, ¾e modré hrany le¾í na kostøe a èervené naopak le¾í mimo kostru. Dále pak z lemmatu 3 víme, ¾e v¹echny hrany jsou obarvené.
+Právì v¹echny modré hrany tvoøí kostru.
+
+\endalgo
+
+\qed
+
+\s{Poznámka:}
+Dualita èerveného a modrého pravidla.
+
+Mezi èervenými a modrými hranami existuje dualita obdobnì jako u rovinných grafù existuje stìnovì-vrcholová dualita nebo jako existuje dualita u grafových matroidù.
+Pro ka¾dý grafový matroid $\exists$ duální matroid. Kostry $\leftrightarrow$ cykly, kostra $\leftrightarrow$ komplement kostry duálního grafu.
+
+\h{Klasické algoritmy na hledání MST}
+
+\s{Kruskalùv:} (hladový algoritmus).
+
+\itemize\ibull
+\:setøídíme hrany podle vah (i zde nám staèí uspoøádání na hranách), kostra je na zaèátku prázdná (ka¾dý vrchol je v samostatné komponentì souvislosti)
+\:bereme hrany ve vzestupném poøadí
+\:pro ka¾dou hranu $e$ se podíváme jaké komponenty spojuje - spojuje-li rùzné komponenty, zaøadím hranu do stromu a obì komponenty slouèíme, jinak hranu zahodíme
+\endlist
+
+Z hlediska na¹eho Èervenomodrého meta-algoritmu se lze na Kruskala dívat tak, ¾e si pìstujeme modrý les, a pokud hrana spojuje dva stromeèky, omodøím ji a
+nechám stromeèky srùst. Naopak spojuje-li hrana $e = (x,y)$ stejné komponenty, znamená to, ¾e mezi $x$ a $y$ vede modrá cesta $T[x,y]$ a $e$ tedy uzavírá cyklus
+na kterém jsou v¹echny hrany modré. Díky pøedchozímu uspoøádání víme, ¾e $e$ bude urèitì tì¾¹í, ne¾ v¹echny hrany na modré cestì a tak ji mù¾eme s klidným svìdomím
+obarvit na èerveno (tj. zahodit).
+
+Tì¾i¹tì implementace Kruskala spoèívá v efektivním sluèování komponent (tzv. Union-find problem). Pøi efektivní implementaci lze dosáhnou èasové slo¾itosti
+$\O(m \log{n} + m \alpha(n))$
+
+\s{Borùvkùv:}
+
+Opìt si budeme pìstovat modrý les, av¹ak tentokrát jej budeme roz¹iøovat ve fázích. V jedné fázi nalezneme ke ka¾dému stromeèku nejlevnìj¹í incidentní hranu
+a v¹echny tyto nalezené hrany naráz pøidáme (aplikujeme nìkolik modrých pravidel najednou).
+
+Poèet stromeèku klesá exponenciálnì $\Rightarrow$ ka¾dou komponentu (resp. její incidentní hrany) si mohu dovolit prohledávat lineárnì.
+Pokud na udr¾ování incidentních hran pou¾ijeme binomiální haldy, dostaneme èasouvou slo¾itost $\O(m \log{n})$. Vzhledem k povaze algoritmu pùjde ka¾dá
+fáze dobøe paralelizovat.
+
+\s{Jarníkùv:}
+
+Jarníkùv algoritmus je podobný Borùvkovi, ale s tím rozdílem, ¾e nenecháme rùst celý les, ale jen jeden modrý strom. To co se v Borùvkovi nazývalo celou fází,
+bude zde obyèejný krok. V ka¾dém kroku vyhledáme nejlevnìj¹í incidentní hranu a pøidáme ji do stromu. Kroky opakujeme, dokud se strom nerozroste pøes v¹echny vrcholy.
+Jarník se sice nedá rozumnì paralelizovat, av¹ak pøi ¹ikovné implementaci slibuje stejnou èasovou slo¾itost jako Borùvka ($\O(m \log{n})$).
+
+\s{Poznámka:}
+pro celoèíselné váhy hran z rozsahu $\{1,\ldots k\}$, se umí algoritmus se slo¾itostí $\O(m k)$.
+
+\bye
--- /dev/null
+P=5-mst
+
+include ../Makerules
--- /dev/null
+\input ../sgr.tex
+
+\prednaska{6}{Vylep¹ení Borùvkova a Jarníkova algoritmu}{zapsali Petr ©koda a Tomá¹ Gavenèiak}
+
+\h{Upravená verze Borùvkova algoritmu pro rovinné grafy}
+
+Vyjdeme z my¹lenky, ¾e mù¾eme po ka¾dém kroku pùvodního Borùvkova algoritmu vzniklé komponenty
+souvislosti grafu zkontrahovat do jednoho vrcholu a tím získat men¹í graf, který mù¾eme
+znovu rekurzivnì zmen¹ovat. To funguje obecnì, ale uká¾eme, ¾e pro rovinné grafy tak dosáhneme
+lineární èasové slo¾itosti.
+
+\s{Pozorování:}
+Pokud $F \subseteq {\rm MST}(G)$ (kde ${\rm MST}(G)$ je minimální kostra grafu~$G$), $G'$~je graf vzniklý
+z~$G$ kontrakcí podél hran z~$F$, pak kostra grafu~$G$, která vznikne z~${\rm MST}(G')$ zpìtným
+expandováním kontrahovaných vrcholù, je ${\rm MST}(G)$.
+To nás vede k následujícímu algoritmu:
+
+\s{Algoritmus: MST v rovinných grafech}
+\algo
+\:Ke ka¾dému vrcholu najdeme nejlevnìj¹í incidentní hranu -- dostaneme mno¾inu hran $F \subseteq E$.
+\:Graf zkontrahujeme podle $F$ následovnì:
+\::Prohledáme do ¹íøky graf $(V(G), F)$ a pøiøadíme vrcholùm èíslo komponenty, ve které jsou.
+\::Pøeèíslujeme hrany v~$G$ podle èísel komponent.
+\:Odstraníme násobné hrany:
+\::Setøídíme hrany lexikograficky pomocí pøihrádkového tøídìní (násobné hrany jsou nyní pospolu).
+\::Projdeme posloupnost hran a z~ka¾dého úseku multihran odstraníme v¹echny a¾ na nejlevnìj¹í hranu.
+\:Pokud stále máme netriviální graf, opakujeme pøedchozí kroky.
+\endalgo
+
+\s{Èasová slo¾itost:}
+Oznaème si $n_i$ a $m_i$ poèet vrcholù a hran na~poèátku $i$-té iterace.
+Ka¾dý z~krokù 1--7 trvá $\O(m_i)$, proto i celý cyklus algoritmu trvá $\O(m_i)$.
+Poèet vrcholù grafu klesá s~ka¾dým cyklem exponenciálnì: $n_i \leq n / 2^i$.
+Na~zaèátku ka¾dého cyklu je graf rovinný (kontrakcí hrany v~rovinném grafu se rovinnost
+zachovává) a poèet hran rovinného grafu je lineární v poètu vrcholù, tak¾e
+platí $m_i < 3n_i$. Celkovou èasovou slo¾itost dostaneme jako souèet doby trvání
+v¹ech cyklù: $\O(\sum_i m_i) = \O(\sum_i n_i) = \O(n)$.
+
+\h{Minorovì uzavøené tøídy}
+
+Pøedchozí algoritmus ve~skuteènosti funguje v~lineárním èase i pro obecnìj¹í tøídy grafù,
+ne¾ jsou grafy rovinné. Tím správným universem jsou minorovì uzavøené tøídy:
+
+\s{Definice:}
+Graf $H$ je {\I minorem} grafu $G$ $\equiv$ $H$ lze z $G$ získat
+mazáním vrcholù èi hran a kontrahováním hran. Znaèíme $H \preceq G$.
+
+\s{Pozorování:}
+$H \subseteq G \Rightarrow H \preceq G$.
+
+\s{Definice:}
+Tøída grafù $\cal C$ je {\I minorovì uzavøená} $\equiv \forall G,H: G \in {\cal C}, H \preceq G \Rightarrow H \in {\cal C}$.
+
+\s{Vìta:} (Robertson, Seymour)
+Pokud je $\cal C$ minorovì uzavøená tøída grafù, existuje koneèná mno¾ina grafù $Z$ taková,
+¾e pro ka¾dý graf $G$ platí:
+$$G \not\in {\cal C} \iff \exists H \in Z: H \preceq G.$$
+(Èili ka¾dou minorovì uzavøenou tøídu lze charakterizovat {\I koneèným} poètem zakázaných minorù.
+To není samo sebou, dokazuje se to dosti pracnì, ale plyne z~toho spousta zajímavých dùsledkù.)
+
+\s{Pozorování:} Napøíklad pro rovinné grafy jsou tìmi zakázanými minory právì
+$K_{3,3}$ a $K_5$. To plyne z~Kuratowského vìty: jedna implikace je triviální,
+druhá dá trochu práce: pokud je $K_{3,3} \preceq G$, najde se v~$G$ podgraf
+isomorfní nìjakému dìlení~$K_{3,3}$; pokud je $K_5 \preceq G$, podgraf isomorfní
+dìlení~$K_5$ se v~$G$ najít nemusí, ale pokud se nenajde, najde se tam podgraf isomorfní
+dìlení $K_{3,3}$. (Zkuste si sami.)
+
+\s{Vìta:} (Robertson, Seymour)
+Pokud je tøída grafù $\cal C$ minorovì uzavøená a netriviální (alespoò jeden graf v~ní le¾í
+a alespoò jeden nele¾í), pak $\exists c > 0: \forall G \in {\cal C}:
+\vert E(G) \vert \leq c\cdot \vert V(G) \vert$.
+
+\s{Dùsledek:}
+Jeliko¾ v¹echny grafy vygenerované pøedchozím algoritmem jsou minory grafu ze~vstupu,
+mù¾eme pro odhad jejich hustoty pou¾ít pøedchozí vìtu a dostaneme tak, ¾e pøedchozí
+algoritmus má lineární èasovou slo¾itost dokonce pro ka¾dou netriviální minorovì uzavøenou
+tøídu grafù.
+
+%%%Tomas Gavenciak
+
+\h{Jarníkùv algoritmus s Fibonacciho haldou}
+
+Pùvodní Jarníkùv algoritmus s~haldou má díky ní slo¾itost $\O(m\log n)$, to zlep¹íme pou¾itím
+Fibonacciho haldy $H$, do~které si budeme ukládat trojice $(v,w,w(vw))$ vrcholù $v$ sousedících
+s~dosavadní podkostrou $T$ pøes hranu $vw$, $w\in T$, která bude navíc nejlevnìj¹í mo¾ná.
+
+\newcount\algcnt
+\s{Algoritmus: Jarníkùv algoritmus~\#2 (Fredman, Tarjan)}
+\algo
+\:Zaèneme libovolným vrcholem $v_0$: $T=\{v_0\}$.
+\:Do~haldy $H$ umístíme v¹echny sousedy $v_0$ spolu s pøíslu¹nými hranami.
+\:Opakuji dokud $H\neq\emptyset$:
+\::$(v,w,w(vw))=\<DeleteMin>(H)$
+\::$T:=T\cup\{vw\}$
+\::Pro v¹echny sousedy $u\in E\backslash T$ vrcholu $v$ upravím haldu:
+\:::Pokud je $u$ v~$H$ nový, pøidáme jej spolu s~nejlevnìj¹í hranou vedoucí z~$u$ do~$T$.
+\:::Pokud u¾ $u$ v~$H$ je a $uv$ je levnìj¹í ne¾ pùvodní nejlevnìj¹í hrana z~$u$
+do~$T$, nahradím jeho záznam v~$H$ za~$(u,v,w(uv))$ a provedu $\<DecreaseKey>(u,w(uv))$.
+\global\algcnt=\itemcount
+\endalgo
+
+Správnost algoritmu pøímo plyne ze~správnosti Jarníkova algoritmu.
+
+\s{Èasová slo¾itost:}
+Slo¾itost tohoto algoritmu bude $\O(m+n\log n)$, nebo» vnitøní cyklus se provede
+nanejvý¹ $n$-krát, za~\<DeleteMin> v~nìm tedy zaplatíme $\O(n\log n)$, za~pøidávání
+vrcholù do~$H$ a~nalezání nejlevnìj¹ích hran zaplatíme $\O(m)$ (na~ka¾dou hranu takto
+sáhneme nanejvý¹ dvakrát), za~sni¾ování vah vrcholù v~haldì rovnì¾ pouze $\O(m)$
+(nanejvý¹ $m$-krát provedu porovnání vah a \<DecreaseKey> v~$\the\algcnt.$ za~$\O(1)$).
+
+Toto zlep¹ení je dùle¾itìj¹í, ne¾ by se mohlo zdát, proto¾e nám pro grafy s~mnoha hranami
+(konkrétnì pro grafy s~$m=\Omega(n\log n)$) dává lineární algoritmus.
+
+\h{Kombinace Jarníkova a Borùvkova algoritmu}
+
+K~dal¹ímu zlep¹ení dojde, kdy¾ nejprve spustíme $\log\log n$ cyklù Borùvkova algoritmu
+s~kontrahováním vrcholù, tímto dojde k~velkému sní¾ení poètu vrcholù.
+
+\s{Algoritmus: Jarníkùv algoritmus~\#3 (pùvod neznámý)}
+\algo
+\:Provedeme $\log\log n$ cyklù upraveného Borùvkova algoritmu s~kontrahováním hran popsaného vý¹e.
+\:Pokraèujeme Jarníkovým algoritmem~\#2.
+\endalgo
+
+\s{Èasová slo¾itost:}
+Slo¾itost první èásti je $\O(m\log\log n)$.
+Poèet vrcholù se po~první èásti algoritmu sní¾í na~$n'\leq n/\log n$ a slo¾itost druhé èásti bude
+tedy nanejvý¹ $\O(m+n\log n'/\log n)=\O(m)$. Nyní ji¾ máme lineární algoritmus i~pro grafy
+s~$m\geq n\log\log n$.
+
+\h{Jarníkùv algoritmus s~omezením velikosti haldy}
+
+Je¹tì vìt¹ího zrychlení dosáhneme, omezíme-li Jarníkovu algoritmu \#2 vhodnì
+velikost haldy a takto budeme bìhem jednoho Jarníkova algoritmu skládat pouze
+jednotlivé podkostøièky zastavené v rùstu pøeteèením haldy, podle kterých
+graf následnì skontrahujeme a budeme pokraèovat s mnohem men¹ím grafem.
+
+\s{Algoritmus: Jarníkùv algoritmus~\#4 (Fredman, Tarjan)}
+\algo
+\:Opakuji, dokud mám netriviální $G$ (s alespoò jedou hranou).
+\::$t=\vert V_G\vert$.
+\::Zvolím $k=2^{2m/t}$ podle aktuálního $t$.
+\::$T=\emptyset$
+\::Opakuji, dokud existují vrcholy mimo $T$:
+\:::Najdu vrchol $v_0$ mimo $T$.
+\:::Spustím Jarníkùv alg. \#2 pro celý graf od $v_0$, zastavím ho, pokud:
+\global\algcnt=\itemcount
+\::::$\vert H\vert\geq k$ (pøekroèena vel. haldy) nebo
+\::::$H=\emptyset$ (do¹li sousedé) nebo
+\::::do $T$ jsem pøidal hranu oboustrannì incidentní s~hranami v~$T$ (pøipojil
+jsem novou podkostru k~nìjaké u¾ nalezené).
+\::Skontrahuji $G$ podle podkoster nalezených v~$T$.
+\endalgo
+
+\s{Pozorování:}
+Ka¾dá z~nalezených podkoster v~$T$ je incidentní s~alespoò $k$ hranami (a~nebo
+algoritmus u¾ konèí).
+Jak to vypadá pro jednotlivá ukonèení:
+\numlist\ndotted
+\itemcount=\algcnt
+\:$\vert H\vert\geq k$ -- bylo u¾ pøidáno dost vrcholù.
+\:$H=\emptyset$ -- nalezena celá kostra, konèím.
+\:Pøipojím se k~u¾ existující podkostøe -- jen ji zvìt¹ím.
+\endlist
+
+\s{Èasová slo¾itost:}
+Dùsledkem pozorování je, ¾e poèet podkoster v~jednom prùchodu je nanejvý¹
+$2m/k$. Pro $t'$ a $k'$ v následujícím kroku potom platí $t'\leq 2m/k$ a $k'=2^{2m/t'}\geq 2^k$,
+prùchodù bude tedy nanejvý¹ $\log^* n$\foot{$\log^* n$ je inverzní funkce k~\uv{vì¾i
+z~mocnin}, èili $\min\{i:\log^{(i)} n<1 \}$, kde $\log^{(i)} n$ je $i$-krát iterovaný
+logaritmus.}, proto¾e prùchod s~$k>n$ bude u¾ urèitì poslední.
+Jeden vnìj¹í prùchod trvá $\O(m+t\log k)$, zvolím-li tedy $k=2^{2m/t}$, potom bude mít
+jeden prùchod slo¾itost $\O(m)$. Celková slo¾itost bude $\O(m\log^{*}n)$.
+Podrobnìj¹í analýza pak dá je¹tì o~nìco lep¹í výsledek, a~to $\O(m\beta(m,n))$, kde
+$\beta(m,n)=\min\{i:\log^{(i)}n<m/n\}$, co¾ opìt dává lineární algoritmus pro
+grafy s~$m\geq n\log^{(k)}n$ pro libovolnou konstantu $k$ ($\beta(m,n)$ tehdy vyjde konstantní).
+
+
+%\newbox\tombox\newdimen\tomwd
+%\setbox\tombox=\hbox{$\log\log\log\dots\log$}
+%\tomwd=\wd\tombox
+%\raise 7pt\hbox{$\underbrace{\box\tombox}\kern-\tomwd
+%\lower 16pt\hbox to\tomwd{$\hfill\log^{*}n$\hfill} n<1$},
+%\message{dim: \the\tomwd, \the\tombox}
+
+\h{Dal¹í výsledky}
+
+Chazelle popisuje algoritmus se slo¾itostí $\O(m\alpha(m,n))$. Podle Pettieho je mo¾né dosáhnout
+a¾ optima, tedy slo¾itosti $\O(T(m,n))$, kde $T$ je hloubka optimálního rozhodovacího stromu
+pro grafy na~$n$ vrcholech s $m$ hranami (není ale známo, jak ho sestrojit, ani jak je hluboký);
+zajímavé je, ¾e tento algoritmus funguje i na Pointer Machine, tak¾e pokud existuje lineární
+algoritmus na~MST, nepotøebuje sílu RAMu.\foot{O výpoèetních modelech viz pøí¹tí pøedná¹ka.}
+
+\bye
--- /dev/null
+P=6-borjar
+
+include ../Makerules
--- /dev/null
+\input ../sgr.tex
+
+\long\def\onecol#1{\hbox to 0.5\hsize{\vtop{\hsize=0.45\hsize #1}\hss}}
+\long\def\twocol#1#2{\par\line{\onecol{#1}\hss\onecol{#2}}}
+\def\austere{\parskip=0pt\parindent=0pt}
+\def\lines{\obeylines\austere}
+\def\\{\hfil\break}
+\def\rack#1#2{\setbox0=\hbox{#1}\hbox to \wd0{#2}}
+
+% Alik
+% ----
+\def\0{{\bf 0}}
+\def\1{{\bf 1}}
+\def\9{\rack{\0}{\hss$\cdot$\hss}}
+\def\opnot{\mathop{\lnot}}
+\def\shl{\mathop{<\!<}}
+\def\shr{\mathop{>\!>}}
+\def\opdiv{\mathop{/}}
+\def\opmod{\mathop{\%}}
+\def\ALIK{{\sc Alik}}
+
+\def\alik#1{%
+\medskip
+\halign{\qquad $ ##$&\hbox to 0.4\hsize{${}##$\hfil}&\hfil $ ##$&\hbox to 0.4\hsize{${}##$ \hss}\cr
+#1}
+\medskip
+}
+
+\prednaska{7}{Výpoèetní modely}{zapsal Zdenìk Vilu¹ínský }
+
+\h{Druhy výpoèetních modelù}
+
+\s{Poznámka:} V dostateènì silném výpoèetním modelu lze v¹e (a¾ na
+vyjímky) spoèítat v konstantním èase.
+
+\s{Church - Turingova teze:} V¹echny výpoèetní modely které nejsou
+absurdní jsou ekvivalentní.
+
+ Toto tvrzení ale neplatí a¾ do tìch nejmen¹ích detailù, které by nás zajímaly. Spoèítáme to samé,
+ale pøi zkoumání jemnìj¹ích slo¾itostí ne¾ polynomiálních nejsou
+modely stejnì silné.
+
+Následující modely se li¹í v tom, zda vìøíme, ¾e lze pole
+indexovat v konstantním èase nebo ne.
+
+\s{Pointer Machine:} Uznává 2 typy dat:
+
+\algo
+\: Èísla - jsou omezená parametry modelu
+\: Pointry -
+ukazují na "krabièky"
+\endalgo
+
+Pamì» tohoto modelu je slo¾ená z pevného poètu registrù na èísla a
+na pointry a neomezeného poètu krabièek. Ka¾dá krabièka má pevný
+poèet polo¾ek na èísla a pointry a na krabièku se lze odkázat
+pointrem. Aritmetika není a¾ na triviální pøípady v konstantním
+èase.
+
+\s{Random Access Machine:} Je rodinka modelù, které se li¹í v
+rùzných detailech. Ov¹em dost podstatných. V¹echny mají spoleèné
+to, ¾e existuje pouze jeden datový typ - {\bf èísla}, mají pevný
+poèet registrù (do jednoho registru se vejde jedno èíslo) a pamì»,
+co¾ je pole indexované èísly, jeho¾ polo¾ky jsou èísla. Instrukce
+jsou operace s èísly.
+
+Tento model má dvì nejasnosti:
+
+\algo
+\: Velikost èísel
+\: Cena operací
+\endalgo
+
+\h {Typy Random Access Machine (RAM)}
+
+\s{1) Logaritmická cena instrukcí}
+
+Èasová slo¾itost instrukce roste s poètem bitù èísel, na kterých
+instrukce pracuje. Dojde tím k odstranìní absurdit, ale ¹patnì se
+poèítá $\O(x)$.
+
+\s{2) Omezení velikosti slova; Operace v $\O(1)$}
+
+$W$ - ¹íøka slova. Mám li vstup délky $n$, pak $W$ je alespoò
+$\log n$. Mù¾eme BÚNO pøedpokládat, ¾e $W = \O(\log n)$. Není to
+zcela ekvivalentní prvnímu RAM, ale dovedeme na nìj pøevést se
+zpomalením $\O(\log n)$. Máme k dispozici nìkolik modelù operací:
+
+\algo
+\:WordRAM - +, -, *, /, \%, <<, >>, \&,\^{}
+
+\:AC$^0$ RAM - Jsou funkce spoèitatelné hradlovou sítí, která má
+polynomiální poèet hradel, libovolný poèet vstupù a konstantní
+hloubku.
+
+\:AC$^k$ - Jsou funkce spoèitatelné hradlovou sítí, která má
+polynomiální poèet hradel, libovolný poèet vstupù a hloubku $
+\O(\log^k n)$.
+
+\:MC$^k$ - Jsou funkce spoèitatelné hradlovou sítí, která má
+polynomiální poèet hradel a má nejvý¹e 2 vrstvy.
+
+\: Word $\cap$ AC$^0$ - Mù¾eme v¹e, kromì *, /, \%. Nejedná se ale
+o pøíli¹ zajímavý model .
+\endalgo
+
+Zamìøíme se pøevá¾nì na WordRAM, ale ve¹keré operace lze vyrobit i
+v AC$^0$RAM.
+
+\s{Poznámka:} Ve¹keré Pointer-Machine polynomiání algoritmy, jsem
+schopen na RAM spoèíst v asymptoticky stejném èase.
+
+\h {Van Emde-Boas Trees}
+
+Struktura, která si pamatuje mno¾inu uzlù $X$ z nìjakého omezeného
+universa $U$, $X \subseteq \{0..U-1\}$ a podporují "stromové
+operace" v èase $\O(\log\log U)$. Stromovými operacemi jsou
+my¹leny základní mno¾inové operace a operace pracující s
+uspoøádáním. V takovéto struktuøe pak mají operace slo¾itost:
+
+\settabs 5 \columns \+ & VEBT & Best Known\cr \+ Tøídìní &
+$\O(n\log\log U)$ & $\O(n\log\log n)$ \cr \+ MST & $\O(m\log\log
+U)$ & $\O(m)$ - pro integerovì ohodnocené\cr \+ Dijkstra &
+$\O(m\log\log U)$ & $\O(m+n\log\log U)$, neorientovanì $\O(m)$\cr
+
+
+VEBT pro universum velikosti $U$ (WLOG $U=2^{2^K}$) obsahuje:
+\algo
+
+\:min, max - alespoò jedno nesmí být ulo¾eno rekurzivnì v
+podstromech
+
+\:pøihrádky $P_0$ -- $P_{\sqrt{U}}$, pøièem¾ $x$ padne do
+$P_{\lfloor x/\sqrt{U} \rfloor}$ a pøihrádky jsou ulo¾ené jako
+VEBT($\sqrt{U}$)
+
+\:"sumární" VEBT($\sqrt{U}$) obsahující èísla neprázdných
+pøihrádek
+\endalgo
+
+\s{Find:} - pøímoèaøe v $\O(\log\log U)$
+
+\s{Insert:} Slo¾itost $\O(\log\log U)$
+\algo
+
+\:o¹etøení triviálních stromù
+
+\: je-li tøeba, swap s min/max
+
+\: padne do pøihrádky $P_i$, která je buï
+
+\:: prázdná $\Rightarrow$ update sumárního stromu + zalo¾ení
+triviálního stromu pro pøihrádku
+
+\:: nìco v ní je $\Rightarrow$ zanoøím do podstromu
+\endalgo
+
+\s{Delete:} - analogicky
+
+\s{FindMin:} - Slo¾itost $\O(1)$
+
+\s{Succ:} - Slo¾itost $\O(\log\log U)$
+\algo
+
+\: porovnání $x$ s min a max
+
+\:: $x$ = min $\Rightarrow$ i=FindMin(sumarni strom); Succ =
+FindMin($P_i$) \:: $x$ = max $\Rightarrow$ Succ = 0
+
+\: $x \in P_i$
+
+\:: $x \not =$ max $P_i$ $\Rightarrow$ vnoøím do $P_i$
+
+\:: $x =$ max $P_i$ $\Rightarrow$ j=Succ(i) v sumárním stromu;
+Succ = FindMin($P_j$)
+\endalgo
+
+Na operace mám hezké slo¾itosti, ale s inicializací to tak pìkné
+není. Nicménì z následujícího vyplyne, ¾e není tøeba inicializaci
+psát.
+
+\h{Modely inicializace}
+
+\s{1) "Pøi odchodu zhasni":} Pøidáme axióm, ¾e na zaèátku je celá
+pamì» vynulovaná. Za sebou pak pamì» uklidíme.
+
+\s{2) Neinicializovano:} V pamìti je cokoliv, musím se postarat
+sám
+
+\s{Vìta:} Buï $P$ program pro WordRAM s nulami inicializované
+pamìti bì¾ící v èase $T(n)$. Pak existuje program $P$' pro WordRAM
+s {\bf neinicializovanou} pamìtí poèítající toté¾ v
+èase$\O(T(n))$.
+
+\s{Dùkaz:} Bìhem výpoètu si budeme pamatovat, k terých pamì»ových
+buòkách u¾ nìco máme. Prokládaòe ulo¾íme do pamìti 2 pole:
+\itemize\ibull \:M - pamì» $P$ \:L - seznam pou¾itých bunìk v M a
+L[0] = délka L
+\endlist
+
+Redukci vylep¹íme tím, ¾e zalo¾íme dal¹í prolo¾ené pole R, ve
+kterém $i$-tá polo¾ka je buï neinicializovaná nebo øíká, na které
+pozici v L je $i$-tá pamì»ová buòka pùvodního stroje.R[$i$] = $j$:
+L[$j$] = $i$ $\vee$ neinicializováno pokud $\not \exists$ takové $j$.
+
+Tak¾e jsem schopen v konstantním èase rozhodnout,
+jestli u¾ do buòky bylo psáno a ve stejném èase teké buòku pøidat.
+
+\qed
+
+\s{3) Zákaz dìr:} $\Rightarrow$ randomizované
+
+\s{Dùsledek:}Ani v jednom z pøedchozích pøípadù nemusíme psát
+inicializaci
+
+\h{Operace s RAM}
+
+Dá se s jistými omezeními pou¾ívat jako vektorový poèítaè.
+ \itemize\ibull
+
+\:Nejpravìj¹í jednièka:
+\alik{
+ & & x&=\0\1\9\9\9\0\1\1\0\0\9\9\9\0 \cr
+ & & x - 1&=\0\1\9\9\9\0\1\0\1\1\9\9\9\1 \cr
+ & & x \land (x - 1)&=\0\1\9\9\9\0\1\0\0\0\9\9\9\0 \cr
+ & & x \oplus (x \land (x - 1))&=\0\0\9\9\9\0\0\1\0\0\9\9\9\0 \cr}
+\endlist
+
+Pro následující sadu vektorových operací vezmeme vektor, a
+prolo¾íme jeho jednotlivé prvky zleva nulami. \itemize\ibull
+
+ \alik{ & & \0 x_{n-1} \0 x_{n-2} \9\9\9 \0 x_1\0 x_0 \cr}
+
+\: Replikace:
+
+Replikace $x$ vytvoøí vektor samých $x$
+ \alik{ & & x*(\0\0\9\9\9\0\0\1)^n \cr}
+
+\: Seètení v¹ech slo¾ek $(Sum)$:
+
+a) sèítáním modulo $1^{n+1}$
+
+b) násobením vhodnou konstantou
+\def\.{\hbox to 3pt{\hfil}}
+\def\z{\rack{$x_1$}{\hss\1\hss}}
+\def\y{\rack{$x_{n-1}$}{\hss\1\hss}}
+\def\xa{\rack{$x_1$}{\hss\.\hss}}
+\def\xb{\rack{$x_{n-1}$}{\hss.\hss}}
+\def\|{\hbox to 3pt{\hfil \vrule height 8pt depth 2pt \hfil}}
+
+\alik{ & & x_{n-1} x_{n-2} \9\9\9\9 x_1 x_0 \cr
+ & & * \y\y \9\9\9\9 \z\z \cr
+ & & x_{n-1} x_{n-2} \9\9\9\9 x_1 x_0 \cr
+ & & x_{n-1} x_{n-2} \xb\9\9 x_1 x_0 \xa \cr
+ & & \| \xa\xa\xa\xa\xa\xa\xa \cr
+ & & \| \xa\xa\xa\xa\xa\xa\xa \cr
+ & & \9\9\9 x_1x_0\xa\xa\xa\xa\xa\xa\xa \cr
+ }
+Výsledkem je vektor v¹ech èásteèných souètù.
+
+\: Paralelní porovnání $(Cmp)$:
+
+Compare x $\leq$ y $\equiv z_i = 1 \Leftrightarrow x_i \leq y_i$
+\alik{ & & \1 y_{n-1} \1 y_{n-2} \9\9\9 \1 y_1\1 y_0 \cr & & - \0
+x_{n-1} \0 x_{n-2} \9\9\9 \0 x_1\0 x_0 \cr \cr}
+
+U vektoru y nahradíme prolo¾ené nuly jednièkami a odeèteme vektor
+x. Ve výsledku pak bude na místì prolo¾eného bitu jednièka právì
+tehdy, kdy¾ $x_i \leq y_i$
+
+\: Rank ($x$,y):
+
+Mám nìjaké èíslo $x$ a vektor y, Rank je poèet slo¾ek vektoru,
+které jsou men¹í nebo rovny $x$. \alik{ & &
+Sum(Cmp((xx\9\9\9xx),y)) \cr}
+
+Nareplikuji vektor samých $x$ a ten porovnám s vektorem y.
+Výsledný vektor seètu.
+
+\: Zatøídìní:
+
+Zatøídit novou hodnotu do setøídìného vektoru èísel lze také v
+konstantním èase. Nejprve pomocí Rank najdu místo, kam má pøijít,
+pak nìjakým konstantním poètem operací vektor "roz¹oupnu" o ¹íøku
+slo¾ky doleva a vlo¾ím novou hodnotu.
+
+\: Unpack:
+
+Dostane $k$-bitové èíslo a vyrobí $k$-slo¾kový vektor, jeho¾
+slo¾ky jsou bity pùvodního èísla. Zreplikuji $x$ na vektor \alik{
+& & \0x\0x\9\9\9\0x \cr} a provedu AND s vhodnou bitovou maskou,
+konkrétnì \alik{ & &
+\1\0\9\9\9\0\|\9\9\9\|\0\9\9\9\0\1\0\|\0\9\9\9\0\1 \cr} Tím
+vytvoøím vektor, který má v ka¾dé slo¾ce nulu nebo nenulu, podle
+toho, jestli bit odpovídající této slo¾ce je nulový nebo nenulový,
+ale je na ¹patné pozici. Vektor teï prolo¾ím jednièkami místo nul
+a odeètu jednièku v ka¾dé slo¾ce. Tam kde byla ve slo¾ce nula, tak
+si vypùjèila z jednièky od prostrkání, kde byla nenula tam se
+nevypùjèilo. Prokládací bity jsou teï negace tìch, které tam
+potøebuji mít. Pak staèí znegovat, provést Shift na správné místo
+a AND k odstranìní nepoøádku.
+
+\: Pack:
+
+Dìlá opak ne¾ unpack, dostane vektor jednièek a nul, ze kterého
+vytvoøí $k$-bitové èíslo. Pøerozdìlím slo¾ky vektoru tak, jako by
+byli o bit krat¹í a provedu $Sum$ - Tím se jednotlivé bity posunou
+a posèítá se to, jak potøebuju.
+
+\endlist
+
+Pro dal¹í operace pøedpokládáme, ¾e pro $W$-bitová èísla máme
+slova dlouhá alespoò $b^2$.
+
+\itemize\ibull
+
+\: Poèet jednièek:
+
+Zjistí poèet jednièkových bitù. Provedu Unpack, èím¾ dostanu
+vektor, jeho¾ slo¾ky jsou bity pùvodního èísla a provedu $Sum$.
+
+\: Zrcadlení:
+
+Chci dostat zrcadlovì symetrické èíslo ke vstupnímu. Nejprve
+provedu Replicate. V nejni¾¹í kopii si nechám nejvýznamìj¹í bit, v
+dal¹í druhý, a¾ v poslední nejni¾¹í bit. Poté posèítám s o jedna
+men¹í velikostí bloku.
+
+\: LSB (Least Significant Bit):
+
+Poøadové èíslo první jednièky zprava. Izoluji tento bit pomocí
+nalezení nejpravìj¹í jednièky, \alik{& & \0\9\9\9\0\1\0\9\9\9\0
+\cr} odeètu 1 a seètu jednièky. \alik{& & \0\9\9\9\0\0\1\9\9\9\1
+\cr}
+
+\: MSB (Most Significant Bit):
+
+Provede se pøes zrcadlení.
+
+\: MSB pro normální délu slova:
+
+Rozdìlím èíslo na bloky velikosti $\lfloor\sqrt{w}\rfloor$,
+nejdøíve v ka¾dém bloku zjistím, zda v nìm nìco je nebo ne -
+$Cmp(x,0)$, provedu Pack, èím¾ dostanu jedno slovo $y$ odmocninové
+délky, které øíká, které bloky jsou prázdné a které ne. Dále
+pustím MSB kvadratické na $y$, èím¾ dostanu blok $B_i$, ve kterém
+je nejpravìj¹í jednièka a nakonec pustím MSB kvadratické na blok
+$B_i$. Tím ji najdu uvnitø bloku.
+
+\endlist
+\bye
--- /dev/null
+P=7-ram
+
+include ../Makerules
--- /dev/null
+\def\X{{\cal X}}
+\def\B{{\cal B}}
+\def\and{\ \&\ }
+\def\msb{{\rm MSB}}
+\def\rank{{\rm Rank}}
+
+\def\pnromanp{(\nroman)}
+\def\pnromanap{(\nroman ')}
+
+% Vlozeni obrazku {obrazek}{popisek}
+\def\nosizefigure#1#2{\bigskip\vbox{\centerline{\epsfbox{#1}}\smallskip\centerline{#2}}\bigskip}
+
+\input ../sgr.tex
+
+\prednaska{8}{Q-Heaps \& Union-Find problem}{zapsali Cyril Strejc a Ale¹
+©nupárek}
+
+Na minulém semináøi jsme si mimo jiné ukázali výpoèetní model RAM a nahlédli
+jsme, ¾e pomocí RAMu mù¾eme celkem snadno (v konstatním èase) simulovat
+vektorový poèítaè. A kdy¾ u¾ máme vektorový poèítaè, pojïme si ukázat, jaké
+datové struktury bychom s ním mohli vytváøet. Vektor ukládáme v $\O(w)$ slovech.
+V dal¹ím budeme BÚNO
+pøedpokládat, ¾e hodnoty, které do stuktur ukládáme, jsou rùzné. Svým sna¾ením
+smìrujeme ke strukturám, které zvládnou operace {\it Insert} a {\it Delete} v
+kostantním èase s rozumnou konstantou.
+
+\s{Znaèení:}
+\itemize\ibull
+\: $w$, ¹íøka slova daného stroje (RAMu)
+\: $n$, velikost vstupu
+\endlist
+Abychom mohli indexovat vstup, dále pøedpokládejme, ¾e $w = \O(\log n)$.
+
+\h{Word-Encoded B-Tree}
+V podstatì pùjde o obyèejný B-strom s daty v listech. Faktor stromu oznaème $B$.
+Ukládáme $k$-bitové hodnoty a chceme aby $Bk=\O(w)$ -- tehdy se nám uzel stromu
+vejde do vektoru a my budeme dostateènì rychle umìt operace s uzly, které do
+B-stromu potøebujeme, t.j. dìlení, spojování, vyhledávání. Té¾ ukládáme pointry
+na syny. Je tøeba si trochu rozmyslet, ¾e v ka¾dém uzlu se pointry na syny také
+povede ``scucnout'' do vektoru. Jen¾e poèet prvkù ($\vert T\vert$), které se do
+stromu vejdou je øádovì $B^h$, kde $h$ je hloubka stromu, a tudí¾ $\log\vert
+T\vert = \O(h)$. Proto se pointry do vektoru pìknì vejdou.
+
+\h{Q-Heaps}
+
+Q-Heap je teoreticky zajímavá datová struktura, v praxi neimplementovatelná.
+Narozdíl od Word-Encoded B-Tree není takové omezení na velikost ukládaného èísla
+-- maximální velikost èísla odpovídá velikost slova.
+
+\s{Znaèení:}
+\itemize\ibull
+\: $k = \O(w^{1/4})$
+\: $r\le k$, poèet prvkù v Q-haldì
+\: $x_1 < \ldots < x_r$, prvky (o $w$ bitech) v Q-haldì
+\: $\X = (x_1,\ldots,x_r)$
+\: $c_i = \msb(x_i \oplus x_{i+1})$, nejvy¹¹í bit, na kterém se li¹í $x_i$ a
+$x_{i+1}$
+\: $\rank_\X(x)$, poèet prvkù mno¾iny $\X$, které jsou men¹í ne¾ $x$.
+Samotné $x$ ov¹em nemusí být prvek mno¾iny $\X$.
+\endlist
+
+Haldové operace budeme umìt v $\O(1)$, nicménì bude nutný preprocesing v èase
+$\O(2^{k^4})$, co¾ ov¹em není tak hrozné, jak na prní pohled vypadá, jeliko¾
+$k = \O(\root 4 \of {\log n})$, èili nakonec ho stihneme v lineárním èase.
+
+A jak to tedy vypadá? Prvky $x_1,\ldots,x_r$ budeme ukládat do tzv. trie.
+{\it Trie} je binární strom s prvky (daty) v listech, v na¹em pøípadì s
+hodnotami $x_1,\ldots,x_r$. Popí¹u konstrukci trie. Oznaèím $c = \max(c_i)$.
+Mno¾inu $\X$ rozdìlíme na dvì mno¾iny, podle hodnot prvkù na $c$-tém bitu v
+binárním zápisu: $$ \X_0 = \{ x_i \vert x_i \in \X\and x_i[c]=0 \}$$
+$$ \X_1 = \{x_i \vert x_i \in \X\and x_i[c]=1\}$$
+
+Tedy $$\forall x \in \X_1, \forall y \in \X_2: x < y$$
+
+Èíslo $c$ -- øíkejme mu znaèka -- vlo¾íme do koøene právì budovaného (pod)stromu
+a jako levý podstrom pøipojím strom, který vznikne stejnou oparací na mno¾inì
+$\X_ 0$, prièem¾ $c$ vybíráme jen pøes prvky $\X_0$. Pravý podstrom vznikne
+stejnì z mno¾iny $\X_1$. Pokud je ji¾ mno¾ina pouze jednoprvková, dále
+nepokraèujeme v dìlení a zbývající prvek vlo¾íme jako list. Hodnoty (data) jsou
+tak ulo¾eny v listech a ve vnitøích uzlech jsou znaèky.
+
+\s{Pøíklad:}
+
+\nosizefigure{trie.eps}{Trie. Ohodnocení hran je pouze pro názornost -- není
+souèástí trie.}
+
+\>Pozorování:
+\itemize\ibull
+\: Tvar stromu (oznaème ho $T$) je jednozaène urèen vektrorem
+$\vec c=(c_1,\ldots,c_r)$.
+\: Prvky jsou v trii umístìny vzestupnì zleva doprava.
+\endlist
+
+To je sice v¹echno moc pìkné, ale pro aplikaci v Q-Heapech budeme muset umìt
+poèítat $\rank_\X(x)$ v konstatním èase. Jak tedy na to?
+
+\s{Lemma 1:} $\rank_\X(x)$ je urèen jednoznaènì:
+\numlist\pnromanp
+\: stromem $T$
+\: èíslem $i$: x vede v $T$ do $x_i$
+\: vztahem mezi $x$ a $x_i$ -- my¹leno vzhledem k relacím $<,>,=$
+\: $\msb(x \oplus x_i)$
+\endlist
+
+\s{Zdùvodnìní.} Pokud $x=x_i$, tak zjevnì $\rank_\X(x) = i$. Nech» tedy $x\neq
+x_i$. Hodnoty znaèek klesají ve smìru od koøene k
+listùm a na cestì od koøene k $x_i$ se v¹echny bity v $x_i$ na pozicích urèených
+znaèkami shodují s bity v $x$ -- tak jsme k danému $x$ ve (ii) vybrali $x_i$.
+Vezmeme do ruky $x$ a vydáme se na cestu od koøene do místa, kde bychom
+oèekávali $x$, kdyby v trii bylo. Jeliko¾ $\msb(x \oplus x_i)$ urèitì není
+znaèka na cestì z koøene do $x_i$ (v tom pøípadì by cesta podle znaèek nevedla
+do $x_i$ -- $x$ se na této pozici od $x_i$, vydali bychom se tedy do opaèné
+(shodující) vìtve), tak na této cestì je $\msb(x \oplus x_i)$ ostøe mezi
+nìjakými znaèkami, co¾ mi urèí hranu -- breakpoint, pod kterou u¾ jsou v¹echny
+hodnoty v listech buï men¹í (pokud $x>x_i$), nebo vìt¹í (pokud $x<x_i$), proto¾e
+na bitech vy¹¹ích, ne¾ je znaèka následující po breakpointu, mají v¹echny prvky
+v podstromì stejnou èíslici.
+
+\s{Pøíklad:} Vezmìme mno¾inu $\X=\{x_1,x_2,\ldots,x_6\}$ z pøedchozího pøíkladu
+a poèítejme $\rank_\X(011001)$. Bod zlomu -- breakpoint -- je oznaèen puntíkem.
+A $x>x_i$, tedy celý podstrom je men¹í ne¾ $x$ a tudí¾ $\rank_\X(011001)=4$.
+
+\s{Znaèení:}
+\itemize\ibull
+\: $\B = \{c_1,\ldots,c_k\}$
+\: $\varphi: \{1,\ldots,k\} \to \B, \varphi(i):=c_i$
+\endlist
+
+\s{Lemma 1':} $\rank_\X(x)$ lze spoèítat v konstatním èase z:
+\numlist\pnromanap
+\: $\B,\varphi$ -- tyto objekty urèují strom $T$
+\: $\equiv$ (ii), tj. èísla $i$: x vede v $T$ do $x_i$
+\: $\equiv$ (iii), tj. vztahu mezi $x$ a $x_i$ -- my¹leno vzhledem k relacím
+$<,>,=$
+\: $\rank_\B(\msb(x \oplus x_i))$
+\endlist
+a v¹echny vý¹e uvedené body lze té¾ spoèítat v konstatním èase.
+
+\s{Zdùvodnìní.}
+
+\numlist\pnromanap
+\: Nic se nepoèítá, tak je strom ulo¾en.
+\:
+\endlist
+
+\>A teï konkrétnì, jak bude Q-Heap realizována:
+
+\>Pamatujeme si:
+\itemize\ibull
+\: $\check x_1,\ldots,\check x_2$, hodnoty v libovolném poøadí
+\: $\rho:$ permutace na $\{1,\ldots,k\}$ urèující $x_i = \check x_{\rho(i)}$ pro
+$1 \le i \le r$, platí $x_1 < \ldots < x_r$
+\: $r$, poèet prvkù
+\: $B$, mno¾ina ``zajímavých'' bitových pozic, reprezentovaná jako vektor
+\: $R$, vektor popisující funkci $\varphi: \{1,\ldots,k\} \to \{1,\ldots,k\}$, t.¾.
+$B[\varphi(i)] = c_i$
+\endlist
+
+\> A teï si ji¾ uká¾eme dvì základní operace -- {\it insert} a {\it delete}.
+
+\s{Insert(x)}
+
+\algo
+\: $i = \rank_\X(x)$
+\: porovnáme $x,x_i$
+\: zaøadíme $x$ do $x_1,\ldots,x_r, r++$
+\: spoèítáme $c_i',c_{i+1}'$
+\: update $B,R$
+\endalgo
+
+\s{Delete(x)}
+
+\algo
+\: $i = \rank_\X(x)$, pøedpokládáme $x=x_i$, tedy ¾e prvek v haldì je
+\: sma¾eme $x_i$ (uvolnìní $\check x_{\rho(i)}$, se¹oupnutí $\rho$); $r--$
+\: rozhodneme, jestli zmizí $c_i$, nebo $c_{i+1}$
+\: update $B$ (dle výsledku pøedchozího kroku), update $R$ (se¹oupnutí)
+\endalgo
+
+
+
+\h{Union-Find problem}
+Na Uion-Find problem (UF) mù¾eme nahlí¾et ze více stran (udr¾ování ekvivalencí,
+inkrementální souvislost grafu). K èemu je to dobré?
+Napøíklad v Kruskalovì algoritmu pro hledání minimálních koster v grafu.
+\h{Udr¾ování tøíd ekvivalence }
+Mìjme nìjakou mno¾inu M, která se dá rozlo¾it na k ekvivalentních podmon¾in (tøídy ekvivalence). Na mno¾iòe M chceme provádìt
+dvì operace: jsou p,q z M ekvivalentní (ve stejné tøídì ekvivalence)? (find(p,q)) Dále chceme umìt spojit dvì tøídy ekvivalence do jedné (union(p,q)).
+\h{Inkrementální idr¾ování komponent souvislosti grafu}
+Jedná se o pøipad podobný udr¾ování tøíd ekvivalence. Tentokrát mnou¾inou M bude mno¾ina vrcholù V grafu G (V,E).
+Za »øídy ekvivalence budou jednotlivé komponenty souvislosti grafu G. Operace find nám øekne o dvou vrcholech nachází-li
+se ve stejné komponen»e, èi nikoliv. Operace union nám spojí dvì komponenty do jedné pøidáním hrany. Pokud pøipustíme mazání hran,
+øe©ení problému je výraznì te¾¹í.
+
+\h{Bì¾ná implementace }
+Ka¾dé tøídì ekvivalence pøiøadíme unikátní barvu, kterou obarvíme její prvky. Pro operaci find staèí porovnat barvy prvkù.
+Pro operaci union dvou komponent je nutné pøebarvit obì na stejnou barvu. Budeme pøebarvovat
+v¾dy tu men¹í pak celková operace union bude trvat $\O(n\log(n) + m)$. (Ka¾dé pøebarvení minimálnì zdvojnásobí velikost nové komponenty.)
+\h {Chytøej¹í implementace }
+Budeme-li prvky za stejné tøídy reprezentovat ve jednom stromì (synové mají ukazatel svého otce), operaci find
+provedeme prùchodem ze zadaných prvkù do koøene. Prvky jsou ve stejné komponentì souvislosti, pokud mají stejného otce.
+Operace union bude spojení dvou stromù do jednoho (otec jednoho stromu se zavìsí pod listy druhého).
+Takto definovaný union nemusí garantovat logaritmickou slo¾itost pro find,
+v extrémním pøipadì mù¾e strom mít lineární hloubku vzhledem k poètu vrcholù.
+
+Degeneraci stromu lze zabránit pøidánim podmínky urèující, jaký strom bude dole.
+První mmo¾nost (union by size) je povìsit dolu vìtsí strom.
+Druhá mo¾nost (union by rank) je ka¾dému vrcholu nastavit na zaèátku nìjaký rank (tøeba 1). Pokud je $r1<r2$ pak strom s r1 povìsíme dolu.
+Pokud $r1=r2$ pak ve slouèeném stromù zvý¹íme v koøeni rank o 1. Informace o ranku nám zabere ménì pamìti ne¾ o velikosti stromu.
+
+Dal¹i mo¾nost pro zkrácení vý¹ky stromu je path compression. Pøi operaci find si pamatujeme pro¹lé vrcholy a pøesmìrujeme
+jejich ukazatele na otce na koøen.
+
+\s{Vìta: } UF s Union by size nebo Union by rank s path compression mají amortizovanì
+slo¾itost $\O(n\alpha(n))$, kde $\alpha(n)$ je inverzní funkce k Ackermanovì funkci. ($\alpha(poèet atomù ve vesmíru)=5$)
+
+\h {Microtree/Macrotree decomposition}
+
+\s{Otázka: } Je mo¾né provádìt UF v lep¹ím èase kdy¾ bude pøedem znám výsledný strom?
+
+Cíl: UF lze za pomoci preprocesingu v èase $\O(n)$ zvládnout amortizovanì v èase $\O(1)$ na U,F.
+
+\h{Clusterizace}
+\s {Definice: } Nech» G je je graf kde $\forall v \in V(G): deg(v)<=3$ a $c \in N$ pak c-clusterizací grafu G je rozklad
+G na podgrafy na podgrafy G1,G2 ... Gk s následujícimi vlastnostmi:
+\itemize\ibull
+\: $\forall v \in V \exists !\ i: v \in C_i$
+\: $\forall i\ C_i\leq C$
+\: $\forall i$ vnìj¹i stupeò $C\leq3$, pokud $C_i=3$ pak $\|C_i\|=1$
+\: ¾ádné dva sousední clustery nelze spojit
+\endlist
+
+Co udìlat se stromem, ve kterém existuje vrchol se stupnìm vìt¹im ne¾ 3? Pou¾ijeme French trick.
+V¹echny vrcholy s vy¹¹im stupnìm jak 3, rozdìlíme na více vrcholù se stupnìm 3 spojených cestou.
+Tato ùprava nám asymptoticky nezmìní èasovou slo¾itost UF o proti pùvodnímu stromu.
+
+
+\s{Lemma: } Pro ka¾dou C-clusterizaci n-vrcholového stromu je $\O(n/C)$ clusterù.
+
+\s{Vìta: } Pro ka¾dé C lze C-clusterizaci (splòujíci pøedchozí lemma) najít v èase $\O(n)$.
+
+\s{Dùkaz: } za pomoci DFS ...
+\h{Jednodu¹¹í clusterizace}
+Pøedpokládejme, ¾e máme zakoøeòené stromy
+\s{Definice: } Koøeny makrostromù jsou nejvy¹¹i vrcholy, pod kterými je maximálnì $\log(n)$ listù.
+Celá clusterizace se nám rozpadne na následujíci pøipady:
+\itemize\ibull
+\:N-strom: (má stupeò 1)
+\:M-strom: (má stupeò 2)
+\:cestu: ka¾dá cesta se popí¹e bitovým polem, jednotlivé úseky se poí¹í jednotlivými slovy
+\endlist
+poèet listù v makrostromu je maximálnì $n/\log(n)$ z toho vyplývá, ¾e vý¹ka makrostromu $\O(n/log(n))$
+
+\s{Co si potøebujeme pamatovat}
+\itemize\ibull
+\: $\log(n)$ na clusterizaci
+\: makrostrom G s $C_i$ zkontrahovanými pro deg 1,3 do vrcholu a deg 2 do hrany, jeho velikost je $\O(n/\log(n))$
+\:mikrostromy, bitovì popsány
+\endlist
+
+\h{Makrostromy}
+V makrostromu jsou takové hrany, které vedly v pùvodním stromu mezi clustery, vrcholy budou hranièní
+vrcholy pøislu¹ných clusterù, cluster stupnì 1 je v novém stromì list, cluster stupnì 2 nahradíme
+hranou mezi hranièními vrcholy a cluster stupòe 3 má jeden vnìj¹í vrchol, který je také v novém stromì.
+
+\s{find(x,y)}
+\algo
+nech» $i,j: x \in C_i, y \in C_j$
+\:pokud $i=j$, ptáme se v $C_i$ (mikro-find(i,j))
+\:pokud $i\not = j$, najdeme hranièní vrcholy pøíslu¹ných clusterù, tj. $h_i \in C_i, h_j \in C_j$ a provedeme makro-find($h_i$,$h_j$)
+ find($i$,$j$):=makro-find($h_i$,$h_j$) and mikro-find($i$,$h_i$) and mikro-find($j$,$h_j$);
+\endalgo
+
+\s{delete(x,y)}
+\algo
+\: pokud $x,y$ je makro-hrana, sma¾eme ji
+\: pokud $x,y \in C_i$ pak mikro-delete v $C_i$, pokud je $deg(C_i)=2$ a find($h_1$,$h_2$)==FALSE ($h_1, h_2$hranièní vrcholy),
+pak makro-delete hrany odpovýdajíci $C_i$. ($C_i$ je neprùchodná)
+\endalgo
+\h{Mikrostromy}
+\itemize\ibull
+\: strom zakoøením a oèísluji mu hrany $0..h < \log(n)$
+\: pamatujeme si: \itemize\ibull
+ \:$\forall v \in C_i$ mno¾inu hran $P_v$ na cestì do koøene v (bitový vektor)
+ \:mon¾inu smazaných hran, operace delete pøidá hranu do mno¾iny
+ \:hranièní vrcholy
+ \endlist
+\: $P_x \oplus P_y$ je cesta z x do y
+\endlist
+\bye
+
--- /dev/null
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+%%BeginResource: font CMR10
+%!PS-AdobeFont-1.1: CMR10 1.00B
+%%CreationDate: 1992 Feb 19 19:54:52
+
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.00B) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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+/Weight (Medium) readonly def
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+end readonly def
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+dup 124 /emdash put
+dup 125 /hungarumlaut put
+dup 126 /tilde put
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+dup 128 /suppress put
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+%%BeginResource: font CMR7
+%!PS-AdobeFont-1.1: CMR7 1.0
+%%CreationDate: 1991 Aug 20 16:39:21
+
+% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
+
+11 dict begin
+/FontInfo 7 dict dup begin
+/version (1.0) readonly def
+/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
+/FullName (CMR7) readonly def
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+/Weight (Medium) readonly def
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+/isFixedPitch false def
+end readonly def
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+dup 92 /quotedblleft put
+dup 93 /bracketright put
+dup 94 /circumflex put
+dup 95 /dotaccent put
+dup 96 /quoteleft put
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+dup 123 /endash put
+dup 124 /emdash put
+dup 125 /hungarumlaut put
+dup 126 /tilde put
+dup 127 /dieresis put
+dup 128 /suppress put
+dup 160 /space put
+readonly def
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\ No newline at end of file
--- /dev/null
+\input ../sgr.tex
+
+\def\leaderfill{\leaders\hbox to .5em{\hss.\hss}\hfill}
+\def\itemitemitem{\par\indent\indent \hangindent3\parindent \textindent}
+\dimen0 = \hsize
+\divide \dimen0 by 6
+\def\vs#1#2{#1~{\rm vs.}~#2}
+
+\prednaska{9}{Suffixové stromy} {Tomá¹ Mikula \& Jan Král}
+
+\s {Definícia:}
+\+\indent&\hbox to \dimen0 {$\Sigma$\leaderfill}&koneèná abeceda (mno¾ina znakov)\cr
+\+&$\Sigma^*$\leaderfill&mno¾ina v¹etkých slov nad $\Sigma$\cr
+\+&$\epsilon$\leaderfill&prázdne slovo\cr
+\+&$\alpha\beta$\leaderfill&zre»azenie slov $\alpha,\beta\in\Sigma^*$\cr
+\+&$\alpha^n$\leaderfill&$n=0: \alpha^0=\epsilon$\cr
+\+&&$n>0: \alpha^n=\alpha\alpha^{n-1}$\cr
+\+&$\alpha^{\rm R}$\leaderfill&slovo $\alpha$ odzadu\cr
+\settabs\+\indent&$\alpha$ je {\it (vlastným) podslovom} $\beta$ &\cr
+\+&$\alpha$ je {\it (vlastným) prefixom} $\beta$ &$\equiv \exists\gamma(\not=\epsilon):\beta=\alpha\gamma$\cr
+\+&$\alpha$ je {\it (vlastným) sufixom} $\beta$ &$\equiv \exists\gamma(\not=\epsilon):\beta=\gamma\alpha$\cr
+\+&$\alpha$ je {\it (vlastným) podslovom} $\beta$ &$\equiv \alpha$ je prefixom nejakého sufixu $\beta$ (a $\alpha\not=\beta$)\cr
+
+\h{Trie}
+\s{Definícia:} {\it Trie ($\Sigma$-strom)} pre $X\subset\Sigma^*, \vert X\vert<\infty$ je orientovaný graf $G(V,E)$:\par
+$V = \lbrace\alpha\vert\alpha\hbox{ je prefixom nejakého }\beta\in X\rbrace$\par
+$(\alpha,\beta)\in E \equiv \exists x\in\Sigma:\beta=\alpha x$\par
+
+\s{Pozorovanie:} Trie je strom s koreòom $\epsilon$. Listy sú slová z $X$, ktoré nie sú vlastným prefixom iných slov z $X$.
+
+\s{Definícia:} {\it Komprimovaná trie ($\Sigma^+$-strom)} vznikne z trie nahradením nevetviacich sa ciest hranami (oznaèenými zre»azením znakov príslu¹nej cesty).
+
+Vrcholom, ktoré padli za obe» kompresii budeme hovori» {\it skryté vrcholy}.\par
+Mô¾e sa sta», ¾e nejakému slovu z $X$ odpovedá skrytý vrchol. Aby sme sa tomuto vyhli, pridáme na koniec ka¾dého slova ¹peciálny znak $\$\notin\Sigma$. Teraz platí\par
+\s{Pozorovanie:} Listy (komprimovanej) trie sú práve slová z $X$.
+
+\tabskip=0pt plus 1fil
+\halign to\hsize{
+\hfil#\hfil&\hfil#\hfil&\hfil#\hfil\cr
+\epsfbox{trie.eps}&\epsfbox{trie-c.eps}&\epsfbox{trie-cd.eps}\cr
+Trie pre $\{\hbox{AULA, AUTO, AUTOBUS, BUS}\}$ & \dots komprimovaná & \dots odolarovaná\cr
+}
+\smallskip
+
+\h{Sufixové stromy}
+\s{Definícia:} {\it Sufixový strom} pre slovo $\sigma\in\Sigma^*$ je komprimovaná trie pre $X=\lbrace {\rm sufixy~}\sigma\rbrace$.
+
+\s{Pozorovanie:} Listy sú sufixy $\sigma$, ktoré nie sú prefixom iného sufixu. Po pridaní znaku \$ na koniec ka¾dého sufixu sú listy práve v¹etky sufixy. Vnútorné vrcholy sú {\it vetviace podslová} $\sigma$ (tj. $\alpha$ podslová $\sigma$, ktoré sú prefixom aspoò dvoch sufixov a $\alpha x$ je prefixom men¹ieho poètu sufixov pre ka¾dé $x\in\Sigma$).
+
+V listoch si budeme znaèi» index do $\sigma$, kde daný sufix zaèína.
+
+\figure{st-barbara.eps}{Sufixový strom pre slovo BARBARA}{0pt}
+
+\s{Lemma:} Sufixový strom pre slovo $\sigma$ då¾ky $n$ je reprezentovateµný v priestore $\O(n)$.\par
+\s{Dôkaz:} Strom má $\O(n)$ listov a ka¾dý vnútorný vrchol má výstupný stupeò aspoò $2$ a teda v¹etkých vrcholov je $\O(n)$. Keï¾e sa jedná o strom, hrán je tie¾ len $\O(n)$. Nálepky na hranách budú reprezentované ako 2 indexy do $\sigma$ oznaèujúce zaèiatok a koniec nejakého úseku odpovedajúceho písmenkám na hrane.\qed
+
+\s{Veta:} Sufixový strom pre slovo då¾ky $n$ sa dá zostroji» v èase $\O(n)$.\par
+\s{Dôkaz:} Budú uvedené 2 kon¹trukcie v lineárnom èase.\qed
+
+\s{Aplikácie} (alebo èo doká¾eme v èase $\O(n)$):\par
+\item{1.} inverzné vyhµadávanie (tj. vyhµadávanie µubovoµného výrazu v predspracovanom texte (nie predspracovaného výrazu (napr. koneèný automat) v µubovoµnom texte))
+\itemitem{-}v¹etky výskyty; ak u¾ máme strom postavený, doká¾eme nájs» v¹etkých $k$ výskytov podslova då¾ky $m$ v èase $\O(m+k)$ (vo vnútorných vrcholoch si budeme naviac udr¾iava» pointre na lexikograficky najmen¹í a najväè¹í list podstromu a v listoch pointer na ïal¹í list v lex. poradí)
+\itemitem{-}poèet výskytov (v ka¾dom vnútornom vrchole budeme ma» predpoèítaný poèet listov, ktoré sa pod ním nachádzajú)
+\item{2.} najdlh¹ie opakujúce sa podslovo
+\itemitem{$\cong$} najhlb¹ie vetvenie (písmenkovo najhlb¹í vnútorný vrchol)
+\item{3.} histogram $k$-podslov (èetnos» slov då¾ky $k$)
+\itemitem{-} odre¾eme strom v písmenkovej håbke $k$ a spoèítame odrezané vetvièky
+\item{4.}najdlh¹í spoloèný podre»azec slov $\alpha$ a $\beta$ ($LCS(\alpha, \beta)$)
+\itemitem{-}postavíme strom pre slovo $\alpha\$_1\beta\$_2$; najhlb¹í vrchol, pod ktorým je $\alpha$-ový i $\beta$-ový sufix ($\$_1$ i $\$_2$) odpovedá $LCS(\alpha,\beta)$
+\item{5.}najdlh¹ie palindromické podslovo ($\beta$ je palindromické $\equiv \beta = \beta^{\rm R}$)
+\itemitem{-} Postavíme sufixový strom pre $\alpha\$_1\alpha^R\$_2$. Postupne prechádzame cez v¹etky potenciálne stredy palindromu (je ich $2n-1$) a poèítame:
+\itemitemitem{$\bullet$}$LCP(\alpha[i:], \alpha^{\rm R}[n-(i+1):])$, keï uva¾ujeme ako stred palindromu $i$-té písmenko (èíslované od 0) (tj. palindrom nepárnej då¾ky)
+\itemitemitem{$\bullet$}$LCP(\alpha[i+1:], \alpha^{\rm R}[n-(i+1)])$, keï uva¾ujeme stred palindromu medzi $i$-tým a $(i+1)$-vým písmenkom (tj. palindrom párnej då¾ky).
+\itemitem{}$LCP(\alpha,\beta)$ je najdlh¹í spoloèný prefix $\alpha$ a $\beta$ a v strome je to najhlb¹í spoloèný predchodca ($LCA$) listov odpovedajúcich $\alpha$ a $\beta$.
+\itemitem{}Pritom treba ¹ikovne liez» po strome, aby nájdenie ïal¹ej odpovedajúcej si dvojice listov bolo kon¹tantné.
+\itemitem{}Na nájdenie $LCA$ pou¾ijeme nejakú ¹truktúru, ktorá to po lineárnom predvýpoète zvládne v kon¹tantnom èase.
+\item{6.}Burrows-Wheeler transform (BWT)
+
+\h{Rekurzívna kon¹trukcia sufixového stromu v lineárnom èase}
+\s{Definícia:} {\it Suffix Array} $A$ pre slovo $\sigma$ je permutácia na sufixoch slova $\sigma$, ktorá ich zotriedi lexikograficky:\hfil\break
+$\sigma[A[i]:]<_{lex}\sigma[A[i+1]:]$, kde $\sigma[j:]$ znaèí podslovo $\sigma$ poènúc $j$-tým písmenkom a¾ do konca.
+
+\s{Definícia:} {\it LCP Array} $L$ pre slovo $\sigma$:\hfil\break
+$L[i]=\vert LCP(\sigma[A[i]:], \sigma[A[i+1]:])\vert$, kde $LCP(\alpha, \beta)$ je najdlh¹í spoloèný prefix $\alpha$ a $\beta$.
+
+\s{Definícia:} {\it Kartézsky strom} pre postupnos» kµúèov $x_1\ldots x_n$ je strom, ktorý je haldou podµa hodnôt kµúèov a vyhµadávacím stromom podµa poradia kµúèov v postupnosti. Pritom vrcholy s rovnakou hodnotou, ktoré by spolu susedili, zlúèime do jedného.
+
+Niektoré vrcholy majú voµné miesta, kde by mohol by» pripojený podstrom. Týmto voµným miestam budeme hovori» {\it voµné sloty}.
+
+\tabskip=0pt plus 1fil
+\halign to\hsize{
+\hfil#\hfil&\hfil#\hfil\cr
+\epsfbox{ct-35120024224.eps}&\epsfbox{ct-35120024224-fs.eps}\cr
+Kartézsky strom pre $(3,5,1,2,0,0,2,4,2,2,4)$ & \dots s vyznaèenými voµnými slotmi\cr
+}
+\smallskip
+
+\s{Lemma:} Kartézsky strom pre $x_1\ldots x_n$ sa dá postavi» v èase $\O(n)$.\qed
+\s{Lemma:} Sufixový strom (ST) pre $\sigma$ a $(A,L)$ pre $\sigma$ sú lineárne izomorfné.\par
+\s{Dôkaz:}\par
+${\rm ST}\longrightarrow(A,L)$:
+\itemitem{}Prechádzame strom zµava doprava (tj. lexikograficky). V¾dy, keï sme v liste, opí¹eme do $A[i]$ index do $\sigma$, ktorý sme v liste na¹li; $i$ inicializujeme na $-1$ a pri ka¾dej náv¹teve listu zväè¹ujeme o $1$ (e¹te pred zápisom do $A[i]$).
+\itemitem{}Medzi náv¹tevou 2 susedných listov v¾dy najskôr stúpame a potom sa znova ponoríme. Najvy¹¹í vrchol, po ktorý sme vystúpili medzi náv¹tevou listov $\alpha$ a $\beta$, odpovedá $LCP(\alpha,\beta)$, jeho håbka odpovedá $\vert LCP(\alpha, \beta)\vert$. Do $L[i]$ teda zapí¹eme håbku tohto najhlb¹ieho spoloèného predka, ktorú si poèítame v rekurzii.
+
+$(A,L)\longrightarrow{\rm ST}$:
+\itemitem{}Staviame kartézsky strom pre $L$ a do voµných slotov privesujeme listy. Tento strom má tvar sufixového stromu pre $\sigma$. Indexy v listoch urèujeme podµa $A$. Nálepky na hranách urèujeme podµa indexov v listoch podstromu a (písmenkovej) håbky vrcholov (ktorú pre vnútorné vrcholy máme v $L$ a pre listy vypoèítame ako $\vert\sigma\vert - A[i]$).\qed
+
+\medskip
+Ná¹ algoritmus bude kon¹truova» $(A,L)$, ktoré u¾ teraz vieme v lineárnom èase previes» na sufixový strom.
+
+\algo
+\:(Z abecedy vyhodíme nepou¾ité znaky.)
+\itemitem{}Pou¾ité znaky zotriedime priehradkovým triedením v èase $\O(n)$ a premenujeme na $1,\ldots,n'$, $n'\le n$.
+
+\:Definujeme slová $\sigma_0$, $\sigma_1$, $\sigma_2$ nasledujúcim spôsobom:
+%\settabs\+\itemitemitem{}&$\sigma_2[i] := (\sigma[3i+2],$ &$\sigma[3i+3],$ &$\sigma[3i+4])$\cr
+$$\eqalign{
+\sigma_0[i] &:= (\sigma[3i],\sigma[3i+1],\sigma[3i+2])\cr
+\sigma_1[i] &:= (\sigma[3i+1],\sigma[3i+2],\sigma[3i+3])\cr
+\sigma_2[i] &:= (\sigma[3i+2],\sigma[3i+3],\sigma[3i+4])\cr
+}$$
+\itemitem{}$\sigma_0$, $\sigma_1$, $\sigma_2$ sú slová då¾ky $\approx{n\over3}$ nad abecedou $[n']^3$. (Opä» ale z nej pou¾ijeme najviac $n$ znakov.)
+
+\:Zavoláme sa rekurzívne na $\sigma_0\sigma_1$, získame $(A_{01},L_{01})$. Z $A_{01}$ doká¾eme jedným priechodom vyrobi» $A_0$, $A_1$, $A_{01}^{-1}$. ($A_{01}^{-1}$ je inverzná permutácia k $A_{01}$)
+
+\:Dopoèítame $A_2$.\par
+\itemitem{}Vyu¾ijeme, ¾e $\sigma_2[i:]\approx\sigma[3i+2:]=\sigma[3i+2]\sigma[3i+3:]\approx\sigma[3i+2]\sigma_0[i+1:]$.
+\itemitem{}Pre slová $\sigma_2[i:]\approx\sigma[3i+2]\sigma_0[i+1:]$ máme usporiadanie podµa druhej èasti (tj. podµa $\sigma_0[i+1:]$) v $A_0$. V tomto poradí ich nahád¾eme do priehradok podµa prvého znaku prvého znaku (tj. podµa $\sigma[3i+2]$) a vysypeme do $A_2$.
+
+\:$A_0,A_1,A_2\buildrel merge\over\longrightarrow A$
+\itemitem{}Nech $a$, $b$, $c$, sú aktuálne pozície v $A_0$, $A_1$, $A_2$ (v tomto poradí) pri zlievaní. Nech $i=A_0[a]$, $j=A_1[b]$, $k=A_2[c]$. Potrebujeme porovna» slová $\sigma_0[i:]$, $\sigma_1[j:]$, $\sigma_2[k:]$ v kon¹tantnom èase.
+$$\eqalign{
+\sigma_0[i:] < \sigma_1[j:] &\equiv A_{01}^{-1}[i] < A_{01}^{-1}[\vert\sigma_0\vert+j]\cr
+\sigma_0[i:] < \sigma_2[k:] &\equiv \sigma[3i]\,\sigma_1[i:] < \sigma[3k+2]\,\sigma_0[k+1:]\cr
+&\Leftrightarrow (\sigma[3i] < \sigma[3k+2]) \vee (\sigma[3i] = \sigma[3k+2] \wedge \sigma_1[i:] < \sigma_0[k+1:])\cr
+\sigma_1[j:]<\sigma_2[k:] &\equiv \sigma[3j+1]\,\sigma[3j+2]\,\sigma_0[j+1:] < \sigma[3k+2]\,\sigma[3k+3]\,\sigma_1[k+1:]
+}$$
+
+\:Dopoèítame $L$.
+\itemitem{} Pripomeòme definíciu $L[i]=\vert LCP(\sigma[A[i]:], \sigma[A[i+1]:])\vert$.
+\itemitem{} Nech
+\tabskip=3\parindent
+\halign{\hfil#\tabskip=0pt&#\hfil&#\hfil\cr
+$\sigma[A[i]:]$ &$\approx \sigma_0[j:]$,\quad&tj. $A[i]=3j$ pre nejaké $j$,\cr
+$\sigma[A[i+1]:]$ &$\approx \sigma_0[k:]$,&tj. $A[i+1]=3k$ pre nejaké $k$.\cr
+}
+\itemitem{} Oznaème tento prípad $\vs{\sigma_0}{\sigma_0}$. Potom $L[i]=L_{01}[A_{01}^{-1}[A[i] \div 3]]$.
+\itemitem{} Prípad $\vs{\sigma_0}{\sigma_1}$ vyrie¹ime rovnako.
+\itemitem{} Prípady $\vs{\sigma_1}{\sigma_0}$ a $\vs{\sigma_1}{\sigma_1}$: $L[i]=L_{01}[A_{01}^{-1}[A[i] \div 3 + \vert\sigma_0\vert]]$
+\+\cleartabs\itemitem{} &$\vs{\sigma_2}{\sigma_2}$: &a$)$ lí¹ia sa prvým znakom $\Rightarrow L[i]=0$\cr
+\+&&b$)$ $L[i]=1+(\vs{\sigma_0}{\sigma_0})$\cr
+\+&$\vs{\sigma_2}{\sigma_0}$: &a$)$ lí¹ia sa prvým znakom $\Rightarrow L[i]=0$\cr
+\+&&b$)$ $L[i]=1+(\vs{\sigma_0}{\sigma_1})$\cr
+\+&$\vs{\sigma_2}{\sigma_1}$: &a$)$ lí¹ia sa prvým znakom $\Rightarrow L[i]=0$\cr
+\+&&b$)$ lí¹ia sa druhým znakom $\Rightarrow L[i]=1$\cr
+\+&&c$)$ $L[i]=2+(\vs{\sigma_1}{\sigma_0})$\cr
+
+\endalgo
+
+\s{Odhad èasovej zlo¾itosti}\par
+$T(n) = T({2\over3}n) + \O(n) = \O(n)$
+
+\h{Ukkonenova inkrementální konstrukce}
+
+Pomocí Ukkonnenovy konstrukce doká¾eme sestrojit suffixový strom v lineárním èase. Algoritmus je navíc inkrementální, v ka¾dém kroku tak ze stávajícího suffixového stromu ($ST$ pro $\sigma$) vyrobí strom odpovídající roz¹íøenému slovu ($\sigma a$). Vychází pøitom ze stromu pro prázdné slovo (prázdný strom), který postupnì roz¹iøuje.
+
+Roz¹íøení suffixového stromu doká¾e algoritmus provést v amortizovanì konstantním èase.
+
+\s{Pozorování:}Následující pozorování platí pro roz¹iøování suffixového stromu pro $\sigma$ na suffixový strom pro $\sigma a$:
+\numlist\ndotted
+\:Pokud bylo slovo vìtvící v $\sigma$, pak bude stale vìtvící
+\:Pokud byl uzel nevnoøený, bude stále nevnoøený
+\:listy zùstávají jako listy (musím na hranu, ktera do nìj vede pøipsat $a$) -- øe¹íme otevøenými hranami (ne nálepka odsud sem, ale odsud dokonce) - slo¾itost - $O(0)$
+\:vnitøní vrcholy $\to$ vnitøní vrcholy
+\:Mohou vznikat nové vrcholy (listy i vnitøní), slovo nevìtvící se tak mù¾e stát vìtvícím
+\endlist
+
+\s{Definice:Aktivni suffix ($\alpha(\sigma)$)} nazýváme nejdel¹í vnoøený suffix $\sigma$
+
+\s{Lemma:} Pro ka¾dé $\sigma$, $a$ platí: $\alpha(\sigma a)$ je suffixem $\alpha(\sigma)a$
+
+\s{Dùkaz:} $\alpha(\sigma a)$, $\alpha(\sigma)a$ jsou suffixy $\sigma a$, z toho plyne, ¾e staèí porovnat délky abychom lemma ovìøili.
+Platí: $ \vert\alpha(\sigma a)\vert \le \vert\alpha(\sigma)a\vert$ (pøidáním písmene nemù¾u prodlou¾it prefix o více ne¾ jeden znak).
+
+\s{Definice:Zralé suffixy} jsou ty prefixy $\alpha$, pro ktere platí:
+$\forall \sigma, a,\beta: \beta a$ suffix $\sigma a$ je {\it zralý} právì kdy¾ $ \vert\alpha(\sigma)a\vert \ge \vert\beta a\vert > \vert\alpha(\sigma a)\vert$
+
+Update je $O(1)$ amortizovanì na $1$ roz¹íøení $\sigma$
+
+$\alpha(\sigma)$ reprezentuji jako kanonický (neexistuje jiný referenèní pár s krat¹ím $\alpha$) referenèní pár $(v, \alpha)$ kde $v$ je vrchol $ST$ a $\alpha$ ke cesta kudy dál.
+
+\s{Definice: $back[v]$} je nejdel¹í vlastní suffix $v$, ke kterému existuje vrchol.
+
+\algo
+\:Vstup: $\sigma, a, (v, \alpha)$
+
+// 1. Zalo¾ení listù/Podrozdìlení hran
+\:if $\alpha = \epsilon$:
+\::if $\exists (v, a* )$
+\:::konec
+\::else:
+\:::zalo¾ím list
+\:else if $\alpha \ne \epsilon$:
+\::if $\exists (v, \alpha a*)$:
+\:::konec
+\::else:
+\:::podrozdìlím hranu na které le¾í $\alpha$ // za $\alpha$ vlo¾ím odboèku na $a$, kam dám list a pokraèuju
+
+// 2. Pøechod na dal¹í vrchol
+\:zkrátim $\alpha$ zleva o znak
+\:if $v = \epsilon$:
+\::pøejdu do $( \epsilon, \alpha bez 1. znaku)$
+\:else:
+\::pøejdu do $( back[v], \alpha )$
+
+// 3.
+\:kanonikalizuji // "jdu z $v$ pøes alpha do co nejdal"
+\endalgo
+
+
+\s{Èasová slo¾itost:}
+\numlist\ndotted
+\:Zalo¾ení listu/Podrozdìlení hrany -- $O(1)$
+\:Pøechod na dal¹í vrchol -- $O(1)$ za pøedpokladu existence $back(v)$
+\:Kanonikalizace -- amortizovanì za $O(1)$
+\:Update $back[v]$ -- jen pro vnitøní vrcholy, které vznikly v 1. podrozdìlením hrany. Ta hrana vede do vrcholu, který nav¹tíví pøí¹tí update v kroku 2. a buï tam ten vrchol je, nebo ho v tom kroku zakláda, tedy -- $O(1)$.
+\endlist
+
+\bye
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+%%EOF
--- /dev/null
+P=9-suffix
+
+include ../Makerules
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: GIMP PostScript file plugin V 1.17 by Peter Kirchgessner
+%%Title: ST.eps
+%%CreationDate: Fri Jun 9 21:22:13 2006
+%%DocumentData: Clean7Bit
+%%LanguageLevel: 2
+%%Pages: 1
+%%BoundingBox: 14 14 200 137
+%%EndComments
+%%BeginProlog
+% Use own dictionary to avoid conflicts
+10 dict begin
+%%EndProlog
+%%Page: 1 1
+% Translate for offset
+14.173228346456687 14.173228346456687 translate
+% Translate to begin of first scanline
+0 121.99273036816345 translate
+184.98897637795278 -121.99273036816345 scale
+% Image geometry
+185 122 8
+% Transformation matrix
+[ 185 0 0 122 0 0 ]
+% Strings to hold RGB-samples per scanline
+/rstr 185 string def
+/gstr 185 string def
+/bstr 185 string def
+{currentfile /ASCII85Decode filter /RunLengthDecode filter rstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter gstr readstring pop}
+{currentfile /ASCII85Decode filter /RunLengthDecode filter bstr readstring pop}
+true 3
+%%BeginData: 15749 ASCII Bytes
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+12 scalefont
+setfont
+0 0 0 setrgbcolor
+newpath
+0 0 moveto
+( S) show
+grestore
+grestore
+grestore
+showpage
--- /dev/null
+all:
+
+upload:
+ tar czvvf - Makerules sgr.tex ~/texmf/tex/mj/mjmac.tex | ssh atrey 'cat >WWW/vyuka/sgr-2005/macros.tar.gz'
+
+.SECONDARY:
--- /dev/null
+all: $P.ps
+
+%.dvi: %.tex ../sgr.tex
+ csplain $<
+
+%.pdf: %.tex ../sgr.tex
+ pdfcsplain $<
+
+%-700.tex: %.tex
+ ( echo '\magnification=700' ; cat $< ) >$@
+
+%.ps: %.dvi
+ dvips -D600 -o $@ -O-0.5in,-0.5in -t a4 $<
+
+%-a5.ps: %-700.dvi
+ dvips -D600 -o $@ -O-1in,-1in -T148mm,210mm $<
+
+%-booklet.ps: %-a5.ps
+ psbook <$< | pstops '2:0L(210mm,0)+1L(210mm,148mm)' | sed 's/^%%BoundingBox: .*/%%BoundingBox: 0 0 595 842/' >$@
+
+%-2in1.ps: %-a5.ps
+ pstops '2:0L(212mm,0mm)+1L(212mm,150mm)' <$< | sed 's/^%%BoundingBox: .*/%%BoundingBox: 0 0 595 842/' >$@
+
+mostlyclean:
+ rm -f *.dvi *.log *~ core *.o *-700.tex *-a5.ps
+
+clean:: mostlyclean
+ rm -f *.ps *.pdf
+
+upload:
+ make clean
+ ( cd .. && tar czvvf - $P ) | ssh atrey "cat >WWW/vyuka/sgr-2005/$P.tar.gz"
+ make $P.ps
+ scp -C $P.ps atrey:WWW/vyuka/sgr-2005/$P.ps
+ make clean
+
+.SECONDARY:
--- /dev/null
+P=all
+X=$(shell for a in 1 2 3 4 5 6 7 8 9 ; do echo ../$$a-*/$$a-*.tex ; done)
+
+include ../Makerules
+
+all.tex: $(X) preprocess
+ ./preprocess $(X) >$@
+
+clean::
+ rm -f all.tex
--- /dev/null
+#!/usr/bin/perl
+
+use warnings;
+use strict;
+
+print <<EOF ;
+\\input ../sgr.tex
+EOF
+foreach my $f (@ARGV) {
+ my $d = $f;
+ $d =~ s@/[^/]*$@/@;
+ open X, $f or die;
+ while (<X>) {
+ /^\\input .*sgr\.tex/ && next;
+ /^\\bye/ && last;
+ s@\\(figure|epsfbox){([^}]+)}@\\$1\{$d$2}@g;
+ print;
+ }
+ close X;
+ print "\\bigbreak\\bigbreak\n";
+}
+print <<EOF ;
+\\bye
+EOF
--- /dev/null
+% Makra pro sazbu poznamek ze Seminare z grafovych algoritmu
+
+\input mjmac.tex
+\input epsf.tex
+
+\language=\czech
+\chyph
+
+% A4 s 0.5in okraji
+\hsize=184.6mm
+\vsize=271.6mm
+
+\parindent=0.25in
+
+% Zacatek prednasky {cislo prednasky}{jmeno prednasky}{jmeno zapisovatele}
+\def\prednaska#1#2#3{%
+\line{{\Large\bf #1. #2} \hfil {\it (#3)}}
+\medskip
+\hrule
+\medskip
+}
+
+% Nadpis {text}
+\def\h#1{\bigbreak\leftline{\bf #1}\smallskip}
+
+% Zvyrazneny zacatek odstavce coby podnadpis (napr. vety apod.)
+\def\s#1{\noindent {\bo #1}}
+
+% Ctverecek na konci dukazu
+%\def\qed{{\parfillskip=0pt\quad\hfil\hbox{\I QED} \par}}
+\def\qed{\hfill\allowbreak\hfill\nobreak $\heartsuit$\par}
+
+% Poznamky pod carou
+\newcount\footcnt
+\footcnt=0
+\def\foot#1{\global\advance\footcnt by 1{\parindent=0.25in\parskip=0pt\footnote{$^{[\the\footcnt]}$}{#1}}}
+
+% \noindent se casto hodi, tak na nej mame zkratku
+\let\>=\noindent
+
+% Sazba algoritmu:
+% \algo
+% \:krok
+% \:krok
+% \::vnoreny krok
+% \endalgo
+
+\def\algo{
+\interlistskip
+\begingroup
+\let\:=\algoitem
+\let\*=\algohang
+\parskip=1pt plus 1pt minus 0.3pt
+\itemcount=0
+}
+\def\endalgo{\interlistskip\endgroup}
+\def\algoitem{\par
+\parindent=2em
+\hangindent=4em
+\hangafter=1
+\advance\itemcount by 1
+\leavevmode\hbox to 2em{\hss \the\itemcount. }%
+\futurelet\next\algoitemh}
+\def\algoitemh{\ifx\next:\let\next=\algohang\else\let\next=\relax\fi\next}
+\def\algohang:{\advance\hangindent by 2em \hskip 2em\futurelet\next\algoitemh}
+
+% Nekolikapismenkova promenna (mozno pouzit v textovem i math modu)
+\def\<#1>{\hbox{\it #1}}
+
+% Asymptoticke O-cko
+\def\O{{\cal O}}
+
+% Vlozeni obrazku {obrazek}{popisek}{sirka}
+\def\figure#1#2#3{\bigskip\vbox{\centerline{\epsfxsize=#3\epsfbox{#1}}\smallskip\centerline{#2}}\bigskip}
projects / ga.git / commitdiff