Prevody: Prepis 3D-parovani a vylepseni schematu prevodu

diff --git a/8-prevody/8-prevody.tex b/8-prevody/8-prevody.tex
index e4e33e9248821ba0faae07b9f317e0d395f2a045..e53730d3d82cab6ec1a3090d6fd033fffd74cf61 100644 (file)
--- a/8-prevody/8-prevody.tex
+++ b/8-prevody/8-prevody.tex
@@ -357,79 +357,77 @@ je jen zkratka za $\lnot a\lor b$).
 \prob{3D-párování}
 
 \inp Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a
 \prob{3D-párování}
 
 \inp Tøi mno¾iny, napø. $K$ (kluci), $H$ (holky), $Z$ (zvíøátka) a

-mno¾ina~$T$ kompatibilních trojic (tìch, kteøí se spolu snesou),
-tedy $T \subseteq K\times H\times Z$.
+mno¾ina $T \subseteq K\times H\times Z$ kompatibilních trojic (tìch, kteøí se
+spolu snesou).

 
 

-\outp Perfektní podmno¾ina trojic -- tj. taková podmno¾ina trojic, v~ní¾ se ka¾dý
-prvek mno¾in $K$, $H$ a~$Z$ vyskytuje právì jednou.
+\outp Zda existuje perfektní podmno¾ina trojic, tedy taková, v~ní¾ se ka¾dý
+prvek mno¾in $K$, $H$ a~$Z$ úèastní právì jedné trojice.

 
 \figure{3d_parovani.eps}{Ukázka 3D-párování}{3in}
 
 \s{Pøevod 3,3-SATu na 3D-párování:}
 
 \figure{3d_parovani.eps}{Ukázka 3D-párování}{3in}
 
 \s{Pøevod 3,3-SATu na 3D-párování:}

-Kdy¾ chceme ukázat, ¾e na
-nìco se dá pøevést SAT, potøebujeme obvykle dvì vìci:
-konstrukci, která bude simulovat promìnné, tedy nìco, co nabývá dvou stavù
-\<true>/\<false>; a nìco, co bude reprezentovat klauzule a umí zaøídit, aby
-ka¾dá klauzule byla splnìna alespoò jednou promìnnou.
+Uva¾ujme trochu obecnìji. Pokud chceme ukázat, ¾e se na nìjaký problém dá pøevést
+SAT, potøebujeme obvykle dvì vìci: Jednak konstrukci, která bude simulovat
+promìnné, tedy nìco, co nabývá dvou stavù \<true>/\<false>. Pak také
+potøebujeme cosi, co umí zaøídit, aby ka¾dá klauzule byla splnìna alespoò
+jednou promìnnou. Jak to provést u~3D-párování?

 
 

-Uva¾ujme takovouto konfiguraci:
+Uva¾ujme následující konfiguraci:

 
 \fig{3d.eps}{4cm}
 
 
 \fig{3d.eps}{4cm}
 

-4 zvíøátka, 2 kluci, 2 dívky a 4 trojice, které oznaèíme $A, B, C, D$. Je¹tì
-pøedpokládáme, ¾e zvíøátka se mohou úèastnit nìjakých jiných trojic, ale tito
-ètyøi lidé se vyskytují pouze v~tìchto ètyøech trojicích a~nikde jinde.
-V¹imneme si, ¾e existují právì dvì mo¾nosti, jak tento obrázek spárovat.
-Abychom spárovali kluka $k_1$, tak si musíme vybrat $A$ nebo $B$. Kdy¾ si
-vybereme $A$, $k_1$ i $d_2$ u¾ jsou spárovaní tak¾e si nesmíme vybrat $B$ ani
-$D$. Pak jediná mo¾nost, jak spárovat $d_1$ a~$k_2$ je $C$. Jedna mo¾nost je
-tedy vybrat si $A$ a $C$ a jeliko¾ je obrázek symetrický, tak kdy¾ vybereme
-místo $A$ trojici $B$, dostaneme $B$ a~$D$. V¾dy si tedy vybereme dvì protìj¹í
-trojice v~obrázku.
-
-Tyto konfigurace budeme pou¾ívat k~reprezentaci promìnných. Pro ka¾dou
-promìnnou si nakreslíme takový obrázek a~to, ¾e $A$ bude spárované s~$C$, bude
-odpovídat tomu, ¾e $x=1$, a~spárování $B$ a~$D$ bude odpovídat $x=0$. Pokud
-jsme pou¾ili $A$ a~$C$, zvíøata se sudými èísly, tj. $z_2$ a~$z_4$, horní
-a~dolní, jsou nespárovaná a~pokud jsme pou¾ili $B$ a~$D$, zvíøátka $z_1$
-a~$z_3$ zùstala nespárovaná. Pøes tato nespárovaná zvíøátka mù¾eme pøedávat
-informaci, jestli promìnná $x$ má hodnotu \<true> nebo \<false> do dal¹ích
-èástí vstupu.
-
-Zbývá vymyslet, jak reprezentovat klauzule. Klauzule jsou trojice popø. dvojice
-literálù, napø. $\kappa = (x \lor y \lor \lnot r)$, kde potøebujeme zajistit,
-aby $x$ bylo nastavené na $1$ nebo $y$ bylo nastavené na~$1$ nebo $r$ na $0$.
+\>V~ní se nacházejí 4~zvíøátka ($z_1$ a¾ $z_4$), 2~kluci ($k_1$ a $k_2$),
+2~dívky ($d_1$ a $d_2$) a 4~trojice ($A$, $B$, $C$ a~$D$). Zatímco zvíøátka
+se budou moci úèastnit i jiných trojic, kluky a dìvèata nikam jinam nezapojíme.
+
+V¹imneme si, ¾e existují právì dvì mo¾nosti, jak tuto konfiguraci spárovat.
+Abychom spárovali kluka $k_1$, tak musíme vybrat buï trojici~$A$ nebo~$B$.
+Pokud si vybereme~$A$, $k_1$ i $d_2$ u¾ jsou spárovaní, tak¾e si nesmíme vybrat
+$B$ ani~$D$. Pak jediná mo¾nost, jak spárovat $d_1$ a~$k_2$, je pou¾ít~$C$.
+Naopak zaèneme-li trojicí~$B$, vylouèíme $A$ a~$D$ a pou¾ijeme~$D$ (situace
+je symetrická).
+
+V¾dy si tedy musíme vybrat dvì protìj¹í trojice v~obrázku a druhé dvì nechat
+nevyu¾ité. Tyto mo¾nosti budeme pou¾ívat k~reprezentaci promìnných. Pro ka¾dou
+promìnnou si poøídíme jednu kopii obrázku. Volba $A+C$ bude odpovídat nule
+a nespáruje zvíøátka $z_2$ a~$z_4$. Volba $B+D$ reprezentuje jednièku a nespáruje
+$z_1$ a~$z_3$. Pøes tato nespárovaná zvíøátka mù¾eme pøedávat informaci o~hodnotì
+promìnné do~klauzulí.
+
+Zbývá vymyslet, jak reprezentovat klauzule. Mìjme klauzuli tvaru øeknìme
+$(x \lor y \lor \lnot r)$. Potøebujeme zajistit, aby $x$ bylo
+nastavené na $1$ nebo $y$ bylo nastavené na~$1$ nebo $r$ na $0$.

 
 \fig{klauzule.eps}{4cm}
 
 
 \fig{klauzule.eps}{4cm}
 

-Pro takovouto klauzuli si poøídíme dvojici kluk-dívka, kteøí budou figurovat
-ve tøech trojicích se tøemi rùznými zvíøátky, co¾ mají být volná zvíøátka
-z~obrázkù pro pøíslu¹né promìnné (podle toho, má-li se promìnná vyskytnout
-s negací nebo ne). A~zaøídíme to tak, aby ka¾dé zvíøátko bylo
-pou¾ité maximálnì v~jedné takové trojici, co¾ jde proto, ¾e ka¾dý literál se
-vyskytuje maximálnì dvakrát a~pro ka¾dý literál máme dvì volná zvíøátka,
-z~èeho¾ plyne, ¾e zvíøátek je dost pro v¹echny klauzule. Pro dvojice se postupuje
-obdobnì.
-
-Je¹tì nám ale urèitì zbude $2p-k$ zvíøátek, kde $p$ je poèet promìnných, $k$
-poèet klauzulí -- ka¾dá promìnná vyrobí 4 zvíøátka, klauzule zba¹tí jedno
-a samotné ohodnocení 2 zvíøátka -- tak pøidáme je¹tì $2p-k$ párù
-kluk-dìvèe, kteøí milují
-v¹echna zvíøátka, a~ti vytvoøí zbývající trojice.
-
-Pokud formule byla splnitelná, pak ze splòujícího ohodnocení mù¾eme vyrobit
-párování s~na¹í konstrukcí. Obrázek pro ka¾dou promìnnou spárujeme podle
-ohodnocení, tj. promìnná je $0$ nebo $1$ a~pro ka¾dou klauzuli si vybereme
-nìkterou z~promìnných, kterými je ta klauzule splnìna. Funguje to ale
-i~opaènì: Kdy¾ nám nìkdo dá párovaní v~na¹í konstrukci, pak z nìho doká¾eme
+Pro takovouto klauzuli si poøídíme novou dvojici kluk a dívka, kteøí budou figurovat
+ve~tøech trojicích se tøemi rùznými zvíøátky, co¾ budou volná zvíøátka
+z~obrázkù pro pøíslu¹né promìnné. Zvolíme je tak, aby byla volná pøi správném
+nastavení promìnné. Doká¾eme pøitom zaøídit, ¾e ka¾dé zvíøátko bude pou¾ité
+v~maximálnì jedné klauzuli, nebo» ka¾dý literál se vyskytuje nejvý¹e dvakrát
+a máme pro nìj dvì volná zvíøátka.
+
+Je¹tì nám urèitì zbude $2p-k$ zvíøátek, kde $p$ je poèet promìnných a $k$
+poèet klauzulí. Ka¾dá promìnná toti¾ dodá 2~volná zvíøátka a ka¾dá klauzule
+pou¾ije jedno z~nich. Pøidáme proto je¹tì $2p-k$ párù lidí, kteøí milují úplnì
+v¹echna zvíøátka; ti vytvoøí zbývající trojice.
+
+Snadno ovìøíme, ¾e celý pøevod pracuje v~polynomiálním èase, rozmysleme si je¹tì,
+¾e je korektní.
+
+Pokud formule byla splnitelná, z~ka¾dého splòujícího ohodnocení mù¾eme vyrobit
+párování v~na¹í konstrukcí. Obrázek pro ka¾dou promìnnou spárujeme podle
+ohodnocení (buï $A+C$ nebo $B+D$). Pro ka¾dou klauzuli si vybereme trojici,
+která odpovídá nìkterému z~literálù, jimi¾ je klauzule splnìna.
+
+A~opaènì: Kdy¾ nám nìkdo dá párovaní v~na¹í konstrukci, doká¾eme z~nìj

 vyrobit splòující ohodnocení dané formule. Podíváme se, v~jakém stavu je
 promìnná, a~to je v¹echno. Z~toho, ¾e jsou správnì spárované klauzule, u¾
 okam¾itì víme, ¾e jsou v¹echny splnìné.
 
 vyrobit splòující ohodnocení dané formule. Podíváme se, v~jakém stavu je
 promìnná, a~to je v¹echno. Z~toho, ¾e jsou správnì spárované klauzule, u¾
 okam¾itì víme, ¾e jsou v¹echny splnìné.
 

-Zbývá ovìøit, ¾e na¹e redukce funguje v~polynomiálním èase. Pro ka¾dou klauzuli
-spotøebujeme konstantnì mnoho èasu, $2p-k$ je také polynomiálnì mnoho a~kdy¾ v¹e
-seèteme, máme polynomiální èas vzhledem k~velikosti vstupní formule. Tím je
-pøevod hotov.
+Ukázali jsme tedy, ¾e na 3D-párování lze pøevést 3,3-SAT, a~tedy i obecný SAT.
+Pøevod v~opaèném smìru ponecháme jako cvièení, mù¾ete ho provést podobnì,
+jako jsme na SAT pøevádìli nezávislou mno¾inu.

 
 \figure{prevody.eps}{Pøevody mezi problémy}{3in}
 
 
 \figure{prevody.eps}{Pøevody mezi problémy}{3in}
 

diff --git a/8-prevody/prevody.eps b/8-prevody/prevody.eps
index 157646c6221a007b8f1a2f659ea48e519451aea8..c8ffeddb68b8ad23207cddb4e525d87838ab0043 100644 (file)
--- a/8-prevody/prevody.eps
+++ b/8-prevody/prevody.eps
@@ -1,2055 +1,1382 @@
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