From 11ec08cd75409ba24d3cba838d4f2e6e43573491 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Mon, 22 Jan 2007 18:06:31 +0100
Subject: [PATCH] Korektury a novy obrazek. Kapitolu o parovanich jsem presunul
 az na konec, protoze v ni budou potreba rezy, ale jeste na jeji zmeny
 nedoslo.

---
 1-toky/1-toky.tex       |   88 +-
 1-toky/podrozdeleni.eps |  262 ----
 1-toky/vertex-split.eps | 3181 +++++++++++++++++++++++++++++++++++++++
 1-toky/vertex-split.vrr |   37 +
 1-toky/vrchol.eps       |  196 ---
 5 files changed, 3263 insertions(+), 501 deletions(-)
 delete mode 100644 1-toky/podrozdeleni.eps
 create mode 100644 1-toky/vertex-split.eps
 create mode 100644 1-toky/vertex-split.vrr
 delete mode 100644 1-toky/vrchol.eps

diff --git a/1-toky/1-toky.tex b/1-toky/1-toky.tex
index 6788966..eb7dae4 100644
--- a/1-toky/1-toky.tex
+++ b/1-toky/1-toky.tex
@@ -32,7 +32,7 @@ se nikde mimo tato dv
   [co do~vrcholu pøiteèe, co odteèe a jaký je v~nìm pøebytek].
 \:{\I Tok} je ohodnocení $f:\, E\rightarrow {\bb R}$, pro které platí:
   \itemize\ibull
-  \:$\forall e: 0 \le f(e) \le w(e)$, \quad {\I (dodr¾uje kapacity)}
+  \:$\forall e: 0 \le f(e) \le c(e)$, \quad {\I (dodr¾uje kapacity)}
   \:$\forall v\ne s,t: f^\Delta(v)=0$. \quad {\I (Kirchhoffùv zákon)}
   \endlist
 \:{\I Velikost toku:} $\vert f \vert = -f^\Delta(s)$.
@@ -44,7 +44,7 @@ se nikde mimo tato dv
   $f^+(C)=\sum_{e\in C^+} f(e)$,
   $f^-(C)=\sum_{e \in C^-} f(e)$,
   $f^\Delta(C)=f^+(C)-f^-(C)$.
-\:{\I Cirkulace} je nulový tok, èili $\forall v: f^\Delta(v)=0$.
+\:{\I Cirkulace} je tok nulové velikosti, èili $f$ takové, ¾e $f^\Delta(v)=0$ pro v¹echna~$v$.
 \endlist
 
 %%%\figure{graf.eps}{Pøíklad orientovaného grafu ze zvoleným zdrojem a stokem.}{0.4\hsize}
@@ -56,7 +56,7 @@ velikosti) a {\I minim
 
 \proof Existence minimálního øezu je triviální, proto¾e øezù v~ka¾dé síti je koneènì mnoho;
 pro toky v~sítích s~reálnými kapacitami to ov¹em není samozøejmost a je k~tomu potøeba trocha matematické
-analýzy (v~prostoru v¹ech ohodnocení hran tvoøí toky kompaktní mno¾inu, velikost toku je lineární funkce
+analýzy (v~prostoru v¹ech ohodnocení hran tvoøí toky kompaktní mno¾inu, velikost toku je lineární funkce,
 a~tedy i spojitá, proèe¾ nabývá maxima). Pro racionální kapacity dostaneme tuto vìtièku jako dùsledek
 analýzy Ford-Fulkersonova algoritmu.
 \qed
@@ -71,12 +71,12 @@ Stejn
 \s{Lemma:} Pro ka¾dý øez $C$ platí, ¾e $\vert f\vert = -f^\Delta(C) \le \vert C \vert$.
 
 \proof První èást indukcí: ka¾dý øez mù¾eme získat postupným pøidáváním vrcholù do~triviálního øezu $\{s\}$
-[tedy pøesouváním vrcholù zprava doleva], a~to, jak uká¾e jednoduchý rozbor pøípadù, nezmìní $f^\Delta$.
+[èili pøesouváním vrcholù zprava doleva], a~to, jak uká¾e jednoduchý rozbor pøípadù, nezmìní $f^\Delta$.
 Druhá èást: $-f^\Delta(C) = f^-(C) - f^+(C) \le f^-(C) \le \vert C \vert.$ \qed
 
-Velikost ka¾dého toku mù¾eme tedy omezit kapacitou libovolného øezu. Kdybychom na¹li tok a øez stejné
-velikosti, mù¾eme si tedy být jisti, ¾e tok je maximální a øez minimální. To není náhoda, platí toti¾
-následující:
+Víme tedy, ¾e velikost ka¾dého toku lze omezit kapacitou libovolného øezu. Kdybychom na¹li tok a øez stejné
+velikosti, mù¾eme si proto být jisti, ¾e tok je maximální a øez minimální. To není náhoda, platí toti¾
+následující vìta:
 
 \s{Vìta (Ford, Fulkerson):} V~ka¾dé síti je velikost maximálního toku rovna velikosti minimálního øezu.
 
@@ -97,18 +97,18 @@ v~na
 
 \itemize\ibull
 \:{\I Rezerva} hrany $e=(v,w)$ pøi toku~$f$ se definuje jako $r(e) = (c(e) - f(e)) + f(e^\prime)$, kde $e^\prime=(w,v)$.
-  Pro úèely tohoto algoritmu budeme pøedpokládat, ¾e ke~ka¾dé hranì hrana opaèná existuje; pokud ne, dodefinujeme
-  si ji s~nulovou kapacitou.
-\:{\I Zlep¹ující cesta} je orientovaná cesta taková, ¾e v¹echny její hrany mají nenulovou rezervu.
+  Pro úèely tohoto algoritmu budeme pøedpokládat, ¾e ke~ka¾dé hranì existuje hrana opaèná; pokud ne, dodefinujeme
+  si ji a dáme jí nulovou kapacitu.
+\:{\I Zlep¹ující cesta} je orientovaná cesta, její¾ v¹echny hrany mají nenulovou rezervu.
 \endlist
 
 \ss{Algoritmus:}
 
 \algo
-\:$f \leftarrow \<nulový tok>$
-\:while $\exists$ zlep¹ující cesta $P$ z~$s$ do~$t$
-\::$m \leftarrow \min_{e\in P} r(e)$
-\::zvìt¹i tok $f$ podél~$P$ o~$m$ (ka¾dé hranì $e\in P$ zvìt¹i $f(e)$, pøípadnì zmen¹i $f(e^\prime)$, podle toho, co jde).
+\:$f \leftarrow \<nulový tok>$.
+\:Dokud existuje zlep¹ující cesta $P$ z~$s$ do~$t$:
+\::$m \leftarrow \min_{e\in P} r(e)$.
+\::Zvìt¹íme tok $f$ podél~$P$ o~$m$ (ka¾dé hranì $e\in P$ zvìt¹íme $f(e)$, pøípadnì zmen¹íme $f(e^\prime)$, podle toho, co jde).
 \endalgo
 
 \s{Analýza:} Nejdøíve si rozmysleme, ¾e pro celoèíselné kapacity algoritmus v¾dy dobìhne: v~ka¾dém kroku
@@ -120,11 +120,11 @@ dob
 Uva¾me nyní situaci po~zastavení algoritmu. Funkce~$f$ je urèitì tok, proto¾e jím byla po~celou dobu
 bìhu algoritmu. Prozkoumejme teï mno¾inu $C$ vrcholù, do~nich¾ po~zastavení algoritmu vede zlep¹ující cesta ze~zdroje.
 Jistì $s\in C$, $t\not\in C$, tak¾e tato mno¾ina je øez. Navíc pro ka¾dou hranu $e\in C^-$ musí
-být $f(e)=c(e)$ a pro ka¾dou $e\in C^+$ je $f(e)=0$, proto¾e jinak by rezerva hrany~$e$ nebyla
+být $f(e)=c(e)$ a pro ka¾dou $e'\in C^+$ je $f(e')=0$, proto¾e jinak by rezerva hrany~$e$ nebyla
 nulová. Tak¾e $f^-(C) = \vert C \vert$ a $f^+(C) = 0$, èili $\vert f\vert = \vert C \vert$.
 
-Na¹li jsme tedy k~toku, který algoritmus vydal, øez stejné velikosti, tak¾e, jak u¾ víme,
-tok je maximální a øez minimální. Tím jsme také dokázali Ford-Fulkersonovu vìtu\foot{Dokonce
+Na¹li jsme tedy k~toku, který algoritmus vydal, øez stejné velikosti, a~proto, jak u¾ víme,
+je tok maximální a øez minimální. Tím jsme také dokázali Ford-Fulkersonovu vìtu\foot{Dokonce
 jsme ji dokázali i pro reálné kapacity, proto¾e mù¾eme algoritmus spustit rovnou na maximální tok místo
 nulového a on se ihned zastaví a vydá certifikující øez.} a existenci maximálního toku. Navíc algoritmus nikdy
 nevytváøí z~celých èísel necelá, èim¾ získáme:
@@ -137,33 +137,9 @@ do~
 v¹echny kapacity jednotkové, snadno nahlédneme, ¾e staèí $\O(nm)$. Edmondsùv a Karpùv
 odhad nebudeme dokazovat, místo toho si v~pøí¹tí kapitole pøedvedeme efektivnìj¹í algoritmus.
 
-\h{Maximální párování v bipartitním grafu}
-
-Jedním z~problémù, které lze snadno pøevést na~hledání maximálního toku, je nalezení
-{\I maximálního párování} v~bipartitním grafu\foot{Párování je mno¾ina hran taková, ¾e ¾ádné
-dvì nemají spoleèný vrchol. Maximálním míníme vzhledem k~poètu hran, nikoliv vzhledem k~inkluzi.}.
-
-Bipartitní graf $(A\cup B, E)$ pøevedeme na sí» obsahující v¹echny pùvodní vrcholy
-a navíc dva nové vrcholy $s$ a~$t$, dále v¹echny pùvodní hrany orientované z~$A$ do~$B$,
-nové hrany z~$s$ do~v¹ech vrcholù partity~$A$ a ze~v¹ech vrcholù partity~$B$ do~$t$.
-Kapacity v¹ech hran nastavíme na jednièky:
-
-\fig{bipartitni.eps}{0.4\hsize}
-
-Nyní si v¹imneme, ¾e ke~ka¾dému párování existuje celoèíselný tok stejné velikosti a naopak.
-Tak¾e najdeme maximální celoèíselný tok (tøeba F-F algoritmem) a do~párování umístíme
-právì hrany, po~kterých nìco teèe.
-
-Podobnì mù¾eme najít souvislost mezi øezy v~této síti a {\I vrcholovými pokrytími}
-zadaného grafu -- to jsou mno¾iny vrcholù takové, ¾e se dotýkají ka¾dé hrany.
-Tak z~F-F vìty získáme:
-
-\s{Vìta (König):} V~ka¾dém bipartitním grafu je velikost maximálního párování
-rovna velikosti minimálního vrcholového pokrytí.
-
 \h{Øezy, separátory a $k$-souvislost}
 
-Teorie tokù nám také poslou¾í ke~zkoumání násobné souvislosti grafù.
+Teorie tokù nám rovnì¾ poslou¾í ke~zkoumání násobné souvislosti grafù.
 
 \s{Definice:} Pro ka¾dý neorientovaný graf $G$ a libovolné jeho vrcholy $s,t$ zavedeme:
 \itemize\ibull
@@ -179,7 +155,7 @@ Teorie tok
   mají alespoò~$k$ vrcholù.
 \endlist
 
-V¹imnìte si, ¾e nesouvislý graf má øez i separátor velikosti~0, tak¾e vrcholová i hranová 1-souvislost
+V¹imnìte si, ¾e nesouvislý graf má øez i separátor nulové velikosti, tak¾e vrcholová i hranová 1-souvislost
 splývají s~obyèejnou souvislostí pro v¹echny grafy o~alespoò dvou vrcholech. Hranovì 2-souvislé
 jsou právì (netriviální) grafy bez {\I mostù,} vrcholovì 2-souvislé jsou ty bez {\I artikulací.}
 
@@ -224,6 +200,8 @@ a p
 Toky nyní odpovídají vrcholovì disjunktním cestám, øezy v~síti separátorùm.
 \qed
 
+\figure{vertex-split.eps}{Rozdìlení vrcholu}{\epsfxsize}
+
 %\figure{vrchol.eps}{Vrchol který chceme podrozdìlit.}{0.1\hsize}
 %\figure{podrozdeleni.eps}{Výsledek podrozdìlení vrcholu.}{0.15\hsize}
 
@@ -239,4 +217,28 @@ dv
 Neorientovaný graf~$G$ je vrcholovì $k$-souvislý právì tehdy, kdy¾ mezi ka¾dými dvìma
 vrcholy existuje alespoò~$k$ vrcholovì disjunktních cest.
 
+\h{Maximální párování v bipartitním grafu}
+
+Jedním z~problémù, které lze snadno pøevést na~hledání maximálního toku, je nalezení
+{\I maximálního párování} v~bipartitním grafu\foot{Párování je mno¾ina hran taková, ¾e ¾ádné
+dvì nemají spoleèný vrchol. Maximálním míníme vzhledem k~poètu hran, nikoliv vzhledem k~inkluzi.}.
+
+Bipartitní graf $(A\cup B, E)$ pøevedeme na sí» obsahující v¹echny pùvodní vrcholy
+a navíc dva nové vrcholy $s$ a~$t$, dále v¹echny pùvodní hrany orientované z~$A$ do~$B$,
+nové hrany z~$s$ do~v¹ech vrcholù partity~$A$ a ze~v¹ech vrcholù partity~$B$ do~$t$.
+Kapacity v¹ech hran nastavíme na jednièky:
+
+\fig{bipartitni.eps}{0.4\hsize}
+
+Nyní si v¹imneme, ¾e ke~ka¾dému párování existuje celoèíselný tok stejné velikosti a naopak.
+Tak¾e najdeme maximální celoèíselný tok (tøeba F-F algoritmem) a do~párování umístíme
+právì hrany, po~kterých nìco teèe.
+
+Podobnì mù¾eme najít souvislost mezi øezy v~této síti a {\I vrcholovými pokrytími}
+zadaného grafu -- to jsou mno¾iny vrcholù takové, ¾e se dotýkají ka¾dé hrany.
+Tak z~F-F vìty získáme:
+
+\s{Vìta (König):} V~ka¾dém bipartitním grafu je velikost maximálního párování
+rovna velikosti minimálního vrcholového pokrytí.
+
 \bye
diff --git a/1-toky/podrozdeleni.eps b/1-toky/podrozdeleni.eps
deleted file mode 100644
index ee2fd3f..0000000
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+%%BeginResource: font CMSY7
+%!PS-AdobeFont-1.1: CMSY7 1.0
+%%CreationDate: 1991 Aug 15 07:21:52
+
+% 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 (CMSY7) readonly def
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+/Weight (Medium) readonly def
+/ItalicAngle -14.035 def
+/isFixedPitch false def
+end readonly def
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+dup 123 /paragraph put
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+%%EndResource
+%%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
+/FamilyName (Computer Modern) readonly def
+/Weight (Medium) readonly def
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+/isFixedPitch false def
+end readonly def
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+dup 168 /Upsilon put
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+dup 124 /emdash put
+dup 125 /hungarumlaut put
+dup 126 /tilde put
+dup 127 /dieresis put
+dup 128 /suppress put
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