From 09d7c85a44c07040d164d8293593fdb84fc9bba7 Mon Sep 17 00:00:00 2001
From: Martin Mares <mj@ucw.cz>
Date: Sat, 12 May 2007 12:16:11 +0200
Subject: [PATCH] Pridana prvni verze kapitoly o grafovych algoritmech.

---
 8-grafy/8-grafy.tex      | 271 +++++++++++++++++++++++++++++++++++++++
 8-grafy/Makefile         |   3 +
 8-grafy/img1_stvorec.eps | 121 +++++++++++++++++
 8-grafy/img2_dfs.eps     | 211 ++++++++++++++++++++++++++++++
 8-grafy/img3_dfs.eps     | 264 ++++++++++++++++++++++++++++++++++++++
 8-grafy/img4_susedia.eps | 245 +++++++++++++++++++++++++++++++++++
 8-grafy/img5_dfso.eps    | 259 +++++++++++++++++++++++++++++++++++++
 8-grafy/img6_BFS.eps     | 223 ++++++++++++++++++++++++++++++++
 8-grafy/img7_hrany.eps   | 179 ++++++++++++++++++++++++++
 8-grafy/img8_kvz.eps     | 165 ++++++++++++++++++++++++
 all/Makefile             |   2 +-
 11 files changed, 1942 insertions(+), 1 deletion(-)
 create mode 100644 8-grafy/8-grafy.tex
 create mode 100644 8-grafy/Makefile
 create mode 100644 8-grafy/img1_stvorec.eps
 create mode 100644 8-grafy/img2_dfs.eps
 create mode 100644 8-grafy/img3_dfs.eps
 create mode 100644 8-grafy/img4_susedia.eps
 create mode 100644 8-grafy/img5_dfso.eps
 create mode 100644 8-grafy/img6_BFS.eps
 create mode 100644 8-grafy/img7_hrany.eps
 create mode 100644 8-grafy/img8_kvz.eps

diff --git a/8-grafy/8-grafy.tex b/8-grafy/8-grafy.tex
new file mode 100644
index 0000000..2feed6d
--- /dev/null
+++ b/8-grafy/8-grafy.tex
@@ -0,0 +1,271 @@
+\input mjmac.tex
+\input lecnotes.tex
+\input epsf.tex
+
+\prednaska{8}{Základní grafové algoritmy}{(Zapsali M. Petr, O.Tichý, ¥. Magic)}
+
+\h{Reprezentace grafu}
+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{img1_stvorec.eps}{}{\epsfxsize}
+
+\s{1. seznam hran}
+
+Jednodu¹e vypí¹eme seznam v¹ech dvojic vrcholù grafu spojených hranou.
+
+AB, BC, DC, CA, BC
+\medskip
+
+\s{2. matice sousednosti}
+
+Matice sousednosti je pole $A$ o velikosti $n \times n$, jeho¾ prvky na
+souøadnicích $i, j$ jsou dány následujícím pøedpisem:
+
+$$ A_{i,j} = \left\{ \matrix {0 \Leftrightarrow \{i,j\} \notin E \cr
+            			1 \Leftrightarrow \{i,j\} \in E  \cr
+				}
+\right.$$
+
+Ná¹ graf z obrázku vý¹e by tedy v maticové reprezentaci vypadal takto:
+
+$$\bordermatrix{
+  & A & B & C & D\cr
+A & 0 & 1 & 1 & 0\cr
+B & 1 & 0 & 1 & 1\cr
+C & 1 & 1 & 0 & 1\cr
+D & 0 & 1 & 1 & 0\cr
+}$$
+
+S touto matici se pracuje velmi snadno (napø. v¹echny sousedy i-tého vrcholu
+zjistíme jednodu¹e tak, ¾e projdeme i-tý øádek matice, co¾ pøedstavuje èasovou
+slo¾itost $O(n)$), ale má i jednu zøejmou nevýhodu: její velikost je v¾dy
+kvadratická bez ohledu na to, jak "øídký" je graf. U grafu s mnoha vrcholy, ale
+s malým poètem hran, tedy budeme zbyteènì plýtvat místem v pamìti.
+
+Nabízí se proto tøetí, úspornìj¹í, zpùsob, a to:
+\medskip
+
+\s{3. seznam sousedù}
+
+%$$\matrix{
+%A&\rightarrow&BC\quad \cr 
+%B&\rightarrow&ACD\cr
+%C&\rightarrow&ABD\cr
+%D&\rightarrow&BC\quad \cr
+%}$$
+
+V pamìti poèítaèe by se dal seznam sousedù uchovávat dvìma poli: polem vrcholù
+$V$ grafu, jeho¾ prvky postupnì pro ka¾dý vrchol udávají index na zaèátek
+odpovídajícího úseku v poli $E$, ve kterém by byli ulo¾eni jeho sousedé.
+\figure{img4_susedia.eps}{Znázornìní polí seznamu sousedù.}{\epsfxsize}
+
+Na tuto reprezentaci u¾ staèí prostor $O(n + m)$, co¾ u¾ je, na rozdíl od
+pøedchozího kvadratického prostoru, docela pøíjemné.
+
+Je¹tì dodejme, ¾e aèkoliv jsme jednotlivé reprezentace ukazovali na pøíkladì
+neorientovaného grafu, orientované grafy se dají reprezentovat prakticky stejnì. U
+orientovaných grafù je toti¾ jediný rozdíl: existuje-li hrana z vrcholu $u$ do
+vrcholu $v$, neznamená to, ¾e existuje i hrana z $v$ do $u$ (tedy $(u, v)$ a $(v, u)$
+tvoøí uspoøádané dvojice).
+
+\h{Prùchod grafem}
+
+\s{Definice:} Vrchol je dosa¾itelný, pokud do nìj vede cesta z poèáteèního vrcholu.
+
+\s{Efektivní prùchod by mìl splòovat tyto podmínky:}
+\itemize\ibull
+\:ka¾dou hranu projdu právì jednou (resp. maximálnì dvakrát - 1x tam, 1x zpìt)
+\:hranou se vracím zpìt, a¾ kdy¾ u¾ z vrcholu, na kterém momentálnì jsem, nevede
+  dosud neprozkoumaná hrana
+\:pokud jsem do¹el po hranì do ji¾ døíve nav¹tíveného vrcholu, hned se vrátím
+\endlist
+
+\s{Koneènost}
+
+\proof
+Plyne z prvního bodu.
+\qed
+
+\s{Korektnost} (algoritmus projde celý graf)
+
+\proof
+Sporem: Nech» existuje vrchol, který je dosa¾itelný, ale algoritmus ho nepro¹el.
+Pak jsme ale pøi prùchodu narazili na $\{u, v\}\in E$ takovou, ¾e vrchol $u$ je
+dosa¾itelný a $w$ dosa¾itelný není. Tzn., ¾e $\{u, w\}$ tvoøí neprozkoumanou
+hranu $\Rightarrow$ spor s druhým bodem.
+\qed
+
+\s{Èasová slo¾itost}
+
+Z toho, ¾e ¾ádná hrana ani vrchol se neprojde vícekrát (první podmínka
+efektivního prùchodu), plyne èasová slo¾itost $O(n + m)$.
+
+\>Nyní si nìco povíme o dvou zpùsobech, kterak mù¾eme procházet grafy.
+
+\s{Prùchod do hloubky} (Depth First Search, DFS)
+
+Prùchod do hloubky je prvním základním postupem pøi procházení grafem. Pokud bychom
+chtìli DFS trochu pøiblí¾it, mohli bychom ho pøirovnat k Théseovì postupu
+pøi prùchodu Minotaurovho bludi¹te za pomoci Ariadniny nitì.
+
+\s{Prùchod do ¹íøky} (Breadth First Search, BFS)
+
+Na rozdíl od prohledávání do hloubky vyu¾ívá prohledávání do ¹íøky namísto zásobníku
+frontu. Motivaèní úlohou by mohlo být napø. hledání nejkrat¹í cesty mezi dvìma vrcholy
+v grafu, nebo známý pøevozníkùv problém.
+\figure{img6_BFS.eps}{}{\epsfxsize}
+
+\s{Pø.} Koza, vlk a zelí
+\figure{img8_kvz.eps}{}{\epsfxsize}
+
+Øe¹ení ulohy by se dalo popsat jako hledání cesty grafem stavù od poèáteèního stavu
+(koza, vlk i zelí jsou na jednom bøehu) do koncového (v¹ichni byli bez újmy pøevezeni
+na druhý bøeh øeky).
+
+\h{DFS na neorientovaném grafu}
+
+Pøi prùchodu grafu do hloubky pou¾íváme zásobník. Na zaèátku se v nìm nachází pouze
+vstupní vrchol v. Dokud není zásobník prázdný, tak z nìj odebíráme vrchol; ten oznaèíme
+jako zpracovaný (abychom ho v budoucnu, kdybychom na nìj je¹tì nìkdy narazili,
+nezpracovávali znovu) a do zásobníku pøidáme v¹echny jeho sousedy takové, kteøí
+je¹tì nebyli oznaèeni jako zpracovaní.
+ 
+\>Vrcholy znaèíme tìmito barvami
+\itemize\ibull
+\:bílá - dosud nenav¹tívìné vrcholy (na zaèátku jsou tedy bílé v¹echny)
+\:¹edá - oznaèuje právì zpracovávaný vrchol
+\:èerná - u¾ zpracované vrcholy
+\endlist
+ 
+\>U ka¾dého vrcholu si dále také zapamatujeme èas pøíchodu a èas odchodu
+($in[v]$, $out [v]$).
+
+\s{Algoritmus}
+
+\algo
+\:procedure Projdi(v)
+\::color[v] := ¹edá
+\::time := time + 1
+\::in[v] := time
+\::$\forall$ w $\in$ Sousedi(v): 
+\:::if color[w] = bílá $\Rightarrow$
+\::::Projdi(w)
+\::color[v] := èerná
+\::time := time + 1
+\::out[v] := time
+\::
+\:procedure DFS()
+\::$\forall$ v: color[v] := bílá
+\::time := 0
+\::$\forall$ v $\in$ V :
+\:::if color[v] = bílá $\Rightarrow$
+\::::Projdi(v)
+\endalgo
+
+Implementaci zásobníku jsme v tomto meta-algoritmu nahradili pou¾itím rekurze
+(která ale funguje na podobném principu, kdy se na zásobník vkládají jednotlivá
+volání rekurzivní funkce).
+
+Na následujících obrázcích je znázornìno, jak probíhá prùchod grafu pomocí DFS:
+
+\figure{img2_dfs.eps}{}{\epsfxsize}
+\figure{img3_dfs.eps}{Znázornìní prùbìhu algoritmu.}{\epsfxsize}
+
+\s{Hrany v tomto znázornìní dìlíme na:}
+\itemize\ibull
+\:stromové - v momentì prùchodu vedou do dosud nenav¹tíveného, bílého, vrcholu
+\:zpìtné - ostatní hrany
+\endlist
+
+\s{Pozorování:} plné hrany tvoøí strom (DFS strom, strom prùchodu do hloubky).
+
+\s{Aplikace:}
+\item{(1)} hledání kostry
+\item{(2)} hledání komponent souvislosti
+
+Èasová slo¾itost tìchto aplikací DFS je stejná jako slo¾itost pùvodního 
+meta-algoritmu, tedy $O(n + m)$.
+
+\item{(3)} hledání artikulací
+
+Tuto aplikaci si teï rozebereme tro¹ku podrobnìji. Nejprve v¹ak nìkolik definic.
+
+\s{Definice:} Artikulace je vrchol v grafu, jeho¾ odebráním se graf rozpadne na alespoò
+dvì komponenty souvislosti.
+
+\s{Definice:} Most je hrana v grafu, jejím¾ odebráním se graf rozpadne na dvì
+komponenty souvislosti.
+
+\s{Pozorování}: Most je taková hrana, její¾ oba koncové vrcholy jsou artikulacemi.
+
+\s{Definice:} Graf je 2-souvislý, jestli¾e po odstranìní libovolného vrcholu
+zùstane souvislý.
+
+\s{Definice:} 2-souvislá komponenta je maximální 2-souvislý podgraf.
+
+Jak bychom tedy pøi hledání artikulací v grafu mohli postupovat? Jako první
+se nabízí tento algoritmus:
+
+\s{Hloupý algoritmus}
+
+Berme si postupnì v¹echny vrcholy grafu; pro ka¾dý zjistíme, jestli se jeho
+odebráním zvý¹il poèet komponent. K tomu bychom mohli pou¾ít prùchod do hloubky,
+který jsme si u¾ ukázali. Procedura Projdi(v) projde a oznaèí v¹echny vrcholy
+dosa¾itelné z v, tedy v¹echny vrcholy komponenty obsahující vrchol v. Mù¾eme
+tedy udìlat takovou zmìnu, ¾e si budeme nav¹tívené vrcholy oznaèovat èíslem
+komponenty.
+
+Takový algoritmus by mìl slo¾itost $n \times O(n + m))$ (pro ka¾dý vrchol grafu
+bychom n-krát spou¹tìli DFS), co¾ se nám vùbec nelíbí. Pokusme se tedy najít
+nìco lep¹ího. Nejprve uèiòmì toto pozorování:
+
+\s{Pozorování:} v je artikulace $\Leftrightarrow$ v je koøen DFS stromu a má 
+alespoò dva potomky nebo v není koøen a má syna w takového, ¾e ze ¾ádného
+jeho potomka nevede hrana do pøedchùdce v.
+
+\s{Definice:} Pøípustná cesta je taková cesta, pokud vede pøes stromové hrany
+DFS stromu (mù¾e konèit i skokem zpìt).
+
+\figure{img7_hrany.eps}{Pøíklady pøípustných cest.}{\epsfxsize}
+
+\>Dále si definujme funkci low pro ka¾dý vrchol v, takto:
+
+\s{low(v)} = min(in[w], z v do w vede pøípustná cesta)
+
+\s{Pozorování:} v je artikulace $\Leftrightarrow$ v je koøen DFS stromu a má
+alespoò dva potomky nebo má syna w takového, ¾e platí $low(w) \geq in[w]$.
+
+\s{Jak ale hodnotu funkce low(w) spoèítat?}
+
+low(v) = min(x, y, z)
+
+x = in[v]
+
+y = min(in[w],  $\{w, v\}$ je zpìtná hrana)
+
+z = min(low(w), $\{v, w\}$ je stromová hrana)
+
+\h{DFS na orientovaném grafu}
+        
+\figure{img5_dfso.eps}{Graf a znázornení prùbehu:}{\epsfxsize}
+
+\s{Hrany v dìlíme na: }
+\itemize\ibull
+\:stromové $\{A,B\}$, $\{B,C\}$, $\{B,D\}$
+\:zpìtné: vedou do pøedchùdce v DFS stromì (do ¹edého vrcholu)
+\:pøíèné: vedou do vrcholu v jiném podstromì (do èerného vrcholu)
+\:dopøedné: vedou do svého potomka (do èerného vrcholu v tém¾e podstromì)
+\endlist
+\bigskip
+
+\>Na závìr nìkolik \s{pozorování:}
+
+Intervaly (in[v], out[v]) $\forall v \in V $ tvoøí dobré uzávorkování.
+
+v je následník u $\Leftrightarrow (in[v], out[v]) \leq (int[u], out[u]) $
+
+$\{u,v\}$ je zpìtná hrana $\Leftrightarrow (in[u], out[u]) \leq (in[v], out[v])$
+
+$\{u,v\}$ je pøíèná hrana $\Leftrightarrow int[v] < out[v] < in[u] < out[u]$
+\bye
diff --git a/8-grafy/Makefile b/8-grafy/Makefile
new file mode 100644
index 0000000..fce3887
--- /dev/null
+++ b/8-grafy/Makefile
@@ -0,0 +1,3 @@
+P=8-grafy
+
+include ../Makerules
diff --git a/8-grafy/img1_stvorec.eps b/8-grafy/img1_stvorec.eps
new file mode 100644
index 0000000..13e4481
--- /dev/null
+++ b/8-grafy/img1_stvorec.eps
@@ -0,0 +1,121 @@
+%!PS-Adobe-2.0 EPSF-2.0
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diff --git a/8-grafy/img2_dfs.eps b/8-grafy/img2_dfs.eps
new file mode 100644
index 0000000..deb80c2
--- /dev/null
+++ b/8-grafy/img2_dfs.eps
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+%!PS-Adobe-2.0 EPSF-2.0
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diff --git a/8-grafy/img8_kvz.eps b/8-grafy/img8_kvz.eps
new file mode 100644
index 0000000..c600f5a
--- /dev/null
+++ b/8-grafy/img8_kvz.eps
@@ -0,0 +1,165 @@
+%!PS-Adobe-2.0 EPSF-2.0
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diff --git a/all/Makefile b/all/Makefile
index 9b17b08..35c14c5 100644
--- a/all/Makefile
+++ b/all/Makefile
@@ -1,5 +1,5 @@
 P=ads
-X:=$(shell for a in 1 2 3 4 6 ; do echo ../$$a-*/$$a-*.tex ; done)
+X:=$(shell for a in 1 2 3 4 6 8 ; do echo ../$$a-*/$$a-*.tex ; done)
 
 %universe: all ChangeLog
 
-- 
2.39.2