\def\XOR{{\csc xor}}
\def\and{\mathbin{\&}}
-®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).
+Pomocí poèítaèù øe¹íme stále slo¾itìj¹í a rozsáhlej¹í úlohy a potøebujeme
+k~tomu èím dál víc výpoèetního výkonu. Rychlost a kapacita poèítaèù zatím
+rostla exponenciálnì, tak¾e se zdá, ¾e staèí chvíli poèkat. Jen¾e tento rùst
+se urèitì nìkdy zastaví -- napøíklad není jasné, jestli pùjde 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,
Ta poèítají jednotlivé logické funkce, mezi nejbì¾nìj¹í patøí:
\itemize\ibull
-\:nulární: to jsou konstanty ($\hbox{\csc false}=0$, $\hbox{\csc true}=1$),
-\:unární: identita a negace (\NOT,~$\lnot$),
-\:binární: logický souèin (\AND,~$\and$), souèet (\OR,~$\lor$), \dots
+\:nulární funkce: to jsou konstanty ($\hbox{\csc false}=0$, $\hbox{\csc true}=1$),
+\:unární funkce: identita a negace (\NOT,~$\lnot$),
+\:binární funkce: logický souèin (\AND,~$\and$), souèet (\OR,~$\lor$), \dots
\endlist
Propojením hradel pak vznikne {\I hradlová sí».} Ne¾ vyøkneme formální definici,
\figure{hradlova_sit.eps}{Hradlová sí» -- tøívstupová verze funkce {\I majorita}}{3in}
-Na¹e sí» má tøi vstupy, vnitøní booleovská hradla a jeden výstup. Na výstupu
+Na¹e sí» má tøi vstupy, pìt booleovských hradel 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.
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
+dal¹ích hradel (mohou se vìtvit), nebo na výstupy sítì. Libovolnì, jen nesmíme
vytváøet cykly.
Nyní toté¾ formálnìji:
$I$~({\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.}
+ na více rùzných vstupù jiného 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~$h$;
\>Pøitom jsou splnìny následující podmínky:
\itemize\ibull
-\:$\forall i \in I: \deg^+(i)=0$ {\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).}
+\:Do vstupù nevedou ¾ádné hrany.
+\:Z~výstupù nevedou ¾ádné hrany a do ka¾dého výstupu vede právì jedna hrana.
+\:Do ka¾dého hradla vede tolik hran, kolik je jeho arita.
+\:V¹echny vstupy hradel jsou zapojeny. Tedy pro ka¾dé hradlo~$h$ a ka¾dý
+ jeho vstup $j\in \{1,\ldots,a(h)\}$ existuje právì jedna hrana~$e$, která
+ vede do hradla~$h$ a~$z(e)=j$.
\endlist
\s{Poznámka:} Nìkdy se hradlovým sítím také øíká {\I kombinaèní obvody} a pokud pracují
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
+uspoøádaných podle funkce~$z$. Výstup sítì pouze zkopíruje hodnotu, která do
nìj po~hranì pøi¹la.
Jakmile budou po~nìjakém poètu taktù definované hodnoty v¹ech výstupù, výpoèet
Podle prùbìhu výpoètu mù¾eme vrcholy sítì rozdìlit do vrstev:
-\s{Definice:} {\I $i$-tá vrstva $S_i$} obsahuje právì takové vrcholy~$v$,
+\s{Definice:} {\I $i$-tá vrstva $S_i$} obsahuje právì ty vrcholy~$v$,
pro~které nejdel¹í z~cest z~libovolného vrcholu se~vstupním stupnìm~0
do~$v$ má délku rovnou právì~$i$.
\>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.
+vrstev. Podobnì {\I prostorovou slo¾itost} polo¾íme rovnu poètu hradel v~síti.
\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
+jakýkoliv 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.
-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.
+To v¹ak není ani realistické, ani pìkné. Jak z~toho ven? Pøidáme pravidlo,
+které omezí arity v¹ech hradel nìjakou pevnou konstantou, tøeba dvojkou.
+Budeme tedy pou¾ívat výhradnì nulární, unární a binární hradla.
Poznamenejme je¹tì, ¾e realistický model (by» s~trochu jinými vlastnostmi)
by vznikl také tehdy, kdybychom místo arity omezily typy funkcí, øeknìme
na \AND, \OR{} a \NOT.
\s{Poznámka o~uniformitì:}
-Dodejme, ¾e od~bì¾ných výpoèetním modelùm, jako je tøeba RAM, se hradlové
+Dodejme, ¾e od~bì¾ných výpoèetních modelù, 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.}
A~co myslíme oním pøedpisem pro sestrojení sítì? Bude to pro nás prostì
-nìjaký algoritmus (klasický, neparalelní) bì¾ící v~polynomiálním èase.
+jakýkoliv 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.)
+sítì. To obvykle není ¾ádoucí.)
\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 \OR.
+To znamená vypoèítat $n$-vstupovou funkci \OR.
-\>{\I První øe¹ení:} zapojíme hradla za~sebe (sériovì). Èasová i prostorová
+\>{\I První øe¹ení:} Zapojíme hradla za~sebe (sériovì). Èasová i prostorová
slo¾itost èiní $\Theta(n)$. Zde ov¹em vùbec nevyu¾íváme toho, ¾e by mohlo poèítat více
hradel souèasnì.
-\>{\I Druhé øe¹ení:} Hradla budeme spojovat do~dvojic, pak výsledky z~tìchto dvojic opìt
+\>{\I Druhé øe¹ení:} Hradla budeme spojovat do~dvojic, pak výsledky tìchto dvojic opìt
do~dvojic a tak dále. Díky paralelnímu zapojení dosáhneme èasové slo¾itosti $\Theta(\log n)$,
-prostorová slo¾itost zùstane lineární.
+pøièem¾ prostorová slo¾itost zùstane lineární.
\twofigures{hloupy_or.eps}{Sériové øe¹ení}{1in}{chytry_or.eps}{Paralelní øe¹ení}{3in}
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 $2^i$.
+$x_{n-1}\ldots x_0$ a $y_{n-1}\ldots y_0$, pøièem¾ $i$-tý øád má váhu $2^i$.
-K~seètení se~ihned nabízí pou¾ít starý dobrý \uv{¹kolní algoritmus sèítání pod sebou}, který
+Ihned se nabízí pou¾ít starý dobrý \uv{¹kolní algoritmus sèítání pod sebou}. Ten
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,
$$
kde $z_i$ je $i$-tá èíslice souètu, $\oplus$ znaèí operaci \XOR{} (souèet modulo~2) a~$c_i$ je {\I pøenos}
-z~$(i-1)$-ního øádu do~$i$-tého. Pøenos pøitom nastane tehdy, pokud se~nám potkají
+z~$(i-1)$-ního øádu do~$i$-tého. Pøenos do vy¹¹ího øádu nastane tehdy, pokud se~nám potkají
dvì jednièky pod~sebou, nebo kdy¾ se~vyskytne alespoò jedna jednièka a~k~tomu
pøenos z~ni¾¹ího øádu. To je tehdy, kdy¾ mezi tøemi xorovanými èíslicemi jsou alespoò
dvì jednièky -- k~tomu se nám hodí ji¾ známý obvod pro majoritu:
\figure{hloupe_scitani.eps}{Sèítání ¹kolním algoritmem}{1.5in}
-Ka¾dá krabièka má sice uvnitø konstantní hloubku, ale její výstupy závisí na pøenosu
-vypoèítaném pøedcházející krabièkou. Jednotlivé krabièky tedy musí urèitì le¾et
-v~rùzných vrstvách sítì. Celkovì má tedy sí» $\Theta(n)$ hladin a také $\Theta(n)$
-hradel. Oproti sekvenènímu algoritmu jsme si tedy vùbec nepomohli.
+Ka¾dá krabièka má sama o~sobì konstantní hloubku, ov¹em k~výpoètu potøebuje pøenos
+vypoèítaný pøedcházející krabièkou. Jednotlivé krabièky proto musí le¾et
+v~rùzných vrstvách sítì. Vrstev tedy je $\Theta(n)$ a stejnì tak hradel.
+Oproti sekvenènímu algoritmu jsme si bohu¾el ani trochu nepomohli.
\h{Pøenosy v~blocích}
Jak sèítání zrychlit? To, co nás pøi sèítání brzdí, jsou evidentnì pøenosy mezi
jednotlivými øády. Kdybychom je dokázali spoèítat rychleji, máme vyhráno -- souèet
-u¾ získáme jednoduchým \XOR{}ováním, které zvládneme paralelnì v~èase $\Theta(1)$.
+u¾ získáme jednoduchým xorováním, které zvládneme paralelnì v~èase $\Theta(1)$.
Uva¾ujme tedy nad zpùsobem, jak pøenosy spoèítat paralelnì.
Podívejme se na~libovolný {\I blok} v~na¹em souètu. Tak budeme øíkat èíslùm
0&konstantní {\bo 0}, blok {\I pohlcuje} pøenos \cr
1&konstantní {\bo 1}, blok {\I vytváøí} pøenos \cr
x&identita (znaèíme {\tt <}), blok {\I kopíruje} pøenos \cr
-\neg{x}&negace, uká¾eme, ¾e nenastane \cr
+\neg{x}&negace, uká¾eme, ¾e u~¾ádného bloku nenastane \cr
}}$$
Této funkci budeme øíkat {\I chování} pøíslu¹ného bloku.
\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,
+Blok prvního druhu v¾dy pøedává nulový pøenos, a» u¾ do~nìj vstoupí jakýkoliv
+-- pøenos tedy pohlcuje. 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.
-{\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:
+{\I Vìt¹í bloky} mù¾eme rozdìlit na èásti a podle chování èástí urèit,
+jak se chová celý blok. Mìjme blok~$B$ slo¾ený z~men¹ích podblokù~$H$ (horní èást)
+a~$D$ (dolní). Chování celku závisí na chování èástí takto:
\figure{tabulka_skladani_bloku.eps}{Skládání chování blokù}{1.3in}
Pokud vy¹¹í blok pøenos pohlcuje, pak a»~se u¾~ni¾¹í blok chová jakkoli, slo¾ení
-obou 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. Pojïme si v¹ak
-rozmyslet, jak~bychom takovouto vìc popsali èistì binárnì. Jak tedy tyto tøi stavy
-popisovat pouze nìkolika bity?
-
-Evidentnì nám k tomuto binárnímu zakódování tøí stavù budou staèit bity dva.
-Oznaème si je jako $p$ a $q$. Tato dvojice mù¾e nabývat hned ètyø mo¾ných hodnot,
-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:
+a~¾e~celou tabulku doká¾eme spoèítat jedním jediným hradlem. Pojïme si pøeci jen
+rozmyslet, jak~bychom takovou operaci popsali èistì binárnì.
+
+Tøi stavy mù¾eme zakódovat pomocí dvou bitù, øíkejme jim tøeba $p$ a~$q$.
+Dvojice $(p,q)$ pøitom mù¾e nabývat hned ètyø mo¾ných hodnot, my dvìma z~nich
+pøiøadíme stejný význam:
$$
(1,*) = \hbox{\tt <} \qquad
(0,0) = \hbox{\bo 0} \qquad
(0,1) = \hbox{\bo 1}.
$$
-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.
-
-V~tomto kódování mù¾eme na¹i tabulku popsat následovnì:
+Kdykoliv $p=1$, blok kopíruje pøenos. Naopak $p=0$ odpovídá tomu, ¾e blok
+posílá do~vy¹¹ího øádu konstantní pøenos, a $q$~pak urèuje, jaký.
+Kombinování blokù (skládání funkcí) pak mù¾eme popsat následovnì:
$$\eqalign{
p_B &= p_H \and p_D,\cr
q_B &= (\neg{p_H} \and q_H) \lor (p_H \and q_D).
}$$
+A~prùchod pøenosu blokem (dosazení hodnoty funkce) takto:
+$$
+ c_{j+1} = (p \and c_i) \lor (\neg p \and q).
+$$
+Rozmyslete si, ¾e tyto formule odpovídají vý¹e uvedené tabulce.
\h{Paralelní sèítání}
-Z~popisu chování blokù u¾ je jenom krùèek k~paralelnímu pøedpovídání pøenosù,
+Od~popisu chování blokù je u¾ 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.
+¾e~poèet bitù vstupních èísel je mocnina dvojky; jinak vstup doplníme zleva
+nulami.
-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~$2^i$.
+Algoritmus bude rozdìlen na~dvì èásti:
-V~druhé èásti pak dopoèítává pøenosy, a~to tak, aby v~$i$-tém kroku byly známy
+{\I První èást} spoèítá chování v¹ech {\I pøirozených blokù} -- tak budeme øíkat
+blokùm, jejich¾ velikost je mocnina dvojky a pozice je dìlitelná velikostí
+(bloky té¾e velikosti se tedy nepøekrývají). Nejprve v~konstantním èase stanoví
+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~$2^i$.
+
+{\I Druhá èást} pak dopoèítá 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~prvním kroku pomocí bloku $\left< 0,n/2 \right>$ 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
\:$\Theta(1)$ hladin výpoètu chování blokù velikosti~1.
-\:$\Theta(\log n)$ hladin poèítajících chování pøirozených blokù velikosti~$2^i$.
+\:$\Theta(\log n)$ hladin poèítajících chování v¹ech pøirozených blokù.
\:$\Theta(\log n)$ 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{deleni_bloku.eps}{Výpoèet pøenosu}{2.5in}
-Algoritmus tedy pracuje v~èase $\Theta(\log n)$. Hradel je pou¾ito lineárnì: pøi
-výpoètu chování blokù na jednotlivých hladinách poèet hradel exponenciálnì klesá
+Algoritmus tedy pracuje v~èase $\Theta(\log n)$. Vyu¾ívá k~tomu lineárnì mnoho hradel:
+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 se seètou na $\Theta(n)$.
Je¹tì si rozmysleme, jak rychle by bylo mo¾né èísla násobit. Opìt se inspirujeme
¹kolním algoritmem: pokud násobíme dvì $n$-ciferná èísla $x$ a~$y$, uvá¾íme
-v¹ech~$n$ posunutí èísla~$y$, ka¾dé z~nich vynásobíme pøíslu¹nou èíslicí v~$y$
+v¹ech~$n$ posunutí èísla~$x$, ka¾dé z~nich vynásobíme pøíslu¹nou èíslicí v~$y$
a výsledky posèítáme.
\figure{skolni_nasobeni.eps}{©kolní násobení}{2in}
To v¹e stihneme za 1~takt výpoètu.
Zbývá seèíst $n$~èísel, z~nich¾ ka¾dé má $\Theta(n)$ bitù. Mohli bychom opìt
-pou¾ít osvìdèený trik: sèítat dvojice èísel, pak dvojice souètù dvojic, atd.
+sáhnout po~osvìdèeném triku: sèítat dvojice èísel, pak dvojice tìchto souètù, atd.
Taková sí» by mìla tvar binárního stromu hloubky $\log n$, jeho¾ ka¾dý vrchol
-obsahuje jednu sèítaèku a na tu, jak víme, postaèí $\Theta(\log n)$ hladin.
+by obsahoval jednu sèítaèku, a na tu, jak víme, postaèí $\Theta(\log n)$ hladin.
Celý výpoèet tedy bì¾í v~èase $\Theta(\log^2 n)$.
Jde to ale rychleji, pou¾ijeme-li jednoduchý, témìø kouzelnický trik.
dvoubitové èíslo, tak¾e mù¾eme jeho ni¾¹í bit prohlásit za~$p_i$
a vy¹¹í za~$q_{i+1}$.
-Jinými slovy jsme v¹echna tøi èísla normálnì seèetli, ale místo abychom
+Jinými slovy v¹echna tøi èísla jsme normálnì seèetli, ale místo abychom
pøenosy posílali do vy¹¹ího øádu, vytvoøili jsme z~nich dal¹í èíslo,
které má být k~výsledku èasem pøièteno.
V~na¹em formalismu hradlových sítí bychom mohli komparátor reprezentovat dvojicí
hradel: jedno z~nich by poèítalo minimum, druhé maximum. Hodnoty, které tøídíme,
-bychom prostì pova¾ovali za prvky abecedy.\foot{Komparátorovou sí» mù¾eme také snadno
+bychom pøitom pova¾ovali za prvky abecedy.\foot{Komparátorovou sí» mù¾eme také snadno
pøelo¾it na booleovský obvod. Ka¾dý prvek abecedy reprezentujeme èíslem
o~$b=\lceil\log_2 \Sigma\rceil$ bitech. Zpùsobem podobným paralelní sèítaèce
lze z~booleovských hradel sestrojit komparátor hloubky $\Theta(\log b)$. Zkuste to.}
z~nich pøivedeme na~vstup jiného komparátoru nebo na~výstup sítì. Vìtvení
by nám ostatnì k~nièemu nebylo, proto¾e na~výstupu potøebujeme vydat stejný
poèet hodnot, jako byl na vstupu, a nemáme ¾ádné hradlo, kterým bychom mohli
-pøípadných více vìtví slouèit opìt do~jedné.
+pøípadných vícero vìtví slouèit opìt do~jedné.
\s{Pøíklad:} Zkusíme do øeèi komparátorových sítí pøelo¾it {\I bublinkové tøídìní.}
Z~nìj získáme obvod na~levém obrázku (¹ipky pøedstavují jednotlivé komparátory).
rozdìlit na nìjaké pozici $k\in\{0, \dots, n-1\}$ tak, ¾e prvky $x_0,\ldots,x_k$ tvoøí rostoucí
poslopnost, zatímco prvky $x_k,\ldots,x_{n-1}$ tvoøí posloupnost klesající.
-\s{Definice:} Posloupnost $x_0,\dots,x_{n-1}$ je {\I bitonická}, pokud ji lze získat
-rotací (cyklickým posunutím) nìjaké èistì bitonické posloupnosti. Jinými slovy pokud existuje
+\s{Definice:} Posloupnost $x_0,\dots,x_{n-1}$ je {\I bitonická}, jestli¾e ji lze získat
+rotací (cyklickým posunutím) nìjaké èistì bitonické posloupnosti. Tedy pokud existuje
$0\le j<n$ takové, ¾e posloupnost $x_j,x_{(j+1) \bmod n},\dots, x_{(j+n-1) \bmod n}$
je èistì bitonická.
\itemize\ibull
\:$y_0,\ldots,y_{n/2-1}$ a $y_{n/2},\ldots,y_{n-1}$ jsou bitonické posloupnosti;
-\:$y_i < y_j$, kdykoliv $0\le i<n/2$ a $n/2\le j<n$.
+\:$y_i < y_j$, kdykoliv $0\le i<n/2 \le j<n$.
\endlist
\>Jinak øeèeno, separátor rozdìlí bitonickou posloupnost na dvì polovièní
\proof
Konstrukce bitonické tøidièky je snadná: nejprve separátorem~$S_n$ zadanou bitonickou
-posloupnost rozdìlíme na dvì bitonické posloupnosti délky $n/2$,
-pak ka¾dou z~nich separátorem~$S_{n/2}$ na dvì délky $n/4$, atd.,
-a¾ získáme jednoprvkové bitonické posloupnosti ve~správném poøadí.
+posloupnost rozdìlíme na dvì bitonické posloupnosti délky $n/2$, ka¾dou z~nich pak separátorem~$S_{n/2}$
+na dvì èásti délky $n/4$, atd., a¾ získáme jednoprvkové posloupnosti ve~správném poøadí.
Celkem pou¾ijeme~$\log n$ hladin slo¾ených z~$n$ separátorù, ka¾dá
hladina má pøitom konstantní hloubku.
\qed
Proè separátor separuje? Nejprve pøedpokládejme, ¾e vstupem je èistì bitonická
posloupnost. Oznaème~$m$ polohu maxima této posloupnosti; maximum bez újmy
na obecnosti le¾í v~první polovinì (jinak celý dùkaz provedeme \uv{zrcadlovì}).
-Oznaème dále~$k$ nejmen¹í index, pro který komparátor mezi $x_k$ a~$x_{k+n/2}$
+Oznaème dále~$k$ nejmen¹í index, pro který komparátor zapojený mezi $x_k$ a~$x_{k+n/2}$
hodnoty prohodí, tedy $k=\min \{ i \mid x_i > x_{i+n/2} \}.$
Jeliko¾ maximum je jedineèné, musí platit $x_m > x_{m+n/2}$, tak¾e~$k$
existuje a navíc $0\le k\le m < n/2$. Také platí, ¾e pro ka¾dé~$i$ mezi
$k$ a~$n/2$ u¾ komparátory musí prohazovat, proto¾e od~$x_k$ je posloupnost
-a¾ do konce klesající, tak¾e $x_i > x_{i+n/2}$.
+a¾ do konce klesající, a~proto $x_i > x_{i+n/2}$.
Separátor se tedy chová velice jednodu¹e: první polovina výstupu vznikne
slepením rostoucího úseku $x_0,\ldots,x_{k-1}$ s~klesajícím úsekem $x_{n/2+k},\ldots,x_{n-1}$;
druhou polovinu tvoøí spojení klesajícího úseku $x_{n/2},\ldots,x_{n/2+k-1}$, rostoucího
-úseku $x_k,\ldots,x_m$ a klesajícího úseku $x_m,\ldots,x_{n/2-1}$. První polovina
+úseku $x_k,\ldots,x_m$ a klesajícího úseku $x_m,\ldots,x_{n/2-1}$.
+
+Snadno tedy ovìøíme, ¾e se separátor chová podle definice: První polovina
je èistì bitonická a jeliko¾ $x_{n/2-1} > x_{n/2}$, je druhá polovina bitonická
-(ov¹em obvykle ne èistì).
+(obvykle ne èistì). Navíc víme, ¾e nejvy¹¹ím bodem první èásti je~$x_k$
+a nejni¾¹ím bodem druhé èásti~$x_{n/2+k} > x_k$, tak¾e v¹echny prvky první
+èásti musí být men¹í ne¾ v¹echny v~té druhé.
\figure{sortnet.7}{Ilustrace èinnosti separátoru}{\epsfxsize}
-Doplòme, co se stane, pokud vstup není èistì bitonický. Zde vyu¾ijeme
-toho, ¾e pokud vstup separátoru zrotujeme o~$p$ pozic, dostaneme o~$p$ pozic
-zrotované i obì poloviny výstupu. Podle definice ov¹em pro ka¾dou bitonickou
+Doplòme, co se stane, pokud vstup není èistì bitonický. Zde vyu¾ijeme toho,
+¾e separátor je symetrický, tudí¾ zrotujeme-li jeho vstup o~$p$ pozic, dostaneme
+o~$p$ pozic zrotované i obì poloviny výstupu. Podle definice ov¹em pro ka¾dou bitonickou
posloupnost existuje její rotace, která je èistì bitonická, a~pro ní¾, jak
u¾ víme, separátor funguje. Tak¾e pro neèistou bitonickou posloupnost musí
vydat výsledek pouze zrotovaný, co¾ ov¹em na jeho správnosti nic nemìní.
\qed
-Ukázali jsme tedy paralelní tøídící algoritmus o~slo¾itosti $\Theta(\log^2 n)$
-slo¾ený z~$\Theta(n\log^2 n)$ komparátorù.
-
-Dodejme je¹tì, ¾e existuje i~tøídicí algoritmus, kterému staèí jen $\O(\log n)$
-hladin. Jeho multiplikativní konstanta je v¹ak pøíli¹ veliká, tak¾e je v~praxi
-nepou¾itelný.
+\s{Závìrem:} Nalezli jsme paralelní tøídící algoritmus o~èasové slo¾itosti $\Theta(\log^2 n)$,
+který vyu¾ívá $\Theta(n\log^2 n)$ komparátorù. Dodejme, ¾e jsou známé i tøídící sítì
+hloubky $\Theta(\log n)$, ale jejich konstrukce je mnohem komplikovanìj¹í a dává
+obrovské multiplikativní konstanty, které brání praktickému pou¾ití.
+
+Pomocí dolního odhadu slo¾itosti tøídìní (viz minulý semestr) navíc mù¾eme
+snadno dokázat, ¾e logaritmický poèet hladin je nejni¾¹í mo¾ný. Máme-li toti¾
+libovolnou tøídící sí» hloubky~$h$, mù¾eme ji simulovat po~hladinách a získat
+tak sekvenèní tøídící algoritmus. Jeliko¾ na ka¾dé hladinì mù¾e le¾et nejvý¹e~$n/2$
+komparátorù, ná¹ algoritmus provede maximálnì $hn/2$ porovnání. My ov¹em víme, ¾e pro
+ka¾dý tøídící algoritmus existují vstupy, na~kterých porovná $\Omega(n\log n)$-krát.
+Proto $h=\Omega(\log n)$.
\bye
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+label.bot(btex $<$ etex, .5*z595 + 0.5*z513);
endfig;
label.bot(btex \strut $k$ etex,z2);
label.bot(btex \strut $m$ etex,z8);
label.llft(btex \strut ${n\over 2}$ etex,z42);
-label.bot(btex \strut $k+{n\over 2}$ etex,z3);
+label.bot(btex \strut ${n\over 2}+k$ etex,z3);
label.bot(btex \strut $n$ etex,z72);
endfig;
vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
_n1="cmmi10";
+_s("<",_n1,1.00000,0.0000,0.0000,);
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+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
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+_n1="cmmi10";
_s("S",_n1,1.00000,0.0000,0.0000,);
_n5="cmmi5";
_s("n",_n5,1.00000,7.3045,1.1829,);
vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
_n1="cmmi10";
+_s("<",_n1,1.00000,0.0000,0.0000,);
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+begingroup save _p,_r,_s,_n; picture _p; _p=nullpicture;
+string _n[];
+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+ addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
_s("S",_n1,1.00000,0.0000,0.0000,);
_n5="cmmi5";
_s("n",_n5,1.00000,7.3045,1.1829,);
vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
_n1="cmmi10";
+_s("<",_n1,1.00000,0.0000,0.0000,);
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+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+ addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
_s("S",_n1,1.00000,0.0000,0.0000,);
_n5="cmmi5";
_s("n",_n5,1.00000,7.3045,1.1829,);
vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
_n1="cmmi10";
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+vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
+ addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
+_n1="cmmi10";
_s("M",_n1,1.00000,0.0000,0.0000,);
_n2="cmr7";
_s("4",_n2,1.00000,9.6652,-1.4944,);
string _n[];
vardef _s(expr _t,_f,_m,_x,_y)(text _c)=
addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef;
-_n1="cmmi10";
-_s("k",_n1,1.00000,0.0000,0.0000,);
-_n0="cmr10";
-_s("+",_n0,1.00000,7.7141,0.0000,);
_n3="cmmi7";
-_s("n",_n3,1.00000,18.8722,3.9226,);
+_s("n",_n3,1.00000,1.1955,3.9226,);
interim linecap:=0;
vardef _r(expr _a,_w)(text _t) =
- addto _p doublepath _a withpen pencircle scaled _w _t enddef;_r((18.8722,2.4907)..(23.7971,2.4907), 0.3985,);
+ addto _p doublepath _a withpen pencircle scaled _w _t enddef;_r((1.1955,2.4907)..(6.1204,2.4907), 0.3985,);
_n2="cmr7";
-_s("2",_n2,1.00000,19.3490,-3.4355,);
+_s("2",_n2,1.00000,1.6723,-3.4355,);
+_n0="cmr10";
+_s("+",_n0,1.00000,9.5298,0.0000,);
+_n1="cmmi10";
+_s("k",_n1,1.00000,19.4924,0.0000,);
setbounds _p to (0,-3.4869)--(24.9926,-3.4869)--
(24.9926,8.4682)--(0,8.4682)--cycle;
_p endgroup
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projects / ads2.git / commitdiff
