Jeste potrebuje trochu uhladit a take upravit cast o nasobeni.
\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~nìkolika málo procesorech. Konec koncù i proto, ¾e vnitøní
-architektura procesoru se na¹emu modelu velmi podobá. Budeme se zabývat
-jednoduchým modelem paralelního poèítaèe, toti¾ hradlovou sítí.
+®ivot nám pøiná¹í stále vìt¹í problémy, které obvykle vy¾adují stále více
+výpoèetního výkonu. Rychlost poèítaèù sice posledních pár desetiletí stále
+roste exponenciálnì, ale tento rùst se urèitì nìkdy zastaví (napøíklad proto,
+¾e vesmír je koneèný) a podle v¹eho u¾ k~tomuto bodu jsme docela blízko
+(nará¾íme na nejrùznìj¹í fyzikální limity, napøíklad není jasné, jak vyrábìt
+transistory men¹í ne¾ jeden atom).
+
+Jak si tedy s~obrovskými daty poradíme? Jedna z~lákavých mo¾ností je prostì
+do~výpoètu zapøáhnout více ne¾ jeden procesor. Ostatnì, vícejádrové procesory,
+které dneska najdeme ve~svých stolních poèítaèích, nejsou nic jiného ne¾
+víceprocesorový systém na jednom èipu.
+
+Nabízí se tedy obtí¾nou úlohu rozdìlit na nìkolik èástí, nechat ka¾dý
+procesor (èi jádro) poèítat jednu z~èástí a nakonec jejich výsledky spojit
+dohromady. To se snadno øekne, ale s~výjimkou triviálních úloh u¾ obtí¾nìji
+provede.
+
+Pojïme se podívat na nìkolik zajímavých paralelních algoritmù. Abychom se
+nemuseli zabývat detaily hardwaru konkrétního víceprocesorového poèítaèe,
+zavedeme si pomìrnì abstraktní výpoèetní model, toti¾ hradlové sítì. Tento
+model je daleko paralelnìj¹í ne¾ skuteèný poèítaè, ale pøesto se techniky,
+které si uká¾eme, dají snadno vyu¾ít i prakticky. Konec koncù sama vnitøní
+architektura procesorù se na¹emu modelu velmi podobá.
\h{Hradlové sítì}
-\s{Definice:} {\I Hradlo} je prvek, který umí vyhodnocovat nìjakou funkci
-nad~koneènou abecedou $\Sigma$.
+Hradlové sítì jsou tvoøeny navzájem propojenými {\I hradly.} Ka¾dé hradlo pøitom
+poèítá nìjakou (obecnì libovolnou) funkci $\Sigma^k \rightarrow \Sigma$, kde~$\Sigma$
+je koneèná abeceda (stejná pro celou sí») a~$k$ pøirozené èíslo (poèet vstupù hradla,
+jinak té¾ jeho {\I arita}).
-Obecnì se na~hradlo díváme jako na~funkci $f: {\Sigma}^{k} \rightarrow \Sigma$, která dostane $k$ vstupù
-a~vrátí jeden výstup, pøièem¾ hodnoty, nad~kterými pracuje, budou z~nìjaké koneèné
-abecedy -- tedy z~nìjaké koneèné mno¾iny symbolù $\Sigma$. Písmenku $k$ zde øíkáme {\I arita
-hradla}.
-
-\s{Pøíklad:} Èasto studujeme hradla booleovská (pracující nad abecedou $\{0,1\}$), která poèítají logické funkce.
-
-Z~nich nejèastìji potkáme:
+\s{Pøíklad:} Èasto studujeme hradla {\I booleovská} (pracující nad abecedou $\Sigma=\{0,1\}$).
+Ta poèítají jednotlivé logické funkce, mezi nejbì¾nìj¹í patøí:
\itemize\ibull
-\:nulární: to jsou konstanty (FALSE=0, TRUE=1),
-\:unární: napø. negace (znaèíme~$\lnot$),
-\:binární: logický souèin ({\csc and},~$\land$), souèet ({\csc or},~$\lor$), ...
+\:nulární: to jsou konstanty ($\hbox{\csc false}=0$, $\hbox{\csc true}=1$),
+\:unární: identita a negace ({\csc not},~$\lnot$),
+\:binární: logický souèin ({\csc and},~$\land$), souèet ({\csc or},~$\lor$), \dots
\endlist
-\>Hradla kreslíme tøeba následovnì:
-
-\figure{hradlo_and.eps}{Binární hradlo provádící logickou operaci {\csc 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.}
+Propojením hradel pak vznikne {\I hradlová sí».} Ne¾ vyøkneme formální definici,
+pojïme se podívat na pøíklad jedné takové sítì:
-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 (mohou se vìtvit), nebo na výstupy sítì. Pøitom máme zakázáno vytváøet
-cykly.
+\figure{hradlova_sit.eps}{Hradlová sí» -- tøívstupová verze funkce {\I majorita}}{3in}
-Ne¾ si øekneme formální definici, podívejme se na obrázek.
+Na¹e sí» má tøi vstupy, vnitøní booleovská hradla a jeden výstup. Na výstupu
+je pøitom jednièka právì tehdy, jsou-li jednièky pøítomny na alespoò dvou
+vstupech. Jinými slovy vrací {\I majoritu} ze~vstupù, tedy hodnotu, která
+pøeva¾uje.
-\figure{hradlova_sit.eps}{Hradlová sí» -- tøívstupová verze funkce {\I majorita}.}{3in}
+Obecnì ka¾dá hradlová sí» má nìjaké vstupy, hradla a výstupy.
+Ka¾dý vstup hradla je pøitom pøipojen buïto na~nìkterý ze~vstupù sítì
+nebo na~výstup jiného hradla. Výstupy hradel mohou být propojeny na~vstupy
+dal¹ích hradel (mohou se vìtvit), nebo na výstupy sítì. Pøitom máme zakázáno
+vytváøet cykly.
-Obrazek znázoròuje hradlovou sí», která poèítá, zda je alespoò na~dvou ze~vstupù
-jednièka. Pojïme si ale {\I hradlovou sí»} definovat formálnì.
+Nyní toté¾ formálnìji:
\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}) \hfil\break a~$H$~({\I hradla});
-\:acyklickým orientovaným multigrafem~$(V,E)$, kde~$V = I \cup O \cup H$;
+\:{\I abecedou} $\Sigma$, co¾ je nìjaká koneèná mno¾ina symbolù;
+\:po dvou disjunktními koneènými mno¾inami \hfil\break
+ $I$~({\I vstupy}), $O$~({\I výstupy}) a~$H$~({\I hradla});
+\:acyklickým orientovaným multigrafem~$(V,E)$, kde~$V = I \cup O \cup H$;\foot{Proè
+ potøebujeme multigraf? Napøíklad chceme-li výstup jednoho hradla pøipojit souèasnì
+ na více rùzných vstupù druhého hradla.}
\: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.
+\:zobrazením~$z: E \rightarrow {\bb N}$, je¾ o~hranách vedoucích do hradel øíká,
+ kolikátému argumentu funkce odpovídají;\foot{Na hranách vedoucích do výstupù
+ necháváme hodnotu této funkce nevyu¾itu.}
\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~$e$ konèí v~$h$ a~$z(e)=j$,
- (v¹echny vstupy hradel jsou zapojeny).
+\:$\forall i \in I: \deg^+(i)=0$ {\sl (do~vstupù nic nevede);}
+\:$\forall o \in O: \deg^+(o)=1, \deg^-(o)=0$ {\sl (z~výstupù nic nevede a do~ka¾dého vede právì jedna hrana);}
+\:$\forall h \in H: \deg^+(v)=a(v)$ {\sl (do~ka¾dého hradla vede tolik hran, kolik je jeho arita);}
+\:$\forall h \in H~\forall j \in \{1,\ldots,a(h)\}$ existuje právì jedna hrana~$e$ taková, ¾e $e$~konèí v~$h$ a~$z(e)=j$
+ {\sl (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:
+\s{Poznámka:} Nìkdy se hradlovým sítím také øíká {\I kombinaèní obvody} a pokud pracují
+nad abecedou $\Sigma = \{0,1\}$, pak {\I booleovské obvody}.
+
+\s{Definice:} {\I Výpoèet sítì} postupnì pøiøazuje hodnoty z~abecedy~$\Sigma$
+vrcholùm grafu. Výpoèet probíhá po~{\I taktech.} V~nultém taktu jsou definovány
+pouze hodnoty na~vstupech sítì a v~hradlech arity~0 (konstanty). V~ka¾dém
+dal¹ím taktu pak ohodnotíme vrcholy, jejich¾ v¹echny vstupní hrany vedou
+z~vrcholù s~ji¾ definovanou hodnotou.
+
+Hodnotu hradla~$h$ pøitom spoèteme funkcí~$F(h)$ z~hodnot na jeho vstupech
+uspoøádaných podle funkce~$z(h)$. Výstup sítì pouze zkopíruje hodnotu, která do
+nìj po~hranì pøi¹la.
+
+Jakmile budou po~nìjakém poètu taktù definované hodnoty v¹ech výstupù, výpoèet
+se zastaví a~sí» vydá výsledek -- ohodnocení výstupù.
-\twofigures{hradlo_ternor.eps}{Trojvstupové hradlo \csc or.}{0.5in}{hradlo_ternbior.eps}{Jeho nahrazení 2-vstupovými hradly.}{0.6in}
+Podle prùbìhu výpoètu mù¾eme vrcholy sítì rozdìlit do vrstev:
+\s{Definice:} {\I $i$-tá vrstva $S_i$} obsahuje právì takové vrcholy~$v$,
+pro~které nejdel¹í z~cest z~libovolného vrcholu se~vstupním stupnìm~0
+do~$v$ má délku rovnou právì~$i$.
-Nyní bychom je¹tì mìli definovat, co taková hradlová sí» vlastnì poèítá a~jak
-její výpoèet probíhá.
+\figure{vypocet_site.eps}{Prùbìh výpoètu a rozdìlení sítì na vrstvy}{6cm}
-\s{Definice:} {\I Výpoèet sítì} probíhá po~{\I taktech.} V nultém taktu jsou definovány pouze
-hodnoty na~vstupech sítì a na~výstupech hradel 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{Pár pozorování} o~prùbìhu výpoètu:
-\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.
+\itemize\ibull
+\:V~$i$-té vrstvì jsou tedy právì ty vrcholy, které poprvé ohodnotíme v~$i$-tém
+ taktu výpoètu.
+\:Jeliko¾ sí» je acyklická, tak platí, ¾e jakmile vrchol ohodnotíme, u¾ se
+ jeho ohodnocení nikdy nemù¾e zmìnit.
+\:Kdy¾ se vydáme z~libovolného vrcholu proti smìru hran, po~koneènì mnoha
+ krocích dojdeme do~vrcholu s~nulovým vstupním stupnìm (vstupu sítì nebo
+ konstantního hradla). Proto ka¾dý vrchol le¾í v~nìjaké vrstvì.
+\:Vrstvy jsou disjunktní, tak¾e poèet neprázdných vrstev je nutnì koneèný.
+ V~kombinaci s~pøedchozím pozorováním dostaneme, ¾e výpoèet sítì se v¾dy zastaví.
+\:Navíc poslední neprázdná vrstva je právì ta, kde se výpoèet zastaví -- z~ka¾dého
+ dal¹ího vrcholu by toti¾ bylo mo¾né po~hranách dojít do vrcholu s~výstupním
+ stupnìm~0 a jediné takové vrcholy jsou výstupy sítì. Ty by tedy musely také
+ le¾et v~nìkteré z~následujících vrstev, co¾ ov¹em není mo¾né, nebo» výpoèet
+ se u¾ zastavil.
+\endlist
+
+\>To nás motivuje k~následující definici:
+
+\s{Definice:} {\I Èasovou slo¾itost} sítì definujeme jako poèet jejích neprázdných
+vrstev. Podobnì {\I prostorovou slo¾itost} urèíme jako poèet hradel v~síti.
-\figure{vypocet_site.eps}{Výpoèet hradlové sítì.}{6cm}
+\s{Poznámka o~aritách:}
+Kdybychom pøipustili hradla s~libovolnì vysokým poètem vstupù, mohli bychom
+libovolný problém se vstupem délky~$n$ a výstupem délky~$\ell$ vyøe¹it v~jedné
+vrstvì pomocí $\ell$~kusù $n$-vstupových hradel. Ka¾dému bychom prostì
+pøiøadili funkci, která poèítá pøíslu¹ný bit výsledku ze~v¹ech bitù vstupu.
-\>Podle toho, jak sí» poèítá, si ji mù¾eme rozdìlit na~vrstvy:
+To v¹ak není ani realistické, ani pìkné. Jak z~toho ven? Pøijmeme prostì
+omezení, ¾e~arity v¹ech hradel budou omezeny nìjakou pevnou konstantou,
+tøeba dvojkou. Budeme tedy pou¾ívat výhradnì nulární, unární a binární hradla.
-\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~$i$.
+Poznamenejme je¹tì, ¾e realistický model (by» s~trochu jinými vlastnostmi)
+by vznikl také tehdy, kdybychom místo arity omezily typy funkcí, øeknìme
+na {\csc and}, {\csc or} a {\csc not}.
-\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{Poznámka o~uniformitì:}
+Dodejme, ¾e od~bì¾ných výpoèetním modelùm, jako je tøeba RAM, se hradlové
+sítì li¹í jednou podstatnou vlastností -- ka¾dá sí» zpracovává výhradnì vstupy
+jedné konkrétní velikosti. Chceme tedy najít nìjaký obecný pøedpis, který
+pro ka¾dou velikost vstupu sestrojí pøíslu¹nou sí». Takovým výpoèetním modelùm
+se øíká {\I neuniformní.}
-\s{Pøíklad:} Sestrojme sí», která zjistí, zda se mezi jejími~$n$ vstupy
-vyskytuje alespoò jedna jednièka.
+A~co myslíme oním pøedpisem pro sestrojení sítì? Bude to pro nás prostì
+nìjaký algoritmus (klasický, neparalelní) bì¾ící v~polynomiálním èase.
+(Kdybychom dovolili i pomalej¹í algoritmy, mohli bychom bìhem konstrukce
+provádìt nìjaký nároèný pøedvýpoèet a jeho výsledek zabudovat do struktury
+sítì.)
+
+\h{Hledá se jednièka}
+
+Abychom si nový výpoèetní model osahali, zkusme nejprve sestrojit obvod,
+který zjistí, zda se mezi jeho~$n$ vstupy vyskytuje alespoò jedna jednièka.
+Jinými slovy vypoèítat $n$-vstupovou funkci {\csc or}.
\>{\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
+slo¾itost èiní $\Theta(n)$. Zde ov¹em vùbec nevyu¾íváme toho, ¾e by mohlo poèítat více
hradel souèasnì.
-\figure{hloupy_or.eps}{Hradlová sí», která zjistí, zdali 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 pro vstup velikosti 16.}{3in}
-
-Minule jsme si zavedli paralelní výpoèetní model, ve kterém si nyní nìco naprogramujeme \dots
+\twofigures{hloupy_or.eps}{Sériové øe¹ení}{1in}{chytry_or.eps}{Paralelní øe¹ení}{3in}
\h{Sèítání binárních èísel}
-Mìjme dvì èísla $x$ a $y$ zapsané ve~dvojkové soustavì. Jejich èíslice oznaème
-$x_{n-1}\ldots x_0$ a $y_{n-1}\ldots y_0$, kde $i$-tý øád má váhu $2^i$. Nyní budeme chtít tato èísla
-seèíst.
+Zajímavìj¹í úlohou, její¾ paralelizace u¾ nebude tak triviální, je obyèejné
+sèítání dvojkových èísel. Mìjme dvì èísla $x$ a $y$ zapsané ve~dvojkové soustavì. Jejich èíslice oznaème
+$x_{n-1}\ldots x_0$ a $y_{n-1}\ldots y_0$, kde $i$-tý øád má váhu $2^i$.
-K tomuto úèelu se~ihned nabízí pou¾ít starý dobrý \uv{¹kolní algoritmus}, který
-funguje ve~dvojkové soustavì stejnì dobøe jako v~desítkové. Zkrátka sèítáme èísla
-zprava doleva. V¾dy seèteme pøíslu¹né èíslice pod~sebou a~pøièteme pøenos z~ni¾¹ího
-øádu. Tím dostaneme jednu èíslici výsledku a~pøenos do~vy¹¹ího øádu. Formálnì
-bychom to mohli zapsat tøeba takto:
+K~seètení se~ihned nabízí pou¾ít starý dobrý \uv{¹kolní algoritmus sèítání pod sebou}, který
+funguje ve~dvojkové soustavì stejnì dobøe jako v~desítkové. Sèítáme èísla zprava doleva,
+v¾dy seèteme~$x_i$ s~$y_i$ a~pøièteme pøenos z~ni¾¹ího øádu. Tím dostaneme jednu èíslici
+výsledku a~pøenos do~vy¹¹ího øádu. Formálnì bychom to mohli zapsat tøeba takto:
$$
z_i=x_i \oplus y_i \oplus c_i,
$$
kde $z_i$ je $i$-tá èíslice souètu, $\oplus$ znaèí operaci {\csc xor} (souèet modulo~2) a~$c_i$ je {\I pøenos}
z~$(i-1)$-ního øádu do~$i$-tého. Pøenos pøitom nastane tehdy, pokud se~nám potkají
dvì jednièky pod~sebou, nebo kdy¾ se~vyskytne alespoò jedna jednièka a~k~tomu
-pøenos z~ni¾¹ího øádu. Jinými slovy tehdy, kdy¾ ze~tøí xorovaných èíslic jsou alespoò
-dvì jednièky (pomocí obvodu pro majoritu z minulé pøedná¹ky lehce zkonstruujeme):
+pøenos z~ni¾¹ího øádu. To je tehdy, kdy¾ mezi tøemi xorovanými èíslicemi jsou alespoò
+dvì jednièky -- k~tomu se nám hodí ji¾ známý obvod pro majoritu:
$$
\eqalign{
c_{0} &= 0, \cr
}
$$
-Takovéto sèítání sice perfektnì funguje, nicménì je bohu¾el pomìrnì pomalé.
-Pokud bychom stavìli hradlovou sí» podle tohoto pøedpisu, byla by slo¾ená z~nìjakých
-podsítí (\uv{krabièek}), které budou mít na~vstupu $x_i$, $y_i$ a~$c_i$ a jejich výstupem
-bude $z_i$ a~$c_{i+1}$.
-
-\figure{hloupe_scitani.eps}{Sèítání ¹kolním algoritmem.}{1.5in}
+O~tomto pøedpisu snadno doká¾eme, ¾e funguje (zkuste si to), nicménì pokud podle
+nìj postavíme hradlovou sí», bude pomìrnì pomalá. Sí» si mù¾eme pøedstavit tak,
+¾e je slo¾ena z~nìjakých podsítí (\uv{krabièek}), které budou mít na~vstupu $x_i$,
+$y_i$ a~$c_i$ a jejich výstupem bude $z_i$ a~$c_{i+1}$:
-V¹imìme si, ¾e~ka¾dá krabièka závisí na~výstupu té pøedcházející. Jednotlivé
-krabièky tedy musí urèitì le¾et na~rùzných hladinách. Celkovì bychom museli pou¾ít
-$\Theta{(n)}$ hladin a~jeliko¾ je ka¾dá krabièka konstantnì velká, také $\Theta{(n)}$ hradel. To dává
-lineární èasovou i~prostorovou slo¾itost, èili oproti sekvenènímu algoritmu jsme si nepomohli.
+\figure{hloupe_scitani.eps}{Sèítání ¹kolním algoritmem}{1.5in}
-Zamysleme se nad tím, jak by se proces sèítání mohl zrychlit.
+Ka¾dá krabièka má sice uvnitø konstantní hloubku, ale její výstupy závisí na pøenosu
+vypoèítaném pøedcházející krabièkou. Jednotlivé krabièky tedy musí urèitì le¾et
+v~rùzných vrstvách sítì. Celkovì má tedy sí» $\Theta(n)$ hladin a také $\Theta(n)$
+hradel. Oproti sekvenènímu algoritmu jsme si tedy vùbec nepomohli.
\h{Pøenosy v~blocích}
-Jediné, co nás pøi sèítání brzdí, jsou pøenosy mezi jednotlivými øády. Ka¾dý øád,
-aby~vydal souèet, musí poèkat na~to, a¾~dopoèítají v¹echny pøedcházející øády.
-Teprve pak se~toti¾ dozví pøenos. Kdybychom ov¹em pøenosy dokázali spoèítat
-nìjakým zpùsobem paralelnì, máme vyhráno. Jakmile známe v¹echny pøenosy, souèet
-u¾~zvládneme dopoèítat na~konstantnì mnoho hladin -- tedy v~konstantním èase.
+
+Jak sèítání zrychlit? To, co nás pøi sèítání brzdí, jsou evidentnì pøenosy mezi
+jednotlivými øády. Kdybychom je dokázali spoèítat rychleji, máme vyhráno -- souèet
+u¾ získáme jednoduchým {\csc xor}ováním, které zvládneme paralelnì v~èase $\Theta(1)$.
+Uva¾ujme tedy nad zpùsobem, jak pøenosy spoèítat paralelnì.
Podívejme se na~libovolný {\I blok} v~na¹em souètu. Tak budeme øíkat èíslùm
$x_j\ldots x_i$ a $y_j\ldots y_i$ v~nìjakém intervalu indexù $\left<i,j\right>$. Pøenos $c_{j+1}$ vystupující z~tohoto bloku závisí mimo hodnot sèítancù u¾ pouze na~pøenosu $c_{i}$, který do bloku vstupuje.
-\figure{blok_scitani.eps}{Blok souètu.}{3in}
+\figure{blok_scitani.eps}{Blok souètu}{3in}
-Z tohoto pohledu se mù¾eme na blok také dívat jako na nìjakou funkci, která
-dostane jednobitový vstup a vydá jednobitový výstup. To je nám pøíjemné, nebo»
-takových funkcí existují jenom ètyøi typy:
-\numlist\ndotted
-\:$f(x) = 0$, ~~~~(0)
-\:$f(x) = 1$, ~~~~(1)
-\:$f(x) = x$, ~~~~($<$ -- kopírování)
-\:$f(x) = \neg{x}$.
-\endlist
-Jak se za~chvíli uká¾e, poslední pøípad, kdy by~nìjaký blok pøedával opaèný
-pøenos, ne¾ do~nìj vstupuje, navíc nikdy nemù¾e nastat. Pojïme si to rozmyslet.
-Jednobitové bloky se chovají velice jednodu¹e:
+Pro konkrétní sèítance se tedy mù¾eme na blok dívat jako na nìjakou funkci,
+která dostane jednobitový vstup (pøenos zespoda) a vydá jednobitový výstup (pøenos
+nahoru). To je nám milé, nebo» takové funkce existují pouze ètyøi:
+$$\vbox{\halign{$f(x) = #$\hfil &\qquad # \hfil\cr
+0&konstantní {\bo 0}, blok {\I pohlcuje} pøenos \cr
+1&konstantní {\bo 1}, blok {\I vytváøí} pøenos \cr
+x&identita (znaèíme {\tt <}), blok {\I kopíruje} pøenos \cr
+\neg{x}&negace, uká¾eme, ¾e nenastane \cr
+}}$$
+Této funkci budeme øíkat {\I chování} pøíslu¹ného bloku.
-\figure{bloky_1bit.eps}{Tabulka triviálních blokù.}{1.1in}
+{\I Jednobitové bloky} se pøitom chovají velice jednodu¹e:
-Z prvního bloku evidentnì v¾dy vyleze 0, a»~do~nìj vstoupí jakýkoli pøenos.
-Poslední blok naopak sám o~sobì pøenos vytváøí, a»~ji¾ do~nìj vleze jakýkoliv.
-Bloky prostøední se chovají stejnì, a~to tak, ¾e~samy o~sobì ¾ádný pøenos nevytvoøí,
+\figure{bloky_1bit.eps}{Tabulka triviálních blokù}{1.1in}
+
+Blok prvního druhu v¾dy pøedává nulový pøenos, a» u¾ do~nìj vstoupí jakýkoliv.
+Poslední blok naopak sám o~sobì pøenos vytváøí, a» dostane cokoliv.
+Oba bloky prostøední se chovají tak, ¾e samy o~sobì ¾ádný pøenos nevytvoøí,
ale~pokud do~nich nìjaký pøijde, tak~také odejde.
-Mìjme nyní nìjaký vìt¹í blok~$C$ slo¾ený ze~dvou men¹ích podblokù $A$ a~$B$, jejich¾
-chování u¾ známe. Z~toho mù¾eme odvodit, jak se chová celý blok:
+{\I Vìt¹í bloky} mù¾eme rozdìlit na èásti a podle jejich chování urèit,
+jak se chová celý blok. Mìjme blok~$B$ slo¾ený z~men¹ích podblokù~$H$ (horní)
+a~$D$ (dolní), jejich¾ chování u¾ známe. Z~toho mù¾eme odvodit, jak se chová celý blok:
-\figure{tabulka_skladani_bloku.eps}{Skládání chování blokù.}{1.3in}
+\figure{tabulka_skladani_bloku.eps}{Skládání chování blokù}{1.3in}
Pokud vy¹¹í blok pøenos pohlcuje, pak a»~se u¾~ni¾¹í blok chová jakkoli, slo¾ení
-tìchto blokù musí v¾dy pohlcovat. V~prvním øádku tabulky jsou tudí¾ nuly. Analogicky,
+obou blokù musí v¾dy pohlcovat. V~prvním øádku tabulky jsou tudí¾ nuly. Analogicky,
pokud vy¹¹í blok generuje pøenos, tak~ten ni¾¹í na~tom nic nezmìní. V~druhém
øádku tabulky jsou tedy samé jednièky. Zajímavìj¹í pøípad nastává, pokud vy¹¹í blok
kopíruje -- tehdy zále¾í èistì na~chování ni¾¹ího bloku.
V¹imnìme si, ¾e~skládání chování blokù je vlastnì úplnì obyèejné skládání
funkcí. Nyní bychom mohli prohlásit, ¾e~budeme poèítat nad~tøíprvkovou abecedou,
-a~¾e~celou tabulku doká¾eme spoèítat jedním jediným hradlem. Pojïmì si v¹ak
+a~¾e~celou tabulku doká¾eme spoèítat jedním jediným hradlem. Pojïme si v¹ak
rozmyslet, jak~bychom takovouto vìc popsali èistì binárnì. Jak tedy tyto tøi stavy
popisovat pouze nìkolika bity?
kterým pøiøadíme tøi mo¾ná chování bloku. Toto kódování mù¾eme zvolit zcela
libovolnì, ale pokud si ho zvolíme ¹ikovnì, u¹etøíme si dále práci pøi kompozici.
Zvolme si tedy kódování takto:
-
-\itemize\ibull
-\:$(1,*) = <$,
-\:$(0,0) = 0$,
-\:$(0,1) = 1$
-\endlist
-
-Tomu, ¾e blok kopíruje, odpovídá dvojice $p = 1$; $q =$ \<cokoliv>. V~ostatních
+$$
+ (1,*) = \hbox{\tt <} \qquad
+ (0,0) = \hbox{\bo 0} \qquad
+ (0,1) = \hbox{\bo 1}.
+$$
+Tomu, ¾e blok kopíruje, odpovídá dvojice $p = 1$, $q = \<cokoliv.>$ V~ostatních
pøípadech bude~$p$ nulové a~$q$ nám bude øíkat, co je na~výstupu pøíslu¹ného bloku.
Jinými slovy $p = 0$ znamená, ¾e funkce je konstanta, pøièem¾ $q$ øíká jaká; naproti
tomu $p = 1$~znamená, ¾e funkce je identita, a»~u¾~je $q$ cokoli.
-Pojïme si nyní ukázat, jak bude celé skládání blokù vypadat. Rozmysleme si,
-kdy je~$p$ celého dvojbloku jednièkové, tedy kdy celý dvojblok kopíruje. To nastane
-tehdy, pokud kopírují obì jeho èásti, a tedy $p = p_a~\&~p_b$. Dále $q$ bude rovno jednièce,
-pokud $q = (\neg{p_a}~\&~q_a) \lor (p_a~\&~q_b)$.
-
-Skládání chování blokù lze tedy popsat buï ternárnì -- tabulkou, ale lze to
-i~binárnì vý¹e uvedeným pøedpisem.
-
-Nyní si tedy mù¾eme dopøedu vypoèítat chování bloku velikosti jedna, poté
- z~nich skládáním blokù velikosti dva, dál velikosti ètyøi, osm, atd \dots
+V~tomto kódování mù¾eme na¹i tabulku popsat následovnì:
+$$\eqalign{
+ p_B &= p_H \land p_D,\cr
+ q_B &= (\neg{p_H} \land q_H) \lor (p_H \land q_D).
+}$$
\h{Paralelní sèítání}
-Paralelní algoritmus na~sèítání u¾~zkonstruujeme pomìrnì snadno. Bez újmy
-na~obecnosti budeme pøedpokládat, ¾e~poèet bitù vstupních èísel je mocnina dvojky,
-jinak si vstup doplníme nulami, co¾ výsedný èas bìhu algoritmu zhor¹í maximálnì
-konstanta-krát.
-
+Z~popisu chování blokù u¾ je jenom krùèek k~paralelnímu pøedpovídání pøenosù,
+a~tím i k~paralelní sèítaèce. Bez újmy na~obecnosti budeme pøedpokládat,
+¾e~poèet bitù vstupních èísel je mocnina dvojky, jinak si vstup doplníme
+nulami, co¾ výsledný èas bìhu algoritmu zhor¹í maximálnì konstanta-krát.
+
+Algoritmus bude rozdìlen na~dvì èásti. V~první èásti spoèítá chování
+v¹ech {\I pøirozených blokù} -- tak budeme øíkat blokùm, jejich¾ velikost
+je mocnina dvojky a pozice je dìlitelná velikostí. Nejprve v~konstantním
+èase spoèítá chování blokù velikosti~1, ty pak spojí do dvojic, dvojice
+zase do dvojic atd., obecnì v~$i$-tém kroku spoète chování v¹ech pøirozených
+blokù velikosti~$2^i$.
+
+V~druhé èásti pak dopoèítává pøenosy, a~to tak, aby v~$i$-tém kroku byly známy
+pøenosy do~øádù dìlitelných~$2^{\log n - i}$. V~nultém kroku tedy pouze $c_0=0$
+a~$c_n$, který spoèítá z~$c_0$ pomocí chování bloku $\left< 0,n \right>$.
+V~prvním kroku pomocí $\left< 0,n/2 \right>$ dopoèítá $c_{n/2}$,
+v~druhém pomocí $\left< 0,n/4 \right>$ spoèítá $c_{n/4}$ a pomocí
+$\left< n/2,3/4\cdot n \right>$ dostane $c_{3/4\cdot n}$ atd.
+Obecnì v~$i$-tém kroku pou¾ívá chování blokù velikosti $2^{\log n - i}$.
+
+\s{Sèítací sí»} tedy bude vypadat takto:
\algo
-\:Spoèteme chování blokù velikosti~1. ($\O(1)$ hladin)
-\:Postupnì poèítáme chování blokù\foot{myslíme \uv{pøirozenì zarovnané} bloky, tedy takové, jejich¾ poloha je násobkem velikosti} velikosti 2, 4, 8, ..., $2^k$.
- ($\O(\log n)$ hladin, na~nich¾ se skládají bloky)
-\:$c_0 \leftarrow 0$ (pøenos do nejni¾¹ího øádu je v¾dy 0)
-\:Urèíme $c_n$ podle $c_0$ a chování (jediného) bloku velikosti~$n$.
-\:Postupnì dopoèítáme pøenosy na~hranicích dìlitelných $2^k$ \uv{zahu¹»ováním}:
- jakmile víme $c_{2^k}$, mù¾eme dopoèítat $c_{2^k+2^{k-1}}$ podle
- chování bloku $\left< 2^k, 2^k+2^{k-1}\right>$. ($\O(\log n)$ hladin,
- na~nich¾ se dosazuje)
-\:Seèteme: $\forall i: z_i = x_i \oplus y_i \oplus c_i$.
+\:$\Theta(1)$ hladin výpoètu chování blokù velikosti~1.
+\:$\Theta(\log n)$ hladin poèítajících chování pøirozených blokù velikosti~$2^i$.
+\:$\Theta(\log n)$ hladin dopoèítávajících pøenosy \uv{zahu¹»ováním}.
+\:$\Theta(1)$ hladin na samotné seètení: $\forall i: z_i = x_i \oplus y_i \oplus c_i$.
\endalgo
-\figure{1_9_deleni_bloku.eps}{Výpoèet pøenosu.}{2.5in}
+\figure{1_9_deleni_bloku.eps}{Výpoèet pøenosu}{2.5in}
-Algoritmus pracuje v~èase $\O(\log n)$. Hradel je pou¾ito lineárnì: na~jednotlivých
-hladinách kroku~2 poèet hradel exponenciálnì klesá od~$n$ k~1, na~hladinách kroku~5
-exponenciálnì stoupá od~1 k~$n$, tak¾e dohromady se seète na~$\Theta(n)$.
+Algoritmus tedy pracuje v~èase $\Theta(\log n)$. Hradel je pou¾ito lineárnì: pøi
+výpoètu chování blokù na jednotlivých hladinách poèet hradel exponenciálnì klesá
+od~$n$ k~1, bìhem zahu¹»ování pøenosù naopak exponenciálnì stoupá od~1 k~$n$.
+Obì geometrické øady mají souèet $\Theta(n)$.
\h{Paralelní násobení}
Podobnì jako u~sèítání si vzpomeneme na~¹kolní algoritmus -- tentokráte v¹ak pro násobení.
To fungovalo tak, ¾e~jsme si jedno ze~dvou binárních èísel na~vstupu (øíkejme mu tøeba $x$) $n$-krát posouvali. Tam kde pak byly v~èísle~$y$ jednièky, pøíslu¹né kopie $x$ jsme seèetli. Jinými slovy tedy násobení umíme pøevést na~nìjaké posuny (ty lze realizovat pouze \uv{pøedrátováním} -- nic nás nestojí), násobení jedním bitem (co¾ je
{\csc and} ) a~nakonec potøebujeme výsledných~$n$ èísel seèíst.
-\figure{skolni_scitani.eps}{©kolní sèítání.}{2in}
+\figure{skolni_scitani.eps}{©kolní sèítání}{2in}
Jak nyní seèíst $n$ $n$-bitových èísel..? Nabízí se vyu¾ít osvìdèený \uv{stromeèek} -- sèítat dvojice èísel, výsledky pak opìt po~dvojicích seèíst, a¾ na~konci vyjde jediný výsledek.
\figure{stromecek.eps}{Stromeèek}{1.4in}
Seètení v¹ech $n$ èísel tedy zabere $\Theta (\log n)$ hladin.
-Kdy¾ se nyní vrátíme k~násobení, zbývá nám vyøe¹it posouvání a~{\csc and}ování. Uvìdomme si, ¾e to je plnì paralelení a~zvládneme ho za~konstantnì mnoho hladin. Celé násobení tedy zvládneme v~logaritmickém èase.
+Kdy¾ se nyní vrátíme k~násobení, zbývá nám vyøe¹it posouvání a~{\csc and}ování. Uvìdomme si, ¾e to je plnì paralelní a~zvládneme ho za~konstantnì mnoho hladin. Celé násobení tedy zvládneme v~logaritmickém èase.
\bye
+++ /dev/null
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projects / ads2.git / commitdiff
