Grafy: Prepsani casti o BFS

diff --git a/3-grafy/3-grafy.tex b/3-grafy/3-grafy.tex
index 796f6a46f78c756432804248fb549ba279cc4c58..403124345a76c3f15317592026e9cb7472f57e79 100644 (file)
--- a/3-grafy/3-grafy.tex
+++ b/3-grafy/3-grafy.tex
@@ -2,19 +2,28 @@
 
 \prednaska{3}{Prohledání do~¹íøky a do~hloubky}{}
 
-\h{Prohledání do~¹íøky (BFS) {\I Breadth-First Search} }
+\h{Prohledání do~¹íøky {\I Breadth-First Search -- BFS} }
 
 Jde o grafový algoritmus, který postupnì prochází v¹echny vrcholy v~dané komponentì souvislosti.
 Algoritmus nejprve projde v¹echny sousedy poèáteèního vrcholu, poté sousedy sousedù, atd\dots
-Díky tomuto zpùsobu procházení se nìkdy té¾ nazývá \uv{\I algoritmus vlny}, nebo» se z~poèáteèního vrcholu ¹íøí pomyslná vlna, která v~ka¾dém kroku nalezne v¹echny uzly, které mají od~poèáteèního vrcholu stejnou vzdálenost. Algoritmus se tedy skvìle hodí napøíklad pro hledání nejkra¹í cesty mezi dvìma vrcholy v~grafu.
-\figure{praseci-graf.eps}{Praseèí graf}{55mm}
+Díky tomuto zpùsobu procházení se nìkdy té¾ nazývá \uv{\I algoritmus vlny}, nebo» se z~poèáteèního vrcholu ¹íøí pomyslná vlna, která v~ka¾dém kroku nalezne v¹echny uzly, které mají od~poèáteèního vrcholu stejnou vzdálenost. Algoritmus se tedy skvìle hodí napøíklad pro hledání nejkrat¹í cesty mezi dvìma vrcholy v~grafu.
 
+Zatím pøedpokládejme, ¾e graf, se kterým pracujeme, je orientovaný.
+Orientovanou hranu $(u,v)$ z~$u$ do~$v$ budeme obvykle zkracovat jako $uv$.
+Pro neorientované grafy bude v¹e obdobné.
+
+\figure{praseci-graf.eps}{Praseèí graf a prùchod vlny skrz nìj}{55mm}
 
 \s{Popis algoritmu:}
-Na zaèátku vlo¾íme do~fronty $Q$ poèáteèní vrchol $v_0$. Dále si v~poli $Z$ budeme pro ka¾dý vrchol pamatovat znaèku, zda jsme ho ji¾ nav¹tívili, èi nikoli. Pro vrchol $v_0$ si tedy dosazením jednièky zapamatujeme, ¾e je ji¾ nav¹tívený. V~dal¹ím kroku pak zkoumáme frontu $Q$: pokud není prázdná, vezmeme z~ní první vrchol a podíváme se na~v¹echny jeho sousedy $w$. Pokud je¹tì nejsou oznaèené (tedy $Z[w]=0$), tak je oznaèíme (zapamatujeme si, ¾e je pøedáváme ke zpracování a u¾ je nemáme znovu nav¹tìvovat) a pøidáme je do~fronty k~následnému zpracování. Takto cyklus opakujeme, dokud není fronta prázdná.
+Na zaèátku vlo¾íme do~fronty $Q$ poèáteèní vrchol $v_0$. Dále si v~poli~$Z$
+budeme pro ka¾dý vrchol pamatovat znaèku, zda jsme ho ji¾ nav¹tívili ($Z[v]=1$), èi
+nikoli ($Z[v]=0$). Na poèátku jsou v¹echny znaèky nulové, jen vrchol~$v_0$, který
+je oznaèen a vlo¾en do fronty.
 
-\s{Poznámka:}
-Nejprve se podívejme, jak algoritmus pracuje na orientovaném grafu. Pro neorientovaný graf funguje algoritmus analogicky, co¾ si uká¾eme pozdìji.
+V~ka¾dém dal¹ím kroku pak zkoumáme frontu $Q$: pokud není prázdná, vezmeme
+z~ní první vrchol~$u$ a podíváme se na~v¹echny vrcholy~$v$, do nich¾ z~$u$
+vede hrana. Pokud sousedi je¹tì nejsou oznaèení, tak je oznaèíme a pøidáme je do~fronty
+k~následnému zpracování. Toto opakujeme, dokud není fronta prázdná.
 
 \s{Algoritmus:}
 
@@ -22,13 +31,12 @@ Nejprve se pod
 \:$Q \leftarrow \{v_0\}$.
 \:$Z[*] \leftarrow 0, Z[v_0] \leftarrow 1$.
 \:Dokud $Q \not= \emptyset $ opakujeme:
-\::Vyzvedneme vrcholy $u$ z~$Q$.
-\::$\forall w: (u,w)\in E$:
-\:::Je-li $Z[w]=0 \Rightarrow Z[w] \leftarrow 1$, pøidáme $w$ do~$Q$.
+\::Vyzvedneme vrchol $u$ z~$Q$.
+\::$\forall v: uv\in E$:
+\:::Je-li $Z[v]=0 \Rightarrow Z[v] \leftarrow 1$, pøidáme $v$ do~$Q$.
 \endalgo
 
-
-\>{\I Pozorování:} {\I BFS} se zastaví.
+\s{Pozorování:} {\I BFS} se zastaví.
 
 \proof Zpracováváme jen vrcholy, které byly ve~frontì. Ka¾dý vrchol se dostane do~fronty maximálnì jednou. (Ka¾dý je oznaèen max. jednou, znaèky neodstraòujeme.)
 
@@ -36,31 +44,43 @@ Nejprve se pod
 
 \proof 
 \uv{$\Longrightarrow$}: 
-Platí jako invariant po celou dobu bìhu algoritmu. To doká¾eme indukcí dle doby bìhu algoritmu:
+Platí jako invariant po celou dobu bìhu algoritmu. To doká¾eme indukcí dle doby
+bìhu algoritmu:
+
+První krok indukce je triviální, nebo» cesta z~$v_0$ do~$v_0$ existuje v¾dy.
+Nyní si pøedstavme, ¾e oznaèujeme vrchol~$v$ pøes hranu~$uv$. To znamená, ¾e
+vrchol~$u$ ji¾ musel být oznaèený. Dle indukèního pøedpokladu tedy existuje
+cesta z~$v_0$ do~$u$, a tudí¾ pokud k~této cestì \uv{pøilepíme} hranu~$uv$,
+dostáváme sled z~$v_0$ do~$v$, který lze v¾dy zredukovat na cestu.
+
+\uv{$\Longleftarrow$} Sporem: Nech» existuje neoznaèený vrchol $v$ dosa¾itelný
+po nìjaké cestì z~$v_0$. Uva¾me nejkrat¹í cestu $(v_0, v)$: $v_0, v_1 \dots,
+v_k = v$ a vezmìme minimální~$i$ takové, ¾e~$v_i$ není oznaèený. Víme, ¾e~$i>0$
+(nebo»~$v_0$ je urèitì oznaèen u¾ na zaèátku algoritmu). Tudí¾~$v_{i-1}$ je
+oznaèený. Pøi oznaèení jsme ho ov¹em pøidali do fronty, tak¾e jsme ho z~fronty
+museli pozdìji zase vyjmout. Pøi tom jsme ov¹em museli objevit hranu $v_{i-1}v_i$
+a oznaèit vrchol~$v_i$, co¾ je spor.
+\qed
 
-První krok indukce je triviální, nebo» cesta z~$v_0$ do~$v_0$ existuje v¾dy. Nyní si pøedstavme, ¾e oznaèujeme vrchol~$v$ pøes hranu~$uv$. To znamená, ¾e vrchol~$u$ ji¾ musel být oznaèený. Dle indukèního pøedpokladu tedy existuje cesta z~$v_0$ do~$u$, a tudí¾ pokud k~této cestì \uv{pøilepíme} hranu~$uv$, dostáváme sled z~$v_0$ do~$v$, který lze v¾dy zredukovat na cestu.
+Nyní tedy víme, ¾e algoritmus je správný, a máme pøedstavu o tom, jak funguje.
+Podíváme-li se na~nìj podrobnìji, zjistíme, ¾e je hodnì závislý na~tom, jak si
+budeme graf pamatovat. Zanedlouho zároveò zjistíme, ¾e nám reprezentace grafu
+v~pamìti znatelnì ovlivní èasovou (i pamì»ovou) slo¾itost celého algoritmu.
 
-\uv{$\Longleftarrow$} Sporem: Nech» existuje neoznaèený vrchol $v$ dosa¾itelný po nìjaké cestì z~$v_0$. Uva¾me nejkrat¹í cestu $(v_0, v)$: $v_0, v_1 \dots, v_k = v$ a vezmìme minimální~$i$ takové, ¾e~$v_i$ není oznaèený. Víme, ¾e~$i>0$ (nebo»~$v_0$ je urèitì oznaèen u¾ na zaèátku algoritmu). Tudí¾~$v_{i-1}$ je oznaèený. Museli jsme tedy pou¾ít hranu~$v_{i-1}~v_i$ a oznaèit vrchol~$v_i$, co¾ je SPOR. \qed
+\h{Reprezentace grafu v~pamìti}
 
-Nyní tedy víme, ¾e je algoritmus správný, a máme pøedstavu o tom, jak funguje. Podíváme-li se na~nìj podrobnìji, zjistíme, ¾e je hodnì závislý na~tom, jak si budeme graf pamatovat. Zanedlouho zároveò zjistíme, ¾e nám reprezentace grafu v~pamìti znatelnì ovlivní èasovou (i pamì»ovou) slo¾itost celého algoritmu.
+Mìjme nìjaký orientovaný graf~$G$ s~$n$ vrcholy a $m$~hranami. Jak ho reprezentovat?
 
-\h{Reprezentace grafu v~pamìti}
+Vrcholy mù¾eme oèíslovat od~1 do~$n$. Pro ulo¾ení hran máme na~výbìr hned nìkolik zpùsobù.
+Pøedvedeme si je na grafu z~následujícího obrázku:
+\figure{imgn_o4.eps}{Ukázkový graf}{\epsfxsize}
 
-Oznaème vrcholy grafu na~následujícím obrázku písmeny A, B, C, D.
-Pokud bychom chtìli tento graf uchovat v~pamìti poèítaèe, máme na~výbìr
-hned nìkolik zpùsobù, jak to udìlat.
-\figure{imgn_o4.eps}{}{\epsfxsize}
 \s{1. matice sousednosti}
 
-Matice sousednosti pro graf $G$ na~$n$ vrcholech je ètvercové pole $A$ o velikosti $n \times n$, jeho¾ prvky na
-souøadnicích $i, j$ jsou dány následujícím pøedpisem:
+Matice sousednosti pro graf $G$ na~$n$ vrcholech je ètvercová matice $A$ o~rozmìrech $n \times n$,
+taková, ¾e $A_{i,j}$ popisuje, jestli z~vrcholu~$i$ do vrcholu~$j$ vede hrana ($A_{i,j}=1$)
+nebo nikoliv ($A_{i,j}=0$).
 
-$$ A_{i,j} = \left\{ \matrix {1 \Leftrightarrow \{i,j\} \in E  \cr
-                               0 \Leftrightarrow \{i,j\} \notin E \cr                                  
-                               }
-\right.$$
-
-Na~pozicích $i,j$ je jednièka, pokud v~grafu $G$ vede hrana z~vrcholu~$i$ do~vrcholu~$j$, jinak to je nula.
 Ná¹ graf z~obrázku vý¹e by tedy v~maticové reprezentaci vypadal takto:
 
 $$\bordermatrix{
@@ -72,8 +92,14 @@ D & 1 & 1 & 0 & 0\cr
 }$$
 
 S touto maticí se pracuje velmi snadno, napø. v¹echny sousedy $i$-tého vrcholu
-zjistíme jednodu¹e tak, ¾e projdeme $i$-tý øádek matice.
-Má ov¹em dvì zøejmé nevýhody: èasovou a pamì»ovou slo¾itost. Projití sousedù jednoho vrcholu trvá v¾dy $\Theta(n)$, projití sousedù pro v¹echny vrcholy (co¾ potøebujeme v~BFS) pak trvá $\Theta(n^2)$. Velikost matice je v¾dy $n \times n$, bez ohledu na~to, jak \uv{øídký} je graf. U grafu s mnoha vrcholy, ale s malým poètem hran, tedy budeme zbyteènì plýtvat místem v~pamìti. Tato reprezentace je tedy nevýhodná pøedev¹ím pro tøídy grafù jako jsou stromy, které mají $n-1$ hran, nebo rovinné grafy, které mají nejvý¹e $3n-6$ hran.
+zjistíme jednodu¹e tak, ¾e projdeme $i$-tý øádek matice a najdeme v¹echny jednièky.
+Má ov¹em dvì zøejmé nevýhody: èasovou a pamì»ovou slo¾itost. Projití sousedù
+jednoho vrcholu trvá v¾dy $\Theta(n)$, projití sousedù pro v¹echny vrcholy (co¾
+potøebujeme v~BFS) pak trvá $\Theta(n^2)$. Velikost matice je v¾dy $n \times
+n$, bez ohledu na~to, jak \uv{øídký} je graf. U grafu s mnoha vrcholy, ale
+s malým poètem hran, tedy budeme zbyteènì plýtvat místem v~pamìti. Tato
+reprezentace je tedy nevýhodná pøedev¹ím pro tøídy grafù, jako jsou stromy,
+které mají $n-1$ hran, nebo rovinné grafy, které mají nejvý¹e $3n-6$ hran.
 
 \s{Pozorování:} BFS s reprezentací maticí sousednosti bì¾í v~èase: $\Theta(n^2)$.
 
@@ -81,38 +107,57 @@ M
 U¾ jsme si uvìdomili, ¾e ka¾dý vrchol se dostane do~fronty $Q$ nejvý¹e jednou. Pro ka¾dý vrchol ve~frontì potøebujeme projít jeho sousedy, co¾ nám trvá s~reprezentací maticí sousednosti $\Theta(n)$. Vrcholù je celkem $n$, tedy èasová slo¾itost je $\Theta(n^2)$.
 \qed
 
+\smallskip
 \s{2. seznam sousedù}
 
 V~matici sousednosti jsme museli procházet jak hrany, tak nehrany, co¾ bylo zbyteèné. Bylo by tedy výhodnìj¹í pamatovat si pro ka¾dý vrchol pouze jeho sousedy. To mù¾eme zaøídit napøíklad jedním ze~dvou následujících zpùsobù:
 
-Uchovávejme pole indexované vrcholy tak, ¾e v~ka¾dém prvku pole je ukazatel na~spojový seznam sousedù tohoto vrcholu. Tedy $L(v)=\{w: vw \in E(G)\}$.
+Uchovávejme pole indexované vrcholy tak, ¾e v~ka¾dém prvku pole je ukazatel
+na~spojový seznam sousedù tohoto vrcholu. Tedy $L(v)=\{w: vw \in E(G)\}$.
 
-Pokud se nám nebude chtít pracovat se spojovými seznamy, mù¾eme vyu¾ít reprezentaci pomocí dvou polí. První pole~$V$ bude opìt indexované vrcholy. V~druhém poli~$E$ budou pro ka¾dý vrchol ulo¾eni jeho sousedé. V~poli~$V$ si pamatujeme pro ka¾dý vrchol~$i$ index do~pole~$E$, kde zaèínají jeho sousedé. K sousedùm vrcholu~$i$ se tedy dostaneme ji¾ snadno - nalezneme je na pozicích $V[i]~\dots~V[i+1]-1$.
-\figure{imgn_nei.eps}{Znázornìní polí seznamu sousedù.}{\epsfxsize}
+Pokud se nám nebude chtít pracovat se spojovými seznamy, mù¾eme vyu¾ít
+reprezentaci pomocí dvou polí. První pole~$V$ bude opìt indexované vrcholy.
+V~druhém poli~$E$ budou pro ka¾dý vrchol ulo¾eni jeho sousedé. V~poli~$V$ si
+pamatujeme pro ka¾dý vrchol~$i$ index do~pole~$E$, kde zaèínají jeho sousedé,
+a navíc dodefinujeme, ¾e $V[n+1]=m+1$.
+K sousedùm vrcholu~$i$ se pak ji¾ dostaneme snadno -- nalezneme je na pozicích
+$V[i], \dots, V[i+1]-1$.
 
+\figure{sousedi.eps}{Znázornìní polí seznamu sousedù}{40mm}
 
-Na tuto reprezentaci u¾ staèí prostor $O(n + m)$, co¾ u¾ je, na~rozdíl od~pøedchozího kvadratického prostoru, docela pøíjemné.
+Na tuto reprezentaci u¾ staèí prostor $\Theta(n + m)$, co¾ u¾ je, na~rozdíl
+od~pøedchozího kvadratického prostoru, docela pøíjemné.
 
-\s{Pozorování:} BFS bì¾í v~èase: $\Theta(n+m)$.
+\s{Pozorování:} BFS s~reprezentací seznamem sousedù bì¾í v~èase $\Theta(n+m)$.
 
 \proof
-Algoritmus vezme ka¾dý vrchol i ka¾dou hranu do~ruky nejvý¹e jednou. Èasová slo¾itost bude tedy:
-$$\Theta\left(n+\sum_{v\in V(G)} {\rm deg}(v)\right) = \Theta(n+m).$$
+Algoritmus zpracuje ka¾dý vrchol nejvý¹e jednou a stráví jím èas lineární
+v~poètu odchozích hran, tedy $\Theta({\rm deg}^-(v))$. Èasová slo¾itost celého
+algoritmu tedy èiní:
+$$\Theta\left(n+\sum_{v\in V(G)} {\rm deg}^-(v)\right) = \Theta(n+m).$$
 \qed
 
 \s{3. orákulum}
 
-Dal¹í mo¾ností reprezentace je pak jakési orákulum, které nám øekne (spoèítá), kam vedou hrany z~daného vrcholu\dots To se hodí napøíklad tehdy, pokud si nepotøebujeme pamatovat celý graf, ale staèí nám naleznout sousedy nìjakého vrcholu, které orákulum jednoznaènì výpoètem doká¾e urèit. Napøíklad pøi øe¹ení známé úlohy odmìøení daného mno¾ství vody za pomocí nádob rùzných objemù mù¾eme tento zpùsob reprezentace grafu pou¾ít. Problém pøevedeme na prohledávání grafu do~¹íøky, kde vrcholùm odpovídají jednotlivé stavy. Stav øíká, kolik vody je zrovna ve které nádobì. Potom nám ji¾ staèí ze zadaného poèáteèního vrcholu (stavu) najít cestu do~cílového. Orákulum je zde vlastnì funkce, které pøedáme vrchol a ona nám vrátí v¹echny vrcholy sousední -- tedy takové stavy, ke kterým dojde, kdy¾ vyzkou¹í v¹echny mo¾nosti pøelití kapaliny z jedné nádoby do~druhé.
+Dal¹í mo¾ností reprezentace je jakési orákulum, které nám na po¾ádání øekne (spoèítá),
+kam vedou hrany z~daného vrcholu. To se hodí napøíklad tehdy, pokud graf vznikl
+výpoètem a nechceme plýtvat pamìtí na jeho ulo¾ení. To se hodí napøíklad u~u¾
+zmínìného hlavolamu \uv{Lloydova osmièka}.
 
-\h{Roz¹íøení algoritmu:}
+\smallskip
+\s{Neorientované grafy}
 
-Abychom mohli vyu¾ít toho, ¾e algoritmus prochází vrcholy grafu ve~vlnì, a jiných hezkých vlastností, tak si dodefinujeme následující oznaèení:
+Chceme-li reprezentovat neorientovaný graf, ulo¾íme ka¾dou hranu v~obou orientacích.
 
-V~poli $D$ bude pro ka¾dý vrchol ulo¾ena vzdálenost od~poèáteèního vrcholu.
-V~poli $P$ si budeme pro ka¾dý vrchol pamatovat jeho pøedchùdce. Dále budeme vyu¾ívat fáze bìhu algoritmu, které budou simulovat onu vlnu:
+\h{Výpoèet vzdáleností}
 
-\s{Definice: {\I Fáze bìhu algoritmu}:} Ve~fázi $F_0$ je zpracováván vrchol $v_0$. Ve~fázi $F_{i+1}$ jsou zpracovávány vrcholy ulo¾ené do~fronty $Q$ bìhem fáze $F_i$.
+Abychom mohli vyu¾ít toho, ¾e algoritmus prochází vrcholy grafu ve~vlnì,
+k~výpoètu vzdáleností, doplníme do nìj dvì pomocná pole:
+$D[v]$ bude øíkat, na~kolik krokù jsme se do~$v$ dostali,
+$P[v]$ bude obsahovat {\I pøedchùdce} vrcholu~$v$, toti¾ vrchol~$u$,
+ze~kterého jsme se do~$v$ dostali po hranì a jeho¾ $D[u]=D[v]-1$.
 
+\smallskip
 \s{Roz¹íøený algoritmus:}
 \algo
 \:$Q \leftarrow \{v_0\}$.
@@ -120,70 +165,87 @@ V~poli $P$ si budeme pro ka
 \:$D[*] \leftarrow \infty, D[v_0] \leftarrow 0$.
 \:Dokud $Q \not= \emptyset $ opakujeme:
 \::Vyzvedneme vrchol $u$ z~$Q$.
-\::Pro ka¾dý vrchol $w$, který je sousedem vrcholu $u$:
-\:::Je-li $Z[w]=0 \Rightarrow Z[w] \leftarrow 1, D[w] \leftarrow D[u]+1, P[w] \leftarrow u$
-\::::Pøidáme $w$ do~$Q$.
+\::Pro ka¾dý vrchol $v$, do~kterého vede hrana z~vrcholu $u$:
+\:::Je-li $Z[v]=0$:
+\::::$Z[v] \leftarrow 1, D[v] \leftarrow D[u]+1, P[v] \leftarrow u$
+\::::Pøidáme $v$ do~$Q$.
 \endalgo
 
-\s{Lemma:} Na~konci BFS pro v¹echny vrcholy dosa¾itelné z~$v_0$ platí, ¾e vrchol $v$ byl zpracován ve~fázi $F_i$ právì tehdy, kdy¾ vzdálenost $v_0$ a $v$ (délka nejkrat¹í cesty z~$v_0$ do~$v$) je rovna $i$. Formálnì zapsáno: $v \in F_i \Leftrightarrow d(v_0,v) = i$. (Kde $d(x,y)$ je délka nejkrat¹í cesty z~$x$ do~$y$).
-
-\proof
-\uv{$\Longrightarrow$}: 
-Dùkaz provedeme indukcí podle $i$ (èísla fáze bìhu algoritmu).
-
-První krok indukce je triviální, nebo» ve~fázi $F_0$ je oznaèen (dle definice) pouze vrchol $v_0$ a to je jediný vrchol vzdálený~0 od~$v_0$. 
+\s{Definice: {\I Fáze bìhu algoritmu}:} Ve~fázi $F_0$ je zpracováván vrchol $v_0$. Ve~fázi $F_{i+1}$ jsou zpracovávány vrcholy ulo¾ené do~fronty $Q$ bìhem fáze $F_i$.
 
-Pokud je vrchol $v$ zpracováván ve~fázi $F_i$, pak musel být zaøazen do~fronty bìhem fáze $F_{i-1}$ jako soused nìjakého vrcholu $u$. Pro vrchol $u$ mù¾eme pou¾ít indukèní pøedpoklad, tedy ¾e délka nejkrat¹í cesty z~$v_0$ do~$u$ je $d(v_0,u)=i-1$. Pak tedy $d(v_0,v)\leq i$, nebo» je¹tì nevíme, zda cesta $v_0, \dots, u, v$ je nejkrat¹í. Kdyby ale existovala nìjaká krat¹í, tedy délky nejvý¹e $i-1$, tak by byl vrchol $v$ objeven u¾ døíve ne¾ ve fázi $F_i$. Proto $d(v_0,v) = i$.
+\s{Pozorování:} Ka¾dý vrchol~$v$ dosa¾itelný z~$v_0$ se úèastní právì jedné fáze,
+a to té s~èíslem~$D[v]$.
 
+\s{Lemma:} Po zastavení BFS pro v¹echny vrcholy dosa¾itelné z~$v_0$ platí, ¾e $D[v]$
+je rovno $d(v_0,v)$, toti¾ vzdálenosti (délce nejkrat¹í cesty) z~$v_0$ do~$v$.
 
-\uv{$\Longleftarrow$}: Ka¾dý dosa¾itelný vrchol padne do~nìjaké fáze (viz. minulé lemma).
+\proof
+Nejprve si uvìdomíme, ¾e kdykoliv je~$v$ oznaèen, vede do nìj z~$v_0$ nìjaký
+sled délky~$v$ (indukcí, stejnì jako jsme pøed chvílí dokazovali, ¾e BFS projde
+v¹echny dosa¾itelné vrcholy). Proto nemù¾e být $D[v]$ men¹í ne¾ $d(v_0,v)$.
+
+Sporem doká¾eme, ¾e nemù¾e být ani vìt¹í. Pøedpokládejme, ¾e existuje nìjaký
+\uv{¹patný} vrchol~$v$, pro který je $D[v] > d(v_0,v)$. Nech» $P$ je nìkterá z~nejkrat¹ích
+cest z~$v_0$ do~$v$. Z~mo¾ných ¹patných vrcholù si vyberme takový, jeho¾ $P$ je nejkrat¹í
+mo¾ná. Jeliko¾ pro vrchol~$v_0$ je zajisté $D[v_0] = d(v_0,v_0) = 0$, musí být $v$
+rùzný od~$v_0$, tak¾e má na~$P$ nìjakého pøedchùdce~$u$. Pro toho ov¹em je vzdálenost
+spoèítána správnì: $D[u] = d(v_0,u) = d(v_0,v)-1$.
+
+Uva¾ujme nyní, co se stalo v~okam¾iku, kdy jsme~$D[u]$ nastavili. Tehdy jsme~$u$
+ulo¾ili do fronty, po èase jsme ho z~fronty zase vytáhli a prozkoumali jsme v¹echny
+vrcholy, do ní¾ vede z~$u$ hrana. Tedy i vrchol~$v$, tak¾e $D[v]$ v~tomto okam¾iku
+nemù¾e být vìt¹í ne¾ $D[u]+1 = d(v_0,v)$, a~to je spor.
 \qed
 
-Ji¾ tedy víme, ¾e vrchol $v_i$, jeho¾ vzdálenost od~vrcholu $v_0$ je $i$, bude zpracován v~$i$-té fázi. Jak ale po~skonèení algoritmu zjistíme, ve které fázi byl zpracován, neboli jak je vzdálený od~startovního vrcholu? Tato informace je právì ulo¾ena v~poli $D$ s indexem $i$ (v~$D[i]$).
-
-Zároveò nás mù¾e zajímat, jak bychom nejkrat¹í cestu z~$v_0$ do~$v_i$ rekonstruovali. Pro tento úèel jsme si zavedli pole $P$. Nejkrat¹í cesta z~$v_0$ do~$v_i$ bude v~obráceném poøadí vypadat: $v_i, P[v_i], P[P[v_i]], P[P[P[v_i]]], \dots, v_0$.
+Víme tedy, ¾e BFS správnì spoèítá délky nejkrat¹ích cest do v¹ech vrcholù grafu.
+Pomocí pøedchùdcù v~poli~$P$ mù¾eme tyto cesty dokonce snadno rekonstruovat:
+pøedposledním vrcholem na nejkrat¹í cestì do~vrcholu~$v$ musí být vrchol $P[v]$,
+jeho pøedchùdcem $P[P[v]]$, \dots, a¾ do~vrcholu~$v_0$.
 
+Pøedchùdci nám tedy kódují strukturu nejkrat¹ích cest do~v¹ech vrcholù.
+Mù¾eme se na nì dívat také následovnì:
 
-\s{Pozorování:} Pokud víme, ¾e $v_0, v_1, \dots, v_{k-1}, v_k$ je nejkrat¹í cesta z~$v_0$ do~$v_k$, pak $v_0,v_1,\dots,v_{k-1}$ je nejkrat¹í cesta z~$v_0$ do~$v_{k-1}$. Neboli prefix nejkrat¹í cesty je nejkrat¹í cesta.
+\s{Definice:} Strom nejkrat¹ích cest je orientovaný strom s~mno¾inou vrcholù
+$W=\{ v\in V(G) \mid \hbox{$v$ dosa¾itelný z~$v_0$} \}$
+a hranami $\{ (P(v),v) \mid v\in W, v\ne v_0 \}$.
 
-\proof
-Kdyby existovala krat¹í cesta z~$v_0$ do~$v_{k-1}$, pak bychom mohli zkrátit i cestu z~$v_0$ do~$v_k$.
+\s{Pozorování:} Koøenem stromu nejkrat¹ích cest je vrchol~$v_0$, cesta v~tomto
+stromu z~$v_0$ do~$v$ (jednoznaènì urèená, je to strom) je pak jednou z~nejkrat¹ích
+cest z~$v_0$ do~$v$ v~pùvodním grafu.
 
+\smallskip
+\s{Komponenty souvislosti}
 
-\s{Pozorování:} BFS u~neorientovaného grafu projde celou komponentu souvislosti.
+V~neorientovaných grafech mù¾eme BFS jednodu¹e pou¾ít na nalezení komponent souvislosti.
+Ji¾ víme, ¾e BFS spu¹tìné z~vrcholu~$v_0$ projde právì ty vrcholy, které jsou z~$v_0$
+dosa¾itelné, co¾ jsou v~neorientovaném grafu pøesnì ty, které le¾í v~té¾e komponentì.
 
-\proof
-Víme, ¾e BFS($v_0$) oznaèí $v$ právì tehdy, kdy¾ existuje cesta z~$v_0$ do~$v$. V~neorientovaném grafu existuje cesta z~$v_0$ do~právì v¹ech vrcholù, které jsou ve~stejné komponentì souvislosti jako $v_0$. Pokud tedy spustíme BFS na~$v_0$, tak se postupnì projdou v¹echny vrcholy této komponenty souvislosti.
-\qed
-
-\s{Pozorování:} Pokud BFS postupnì spou¹tíme na~dosud neobarvené vrcholy v~ neorientovaném grafu, nalezneme nakonec v~èase $\Theta(n+m)$ v¹echny komponenty souvislosti.
+Staèí opakovanì spou¹tìt BFS z~dosud neoznaèených vrcholù, poka¾dé nám oznaèí
+jednu komponentu.
 
 \s{Algoritmus:}
 \algo
-\:Pro v¹echny vrcholy $v \in V(G)$ opakuj:
-\::Pokud je vrchol $v$ neobarvený $ \Rightarrow \<BFS(v)>$.
+\:Pro v¹echny vrcholy $v \in V(G)$ opakujeme:
+\::Pokud je vrchol $v$ neoznaèený:
+\:::Spustíme $\<BFS>(v)$ a pøiøadíme objevené vrcholy nové komponentì.
 \endalgo
 
-\proof
-Ka¾dým spu¹tìním na~dosud neobarvený vrchol neorientovaného grafu obarvíme právì jednu komponentu souvislosti (tu, ve~které je tento vrchol). 
-Postupnì projdeme v¹echny vrcholy od prvního k poslednímu a v¾dy pokud je vrchol neobarvený, spustíme na nìj BFS. Tak nakonec obarvíme v¹echny komponenty souvislosti. Èasová slo¾itost bude stejná jako u~samotného BFS, tedy $\Theta(n + m)$.
-\qed
+To, co jsme o~BFS zjistili, mù¾eme shrnout do následující vìty:
 
-\s{Vìta:} $BFS(v_0)$ v~èase $\Theta(n + m)$ spoète:
+\s{Vìta:} $\<BFS>(v_0)$ v~èase $\Theta(n + m)$ spoète:
 \itemize\ibull
 \:vrcholy dosa¾itelné z~$v_0$
 \:vzdálenosti tìchto vrcholù od~$v_0$
 \:strom nejkrat¹ích cest z~$v_0$
 \endlist
 
-\s{Poznámka:} Algoritmus na prohledávání do~¹íøky bude fungovat i na neorientovaném grafu. Mù¾eme si jednodu¹e pøedstavit, ¾e je to orientovaný graf, kde ka¾dá hrana má oboustrannou orientaci.
-
-
-Prohledávání do~¹íøky ale není jediný algoritmus, který nìjak systematicky prochází graf. Jak u¾ název kapitoly napovídá, budeme se zabývat je¹tì druhým algoritmem, prohledáváním do~hloubky. Podívejme se, jak bude vypadat \dots
+\>Prohledávání do~¹íøky ale není jediný algoritmus, který nìjak systematicky prochází graf. Jak u¾ název kapitoly napovídá, budeme se zabývat je¹tì druhým algoritmem, prohledáváním do~hloubky. Podívejme se, jak bude vypadat \dots
 
-\h{Prohledávání do~hloubky (DFS) {\I Depth-First Search} }
+\h{Prohledávání do~hloubky {\I Depth-First Search -- DFS }}
 
-Tento algoritmus neprochází graf ve~vlnì jako BFS, ale prochází ho rekurzivnì. V¾dy se zanoøí co nejhloubìji a¾ do~listu a pak se o~kus vrátí a opìt se sna¾í zanoøit. Vrcholy, ve kterých u¾ byl, ignoruje.
+Tento algoritmus neprochází graf ve~vlnì jako BFS, nýbr¾ ho prochází
+rekurzivnì. V¾dy se zanoøí co nejhloubìji a¾ do~listu a pak se o~kus vrátí
+a opìt se sna¾í zanoøit. Vrcholy, ve kterých u¾ byl, ignoruje.
 
 Opìt uva¾me nejdøíve graf orientovaný. Následnì si uká¾eme, ¾e v~neorientovaném grafu budou pouze malé zmìny.
 

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+%Copyright: Copyright (URW)++,Copyright 1999 by (URW)++ Design &
+%Copyright:  Development; Cyrillic glyphs added by Valek Filippov (C)
+%Copyright:  2001-2005
+% Generated by FontForge 20070723 (http://fontforge.sf.net/)
+%%EndComments
+
+FontDirectory/NimbusRomNo9L-Regu known{/NimbusRomNo9L-Regu findfont dup/UniqueID known pop false {dup
+/UniqueID get 5020931 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+11 dict begin
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0 ]readonly def
+/FontName /f-0-0 def
+/FontBBox {-168 -281 1031 924 }readonly def
+                     
+/PaintType 0 def
+/FontInfo 9 dict dup begin
+ /version (1.06) readonly def
+ /Notice (Copyright \050URW\051++,Copyright 1999 by \050URW\051++ Design & Development; Cyrillic glyphs added by Valek Filippov \050C\051 2001-2005) readonly def
+ /FullName (Nimbus Roman No9 L Regular) readonly def
+ /FamilyName (Nimbus Roman No9 L) readonly def
+ /Weight (Regular) readonly def
+ /ItalicAngle 0 def
+ /isFixedPitch false def
+ /UnderlinePosition -100 def
+ /UnderlineThickness 50 def
+end readonly def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 1 /one put
+dup 2 /three put
+dup 3 /four put
+dup 4 /five put
+dup 5 /seven put
+dup 9 /A put
+dup 6 /B put
+dup 7 /C put
+dup 8 /D put
+dup 11 /E put
+dup 10 /V put
+readonly def
+currentdict end
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