--- /dev/null
+\input ../lecnotes.tex
+
+\prednaska{4}{Hradlové sítì a paralelní algoritmy}{(zapsal: Petr Jankovský)}
+
+\def\land{\mathbin{\&}}
+
+Výkon poèítaèù nelze zvy¹ovat donekoneèna a pøesto¾e ji¾ pìkných pár let platí,
+¾e se jejich rychlost s~èasem exponenciálnì zvìt¹uje, jednou urèitì narazíme
+pøinejmen¹ím na~fyzikální limity.
+
+Kdy¾ tedy nemù¾eme zvy¹ovat rychlost jednoho procesoru, jak poèítat rychleji?
+Øe¹ením by mohlo být poøídit si procesorù víc. U¾~dnes na~bì¾ném PéCéèku máme
+k~dispozici vícejádrové procesory, díky nim¾ mù¾eme vyu¾ít pararelní poèítání
+a úlohu øe¹it tak, ¾e práci ¹ikovnì rozdìlíme mezi procesory (èi jádra) a
+zamìstnáme je pøi~výpoètu v¹echny.
+
+My se podíváme na~abstraktní výpoèetní model, který je je¹tì paralelnìj¹í.
+Techniky, které si uká¾eme na~tomto modelu, se v¹ak dají pøekvapivì vyu¾ít
+i~pøi~reálném paralelizování na~nìkolik málo procesorù. Budeme se zabývat
+jednoduchým modelem paralelního poèítaèe, toti¾ hradlovou sítí.
+
+\h{Hradlové sítì}
+
+\s{Definice:} {\I Hradlo} je prvek, který umí vyhodnocovat nìjakou funkci
+nad~koneènou abecedou.
+$$f: {\sum}^{k} \rightarrow \sum$$
+
+Obecnì se na~hradlo díváme jako na~funkci, která dostane $k$ vstupù a~vrátí jeden
+výstup, pøièem¾ hodnoty, nad~kterými pracuje, budou z~nìjaké koneèné abecedy --
+tedy z~nìjaké koneèné mno¾iny symbolù $\sum$. Písmenku $k$ zde øíkáme {\I arita hradla}.
+
+\s{Pøíklad:} Èasto studujeme hradla booleovská, která poèítají nìjaké logické funkce.
+
+Z~nich nejèastìji potkáme:
+
+\itemize\ibull
+\:nulární: to jsou konstanty (0, 1, ... -- zkrátka jakákoli konstanta abecedy),
+\:unární: napø. negace (znaèíme~$\lnot$),
+\:binární: logický souèin ({\sc and},~$\land$), souèet ({\sc or},~$\lor$), ...
+\endlist
+
+\>Hradla kreslíme tøeba následovnì:
+
+\figure{hradlo_and.eps}{Binární hradlo provádìjící logickou operaci {\sc and}.}{1in}
+
+Jednotlivá hradla mù¾eme navzájem urèitým zpùsobem propojovat a vytváøet
+z nich {\I hradlové sítì}. Pokud pou¾íváme pouze booleovská hradla, øíkáme takto vytvoøeným
+sítím {\I booleovské obvody}. Pokud pracujeme s~operacemi nad nìjakou obecnìj¹í (ale koneènou)
+mno¾inou symbolù (abecedou), nazývají se {\I kombinaèní obvody.}
+
+Ka¾dá hradlová sí» má nìjaké vstupy, nìjaké výstupy a uvnitø jsou propojovaná
+hradla. Ka¾dý vstup hradla je pøipojen buïto na~nìkterý ze~vstupù sítì nebo
+na~výstup jiného hradla. Výstupy hradel mohou být propojeny na~vstupy dal¹ích
+hradel a mohou se vìtvit, nebo na výstupy sítì. Pøitom máme zakázáno vytváøet
+cykly.
+
+Ne¾ si øekneme formální definici, podívejme se na obrázek.
+
+\figure{hradlova_sit.eps}{Hradlová sí» -- tøívstupová verze funkce {\I majorita}.}{3in}
+
+Obrazek znázoròuje hradlovou sí», která poèítá, zda je alespoò na~dvou ze~vstupù
+jednièka. Pojïme si ale {\I hradlovou sí»} definovat formálnì.
+
+\s{Definice:} {\I Hradlová sí»} je urèena:
+\itemize\ibull
+\:{\I abecedou} $\Sigma$ (to je nìjaká koneèná mno¾ina symbolù, obvykle $\Sigma=\{0,1\}$);
+\:po dvou disjunktními koneènými mno¾inami $I$~({\I vstupy}), $O$~({\I výstupy}) a~$H$~({\I hradla});
+\:acyklickým orientovaným multigrafem~$(V,E)$, kde~$V = I \cup O \cup H$;
+\:zobrazením~$F$, které ka¾dému hradlu $h\in H$ pøiøadí nìjakou funkci~$F(h):
+ \Sigma^{a(h)} \rightarrow \Sigma$, co¾ je funkce, kterou toto hradlo vykonává.
+ Èíslu $a(h)$ øíkáme {\I arita} hradla~$h$;
+\:zobrazením~$z: E \rightarrow {\bb N}$, které ka¾dé hranì vedoucí do~nìjakého
+ hradla pøiøazuje nìkterý ze vstupù tohoto hradla.
+\endlist
+
+\>Pøitom jsou splnìny následující podmínky:
+
+\itemize\ibull
+\:$\forall i \in I: \deg^{in}(i)=0$ (do~vstupù nic nevede);
+\:$\forall o \in O: \deg^{in}(o)=1~\land~\deg^{out}(o)=0$ (z~výstupù nic nevede a do~ka¾dého vede právì jedna hrana);
+\:$\forall h \in H: \deg^{in}(v)=a(v)$ (do~ka¾dého hradla vede tolik hran, kolik je jeho arita);
+\:$\forall h \in H~\forall j: 1\le j\le a(h)$ existuje právì jedena hrana~$e$ taková, ¾e $z(e)=j$, $e$ konèí v $h$
+ (v¹echny vstupy hradel jsou zapojeny).
+\endlist
+
+\s{Pozorování:} Kdybychom pøipustili hradla s~libovolnì vysokým poètem vstupù, mohli
+bychom libovolný problém se vstupem délky~$n$ vyøe¹it jedním hradlem o~$n$~vstupech,
+kterému bychom pøiøadili funkci, která by na¹i úlohu rovnou vyøe¹ila... Tento model,
+v¹ak není ani realistický, ani pìkný. Proto pøijmìme omezení, ¾e~arity v¹ech hradel
+budou omezeny nìjakou pevnou konstantou $k$ (uká¾e se, ¾e nám bude staèit dvojka
+a~vystaèíme si tedy pouze s nulárními, unárními a binárními hradly).
+Následující obrázky ukazují, jak hradla o~více vstupech nahradit dvouvstupovými:
+
+\twofigures{hradlo_ternor.eps}{Trojvstupové hradlo \sc or.}{0.5in}{hradlo_ternbior.eps}{Jeho nahrazení 2-vstupovými hradly.}{0.6in}
+
+
+Nyní bychom je¹tì mìli definovat, co taková hradlová sí» vlastnì poèítá a~jak
+vlastní výpoèet probíhá.
+
+\s{Definice:} {\I Výpoèet sítì} probíhá po~{\I taktech.} V nultém taktu jsou definovány pouze
+hodnoty na~vstupech a na~hradlech arity 0. Mù¾eme si to pøedstavit tak, ¾e na~zaèátku
+nemá ¾ádné hradlo definovánu výstupní hodnotu (a¾ na ji¾ zmínìná hradla
+nulární). V~ka¾dém dal¹ím taktu pak vydají výstup hradla, která na~konci minulého
+taktu mìla definovány v¹echny hodnoty na vstupech. Jakmile budou po~nìjakém
+koneèném poètu taktù definované i hodnoty v¹ech výstupù, sí» se zastaví a~vydá
+výsledek.
+
+\s{Pozorování:} Proto¾e je sí» acyklická, je jasné, ¾e jakmile jednou nìjaké hradlo vydá
+výstup, tak se tento výstup bìhem dal¹ího výpoètu sítì ji¾ nezmìní.
+
+\figure{vypocet_site.eps}{Výpoèet hradlové sítì.}{6cm}
+
+\>Podle toho, jak sí» poèítá, si ji mù¾eme rozdìlit na~vrstvy:
+
+\s{Definice:} {\I $i$-tá vrstva $S_i$} obsahuje právì takové vrcholy~$v$, pro~které nejdel¹í cesta ze~vstupù
+sítì do $v$ má délku rovnou~$i$.
+
+\s{Pozorování:} V¹imnìme si, ¾e v $i$-tém taktu vydají hodnoty právì hradla z $S_i$.
+
+Dává tedy smysl prohlásit za~{\I èasovou slo¾itost} sítì poèet
+jejích vrstev. Podobnì {\I prostorovou slo¾itost} definujeme jako poèet hradel v~síti.
+
+\s{Pøíklad:} Sestrojme sí», která zjistí, zda se mezi jejími~$n$ vstupy
+vyskytuje alespoò jedna jednièka.
+
+\>{\I První øe¹ení:} zapojíme hradla za~sebe (sériovì). Èasová i prostorová
+slo¾itost odpovídají~$\Theta(n)$. Zde ov¹em vùbec nevyu¾íváme toho, ¾e by mohlo poèítat více
+hradel souèasnì.
+
+\figure{hloupy_or.eps}{Hradlová sí», která zjistí, zda-li je na vstupu alespoò jedna jednièka.}{0.7in}
+
+\>{\I Druhé øe¹ení:} Hradla budeme spojovat do~dvojic, pak výsledky z~tìchto dvojic opìt
+do~dvojic a tak dále. Díky paralelnímu zapojení dosáhneme èasové slo¾itosti $\Theta(\log n)$,
+prostorová slo¾itost zùstane lineární.
+
+\figure{chytry_or.eps}{Chytøej¹í øe¹ení stejného problému.}{1.7in}
+
+\bye
--- /dev/null
+P=4-hradla
+
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projects / ads2.git / commitdiff
