\input ../lecnotes.tex
-\def\og{$\buildrel \rightarrow \over G$}
-\def\ogm{\buildrel \rightarrow \over G}
-
\prednaska{2}{Toky v sítích}{(pøedná¹el T. Valla, zapsali K. Vandas a J. Machálek)}
\h{Motivaèní úlohy:}
\:Mìjme orientovaný graf pøedstavující ¾eleznièní sí»; graf má význaèné vrcholy Moskva a Fronta, ka¾dá hrana grafu má kapacitu, kterou mù¾e uvézt. Kolik vojákù je schopna sí» pøevézt z Moskvy a spotøebovat na Frontì?
\endlist
-\s{Definice:} {\I Sí»} je uspoøádaná ètveøice $(\ogm,z,s,c)$, kde \og{} je orientovaný graf, z~a~s~jsou jeho vrcholy (z(droj) a s(tok)) a c je kapacita sítì, kterou pøedstavuje funkce $c:E(\ogm)\to R_{0}^{+}$.
-
-\par\noindent {\sl Intuice:} Toky v sítích pøedstavují rozvr¾ení, jakým suroviny sítí poteèou...
+\s{Definice:} {\I Sí»} je uspoøádaná ètveøice $(G,z,s,c)$, kde $G$ je orientovaný graf, $z$~a~$s$~jsou jeho vrcholy (zdroj a stok) a $c$ je kapacita sítì, kterou pøedstavuje funkce $c:E(G)\to\bb R_{0}^{+}$.
\figure{sit.eps}{Pøíklad sítì. Èísla pøedstavují kapacity jednotlivých hran.}{3in}
-\s{Definice:} {\I Tok} je funkce $E(\ogm)\to R$ taková, ¾e
+\par\noindent {\sl Intuice:} Toky v sítích pøedstavují rozvr¾ení, jakým suroviny sítí poteèou.
+
+\s{Definice:} {\I Tok} je funkce $f:E(G)\to\bb R$ taková, ¾e platí:
\numlist{\ndotted}
-\:Tok ka¾dé hrany je omezen její kapacitou: $0\le f(e)\le c(e)$
-\:\uv{Kirchhoffùv zákon} - nikde se neztrácejí suroviny: $$\sum_{(x,u)\in E}{f(x,u)}=\sum_{(u,x)\in E}{f(u,x)}\quad \forall u\in V(\ogm), u\ne z, u\ne s$$
+\:Tok ka¾dé hrany je omezen její kapacitou: $0\le f(e)\le c(e)$.
+\:Kirchhoffùv zákon -- nikde se neztrácejí suroviny: $$\sum_{(x,u)\in E}{f(x,u)}=\sum_{(u,x)\in E}{f(u,x)}\quad \forall u\in V(G), u\ne z, u\ne s.$$
\endlist
-\s{Poznámka:} Pokud bychom se chtìli v definici toku u bodu 2 vyhnout podmínkám pro z a s, mù¾eme zdroj a stok vzájemnì propojit (pak jde o tzv. cirkulaci).
+\s{Poznámka:} Pokud bychom se chtìli v definici toku u bodu 2 vyhnout podmínkám pro $z$ a $s$, mù¾eme zdroj a stok vzájemnì propojit (pak jde o tzv. cirkulaci).
-\s{Poznámka:} V angliètinì se obvykle zdroj znaèí \uv{s} a stok \uv{t} (zkratky source a sink, ov¹em dvì stejná písmena by byla ponìkud nepraktická\dots)
+\s{Poznámka:} V angliètinì se obvykle zdroj znaèí \uv{$s$} a stok \uv{$t$} (zkratky source a~target).
\figure{tok.eps}{Pøíklad toku. Èísla pøedstavují ohodnocení funkcí toku (v závorkách jsou kapacity hran).}{4in}
-\s{Definice:} {\I Velikost toku} f je: $$w(f)=\sum_{(z,x)\in E}{f(z,x)}-\sum_{(x,z)\in E}{f(x,z)}$$
+\s{Definice:} {\I Velikost toku} $f$ je: $$w(f)=\sum_{(z,x)\in E}{f(z,x)}-\sum_{(x,z)\in E}{f(x,z)}.$$
\s{Vìta:} Pro ka¾dou sí» existuje maximální tok (bez dùkazu).
-\par\noindent {\sl Idea dùkazu:} Doká¾e se pomocí metod matematické analýzy s tím, ¾e mno¾ina tokù je kompaktní a funkce tokù je spojitá\dots
+\par\noindent {\sl Idea dùkazu:} Doká¾e se pomocí metod matematické analýzy s tím, ¾e mno¾ina tokù je kompaktní a funkce velikosti toku je spojitá.
-\>Intuice. Øez grafu je mno¾ina hran oddìlující z(droj) a s(tok).
+\par\noindent {\sl Intuice:} Øez grafu je mno¾ina hran oddìlující zdroj a stok.
-\s{Definice:} {\I Øez} R v síti $(\ogm,z,s,c)$ je $R\subseteq E(\ogm)$ taková, ¾e $\not\kern -.5ex\exists$ cesta ze z do s v grafu $(V(\ogm),E(\ogm)\setminus R)$.
+\s{Definice:} {\I Øez} $R$ v síti $(G,z,s,c)$ je $R\subseteq E(G)$ taková, ¾e neexistuje cesta ze $z$ do $s$ v grafu $(V(G),E(G)\setminus R)$.
-\s{Definice:} {\I Kapacita øezu} $c(R)=\sum_{(u,v\in R)}{c(u,v)}.$
-
-\s{Vìta (Hlavní vìta o tocích, Ford-Fulkerson):} Mìjme S sí». $$\max_{f\hbox{ tok}}{w(f)=\min_{R\hbox{ øez}}{c(R)}}$$
-
-\par\noindent {\sl Intuice:} Uvá¾íme-li mno¾inu kapacit v¹ech øezù, je zdola omezená mno¾inou hodnot tokù.
+\s{Definice:} {\I Kapacita øezu} $c(R)=\sum_{(u,v\in R)}{c(u,v)}$.
-\par\noindent {\sl Idea dùkazu:} Dùkaz provedeme pomocí dokázání dvou neostrých nerovností.
+\s{Vìta (Hlavní vìta o tocích, Ford-Fulkerson):} Mìjme $S$ sí». Platí: $$\max_{f\hbox{ tok}}{w(f)=\min_{R\hbox{ øez}}{c(R)}}$$
\proof
+Dùkaz provedeme pomocí dokázání dvou neostrých nerovností.
-\>Pomocné znaèení: Zaveïme konvenci, ¾e existují-li orientované hrany (u,v), $u\in A$, $v\in B$, znaèíme je S(A,B) (separátor). $f(A,B)=\sum_{(u,v)\in E,u\in A,v\in B}{f(u,v)}.$
+\>Pomocné znaèení: Zaveïme konvenci, ¾e existují-li orientované hrany $(u,v)$, $u\in A$, $v\in B$, znaèíme je $S(A,B)$ (separátor). $f(A,B)=\sum_{(u,v)\in E,u\in A,v\in B}{f(u,v)}.$
{\narrower
-\s{Lemma:} $A\subseteq V(\ogm),z\in A,s\not\in A,f\hbox{ je tok}$. Potom platí, ¾e $$w(f)=f(A,V\setminus~A)-f(V\setminus~A,A)$$
+\par\noindent {\sl Intuice:} Uvá¾íme-li mno¾inu kapacit v¹ech øezù, je zdola omezená mno¾inou hodnot tokù.
+
+\s{Lemma:} $A\subseteq V(G),z\in A,s\not\in A,f\hbox{ je tok}$. Potom platí, ¾e $$w(f)=f(A,V\setminus~A)-f(V\setminus~A,A)$$
\proof
-Dùkaz provedeme pomocí \uv{Kirchhoffova zákonu} a definice velikosti toku:
-$$\sum_{(u,x)\in E}{f(u,x)}-\sum_{(x,u)\in E}{f(x,u)}=0\quad \forall u\in A,u\neq z,u\neq s$$
+Dùkaz provedeme pomocí Kirchhoffova zákonu a definice velikosti toku:
+$$\sum_{(u,x)\in E}{f(u,x)}-\sum_{(x,u)\in E}{f(x,u)}=0\quad\forall u\in A,u\neq z,u\neq s$$
$$\sum_{(z,x)\in E}{f(z,x)}-\sum_{(x,z)\in E}{f(x,z)=w(f)}$$
\>Rovnice seèteme:
$$\sum_{u\in A}{\left(\sum_{(u,x)\in E}{f(u,x)}-\sum_{(x,u)\in E}{f(x,u)}\right)}=w(f)$$
-\s{Poznámka:} Tato rovnice neznamená nic jiného, ne¾ ¾e se hrany vedoucí z A do A jednou pøiètou a jednou odeètou. Projeví se pouze hrany vedoucí dovnitø a ven z $V\setminus A$, tak¾e toky vnitøních hran A se \uv{po¾erou}.
+\s{Poznámka:} Tato rovnice neznamená nic jiného, ne¾ ¾e se hrany vedoucí z~$A$ do $A$ jednou pøiètou a jednou odeètou. Projeví se pouze hrany, které vedou dovnitø a ven z $V\setminus A$, tak¾e toky vnitøních hran $A$ se \uv{po¾erou}.
\>Z toho plyne:
-$$f(A,V\setminus A)-f(V\setminus A,A)=\sum_{u\in A,v\not\in A}{f(u,v)}-\sum_{u\not\in A,v\in A}{f(u,v)}=w(f)$$
+$$f(A,V\setminus A)-f(V\setminus A,A)=\sum_{u\in A,v\not\in A}{f(u,v)}-\sum_{u\not\in A,v\in A}{f(u,v)}=w(f).$$
\qed
-\s{Dùsledek:} Pokud f je tok, R je øez, pak platí: $w(f)\le c(R)$.
+\s{Dùsledek:} Pokud $f$ je tok, $R$ je øez, pak platí: $w(f)\le c(R)$.
\proof
$w(f)=f(A,V\setminus A)-f(V\setminus A,A)\le f(A,V\setminus A)\le c(A,V\setminus A)\le c(R)$
\qed
-}
-\s{Redefinice:} {\I Cesta} je odteï posloupnost navazujících hran, u kterých ignorujeme orientaci. Tyto cesty se obvykle nazývají \uv{zs-cesty}.
+\s{Definice:} {\I Zs-cesta} je posloupnost navazujících hran, u kterých ignorujeme orientaci.
\figure{cesta.eps}{Pøíklad zs-cesty.}{3in}
-\s{Definice:} {\I Cesta} P ze z do s je {\I nasycená}, pokud
+\s{Definice:} {\I Zs-cesta} $P$ ze $z$ do $s$ je {\I nasycená}, pokud
$$\exists\ e \in P\left\{{f(e)=c(e)\dots \hbox{orientovaná po smìru}}\atop{f(e)=0\dots \hbox{orientovaná proti smìru}}\right.$$
-\>Jinak je cesta nenasycená.
+\>Jinak je zs-cesta nenasycená.
-\s{Definice:} {\I Tok} je {\I nasycený}, pokud $\forall$ cesta P je nasycená.
+\s{Definice:} {\I Tok} je {\I nasycený}, pokud $\forall$ zs-cesta $P$ ze $z$ do $s$ je nasycená.
-\s{Vìta:} Tok f je nasycený $\Leftrightarrow$ f je maximální. Navíc pro $\forall$ maximální tok f $\exists$ øez R, ¾e $w(f)=c(R).$
+\s{Vìta:} Tok $f$ je nasycený $\Leftrightarrow$ $f$ je maximální. Navíc pro ka¾dý maximální tok $f$ existuje øez $R$, ¾e $w(f)=c(R)$.
\proof
-\>\uv{$\Leftarrow$} sporem: Mìjme tok f maximální a nenasycený $\Rightarrow \exists$ P nenasycená. Tuto cestu P budeme \uv{vylep¹ovat}.
+\>\uv{$\Leftarrow$} sporem: Mìjme tok $f$ maximální a nenasycený. Existuje tedy zs-cesta $P$ nenasycená. Tuto cestu $P$ budeme vylep¹ovat.
\>Zvolíme:
$$\varepsilon_1=\min_{e\in P,\hbox{ po smìru}}{\{c(e)-f(e)\}}$$
$$\varepsilon_2=\min_{e\in P,\hbox{ proti smìru}}{\{f(e)\}}$$
$$\varepsilon=\min{\{\varepsilon_1,\varepsilon_2\}}>0$$
-\>Poslední ostrá nerovnost vyplývá z definice nenasycené cesty. Nyní vylep¹íme tok f o $\varepsilon$: $f\to f^{'}$:
+\>Poslední ostrá nerovnost vyplývá z definice nenasycené cesty. Nyní vylep¹íme tok $f$ o $\varepsilon$: $f\to f^{'}$:
$$f^{'}(e)=\left\{{\displaystyle f(e)+\varepsilon \dots e\in P\hbox{ po smìru}\hfill}\atop{{\displaystyle f(e)-\varepsilon \dots e\in P\hbox{ proti smìru}\hfill}\atop{\displaystyle f(e) \dots e\not\in P\hfill}}\right.$$
\>Nyní je potøeba ovìøit, ¾e $f^{'}$ je skuteènì tok:
-$$0\le f^{'}(e)\le c(e)\dots\hbox{platí stále díky volbì }\varepsilon $$
+$$0\le f^{'}(e)\le c(e)\dots\hbox{platí stále díky volbì }\varepsilon.$$
-\>Platnost \uv{Kirchhoffova zákonu} ovìøíme rozborem pøípadù:
+\>Platnost Kirchhoffova zákonu ovìøíme rozborem pøípadù:
\figure{kirch.eps}{Rozbor pøípadù.}{4in}
\>$f^{'}$ je tedy tok, ov¹em potom platí:
-$$w(f^{'})=w(f)+\varepsilon\Rightarrow w(f^{'})>w(f)\Rightarrow\hbox{SPOR}$$
+$$w(f^{'})=w(f)+\varepsilon\Rightarrow w(f^{'})>w(f)\Rightarrow\hbox{SPOR.}$$
-\>\uv{$\Rightarrow$}: Uvá¾íme $A\subseteq V(\ogm)$, ¾e $\forall\ v\in{} A\ \exists$ nenasycená cesta ze z do v. $z \in A$, $s \not\in A$. $S(A,V\setminus A)$ je hledaný øez R. Podle lemmatu:
+\>\uv{$\Rightarrow$}: Uvá¾íme $A\subseteq V(G)$, ¾e $\forall\ v\in{} A$ existuje nenasycená cesta ze $z$ do $v$. $z \in A$, $s \not\in A$. $S(A,V\setminus A)$ je hledaný øez $R$. Podle lemmatu:
$$w(f)=f(A,V\setminus A)-f(V\setminus A,A)=c(A,V\setminus A)=c(R)$$
-Víme, ¾e $w(f)\le c(R)$. Staèilo uvá¾it nasycený tok f. Pak jsme sestrojili øez R a~výraz pøe¹el v rovnost $c(R)=w(f)$ (a tedy je tok f maximální).
+Víme, ¾e $w(f)\le c(R)$. Staèilo uvá¾it nasycený tok $f$. Pak jsme sestrojili øez $R$ a~výraz pøe¹el v rovnost $c(R)=w(f)$ (a tedy je tok $f$ maximální).
+
+\qed
+}
+Poslední vìta spolu s dùsledkem lemmatu dokazuje i hlavní vìtu o tocích. Pro ka¾dou sí» máme maximální tok a k nìmu øez stejné kapacity.
\qed
-\figure{nenasyc.eps}{Rozdìlení V(\og) na mno¾inu A a V$\setminus$A v dùkazu hlavní vìty o tocích.}{2.5in}
+\figure{nenasyc.eps}{Rozdìlení $V(G)$ na mno¾inu $A$ a $V\setminus A$ v dùkazu hlavní vìty o tocích.}{2.5in}
-\s{Algoritmus:} (Ford-Fulkerson algoritmus hledání maximálního toku)
+\s{Algoritmus:} (Fordùv-Fulkersonùv algoritmus hledání maximálního toku)
\algo
\:$f(e) := 0\ \forall e \in E$
-\:while $\exists$ zlep¹ující cesta P, vylep¹i P jako v dùkazu hlavní vìty
+\:while $\exists$ zlep¹ující cesta $P$, vylep¹i $P$ jako v dùkazu hlavní vìty
\:f je maximální
\endalgo
\h{Cvièení:}
\itemize\ibull
-\:Je pro pøirozené kapacity F.F. algoritmus koneèný? Ano - v ka¾dém zlep¹ujícím kroku algoritmu se celkový tok zvìt¹í aspoò o 1. Proto¾e máme horní odhad na maximální tok (napø. souèet kapacit v¹ech hran), máme i~horní odhad na dobu bìhu algoritmu.
-\:Je F.F. algoritmus koneèný pro racionální kapacity hran? Ano - v¹echny kapacity vynásobíme spoleèným jmenovatelem a pøevedeme na pøedchozí pøípad (pro obecné kapacity ov¹em takto definovaný F.F. algoritmus nemusí být koneèný).
+\:Je pro pøirozené kapacity F-F algoritmus koneèný? Ano -- v ka¾dém zlep¹ujícím kroku algoritmu se celkový tok zvìt¹í aspoò o 1. Proto¾e máme horní odhad na maximální tok (napø. souèet kapacit v¹ech hran), máme i~horní odhad na dobu bìhu algoritmu.
+\:Je F-F algoritmus koneèný pro racionální kapacity hran? Ano -- v¹echny kapacity vynásobíme spoleèným jmenovatelem a pøevedeme na pøedchozí pøípad (pro obecné kapacity ov¹em takto definovaný F-F algoritmus nemusí být koneèný).
\:Kolik krokù bude muset algoritmus na následující síti maximálnì udìlat, aby úspì¹nì dobìhl? (2M krokù)
\endlist
-\figure{2M.eps}{Pøíklad sítì. Kolik krokù musí maximálnì udìlat F.F. algoritmus?}{2in}
+\figure{2M.eps}{Pøíklad sítì. Kolik krokù musí maximálnì udìlat F-F algoritmus?}{2in}
\bye
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projects / ads2.git / commitdiff