From 54e947b2e7f4313db432474ea066bd300c4d2a36 Mon Sep 17 00:00:00 2001 From: Martin Mares Date: Tue, 18 Dec 2007 16:01:09 +0100 Subject: [PATCH] Nulta verze kapitoly o prevodech. --- 10-prevody/10-prevody.tex | 156 +++++++++++++++++++++++++++++++ 10-prevody/3D parovani.JPG | Bin 0 -> 17230 bytes 10-prevody/3D parovani.eps | Bin 0 -> 1027522 bytes 10-prevody/Doplnok grafu, NM.JPG | Bin 0 -> 10825 bytes 10-prevody/Doplnok grafu, NM.eps | Bin 0 -> 516690 bytes 10-prevody/Klika - Nez.Mnoz.JPG | Bin 0 -> 5278 bytes 10-prevody/Klika - Nez.Mnoz.eps | Bin 0 -> 389327 bytes 10-prevody/Makefile | 3 + 10-prevody/NezMnoz graf.JPG | Bin 0 -> 13299 bytes 10-prevody/NezMnoz graf.eps | Bin 0 -> 391599 bytes 10-prevody/matica.JPG | Bin 0 -> 10468 bytes 10-prevody/matica.eps | Bin 0 -> 310303 bytes 10-prevody/nezavisla mnozina.JPG | Bin 0 -> 4731 bytes 10-prevody/nezavisla mnozina.eps | Bin 0 -> 135784 bytes 14 files changed, 159 insertions(+) create mode 100644 10-prevody/10-prevody.tex create mode 100644 10-prevody/3D parovani.JPG create mode 100644 10-prevody/3D parovani.eps create mode 100644 10-prevody/Doplnok grafu, NM.JPG create mode 100644 10-prevody/Doplnok grafu, NM.eps create mode 100644 10-prevody/Klika - Nez.Mnoz.JPG create mode 100644 10-prevody/Klika - Nez.Mnoz.eps create mode 100644 10-prevody/Makefile create mode 100644 10-prevody/NezMnoz graf.JPG create mode 100644 10-prevody/NezMnoz graf.eps create mode 100644 10-prevody/matica.JPG create mode 100644 10-prevody/matica.eps create mode 100644 10-prevody/nezavisla mnozina.JPG create mode 100644 10-prevody/nezavisla mnozina.eps diff --git a/10-prevody/10-prevody.tex b/10-prevody/10-prevody.tex new file mode 100644 index 0000000..a786154 --- /dev/null +++ b/10-prevody/10-prevody.tex @@ -0,0 +1,156 @@ +\input ../lecnotes.tex + +\prednaska{10}{Pøevody problémù}{(zapsali Martin Chytil, Vladimír Kudelas)} + +Na této pøedná¹ce se budeme zabývat rozhodovacími problémy a pøevody mezi nimy. + +\s{Definice:} {\I Rozhodovací problém} je takový problém, jeho¾ výstupem je v¾dy {\sc ano} nebo {\sc ne} + +\s{Pøíklad:} Je dán bipartitní graf $G$, $k \in N$. Existuje v $G$ párování, které obsahuje alespoò $k$ hran? + +\s{Pøíklad:} Daný problém pøevedeme na jiný: Párování $\rightarrow$ (lze pøevést) $\rightarrow$ hledání maximálního toku. +Tzn. Existuje v 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{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 tahle 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 vedli 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. +Tohle iterujeme dokud to jde. Výsledkem je mno¾ina hran, které patøí do maximálního párování. Tím jsme dané párování nalezli. +Hran je polynomiálne 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, t¾. 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 + ohodnocení) + +\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 +B poèíta v èase $O(n^{kl})$ +$f$ poèíta v polynomiálním èase $\rightarrow$ mù¾e vydat maximálne polynomiální výstup + +\s{Pozorování:} funkce $f$ 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 $(f o g)$ +\endlist + +\h{1.Problém: SAT} +\itemize\ibull +\:splnitelnost logických formulí +\:tj. dosazení $0,1$ do logické formule tak, aby formule platila +\endlist + +Zamìøíme se na speciální formu zadání formulí - konjunktivní normální forma: +$$(\ldots\lor\ldots\lor\ldots\lor\ldots\lor) \& (\ldots\lor\ldots\lor\ldots\lor\ldots\lor) \& \ldots $$ + +{\I Vstup:} formule v konjunktivní normální formì (CNF) +{\I Výstup} $\exists$ dosazení $0/1$ za promìnné t.¾. $\phi(\ldots) = 1$. + +$$ \phi(x, y, \ldots) = (x \lor \lnot y \lor \ldots) \& (\ldots\lor\ldots\lor\ldots\lor\ldots\lor) \& \ldots $$ +\itemize\ibull +\:formule zadána pomocí klauzulí oddìlených \&, +\:ka¾dá klauzule je slo¾ená z literálù oddìlených $\lor$, +\:ka¾dý literál je slo¾ený z promìnných, nebo $\lnot$ promìnných +\endlist + +Uká¾eme, ¾e staèí vyøe¹it jednodu¹¹í problém 3-SAT. + +\h{2.Problém: 3-SAT} +{\I 3-SAT} je SAT, kde ka¾dá klauzule obsahuje nejvý¹e 3 literály + +\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) + +\s {Pøevod SAT na 3-SAT} +- musíme formuli pøevést tak, abychom neporu¹ili splnitelnost +{\I trik pro dlouhé klauzule:} +$$(\alpha \lor \beta) t¾. \vert\alpha\vert + \vert\beta\vert \ge 4$$ +pøepí¹eme na: $$(\alpha \lor x) \& (\beta \lor \lnot x)$$ +$x$ je nová promìnná, kterou nastavíme tak, abychom neovlivnili splnitelnost formule +platí-li: +\itemize\ibull +\:$\alpha \rightarrow x = 0$ (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 my. +Tento trik opakujeme tak dlouho, dokud je to tøeba. + +{\I poznámka} u 3-SAT lze vynutit právì $3$ literály, pro krátké klauzule pou¾ijeme následující trik: +$$(a) \rightarrow (a \lor x) \& (a \lor \lnot x) $$ + +\h{3. Problém: Hledání nezávislé mno¾iny v grafu} +\s{Definice:} {\I Nezávislá mno¾ina:} je tvoøena vrcholy grafu, které spolu nemají spoleènou hranu + +{\I Vstup:} Neorientovaný graf G, $k \in N$ +{\I Výstup:} $\exists A \subseteq V(G)$, $\vert A \vert \ge k$, $u,v \in A \Rightarrow (uv) \not\in E(G)$ ? + +Úlohu øe¹íme tak, ¾e problém 3-SAT pøevedeme tuto úlohu. + +\s{poznámka:} Ka¾dý graf má minimálnì jednu nezávislou mno¾inu, a tou je prázdná mno¾ina. + +\s{øe¹ení úlohy:} Z ka¾dé klauzule vybereme $1$ literál tak, abychom v rùzných klauzulích nevybírali konfliktnì, tj. $x a \lnot x$ + +\s {pøíklad} +$$(x \lor y \lor z) \& (x \lor \lnot y \lor \lnot z) \& (\lnot x \lor \lnot y \lor p) $$ +pro ka¾dou klauzuli sestrojíme graf (trojúhelník) + pøidáme "konfliktní" hrany tj. $x a \lnot x$ + +Princip je takový, ¾e z ka¾dé klauzule si vybereme promìnnou, která danou klauzuli splní a to, aby promìnné, které si vybereme nekolidovali, vyøe¹íme hranami mezi promìnnými a jejich negacemi. + +Existuje NzMna 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 "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 + +pro ka¾dé $(i,j) \in E(G)$ pøidáme klauzuli $(\lnot vi \lor \lnot vj)$ + + +prvek matice $x_{i,j} = 1 \Leftrightarrow i$-tý prvek je vrchol $j$, tj. $\forall i,j$, $x_{ij} \Rightarrow v_j$ + +$\forall j,i,i^{'}, i\ne i^{'} : x_{ij} \Rightarrow x_{i^{'}j}$ + +$\forall i,j,j^{'}, j\ne j^{'} : x_{ij} \Rightarrow x_{ij^{'}}$ + + +\h{5. Problém: 3D -- párování (matching)} +{\I Vstup:} Mno¾ina K, H, Z + mno¾ina kompatibilních 3-ic (ti, kteøí se spolu snesou) +{\I Výstup:} Perfektní podmno¾ina 3-ic +\s{Øe¹ení:} pøes 3,3-SAT (konkrétnìji viz dal¹í pøedná¹ka) + + +\h{3,3-SAT} +\s{Definice:} {\I 3,3-SAT} je speciální pøípad 3-SATu, kde ka¾dá promìnná se vyskytuje v maximálnì 3 literálech + +\s{Pøevod 3-SAT na 3,3-SAT} +Pokud se promìnná $x$ vyskytuje v $k > 3$ literálech, tak nahradíme výskyty novými promìnnými $x_1 \ldots x_k$ a pøidáme klauzule +$$ +(\lnot x_1 \lor x_2) +(\lnot x_2 \lor x_3) +(\lnot x_3 \lor x_4) +\ldots +(\lnot x_{k-1} \lor x_k) +(\lnot x_k \lor x_1) +$$ + +co¾ odpovídá: + +$$ +(x_1 \Rightarrow x_2) +(x_2 \Rightarrow x_3) +(x_3 \Rightarrow x_4) +\ldots +(x_{k-1} \Rightarrow x_k) +(x_k \Rightarrow x_1) +$$ +tímto zaruèíme, ¾e v¹echny promìnné budou mít stejnou hodnotu. + +\bye diff --git a/10-prevody/3D parovani.JPG b/10-prevody/3D parovani.JPG new file mode 100644 index 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