From: Martin Mares Date: Mon, 21 Nov 2011 12:29:24 +0000 (+0100) Subject: Hradla: Korektury X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=1d1065ce0b396612ac81758559122d8e4b7861a4;p=ads2.git Hradla: Korektury --- diff --git a/5-hradla/5-hradla.tex b/5-hradla/5-hradla.tex index 65dcd6c..b096802 100644 --- a/5-hradla/5-hradla.tex +++ b/5-hradla/5-hradla.tex @@ -8,12 +8,11 @@ \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, @@ -43,9 +42,9 @@ jinak t 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, @@ -53,7 +52,7 @@ poj \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. @@ -61,7 +60,7 @@ p 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: @@ -73,7 +72,7 @@ Nyn $I$~({\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~$h$; @@ -85,11 +84,12 @@ Nyn \>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 @@ -102,7 +102,7 @@ dal 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 @@ -110,7 +110,7 @@ se zastav 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~$i$. @@ -138,48 +138,48 @@ do~$v$ m \>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} @@ -187,9 +187,9 @@ prostorov 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: @@ -197,7 +197,7 @@ $$ z_i=x_i \oplus y_i \oplus c_i, $$ 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: @@ -215,16 +215,16 @@ $y_i$ a~$c_i$ a jejich v \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 @@ -239,7 +239,7 @@ $$\vbox{\halign{$f(x) = #$\hfil &\qquad # \hfil\cr 0&konstantní {\bo 0}, blok {\I pohlcuje} pøenos \cr 1&konstantní {\bo 1}, blok {\I vytváøí} pøenos \cr x&identita (znaèíme {\tt <}), blok {\I kopíruje} pøenos \cr -\neg{x}&negace, uká¾eme, ¾e nenastane \cr +\neg{x}&negace, uká¾eme, ¾e u~¾ádného bloku nenastane \cr }}$$ Této funkci budeme øíkat {\I chování} pøíslu¹ného bloku. @@ -247,68 +247,68 @@ T \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 $p = 1$, $q = \$ V~ostatních -pøípadech bude~$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 $2^{\log n - i}$. @@ -316,15 +316,15 @@ Obecn \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 $\Theta(n)$. @@ -332,7 +332,7 @@ Ob 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} @@ -342,9 +342,9 @@ Ve~dvojkov 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. @@ -363,7 +363,7 @@ Pro ka 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. @@ -388,7 +388,7 @@ v 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.} @@ -397,7 +397,7 @@ Je 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). @@ -419,8 +419,8 @@ je mocnina dvojky. 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 $0\le jJinak øeèeno, separátor rozdìlí bitonickou posloupnost na dvì polovièní @@ -450,9 +450,8 @@ s~$\Theta(n\log n)$ kompar \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 @@ -505,36 +504,46 @@ a maximum na~$y_{i+n/2}$. 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 diff --git a/5-hradla/hradlova_sit.eps b/5-hradla/hradlova_sit.eps index 965f4dc..2e71eb0 100644 --- a/5-hradla/hradlova_sit.eps +++ b/5-hradla/hradlova_sit.eps @@ -1,1217 +1,542 @@ %!PS-Adobe-3.0 EPSF-3.0 -%%Creator: inkscape 0.46 +%%Creator: cairo 1.8.10 (http://cairographics.org) +%%CreationDate: Mon Nov 21 12:13:54 2011 %%Pages: 1 -%%Orientation: Portrait -%%BoundingBox: 10 386 593 690 -%%HiResBoundingBox: 10.085321 386.06581 592.01527 689.33641 +%%BoundingBox: 0 0 593 304 +%%DocumentData: Clean7Bit +%%LanguageLevel: 2 %%EndComments +%%BeginProlog +/cairo_eps_state save def +/dict_count countdictstack def +/op_count count 1 sub def +userdict begin +/q { gsave } bind def +/Q { grestore } bind def +/cm { 6 array astore concat } bind def +/w { setlinewidth } bind def +/J { setlinecap } bind def +/j { setlinejoin } bind def +/M { setmiterlimit } 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%*Font: cmmi10 9.96265 9.96265 3a:8000000000000003 %*Font: cmr7 6.97385 6.97385 30:e diff --git a/5-hradla/sortnet.4 b/5-hradla/sortnet.4 index 7fda675..3663559 100644 --- a/5-hradla/sortnet.4 +++ b/5-hradla/sortnet.4 @@ -2,7 +2,7 @@ %%BoundingBox: -1 -1 213 100 %%HiResBoundingBox: -0.19925 -0.19925 212.79729 99.41167 %%Creator: MetaPost 1.208 -%%CreationDate: 2011.11.19:2203 +%%CreationDate: 2011.11.21:1318 %%Pages: 1 %%BeginProlog %%EndProlog diff --git a/5-hradla/sortnet.5 b/5-hradla/sortnet.5 index 8e9dd3b..6409ccb 100644 --- a/5-hradla/sortnet.5 +++ b/5-hradla/sortnet.5 @@ -2,9 +2,9 @@ %%BoundingBox: -1 13 213 114 %%HiResBoundingBox: -0.19925 13.97395 212.79729 113.58487 %%Creator: MetaPost 1.208 -%%CreationDate: 2011.11.19:2203 +%%CreationDate: 2011.11.21:1318 %%Pages: 1 -%*Font: cmmi10 9.96265 9.96265 3a:8000004000000803 +%*Font: cmmi10 9.96265 9.96265 3a:a000004000000803 %*Font: cmmi7 6.97385 6.97385 6e:8 %*Font: cmmi5 4.98132 4.98132 6e:8 %*Font: cmr5 4.98132 4.98132 32:a @@ -171,6 +171,8 @@ newpath 163.60349 61.30731 moveto 167.99107 61.30731 lineto stroke 164.10228 57.16331 moveto (2) cmr5 4.98132 fshow +102.42468 62.49522 moveto +(<) cmmi10 9.96265 fshow 14.56696 32.7118 moveto (S) cmmi10 9.96265 fshow 21.87146 33.89471 moveto @@ -187,6 +189,8 @@ newpath 78.56427 32.9609 moveto 82.95186 32.9609 lineto stroke 79.06306 28.81691 moveto (4) cmr5 4.98132 fshow +45.73187 34.14882 moveto +(<) cmmi10 9.96265 fshow 127.95258 32.7118 moveto (S) cmmi10 9.96265 fshow 135.25708 33.89471 moveto @@ -195,6 +199,8 @@ newpath 135.25708 32.9609 moveto 139.64467 32.9609 lineto stroke 135.75587 28.81691 moveto (4) cmr5 4.98132 fshow +102.42468 34.14882 moveto +(<) cmmi10 9.96265 fshow 184.64539 32.7118 moveto (S) cmmi10 9.96265 fshow 191.94989 33.89471 moveto @@ -203,5 +209,7 @@ newpath 191.94989 32.9609 moveto 196.33748 32.9609 lineto stroke 192.44868 28.81691 moveto (4) cmr5 4.98132 fshow +159.1175 34.14882 moveto +(<) cmmi10 9.96265 fshow showpage %%EOF diff --git a/5-hradla/sortnet.6 b/5-hradla/sortnet.6 index 2372c9e..1c79e55 100644 --- a/5-hradla/sortnet.6 +++ b/5-hradla/sortnet.6 @@ -2,9 +2,9 @@ %%BoundingBox: -1 -1 213 100 %%HiResBoundingBox: -0.19925 -0.19925 212.79729 99.41167 %%Creator: MetaPost 1.208 -%%CreationDate: 2011.11.19:2203 +%%CreationDate: 2011.11.21:1318 %%Pages: 1 -%*Font: cmmi10 9.96265 9.96265 3a:8000104000000803 +%*Font: cmmi10 9.96265 9.96265 3a:a000104000000803 %*Font: cmr7 6.97385 6.97385 30:e8 %%BeginProlog %%EndProlog diff --git a/5-hradla/sortnet.7 b/5-hradla/sortnet.7 index 97baca1..75d7c88 100644 --- a/5-hradla/sortnet.7 +++ b/5-hradla/sortnet.7 @@ -2,10 +2,10 @@ %%BoundingBox: 17 -1 182 110 %%HiResBoundingBox: 17.35184 -0.19925 181.5723 109.33292 %%Creator: MetaPost 1.208 -%%CreationDate: 2011.11.19:2203 +%%CreationDate: 2011.11.21:1318 %%Pages: 1 %*Font: cmr10 9.96265 9.96265 2b:87e00000000003023004 -%*Font: cmmi10 9.96265 9.96265 3a:8000104000005803 +%*Font: cmmi10 9.96265 9.96265 3a:a000104000005803 %*Font: cmr7 6.97385 6.97385 30:e8 %*Font: cmmi7 6.97385 6.97385 6e:8 %%BeginProlog @@ -70,16 +70,16 @@ newpath 90.99202 11.765 moveto 95.91693 11.765 lineto stroke 91.46883 5.83879 moveto (2) cmr7 6.97385 fshow -117.86421 8.37428 moveto -(k) cmmi10 9.96265 fshow -125.57831 8.37428 moveto -(+) cmr10 9.96265 fshow -136.7364 12.29689 moveto +119.05971 12.29689 moveto (n) cmmi7 6.97385 fshow -newpath 136.7364 10.86499 moveto -141.66132 10.86499 lineto stroke -137.21321 4.93878 moveto +newpath 119.05971 10.86499 moveto +123.98462 10.86499 lineto stroke +119.53651 4.93878 moveto (2) cmr7 6.97385 fshow +127.39401 8.37428 moveto +(+) cmr10 9.96265 fshow +137.35661 8.37428 moveto +(k) cmmi10 9.96265 fshow 175.59239 8.37428 moveto (n) cmmi10 9.96265 fshow showpage diff --git a/5-hradla/sortnet.mp b/5-hradla/sortnet.mp index aebdcd1..82c3148 100644 --- a/5-hradla/sortnet.mp +++ b/5-hradla/sortnet.mp @@ -500,10 +500,14 @@ label.llft(btex $n$ etex,z075); label.bot(btex $S_n$ etex,z175); label.bot(btex $S_{n\over 2}$ etex,z335); label.bot(btex $S_{n\over 2}$ etex,z311); +label.bot(btex $<$ etex, .5*z335 + 0.5*z311); label.bot(btex $S_{n\over 4}$ etex,z515); label.bot(btex $S_{n\over 4}$ etex,z555); +label.bot(btex $<$ etex, .5*z515 + 0.5*z555); label.bot(btex $S_{n\over 4}$ etex,z595); +label.bot(btex $<$ etex, .5*z555 + 0.5*z595); label.bot(btex $S_{n\over 4}$ etex,z513); +label.bot(btex $<$ etex, .5*z595 + 0.5*z513); endfig; @@ -672,7 +676,7 @@ label.bot(btex \strut 0 etex,z12); 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; diff --git a/5-hradla/sortnet.mpx b/5-hradla/sortnet.mpx index 58f8d19..092c715 100644 --- a/5-hradla/sortnet.mpx +++ b/5-hradla/sortnet.mpx @@ -360,6 +360,16 @@ 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("<",_n1,1.00000,0.0000,0.0000,); +setbounds _p to (0,-0.3895)--(7.7487,-0.3895)-- + (7.7487,5.3708)--(0,5.3708)--cycle; +_p endgroup +mpxbreak +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,); @@ -394,6 +404,16 @@ 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("<",_n1,1.00000,0.0000,0.0000,); +setbounds _p to (0,-0.3895)--(7.7487,-0.3895)-- + (7.7487,5.3708)--(0,5.3708)--cycle; +_p endgroup +mpxbreak +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,); @@ -411,6 +431,16 @@ 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("<",_n1,1.00000,0.0000,0.0000,); +setbounds _p to (0,-0.3895)--(7.7487,-0.3895)-- + (7.7487,5.3708)--(0,5.3708)--cycle; +_p endgroup +mpxbreak +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,); @@ -428,6 +458,16 @@ 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("<",_n1,1.00000,0.0000,0.0000,); +setbounds _p to (0,-0.3895)--(7.7487,-0.3895)-- + (7.7487,5.3708)--(0,5.3708)--cycle; +_p endgroup +mpxbreak +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("M",_n1,1.00000,0.0000,0.0000,); _n2="cmr7"; _s("4",_n2,1.00000,9.6652,-1.4944,); @@ -556,17 +596,17 @@ 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("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,); 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