Nova verze kapitoly o prevodech.

diff --git a/10-prevody/10-prevody.tex b/10-prevody/10-prevody.tex
index 53abdb2afc20c127c13b72bb8e8d1e8fe42be662..d5e4de090efd2f47a836ed5474cca9472a5c098f 100644 (file)
--- a/10-prevody/10-prevody.tex
+++ b/10-prevody/10-prevody.tex
@@ -6,63 +6,59 @@
 
 \s{Definice:} {\I Rozhodovací problém} je takový problém, jeho¾ výstupem je v¾dy {\sc ano}, nebo {\sc ne}.
 
 
 \s{Definice:} {\I Rozhodovací problém} je takový problém, jeho¾ výstupem je v¾dy {\sc ano}, nebo {\sc ne}.
 

-\s{Pøíklad:} Je dán bipartitní graf $G$, $k \in {\bb N}$. Existuje v $G$ párování, které obsahuje alespoò $k$ hran?
+\s{Pøíklad:} Je dán bipartitní graf $G$ a $k \in {\bb N}$. Existuje v $G$ párování, které obsahuje alespoò $k$ hran?

 
 \s{Pøíklad:} Daný problém pøevedeme na jiný: Párování $\rightarrow$ hledání maximálního toku (¹ipka znamená \uv{lze pøevést}).
 
 \s{Pøíklad:} Daný problém pøevedeme na jiný: Párování $\rightarrow$ hledání maximálního toku (¹ipka znamená \uv{lze pøevést}).

-Tzn. existuje v síti $G$ tok velikosti alespoò $k$?
+Tzn. existuje v nìjaké síti $G'$ tok velikosti alespoò $k$?

 
 

-\s{Obecnì se dá øici:} Pokud daný pro problém umíme rozhodnout, zda platí $\Rightarrow$ umíme najít øe¹ení problému.
+\s{Obecnì se dá øici:} Pokud daný pro rozhodovací problém umíme rozhodnout, zda platí, pak  umíme také najít øe¹ení tohoto problému. Proto¾e jak jinak bychom o daném problému mohli tvrdit, ¾e zaruèenì platí, kdy¾ bychom ho neumìli vyøe¹it. 

 
 \s{Pøíklad:} Mìjme èernou skøíòku (fungující v polynomiálním èase), která odpoví, zda daný graf má nebo nemá perfektní párování. Odebereme hranu a zeptáme se, jestli i tento nový graf má pefektní párovaní. Kdy¾ má, tak tato hrana nebyla potøebná pro párování, vyhodíme ji, proto¾e ji nepotøebujeme. 
 
 \s{Pøíklad:} Mìjme èernou skøíòku (fungující v polynomiálním èase), která odpoví, zda daný graf má nebo nemá perfektní párování. Odebereme hranu a zeptáme se, jestli i tento nový graf má pefektní párovaní. Kdy¾ má, tak tato hrana nebyla potøebná pro párování, vyhodíme ji, proto¾e ji nepotøebujeme. 

-Kdy¾ nemá (hrana patøí do párování), tak si danou hranu poznamenáme a odebereme ji i její vrcholy a také hrany, které vedly do tìchto vrcholù. Toto je korektní krok, proto¾e v pùvodním grafu tyto vrcholy byly navzájem spárované, a tedy nemohou být spárované s~¾ádnými jinými vrcholy.    
-Takto iterujeme, dokud to jde. Výsledkem je mno¾ina hran, které patøí do maximálního párování. Tím jsme dané párování nalezli. 
+Kdy¾ nemá (hrana patøí do ka¾dého párování), tak si danou hranu poznamenáme a odebereme nejen ji a její vrcholy ale také hrany, které do tìchto vrcholù vedly. Toto je korektní krok, proto¾e v pùvodním grafu tyto vrcholy byly navzájem spárované, a tedy nemohou být spárované s~¾ádnými jinými vrcholy.    
+Takto iterujeme, dokud to jde. Výsledkem je mno¾ina hran, které patøí do perfektního párování. Tím jsme dané párování nalezli. 

 Hran je polynomiálnì mnoho a skøíòka funguje v polynomiálním èase, tak¾e algoritmus je polynomiální.
 
 Hran je polynomiálnì mnoho a skøíòka funguje v polynomiálním èase, tak¾e algoritmus je polynomiální.
 

-\s{Definice:} Jsou-li $A$, $B$ rozhodovací problémy, pak øíkáme, ¾e $A$ lze redukovat na $B$ ($A \rightarrow B$) $\Leftrightarrow$ existuje funkce $f$ spoèitatelná v polynomiálním èase taková, ¾e pro $\forall x: A(x) = B(f(x))$.
+\s{Definice:} Jsou-li $A$, $B$ rozhodovací problémy, pak øíkáme, ¾e $A$ lze {\I redukovat} (pøevést) na $B$ (pí¹eme $A \rightarrow B$) $\Leftrightarrow$ existuje funkce $f$ spoèitatelná v polynomiálním èase taková, ¾e pro $\forall x: A(x) = B(f(x))$.

 
 

-\s{Pøíklad:} Bipartitní graf $\rightarrow$ Tok v síti.
-Funkce $f$ je funkce, která vezme bipartitní graf a vyrobí z~nìj regulerní sí» (pøidá zdroj, stok, hrany a ohodnocení).
+\s{Pøíklad:} Hledání maximálního párování v bipartitním grafu $\rightarrow$ Hledání maximálního toku v síti.
+Funkce $f$ je funkce, která vezme bipartitní graf a vyrobí z~nìj sí» (jak správnì vyrobit sí» pro tento pøíklad viz 3.pøedná¹ka - Dinicùv algoritmus).

 
 \s{Nìco málo o slo¾itosti:}
 
 \s{Nìco málo o slo¾itosti:}

-Kdy¾ $A$ lze redukovat na $B$ a $B$ umíme vyøe¹it v èase $\O(\vert $vstup$ \vert^l) = \O(\vert f(x)\vert^l)$
- pro vstup $x: \vert x \vert = n$
-$ \vert f(x)\vert = \O(n^k)$ pro nìjaké $k$,
-$A$ poèítá v~èase $\O(n^{kl})$,
-$f$ poèítá v polynomiálním èase $\rightarrow$ mù¾e vydat maximálnì polynomiální výstup.
+Kdy¾ $A$ lze redukovat na $B$ funkcí $f$ a vstup $A$ je $x$, t¾. $\vert x \vert = n$ a funkce $f$ je spoèitatelná v polynomiálním èase, t¾. $\vert f(x) \vert = \O(n^k)$ pro nìjaké $k$, pak B umíme vyøe¹it v èase $\O(\vert f(x)^l \vert) = \O(n^{kl})$, $f$ poèítá v polynomiálním èase $\rightarrow$ mù¾e vydat maximálnì polynomiální výstup.

 
 

-\s{Pozorování:} Funkce $f$ je:
+$A \rightarrow B$ znamená, ¾e $B$ je alespoò tak tì¾ké jako $A$.
+
+\s{Pozorování:} Pøevoditelnost je:

 \itemize\ibull
 \:reflexivní (úlohu mù¾eme identicky pøevést na tu stejnou), $A \rightarrow A$,
 \:tranzitivní, $A \rightarrow B$ funkcí $f$, $B \rightarrow C$ funkcí $g$, $A \rightarrow C$ slo¾enou funkcí $(g \circ f)$.
 \endlist
 
 \h{1. problém: SAT}
 \itemize\ibull
 \:reflexivní (úlohu mù¾eme identicky pøevést na tu stejnou), $A \rightarrow A$,
 \:tranzitivní, $A \rightarrow B$ funkcí $f$, $B \rightarrow C$ funkcí $g$, $A \rightarrow C$ slo¾enou funkcí $(g \circ f)$.
 \endlist
 
 \h{1. problém: SAT}

-\>Splnitelnost logických formulí, tj. dosazení $0$ èi $1$ do logické formule tak, aby formule platila.
+\>Splnitelnost logických formulí, tj. dosazení $true$ èi $false$ za promìnné v logické formuli tak, aby formule dala výsledek $true$.

 
 

-\>Zamìøíme se na speciální formu zadání formulí,  {\I konjunktivní normální formu} (CNF):
+\>Zamìøíme se na speciální formu zadání formulí,  {\I konjunktivní normální formu} (CNF).

 $$(\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$ 
 
 $$(\ldots\lor\ldots\lor\ldots\lor\ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$ 
 

-\>{\I Vstup:} Formule v konjunktivní normální formì.
-
-\>{\I Výstup} $\exists$ dosazení $0$ a $1$ za promìnné takové, ¾e hodnota formule $\phi(\ldots) = 1$.
+\>{\I Vstup:} Formule $\phi$ v konjunktivní normální formì.

 
 

-$$ \phi(x, y, \ldots) = (x \lor \lnot y \lor \ldots) \land (\ldots\lor\ldots\lor\ldots\lor\ldots) \land \ldots $$
+\>{\I Výstup} $\exists$ dosazení $true$ a $false$ za promìnné takové, ¾e hodnota formule $\phi(\ldots) = true$.

 
 \>Pro formuli platí následující podmínky:
 
 \itemize\ibull
 
 \>Pro formuli platí následující podmínky:
 
 \itemize\ibull

-\:formule je zadána pomocí klauzulí oddìlených $\land$,
-\:ka¾dá klauzule je slo¾ená z literálù oddìlených $\lor$,
-\:ka¾dý literál je buïto promìnná nebo její negace.
+\:{\I formule} je zadána pomocí {\I klauzulí} oddìlených $\land$,
+\:ka¾dá {\I klauzule} je slo¾ená z {\I literálù} oddìlených $\lor$,
+\:ka¾dý {\I literál} je buïto promìnná nebo její negace.

 \endlist 
 
 \>Uká¾eme, ¾e staèí vyøe¹it jednodu¹¹í problém 3-SAT.
 
 \h{2. problém: 3-SAT}
 \endlist 
 
 \>Uká¾eme, ¾e staèí vyøe¹it jednodu¹¹í problém 3-SAT.
 
 \h{2. problém: 3-SAT}

-\s{Definice:} 3-SAT je takový SAT, kde ka¾dá klauzule obsahuje nejvý¹e tøi literály.
+\s{Definice:} 3-SAT je takový SAT, v nìm¾ ka¾dá klauzule obsahuje nejvý¹e tøi literály.

 
 \s{Pøevod 3-SAT na SAT:}
 
 \s{Pøevod 3-SAT na SAT:}

-Platí identita, 3-SAT splòuje vlastnosti SATu, proto 3-SAT = SAT (3-SAT je alespoò tak tì¾ký jako SAT)
+Vstup není potøeba nijak upravovat, 3-SAT splòuje vlastnosti SATu, proto 3-SAT = SAT (3-SAT je alespoò tak tì¾ký jako SAT)

 
 \s {Pøevod SAT na 3-SAT:}
 Musíme formuli pøevést tak, abychom neporu¹ili splnitelnost.
 
 \s {Pøevod SAT na 3-SAT:}
 Musíme formuli pøevést tak, abychom neporu¹ili splnitelnost.


@@ -77,30 +73,32 @@ kde $x$ je nov
 \:$\alpha \Rightarrow x = 0$ (zajistí splnìní druhé poloviny nové formule),
 \:$\beta \Rightarrow x = 1$ (zajistí splnìní první poloviny nové formule),
 \:$\alpha ,\beta / \lnot\alpha ,\lnot\beta \Rightarrow x = 0/1$ (je nám to jedno, celkové øe¹ení nám to neovlivní).
 \:$\alpha \Rightarrow x = 0$ (zajistí splnìní druhé poloviny nové formule),
 \:$\beta \Rightarrow x = 1$ (zajistí splnìní první poloviny nové formule),
 \:$\alpha ,\beta / \lnot\alpha ,\lnot\beta \Rightarrow x = 0/1$ (je nám to jedno, celkové øe¹ení nám to neovlivní).

-
-Hodnota $x$ nám pùvodní formuli nijak neovlivní, proto¾e se v ní nevyskytuje, proto ji mù¾eme nastavit, jak chceme.

 \endlist
 
 \>Tento trik opakujeme tak dlouho, dokud je to tøeba.
 
 \endlist
 
 \>Tento trik opakujeme tak dlouho, dokud je to tøeba.
 

+Nabízí se otázka, proè mù¾eme promìnnou $x$ nastavit, jak se nám zlíbí. Vysvìtlení je prosté, promìnná $x$ nám pùvodní formuli nijak neovlivní, proto¾e se v ní nevyskytuje, proto ji mù¾eme nastavit tak, jak chceme.
+

 \s{Poznámka:} U~3-SAT lze vynutit právì tøi literály, pro krátké klauzule pou¾ijeme následující trik:
 \s{Poznámka:} U~3-SAT lze vynutit právì tøi literály, pro krátké klauzule pou¾ijeme následující trik:

-$$(a) \rightarrow (a \lor x) \land (a \lor \lnot x).$$
+$$(\alpha) \rightarrow (\alpha \lor x) \land (\alpha \lor \lnot x).$$

 
 \h{3. problém: Hledání nezávislé mno¾iny v grafu}
 
 \>Existuje nezávislá mno¾ina vrcholù z~$G$ velikosti alespoò $k$?
 
 
 \h{3. problém: Hledání nezávislé mno¾iny v grafu}
 
 \>Existuje nezávislá mno¾ina vrcholù z~$G$ velikosti alespoò $k$?
 

-\s{Definice:} {\I Nezávislá mno¾ina} (NzMna) je tvoøena vrcholy grafu, které spolu nemají spoleènou hranu.
+\s{Definice:} {\I Nezávislá mno¾ina} (NzMna) je tvoøena vrcholy grafu, které nemají ¾ádnou spoleènou hranu.

 
 

-\>{\I Vstup:} Neorientovaný graf G, $k \in N$.
+\figure{nezmna.eps}{Pøíklad nezávislé mno¾iny.}{3in}

 
 

-\>{\I Výstup:} $\exists A \subseteq V(G)$, $\vert A \vert \ge k$, $u,v \in A \Rightarrow (uv) \not\in E(G)$?
+\>{\I Vstup:} Neorientovaný graf G, $k \in {\bb N}$.

 
 

-\>Úlohu øe¹íme tak, ¾e problém 3-SAT pøevedeme tuto úlohu.
+\>{\I Výstup:} $\exists A \subseteq V(G)$, $\vert A \vert \ge k$: $\forall u,v \in A \Rightarrow uv \not\in E(G)$?

 
 

-\s{Poznámka:} Ka¾dý graf má minimálnì jednu nezávislou mno¾inu, a tou je prázdná mno¾ina.
+\s{Poznámka:} Ka¾dý graf má minimálnì jednu nezávislou mno¾inu, a tou je prázdná mno¾ina. Proto je potøeba zadat i minimální velikost hledané mno¾iny.
+
+\>Uká¾eme, jak tento probém pøevést na 3-SAT.

  
  

-\s{Øe¹ení úlohy:} Z ka¾dé klauzule vybereme jeden literál tak, abychom v rùzných klauzulích nevybírali konfliktnì, tj. $x$ a $\lnot x$.
+\s{Pøevod:} Z ka¾dé klauzule vybereme jeden literál tak, abychom v rùzných klauzulích nevybírali konfliktnì, tj. $x$ a $\lnot x$.

 
 \s{Pøíklad:} 
 $(x \lor y \lor z) \land (x \lor \lnot y \lor \lnot z) \land (\lnot x \lor \lnot y \lor p) $.
 
 \s{Pøíklad:} 
 $(x \lor y \lor z) \land (x \lor \lnot y \lor \lnot z) \land (\lnot x \lor \lnot y \lor p) $.


@@ -109,24 +107,16 @@ $(x \lor y \lor z) \land (x \lor \lnot y \lor \lnot z) \land (\lnot x \lor \lnot
 
 Princip je takový, ¾e z~ka¾dé klauzule si vybereme promìnnou, která danou klauzuli splní, a to, aby promìnné, které si vybereme, nekolidovaly, vyøe¹íme hranami mezi promìnnými a jejich negacemi. 
 
 
 Princip je takový, ¾e z~ka¾dé klauzule si vybereme promìnnou, která danou klauzuli splní, a to, aby promìnné, které si vybereme, nekolidovaly, vyøe¹íme hranami mezi promìnnými a jejich negacemi. 
 

+\figure{nezmna_graf.eps}{Ukázka pøevodu 3-SAT na nezávislou mno¾inu.}{3in}
+

 Existuje nezávislá mno¾ina velikosti rovné poètu klauzulí?
 Pokud ano, tak dostaneme seznam promìnných, pomocí kterých splníme danou formuli.
 
 Existuje nezávislá mno¾ina velikosti rovné poètu klauzulí?
 Pokud ano, tak dostaneme seznam promìnných, pomocí kterých splníme danou formuli.
 

-\h{4. problém: Klika}
-
-\>{\I Vstup:} Graf $G, k \in N$.
-
-\>{\I Výstup:} $\exists$ úplný podgraf grafu $G$ na $k$ vrcholech?
-
-\s{Øe¹ení:} Prohodíme hrany a nehrany $\rightarrow$ hledání nezávislé mno¾iny.
-
-\s{Dùvod:} Pokud existuje úplný graf na $k$ vrcholech, tak v~\uv{invertovaném} grafu tyto vrcholy nejsou spojeny hranou, tj. tvoøí nezávislou mno¾inu.
-
-\s{Pøíklad:} (Viz obrázky.)
-

 \s{Pøevod NzMna na SAT:}
 Máme promìnné $v_1, \ldots , v_n$ pro vrcholy.
 
 \s{Pøevod NzMna na SAT:}
 Máme promìnné $v_1, \ldots , v_n$ pro vrcholy.
 

+\>Nyní uká¾eme, jak pøevést problém hledání nezávislé mno¾iny, na SAT.
+

 \itemize\ibull
 \:Poøídíme si promìnné $v_1, \ldots, v_n$ odpovídající vrcholùm grafu. Promìnná $v_i$ bude
   indikovat, zda se $i$-tý vrchol vyskytuje v~nezávislé mno¾inì.
 \itemize\ibull
 \:Poøídíme si promìnné $v_1, \ldots, v_n$ odpovídající vrcholùm grafu. Promìnná $v_i$ bude
   indikovat, zda se $i$-tý vrchol vyskytuje v~nezávislé mno¾inì.


@@ -144,14 +134,30 @@ M
   x_{i1} \lor x_{i2} \lor \ldots \lor x_{in}$.
 \endlist
 
   x_{i1} \lor x_{i2} \lor \ldots \lor x_{in}$.
 \endlist
 

+\figure{matice.eps}{Výsledná matice.}{3in}
+
+\h{4. problém: Klika}
+
+\>{\I Vstup:} Graf $G, k \in N$.
+
+\>{\I Výstup:} $\exists$ úplný podgraf grafu $G$ na $k$ vrcholech?
+\figure{klika.eps}{Pøíklad kliky.}{3in}
+
+\s{Pøevod:} Prohodíme v grafu $G$ hrany a nehrany $\Rightarrow$ hledání nezávislé mno¾iny.
+
+\s{Dùvod:} Pokud existuje úplný graf na $k$ vrcholech, tak v~\uv{invertovaném} grafu tyto vrcholy nejsou spojeny hranou, tj. tvoøí nezávislou mno¾inu.
+
+\figure{doplnek_nm.eps}{Prohození hran a nehran.}{3in}

 
 \h{5. problém: 3D párování (3D matching)}
 
 
 \h{5. problém: 3D párování (3D matching)}
 

-\>{\I Vstup:} Tøi mno¾iny, napø. K (kluci), H (holky), Z (zvíøátka) a mno¾ina kompatibilních trojic (tìch, kteøí se spolu snesou).
+\>{\I Vstup:} Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a mno¾ina kompatibilních trojic (tìch, kteøí se spolu snesou).
+
+\>{\I Výstup:} Perfektní podmno¾ina trojic - tj. taková podmno¾ina trojic, která obsahuje v¹echna $K$, $H$ a $Z$.

 
 

-\>{\I Výstup:} Perfektní podmno¾ina trojic.
+\>Uká¾eme, jak tento problém pøevést na 3,3-SAT (ov¹em to a¾ na dal¹í pøedná¹ce).

 
 

-\s{Øe¹ení:} Pøes 3,3-SAT (konkrétnìji, viz dal¹í pøedná¹ku).
+\figure{3d_parovani.eps}{Ukázka 3D párování.}{3in}

 
 
 \h{6. problém: 3,3-SAT}
 
 
 \h{6. problém: 3,3-SAT}


@@ -179,6 +185,9 @@ $$
 (x_k \Rightarrow x_1).
 $$
 
 (x_k \Rightarrow x_1).
 $$
 

-Tímto zaruèíme, ¾e v¹echny promìnné budou mít stejnou hodnotu.
+Tímto zaruèíme, ¾e v¹echny promìnné budou mít stejnou hodnotu. Navíc si lze v¹imnout, ¾e ka¾dý literál se vyskytuje nejvíce $2x$. 
+
+\s{Závìr:} Obrázek ukazuje problémy, jimi¾ jsme se dnes zabývali, a vztahy mezi tìmito problémy.
+\figure{prevody.eps}{Pøevody mezi problémy.}{3in}

 
 \bye
 
 \bye

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