From: Martin Mares Date: Fri, 18 Nov 2011 18:04:34 +0000 (+0100) Subject: Oziveni hradlovych siti a bitonickeho trideni, zatim bez uprav X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=8d2a42401e1f2d65d659f649f4d84e0b6ff31703;p=ads2.git Oziveni hradlovych siti a bitonickeho trideni, zatim bez uprav --- diff --git a/5-hradla/1_9_deleni_bloku.eps b/5-hradla/1_9_deleni_bloku.eps new file mode 100644 index 0000000..2c96b1c --- /dev/null +++ b/5-hradla/1_9_deleni_bloku.eps @@ -0,0 +1,3225 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: Mayura Draw, Version 4.3 +%%Title: 1_9_deleni_bloku.ai +%%CreationDate: Mon Oct 29 23:24:27 2007 +%%BoundingBox: -9 -9 276 157 +%%EndComments +%%BeginProlog +%%BeginResource: procset MayuraDraw_ops +%%Version: 4.3 +%%Copyright: (c) 1993-2003 Mayura Software +/PDXDict 100 dict def +PDXDict begin +% width height matrix proc key cache +% definepattern -\> font +/definepattern { %def + 7 dict begin + /FontDict 9 dict def + FontDict begin + /cache exch def + /key 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+%%EOF diff --git a/5-hradla/5-hradla.tex b/5-hradla/5-hradla.tex new file mode 100644 index 0000000..f995d4f --- /dev/null +++ b/5-hradla/5-hradla.tex @@ -0,0 +1,318 @@ +\input ../lecnotes.tex + +\prednaska{5}{Hradlové sítì}{} + +\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. + +\h{Hradlové sítì} + +\s{Definice:} {\I Hradlo} je prvek, který umí vyhodnocovat nìjakou funkci +nad~koneènou abecedou $\Sigma$. + +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: + +\itemize\ibull +\:nulární: to jsou konstanty (FALSE=0, TRUE=1), +\:unární: napø. negace (znaèíme~$\lnot$), +\:binární: logický souèin ({\csc and},~$\land$), souèet ({\csc or},~$\lor$), ... +\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.} + +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. + +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}) \hfil\break 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~$e$ konèí v~$h$ a~$z(e)=j$, + (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 \csc 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 +její výpoèet probíhá. + +\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{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, 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 + +\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. + +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: +$$ +z_i=x_i \oplus y_i \oplus c_i, +$$ +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): +$$ +\eqalign{ +c_{0} &= 0, \cr +c_{i+1} &= (x_i~\&~y_i)\lor(x_i~\&~c_i)\lor(y_i~\&~c_i).\cr +} +$$ + +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} + +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. + +Zamysleme se nad tím, jak by se proces sèítání mohl zrychlit. + +\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. + +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 $\left$. 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} + +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: + +\figure{bloky_1bit.eps}{Tabulka triviálních blokù.}{1.1in} + +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, +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: + +\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, +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 +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: + +\itemize\ibull +\:$(1,*) = <$, +\:$(0,0) = 0$, +\:$(0,1) = 1$ +\endlist + +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. + +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 + +\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. + +\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: $\forall i: z_i = x_i \oplus y_i \oplus c_i$. +\endalgo + +\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~$\Theta(n)$. + +\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} + +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} + +Toto øe¹ení by v¹ak vedlo na~èasovou slo¾itost $\Theta (\log^2 n)$, nebo» sèítat umíme v~èase $\Theta (\log n)$. To je sice dle na¹ich mìøítek +docela efektivní, ale pøekvapivì to jde je¹tì lépe -- toti¾ na~$\Theta (\log n)$ hladin. Této +slo¾itosti dosáhneme malým trikem. + +Vymyslíme obvod konstantní hloubky, který na~vstupu dostane tøi èísla. Odpoví pak dvìma èísly +takovými, ¾e budou mít stejný souèet jako pùvodní tøi èísla. Jinými slovy pomocí +tohoto obvodu budeme umìt seètení tøí èísel pøevést na seètení dvou èísel. + +\figure{obvod.eps}{$x+y+z = p+q$}{0.3in} + +V¹imnìme si, ¾e~kdy¾ sèítáme tøi bity, mù¾e být pøenos do~vy¹¹ího øádu nula èi~jednièka. Vezmeme si tedy bity $x_i$, $y_i$, $z_i$ a~ty seèteme. To nám dá dvoubitový výsledek, pøièem¾ ni¾¹í bit z~tohoto výsledku po¹leme do~èísla~$p$, vy¹¹í do~èísla~$q$. + +\figure{obvod_real.eps}{Redukování sèítání}{1.2in} + +Toto zredukování sèítání nám nyní umo¾ní opìt stavit strom, by» o malièko slo¾itìj¹í. + +\figure{sl_stromecek.eps}{Slo¾itìj¹í stromeèek}{0.9in} + +Pokud jsme mìli na~zaèátku $n$ èísel, po~první redukci nám jich zbývá jen $2/3 \cdot n$ a~obecnì v~$k$-té hladiné $(2/3)^k \cdot n$. Znamená to, ¾e èísel nám ubývá exponenciálnì, tak¾e poèet hladin bude logaritmický. Redukující obvod je pøi tom jen konstantnì hluboký, tak¾e celé redukování zvládneme v~èase $\Theta (\log n)$. Na~konci Slo¾itìj¹ího stromeèku pak máme umístìnou jednu obyèejnou sèítaèku, která zbývající dvì èísla seète v~logaritmickém èase. + +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. + +\bye diff --git a/5-hradla/Makefile b/5-hradla/Makefile new file mode 100644 index 0000000..9d4d631 --- /dev/null +++ b/5-hradla/Makefile @@ -0,0 +1,3 @@ +P=5-hradla + +include ../Makerules diff --git a/5-hradla/blok_scitani.eps b/5-hradla/blok_scitani.eps new file mode 100644 index 0000000..225a323 --- /dev/null +++ b/5-hradla/blok_scitani.eps @@ -0,0 +1,1673 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: inkscape 0.46 +%%Pages: 1 +%%Orientation: Portrait +%%BoundingBox: 79 592 505 727 +%%HiResBoundingBox: 79.471632 592.47642 504.27389 726.62485 +%%EndComments +%%Page: 1 1 +0 842 translate +0.8 -0.8 scale +0 0 0 setrgbcolor +[] 0 setdash +1 setlinewidth +0 setlinejoin +0 setlinecap +gsave [1 0 0 1 0 0] concat +gsave [1 0 0 1 -304.88268 66.42131] concat +gsave +0 0 0 setrgbcolor +newpath +474.79669 99.649963 moveto +470.44611 105.50446 lineto +475.02228 111.68121 lineto +472.69122 111.68121 lineto +469.18927 106.95465 lineto +465.68732 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na~druhý problém, a~to na~problém tøídìní. Ji¾ známe pomìrnì efektivní sekvenèní algoritmy, které dovedou tøídit v~èase $\O(n\log n)$. Byli bychom jistì rádi, kdybychom to zvládli je¹tì rychleji. Pojïme se podívat, zda by nám v tom nepomohlo problém paralelizovat. + +Budeme pøi tom pracovat ve~výpoèetním modelu, kterému se øíká komparátorová sí». Ta je postavená z~hradel, kterým se øíká komparátory. + +\s{Definice:} {\I Komparátorová sí»} je kombinaèní obvod, jeho¾ hradla jsou +komparátory. + +\figure{sortnet.0}{Komparátor}{0.7in} + +Komparátor umí porovnat dvì hodnoty a~rozhodnout, která z~nich je vìt¹í +a~která men¹í. Nevrací v¹ak booleovský výsledek jako bì¾né hradlo, ale má dva +výstupy, pøièem¾ na~jednom vrátí men¹í ze~vstupních hodnot a~na~druhém vìt¹í. + +Výstupy komparátorù se nevìtví. Nemù¾eme tedy jeden výstup \uv{rozdvojit} a~pøipojit ho na~dva vstupy. (Vìtvení by dokonce ani nemìlo smysl, proto¾e zatímco rozdvojit bychom mohli, slouèit u¾~ne. Pokud tedy chceme, aby sí» mìla $n$~vstupù i~$n$~výstupù, rozdvojení stejnì nesmíme provést, i kdybychom jej mìli povolené.) + +\s{Pøíklad:} {\sl Bubble sort} + +Obrázek Bubble 1 ilustruje pou¾ití komparátorù pro tøídìní Bubble sortem. +©ipky pøedstavují jednotlivé komparátory. Výpoèet v¹ak je¹tì mù¾eme vylep¹it. + +\twofigures{sortnet.1}{Bubble 1}{143pt}{sortnet.2}{Bubble 2}{143pt} + +Sna¾íme se výpoèet co nejvíce paralelizovat (viz obrázek Bubble 2). +Jak je vidìt, komparátory na sebe nemusejí èekat. Tím mù¾eme výpoèet urychlit a místo èasu $\Theta{(n^2)}$ docílit èasové slo¾itosti $\Theta{(n)}$. V obou pøípadech je zachován kvadratický prostor. + +Nyní si uká¾eme je¹tì rychlej¹í tøídící algoritmus. Pùjdeme na nìj v¹ak trochu \uv{od lesa}. Nejdøíve vymyslíme sí», která bude umìt tøídit jenom nìco -- toti¾ bitonické posloupnosti. Bez újmy na obecnosti budeme pøedpokládat, ¾e ka¾dé dva prvky na vstupu jsou navzájem rùzné. + +\medskip +\s{Definice:} Øekneme, ¾e posloupnost $x_0,\dots,x_{n-1} $ je {\I èistì bitonická} právì tehdy, kdy¾ +pro nìjaké $x_k, k\in\{0, \dots, n-1\} $ platí, ¾e~v¹echny prvky pøed ním (vèetnì jeho samotného) +tvoøí rostoucí poslopnost, kde¾to prvky stojící za~ním tvoøí poslopnost klesající. +Formálnì zapsáno musí platit, ¾e: +$$x_0\leq x_1\leq \dots \leq x_k \geq x_{k+1}\geq\dots \geq x_{n-1}.$$ + +\s{Definice:} Posloupnost $x_0 \dots x_{n-1}$ je {\I bitonická}, právì kdy¾ $\exists~j\in \{0,\dots ,n-1\}$, pro +které je rotace pùvodní posloupnosti o $j$ prvkù, tedy posloupnost +$$x_j,x_{(j+1) \bmod n},\dots, x_{(j+n-1) \bmod n},$$ èistì bitonická. + +\s{Definice:} {\I Separátor $S_n$} je sí», ve které jsou v¾dy~$i$-tý a~$(i+{n/2})$-tý prvek vstupu +(pro $i=0,\dots, {n/2}-1$) propojeny komparátorem. Minimum se pak stane~$i$-tým, +maximum $(i+{n/2})$-tým prvkem výstupu. +\figure{sortnet.3}{$(y_i, y_{i+{n/2}}) = \(x_i, x_{i+{n/2}})$} {300pt} + +\s{Lemma:} Pokud separátor dostane na~vstupu bitonickou posloupnost, pak jeho výstup $y_0, \dots, y_{n-1}$ +splòuje: + +(i) $y_0,\dots, y_{n/2 -1}$ a~$y_{n/2},\dots, y_{n-1}$ jsou +bitonické posloupnosti, + +(ii) Pro v¹echna $i,j< {n/2}$ platí $y_i < y_{j + {n/2}}$. + +Separátor nám tedy jednu bitonickou posloupnost na~vstupu rozdìlí na~dvì +bitonické posloupnosti, pøièem¾ navíc ka¾dý prvek první posloupnosti ($y_0,\dots, y_{n/2 -1}$) +je men¹í nebo roven prvkùm druhé posloupnosti ($y_{n/2}, \dots, y_{n-1}$). + +\>Ne¾ pøistoupíme k dùkazu lemmatu, uka¾me si, k èemu se nám bude hodit. + + +\s{Definice:} {\I Bitonická tøídièka $B_n$} je obvod sestavený ze separátorù, který dostane-li na vstupu bitonickou posloupnost délky $n$ (BÚNO konstruujeme tøídièku pro $n=2^k$), vydá setøídìnou zadanou posloupnost délky $n$. + +Tøídièka dostane na~vstupu bitonickou posloupnost. Separátor~$S_n$ ji pak dle lemmatu rozdìlí na~dvì bitonické posloupnosti, kdy je ka¾dý prvek z~první posloupnosti men¹í ne¾ libovolný prvek z~druhé. Tyto poloviny pak dal¹í separátory rozdìlí na~ètvrtiny, ..., a¾~na~konci zbudou pouze jednoduché posloupnosti délky jedna (zjevnì setøídìné), které mezi sebou mají po¾adovanou nerovnost -- tedy ka¾dá posloupnost (nebo spí¹e prvek) nalevo je $\leq$ ne¾ prvek napravo od~nìj. + +\centerline{\epsfbox{sortnet.5}} + +Jak je vidìt, bitonická tøídièka nám libovolnou bitonickou posloupnost délky~$n$ +setøídí na~$\Theta(\log n)$ hladin. + +Nyní se dá odvodit, ¾e pokud umíme tøídit bitonické posloupnosti, umíme setøídit v¹echno. +Vzpomeòme si na~tøídìní sléváním -- Merge sort. To funguje tak, ¾e zaène s~jednoprvkovými posloupnostmi, které jsou evidentnì setøídìné, a~poté v¾dy dvì setøídìné posloupnosti slévá do~jedné. Kdybychom nyní umìli paralelnì slévat, mohli bychom vytvoøit i~paralelní Merge sort. Jinými slovy, potøebujeme dvì rostoucí posloupnosti nìjak efektivnì slít do~jedné setøídìné. Uvìdomme si, ¾e to zvládneme jednodu¹e -- staèí druhou z~posloupností obrátit a~\uv{pøilepit} za~první, èím¾ vznikne bitonická posloupnost, kterou poté mù¾eme setøídit na¹í bitonickou tøídièkou. + +\s{Pøíklad:} {\sl Merge sort} + +Bitonická tøídièka se tedy dá pou¾ít ke~slévání setøídìných posloupností. +Uka¾me si, jak s~její pomocí sestavíme souèástky {\I slévaèky} $M_n$: + +Setøídìné posloupnosti $x_0,\dots, x_{n-1}$ a~$y_0,\dots, y_{n-1}$ spojíme do jedné bitonické +posloupnosti $x_0,\dots, x_{n-1},y_{n-1},\dots, y_0$. Z~této posloupnosti vytvoøíme pomocí bitonické tøídièky +$B_{2n}$ setøídìnou posloupnost. Vytvoøíme tedy blok $M_{2n}$, který se ov¹em sestává de~facto pouze z~bloku $B_{2n}$, jeho¾ druhá polovina vstupù je zapojena v~obráceném poøadí. +\figure{sortnet.6}{Paralelní MergeSort.}{3in} + +Nyní se pokusme odhadnout èasovou slo¾itost. Ná¹ MergeSort bude mít øádovì hloubku blokù $\log n$. +V ka¾dém bloku $M_n$ je navíc ukryta bitonická tøídièka s takté¾ logaritmickou hloubkou. +Celková hloubka tedy bude $\log 2 + \log4 + \dots + \log 2^k + \dots + \log n$. +Po seètení nakonec dostáváme výslednou èasovou slo¾itost $\Theta (\log^2 n)$. + +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. + +Vra»me se nyní k dùkazu lemmatu, který jsme na pøedminulé stránce vynechali. + +\h{Dùkaz Lemmatu:} +(i) Nejprve nahlédneme, ¾e lemma platí, je-li vstupem èistì bitonická +posloupnost. Dále BÚNO pøedpokládejme, ¾e vrchol posloupnosti je v~první polovinì (kdyby +byl vrchol za~polovinou, staèilo by zrcadlovì obrátit posloupnost i~komparátory a~øe¹ili +bychom stejný problém). Nyní si definujme $k := \min j: x_j > x_{j+n/2}$. (Pokud +by~takové~$k$ neexistovalo, znamenalo by~to, ¾e vstupní posloupnost je monotónní. +Separátor by tedy nic nedìlal a~pouze zkopíroval vstup na~výstup, co¾ jistì lemma splòuje.) Nyní si v¹imìme, +¾e jakmile jednou zaène platit, ¾e prvek na~levé stranì je men¹í ne¾ na~pravé, bude +nám tato relace platit a¾~do~konce. Oznaème vrchol vstupní posloupnosti jako~$x_m$. +Pak~$k$ bude jistì men¹í ne¾~$m$ a~$k+{n/2}$ bude vìt¹í ne¾~$m$. Mezi~$k$ a~$m$ je tedy +vstupní posloupnost neklesající, mezi $k+{n/2}$ a~$n-1$ nerostoucí. + +Do pozice~$k$ tedy separátor bude pouze kopírovat vstup na~výstup, od~pozice~$k$ +dál u¾~jen prohazuje. Pro ka¾dé~$i$, $(k\leq i\leq {n/2}-1)$ se prvky +$x_i$ a~$x_{i+{n/2}}$ prohodí. Úsek mezi~$k$ a~${n/2}-1$ tedy +nahradíme nerostoucí posloupností, první polovina výstupu tedy bude +(dokonce èistì) bitonická. Podobnì úsek $k+{n/2}$ a¾~$n-1$ nahradíme èistì +bitonickou posloupností. Obì poloviny tedy budou bitonické. + +Dostaneme-li na vstupu obecnou bitonickou posloupnost, pøedstavíme si, +¾e je to èistì bitonická posloupnost zrotovaná o~$r$ prvkù (BÚNO +doprava). Zjistíme, ¾e v~komparátorech se porovnávají tyté¾ prvky, +jako kdyby zrotovaná nebyla. Výstup se od výstupu èistì bitonické +posloupnosti zrotovaného o~$r$ bude li¹it prohozením úsekù $x_0$ a¾ +$x_{r-1}$ a~$x_{n/2}$ a¾ $x_{{n/2}+r-1}$. Obì výstupní +posloupnosti tedy budou zrotované o~$r$ prvkù, ale na jejich +bitoniènosti se nic nezmìní. + +(ii) Z dùkazu (i) pro èistì bitonickou posloupnost víme, ¾e $y_0\dots y_{n/2-1}$ je èistì bitonická a bude rovna $x_0\dots x_{k-1},x_{k+n/2}\dots x_{n-1}$ pro vhodné $k$ a navíc bude mít maximum v $x_{k-1}$ nebo $x_k+{n/2}$. Mezi tìmito body ov¹em ve vstupní posloupnosti urèitì nele¾el ¾ádný $x_i$ men¹í ne¾ $x_k-1$ nebo $x_k+{n/2}$ (jak je vidìt z obrázku) a posloupnost $x_k \dots x_{k-1+{n/2}}$ je rovna $y_{n/2}\dots y_{n-1}$. Pro obecné bitonické posloupnosti uká¾eme stejnì jako v (i). +\qed + +\medskip + +\centerline{\epsfbox{sortnet.7}} + +\bye diff --git a/6-bitonic/Makefile b/6-bitonic/Makefile new file mode 100644 index 0000000..496cb3d --- /dev/null +++ b/6-bitonic/Makefile @@ -0,0 +1,3 @@ +P=6-bitonic + +include ../Makerules diff --git a/6-bitonic/sortnet.0 b/6-bitonic/sortnet.0 new file mode 100644 index 0000000..a2087ec --- /dev/null +++ b/6-bitonic/sortnet.0 @@ -0,0 +1,65 @@ +%!PS +%%BoundingBox: -1 19 60 100 +%%Creator: MetaPost +%%CreationDate: 2008.01.20:2138 +%%Pages: 1 +%*Font: cmr10 9.96265 9.96265 61:c08c01 +%%EndProlog +%%Page: 1 1 + 0 0.3985 dtransform truncate idtransform setlinewidth pop [] 0 setdash + 1 setlinecap 1 setlinejoin 10 setmiterlimit +newpath 0 39.68497 moveto +19.84248 39.68497 lineto +39.68497 39.68497 lineto +59.52745 39.68497 lineto +59.52745 79.36993 lineto +39.68497 79.36993 lineto +19.84248 79.36993 lineto +0 79.36993 lineto +0 39.68497 lineto stroke + 0.3985 0 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+label.bot(btex \strut b etex,z33); +label.top(btex min etex,z22); +label.top(btex max etex,z23); + + +endfig; + +beginfig(1); +v:=u*5mm; +z10=(0v,0v); +z20=(2v,0v); +z30=(4v,0v); +z40=(6v,0v); +z50=(8v,0v); + +z11=(0v,1v); +z12=(0v,2v); +z13=(0v,3v); +z14=(0v,4v); +z15=(0v,5v); +z16=(0v,6v); +z17=(0v,7v); +z18=(0v,8v); +z19=(0v,9v); +z110=(0v,10v); + +z21=(2v,1v); +z22=(2v,2v); +z23=(2v,3v); +z24=(2v,4v); +z25=(2v,5v); +z26=(2v,6v); +z27=(2v,7v); +z28=(2v,8v); +z29=(2v,9v); +z210=(2v,10v); + +z31=(4v,1v); +z32=(4v,2v); +z33=(4v,3v); +z34=(4v,4v); +z35=(4v,5v); +z36=(4v,6v); +z37=(4v,7v); +z38=(4v,8v); +z39=(4v,9v); +z310=(4v,10v); + +z41=(6v,1v); +z42=(6v,2v); +z43=(6v,3v); +z44=(6v,4v); +z45=(6v,5v); +z46=(6v,6v); +z47=(6v,7v); +z48=(6v,8v); +z49=(6v,9v); +z410=(6v,10v); + +z51=(8v,1v); +z52=(8v,2v); +z53=(8v,3v); +z54=(8v,4v); +z55=(8v,5v); +z56=(8v,6v); +z57=(8v,7v); +z58=(8v,8v); +z59=(8v,9v); +z510=(8v,10v); + +z111=(0v,11v); +z211=(2v,11v); +z311=(4v,11v); +z411=(6v,11v); +z511=(8v,11v); + +z1015=(-1v,1.5v); +z1615=(9v,1.5v); +z1035=(-1v,3.5v); +z1635=(9v,3.5v); +z1065=(-1v,6.5v); +z1665=(9v,6.5v); + +pickup pencircle scaled 0.4pt; +draw(z10--z111); +draw(z20--z211); +draw(z30--z311); +draw(z40--z411); +draw(z50--z511); +drawarrow(z11--z21); +drawarrow(z13--z23); +drawarrow(z16--z26); +drawarrow(z110--z210); +drawarrow(z22--z32); +drawarrow(z25--z35); +drawarrow(z29--z39); +drawarrow(z34--z44); +drawarrow(z38--z48); +drawarrow(z47--z57); + +pickup pencircle scaled 0.7pt; +draw(z1015--z1615) dashed withdots scaled 0.7; +draw(z1035--z1635) dashed withdots scaled 0.7; +draw(z1065--z1665) dashed withdots scaled 0.7; + +label.top(btex x1 etex,z111); +label.top(btex x2 etex,z211); +label.top(btex x3 etex,z311); +label.top(btex x4 etex,z411); +label.top(btex x5 etex,z511); + +endfig; + +beginfig(2); +v:=u*5mm; +z13=(0v,3v); +z23=(2v,3v); +z33=(4v,3v); +z43=(6v,3v); +z53=(8v,3v); + +z14=(0v,4v); +z15=(0v,5v); +z16=(0v,6v); +z17=(0v,7v); +z18=(0v,8v); +z19=(0v,9v); +z110=(0v,10v); + +z24=(2v,4v); +z25=(2v,5v); +z26=(2v,6v); +z27=(2v,7v); +z28=(2v,8v); +z29=(2v,9v); +z210=(2v,10v); + +z34=(4v,4v); +z35=(4v,5v); +z36=(4v,6v); +z37=(4v,7v); +z38=(4v,8v); +z39=(4v,9v); +z310=(4v,10v); + +z44=(6v,4v); +z45=(6v,5v); +z46=(6v,6v); +z47=(6v,7v); +z48=(6v,8v); +z49=(6v,9v); +z410=(6v,10v); + +z54=(8v,4v); +z55=(8v,5v); +z56=(8v,6v); +z57=(8v,7v); +z58=(8v,8v); +z59=(8v,9v); +z510=(8v,10v); + +z111=(0v,11v); +z211=(2v,11v); +z311=(4v,11v); +z411=(6v,11v); +z511=(8v,11v); + +pickup pencircle scaled 0.4pt; +draw(z13--z111); +draw(z23--z211); +draw(z33--z311); +draw(z43--z411); +draw(z53--z511); +drawarrow(z14--z24); +drawarrow(z16--z26); +drawarrow(z18--z28); +drawarrow(z110--z210); +drawarrow(z25--z35); +drawarrow(z27--z37); +drawarrow(z29--z39); +drawarrow(z36--z46); +drawarrow(z38--z48); +drawarrow(z47--z57); + +label.top(btex x1 etex,z111); +label.top(btex x2 etex,z211); +label.top(btex x3 etex,z311); +label.top(btex x4 etex,z411); +label.top(btex x5 etex,z511); + +endfig; + + +beginfig(3); +v:=u*5mm; + +z10=(0v,0v); +z15=(0v,5v); +z16=(0v,6v); + +z20=(2v,0v); +z24=(2v,4v); +z26=(2v,6v); + +z30=(4v,0v); +z33=(4v,3v); +z36=(4v,6v); + +z40=(6v,0v); +z42=(6v,2v); +z46=(6v,6v); + +z50=(8v,0v); +z51=(8v,1v); +z56=(8v,6v); + +z60=(10v,0v); +z65=(10v,5v); +z66=(10v,6v); + +z70=(12v,0v); +z74=(12v,4v); +z76=(12v,6v); + +z80=(14v,0v); +z83=(14v,3v); +z86=(14v,6v); + +z90=(16v,0v); +z92=(16v,2v); +z96=(16v,6v); + +z100=(18v,0v); +z101=(18v,1v); +z106=(18v,6v); + +z110=(20v,0v); +z116=(20v,6v); + + +pickup pencircle scaled 0.4pt; +draw(z10--z16); +draw(z20--z26); +draw(z30--z36); +draw(z40--z46); +draw(z50--z56); +draw(z60--z66); +draw(z70--z76); +draw(z80--z86); +draw(z90--z96); +draw(z100--z106); +drawarrow(z15--z65); +drawarrow(z24--z74); +drawarrow(z33--z83); +drawarrow(z42--z92); +drawarrow(z51--z101); + +label.top(btex $x_0$ etex,z16); +label.top(btex $x_1$ etex,z26); +label.top(btex $x_2$ etex,z36); +label.top(btex \dots etex,z66); +label.top(btex $x_{n-2}$ etex,z96); +label.top(btex $x_{n-1}$ etex,z106); + +endfig; + + + +beginfig(4); +v:=u*5mm; + +z1075=(7.5v,0v); + +z10=(0v,1v); +z175=(7.5v,1v); +z115=(15v,1v); + +z20=(0v,2v); +z235=(3.5v,2v); +z211=(11.5v,2v); +z215=(15v,2v); + +z30=(0v,3v); +z335=(3.5v,3v); +z37=(7v,3v); +z38=(8v,3v); +z311=(11.5v,3v); +z315=(15v,3v); + +z40=(0v,4v); +z415=(1.5v,4v); +z455=(5.5v,4v); +z47=(7v,4v); +z48=(8v,4v); +z495=(9.5v,4v); +z413=(13.5v,4v); +z416=(15v,4v); + +z50=(0v,5v); +z515=(1.5v,5v); +z53=(3v,5v); +z54=(4v,5v); +z555=(5.5v,5v); +z57=(7v,5v); +z58=(8v,5v); +z595=(9.5v,5v); +z511=(11v,5v); +z512=(12v,5v); +z513=(13.5v,5v); +z516=(15v,5v); + +z60=(0v,6v); +z61=(1v,6v); +z62=(2v,6v); +z63=(3v,6v); +z64=(4v,6v); +z65=(5v,6v); +z66=(6v,6v); +z67=(7v,6v); +z68=(8v,6v); +z69=(9v,6v); +z610=(10v,6v); +z611=(11v,6v); +z612=(12v,6v); +z613=(13v,6v); +z614=(14v,6v); +z615=(15v,6v); + +z71=(1v,7v); +z72=(2v,7v); +z75=(5v,7v); +z76=(6v,7v); +z79=(9v,7v); +z710=(10v,7v); +z713=(13v,7v); +z714=(14v,7v); + + + + +pickup pencircle scaled 0.4pt; +draw(z10--z115--z215--z20--cycle); +draw(z30--z37--z47--z40--cycle); +draw(z38--z315--z416--z48--cycle); +draw(z50--z53--z63--z60--cycle); +draw(z54--z57--z67--z64--cycle); +draw(z58--z511--z611--z68--cycle); +draw(z512--z516--z615--z612--cycle); +drawarrow(z71--z61); +drawarrow(z72--z62); +drawarrow(z75--z65); +drawarrow(z76--z66); +drawarrow(z79--z69); +drawarrow(z710--z610); +drawarrow(z713--z613); +drawarrow(z714--z614); +drawarrow(z515--z415); +drawarrow(z555--z455); +drawarrow(z595--z495); +drawarrow(z513--z413); +drawarrow(z335--z235); +drawarrow(z311--z211); +drawarrow(z175--z1075); + +endfig; + + +beginfig(5); +v:=u*5mm; + +z075=(7.5v,8v); + +z10=(0v,7v); +z175=(7.5v,7v); +z115=(15v,7v); + +z20=(0v,6v); +z235=(3.5v,6v); +z211=(11.5v,6v); +z215=(15v,6v); + +z30=(0v,5v); +z335=(3.5v,5v); +z37=(7v,5v); +z38=(8v,5v); +z311=(11.5v,5v); +z315=(15v,5v); + +z40=(0v,4v); +z415=(1.5v,4v); +z455=(5.5v,4v); +z47=(7v,4v); +z48=(8v,4v); +z495=(9.5v,4v); +z413=(13.5v,4v); +z416=(15v,4v); + +z50=(0v,3v); +z515=(1.5v,3v); +z53=(3v,3v); +z54=(4v,3v); +z555=(5.5v,3v); +z57=(7v,3v); +z58=(8v,3v); +z595=(9.5v,3v); +z511=(11v,3v); +z512=(12v,3v); +z513=(13.5v,3v); +z516=(15v,3v); + +z60=(0v,2v); +z61=(1v,2v); +z62=(2v,2v); +z63=(3v,2v); +z64=(4v,2v); +z65=(5v,2v); +z66=(6v,2v); +z67=(7v,2v); +z68=(8v,2v); +z69=(9v,2v); +z610=(10v,2v); +z611=(11v,2v); +z612=(12v,2v); +z613=(13v,2v); +z614=(14v,2v); +z615=(15v,2v); + +% ve skutecnosti dle znaceni +% by melo byt z6* ale uz +% obsazeno +z815=(1.5v,2v); +z855=(5.5v,2v); +z895=(9.5v,2v); +z813=(13.5v,2v); + +z715=(1.5v,1v); +z755=(5.5v,1v); +z795=(9.5v,1v); +z713=(13.5v,1v); + +z9=(4v,0v); + +pickup pencircle scaled 0.4pt; +draw(z10--z115--z215--z20--cycle); +draw(z30--z37--z47--z40--cycle); +draw(z38--z315--z416--z48--cycle); +draw(z50--z53--z63--z60--cycle); +draw(z54--z57--z67--z64--cycle); +draw(z58--z511--z611--z68--cycle); +draw(z512--z516--z615--z612--cycle); +drawarrow(z075--z175); +drawarrow(z415--z515); +drawarrow(z455--z555); +drawarrow(z495--z595); +drawarrow(z413--z513); +drawarrow(z235--z335); +drawarrow(z211--z311); +drawarrow(z815--z715); +drawarrow(z855--z755); +drawarrow(z895--z795); +drawarrow(z813--z713); + +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 $S_{n\over 4}$ etex,z515); +label.bot(btex $S_{n\over 4}$ etex,z555); +label.bot(btex $S_{n\over 4}$ etex,z595); +label.bot(btex $S_{n\over 4}$ etex,z513); +label.rt(btex Bitonick\'a t\v r\'\i di\v cka $B_{n}$ etex,z9); + +endfig; + + + +beginfig(6); +v:=u*5mm; + +z1075=(7.5v,0v); + +z10=(0v,1v); +z175=(7.5v,1v); +z115=(15v,1v); + +z20=(0v,2v); +z235=(3.5v,2v); +z211=(11.5v,2v); +z215=(15v,2v); + +z30=(0v,3v); +z335=(3.5v,3v); +z37=(7v,3v); +z38=(8v,3v); +z311=(11.5v,3v); +z315=(15v,3v); + +z40=(0v,4v); +z415=(1.5v,4v); +z455=(5.5v,4v); +z47=(7v,4v); +z48=(8v,4v); +z495=(9.5v,4v); +z413=(13.5v,4v); +z416=(15v,4v); + +z50=(0v,5v); +z515=(1.5v,5v); +z53=(3v,5v); +z54=(4v,5v); +z555=(5.5v,5v); +z57=(7v,5v); +z58=(8v,5v); +z595=(9.5v,5v); +z511=(11v,5v); +z512=(12v,5v); +z513=(13.5v,5v); +z516=(15v,5v); + +z60=(0v,6v); +z61=(1v,6v); +z62=(2v,6v); +z63=(3v,6v); +z64=(4v,6v); +z65=(5v,6v); +z66=(6v,6v); +z67=(7v,6v); +z68=(8v,6v); +z69=(9v,6v); +z610=(10v,6v); +z611=(11v,6v); +z612=(12v,6v); +z613=(13v,6v); +z614=(14v,6v); +z615=(15v,6v); + +z71=(1v,7v); +z72=(2v,7v); +z75=(5v,7v); +z76=(6v,7v); +z79=(9v,7v); +z710=(10v,7v); +z713=(13v,7v); +z714=(14v,7v); + + + + +pickup pencircle scaled 0.4pt; +draw(z10--z115--z215--z20--cycle); +draw(z30--z37--z47--z40--cycle); +draw(z38--z315--z416--z48--cycle); +draw(z50--z53--z63--z60--cycle); +draw(z54--z57--z67--z64--cycle); +draw(z58--z511--z611--z68--cycle); +draw(z512--z516--z615--z612--cycle); +drawarrow(z71--z61); +drawarrow(z72--z62); +drawarrow(z75--z65); +drawarrow(z76--z66); +drawarrow(z79--z69); +drawarrow(z710--z610); +drawarrow(z713--z613); +drawarrow(z714--z614); +drawarrow(z515--z415); +drawarrow(z555--z455); +drawarrow(z595--z495); +drawarrow(z513--z413); +drawarrow(z335--z235); +drawarrow(z311--z211); +drawarrow(z175--z1075); + +label.top(btex $M_8$ etex,z175); +label.top(btex $M_4$ etex,z335); +label.top(btex $M_4$ etex,z311); +label.top(btex $M_2$ etex,z515); +label.top(btex $M_2$ etex,z555); +label.top(btex $M_2$ etex,z595); +label.top(btex $M_2$ etex,z513); + +endfig; + + +beginfig(7); +v:=u*7mm; + +z12=(1v,2v); +z13=(1v,3v); +z16=(1v,6v); +z356=(3.5v,6v); +z40=(5v,1v); +z42=(5v,2v); +z47=(5v,6.5v); +z72=(9v,2v); +z74=(9v,4v); +z76=(9v,6v); + +z100=(0.5v,0.5v); +z101=(1.5v,0.5v); + +z0=whatever[z13,z356]; +z1=whatever[z356,z74]; +z1=z0+4v*right; +z2=whatever[z12,z72]; +z2=z0+whatever*down; +z3=whatever[z12,z72]; +z3=z1+whatever*down; +z4=z356+4v*right; +z5=whatever[z72,z76]; +z5=whatever[z4,z1+4v*right]; +z6=whatever[z40,z47]; +z6=z74+4v*left; + + +pickup pencircle scaled 0.4pt; +draw(z16--z12--z72--z76); +draw(z13--z356--z74); +draw(z40--z47) dashed evenly; +draw(z1--z4) dashed withdots scaled 0.7; +draw(z4--z5) dashed withdots scaled 0.7; +draw(z0--z6) dashed withdots scaled 0.7; +draw(z0--z2) dashed evenly; +draw(z1--z3) dashed evenly; + +draw(z100--z101) dashed withdots scaled 0.7; + +pickup pencircle scaled 3pt; +drawdot(z0); +drawdot(z1); + +label.bot(btex \strut 0 etex,z12); +label.bot(btex $k$ etex,z2); +label.llft(btex \strut ${n\over 2} - 1$ etex,z42); +label.bot(btex \strut $k+{n\over 2}$ etex,z3); +label.bot(btex \strut $n-1$ etex,z72); +label.rt(btex posloupnost prohozen\'a separ\'atorem etex,z101); + +endfig; + + + + + + + +end; diff --git a/6-bitonic/sortnet.mpx b/6-bitonic/sortnet.mpx new file mode 100644 index 0000000..9814d6d --- /dev/null +++ b/6-bitonic/sortnet.mpx @@ -0,0 +1,548 @@ +% Written by DVItoMP, Version 0.64/color (Web2C 7.5.4) +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; +_n0="cmr10"; +_s("a",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,-3.4869)--(4.9813,-3.4869)-- + (4.9813,8.4682)--(0,8.4682)--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; +_n0="cmr10"; +_s("b",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,-3.4869)--(5.5348,-3.4869)-- + (5.5348,8.4682)--(0,8.4682)--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; +_n0="cmr10"; +_s("min",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(16.6045,0.0000)-- + (16.6045,6.6536)--(0,6.6536)--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; +_n0="cmr10"; +_s("max",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(18.5416,0.0000)-- + (18.5416,4.2895)--(0,4.2895)--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; +_n0="cmr10"; +_s("x1",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x2",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x3",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x4",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x5",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x1",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x2",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x3",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x4",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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; +_n0="cmr10"; +_s("x5",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(10.2394,0.0000)-- + (10.2394,6.4204)--(0,6.4204)--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("x",_n1,1.00000,0.0000,0.0000,); +_n2="cmr7"; +_s("0",_n2,1.00000,5.6939,-1.4944,); +setbounds _p to (0,-1.4944)--(10.1633,-1.4944)-- + (10.1633,4.2895)--(0,4.2895)--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("x",_n1,1.00000,0.0000,0.0000,); +_n2="cmr7"; +_s("1",_n2,1.00000,5.6939,-1.4944,); +setbounds _p to (0,-1.4944)--(10.1633,-1.4944)-- + (10.1633,4.2895)--(0,4.2895)--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("x",_n1,1.00000,0.0000,0.0000,); +_n2="cmr7"; +_s("2",_n2,1.00000,5.6939,-1.4944,); +setbounds _p to (0,-1.4944)--(10.1633,-1.4944)-- + (10.1633,4.2895)--(0,4.2895)--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(":",_n1,1.00000,0.0000,0.0000,); +_s(":",_n1,1.00000,4.4278,0.0000,); +_s(":",_n1,1.00000,8.8556,0.0000,); +setbounds _p to (0,0.0000)--(13.2834,0.0000)-- + (13.2834,1.0516)--(0,1.0516)--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("x",_n1,1.00000,0.0000,0.0000,); +_n3="cmmi7"; +_s("n",_n3,1.00000,5.6939,-1.4944,); +_n4="cmsy7"; +_s(char0,_n4,1.00000,10.6188,-1.4944,); +_n2="cmr7"; +_s("2",_n2,1.00000,16.8454,-1.4944,); +setbounds _p to (0,-2.3246)--(21.3148,-2.3246)-- + (21.3148,4.2895)--(0,4.2895)--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("x",_n1,1.00000,0.0000,0.0000,); +_n3="cmmi7"; +_s("n",_n3,1.00000,5.6939,-1.4944,); +_n4="cmsy7"; +_s(char0,_n4,1.00000,10.6188,-1.4944,); +_n2="cmr7"; +_s("1",_n2,1.00000,16.8454,-1.4944,); +setbounds _p to (0,-2.3246)--(21.3148,-2.3246)-- + (21.3148,4.2895)--(0,4.2895)--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("n",_n1,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(5.9799,0.0000)-- + (5.9799,4.2895)--(0,4.2895)--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,); +_n3="cmmi7"; +_s("n",_n3,1.00000,6.1090,-1.4944,); +setbounds _p to (0,-1.4944)--(11.5320,-1.4944)-- + (11.5320,6.8078)--(0,6.8078)--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,); +interim linecap:=0; +vardef _r(expr _a,_w)(text _t) = + addto _p doublepath _a withpen pencircle scaled _w _t enddef; +_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); +_n6="cmr5"; +_s("2",_n6,1.00000,7.8033,-3.8949,); +setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- + (13.3857,6.8078)--(0,6.8078)--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,); +interim linecap:=0; +vardef _r(expr _a,_w)(text _t) = + addto _p doublepath _a withpen pencircle scaled _w _t enddef; +_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); +_n6="cmr5"; +_s("2",_n6,1.00000,7.8033,-3.8949,); +setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- + (13.3857,6.8078)--(0,6.8078)--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,); +interim linecap:=0; +vardef _r(expr _a,_w)(text _t) = + addto _p doublepath _a withpen pencircle scaled _w _t enddef; +_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); +_n6="cmr5"; +_s("4",_n6,1.00000,7.8033,-3.8949,); +setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- + (13.3857,6.8078)--(0,6.8078)--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,); +interim linecap:=0; +vardef _r(expr _a,_w)(text _t) = + addto _p doublepath _a withpen pencircle scaled _w _t enddef; +_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); +_n6="cmr5"; +_s("4",_n6,1.00000,7.8033,-3.8949,); +setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- + (13.3857,6.8078)--(0,6.8078)--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,); +interim linecap:=0; +vardef _r(expr _a,_w)(text _t) = + addto _p doublepath _a withpen pencircle scaled _w _t enddef; +_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); +_n6="cmr5"; +_s("4",_n6,1.00000,7.8033,-3.8949,); +setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- + (13.3857,6.8078)--(0,6.8078)--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,); +interim linecap:=0; +vardef _r(expr _a,_w)(text _t) = + addto _p doublepath _a withpen pencircle scaled _w _t enddef; +_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); +_n6="cmr5"; +_s("4",_n6,1.00000,7.8033,-3.8949,); +setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- + (13.3857,6.8078)--(0,6.8078)--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; +_n0="cmr10"; +_s("Bitonic",_n0,1.00000,0.0000,0.0000,); +_s("k"&char19,_n0,1.00000,31.1333,0.0000,); +_s("a",_n0,1.00000,36.3914,0.0000,); +_s("t",_n0,1.00000,44.6936,0.0000,); +_s(char20,_n0,1.00000,48.0283,0.0000,); +_s("r",_n0,1.00000,48.5680,0.0000,); +_s(char19,_n0,1.00000,51.3631,0.0000,); +_s(char16&"di",_n0,1.00000,52.4700,0.0000,); +_s(char20,_n0,1.00000,63.2629,0.0000,); +_s("ck",_n0,1.00000,63.5397,0.0000,); +_s("a",_n0,1.00000,72.6721,0.0000,); +_n1="cmmi10"; +_s("B",_n1,1.00000,80.9743,0.0000,); +_n3="cmmi7"; +_s("n",_n3,1.00000,88.5311,-1.4944,); +setbounds _p to (0,-1.4944)--(93.9541,-1.4944)-- + (93.9541,6.9185)--(0,6.9185)--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("8",_n2,1.00000,9.6652,-1.4944,); +setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- + (14.1345,6.8078)--(0,6.8078)--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,); +setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- + (14.1345,6.8078)--(0,6.8078)--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,); +setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- + (14.1345,6.8078)--(0,6.8078)--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("2",_n2,1.00000,9.6652,-1.4944,); +setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- + (14.1345,6.8078)--(0,6.8078)--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("2",_n2,1.00000,9.6652,-1.4944,); +setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- + (14.1345,6.8078)--(0,6.8078)--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("2",_n2,1.00000,9.6652,-1.4944,); +setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- + (14.1345,6.8078)--(0,6.8078)--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("2",_n2,1.00000,9.6652,-1.4944,); +setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- + (14.1345,6.8078)--(0,6.8078)--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; +_n0="cmr10"; +_s("0",_n0,1.00000,0.0000,0.0000,); +setbounds _p to (0,-3.4869)--(4.9813,-3.4869)-- + (4.9813,8.4682)--(0,8.4682)--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("k",_n1,1.00000,0.0000,0.0000,); +setbounds _p to (0,0.0000)--(5.5002,0.0000)-- + (5.5002,6.9185)--(0,6.9185)--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; +_n3="cmmi7"; +_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((1.1955,2.4907)..(6.1204,2.4907), 0.3985,); +_n2="cmr7"; +_s("2",_n2,1.00000,1.6723,-3.4355,); +_n7="cmsy10"; +_s(char0,_n7,1.00000,9.5298,0.0000,); +_n0="cmr10"; +_s("1",_n0,1.00000,19.4924,0.0000,); +setbounds _p to (0,-3.4869)--(24.4737,-3.4869)-- + (24.4737,8.4682)--(0,8.4682)--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("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,); +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,); +_n2="cmr7"; +_s("2",_n2,1.00000,19.3490,-3.4355,); +setbounds _p to (0,-3.4869)--(24.9926,-3.4869)-- + (24.9926,8.4682)--(0,8.4682)--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("n",_n1,1.00000,0.0000,0.0000,); +_n7="cmsy10"; +_s(char0,_n7,1.00000,8.1938,0.0000,); +_n0="cmr10"; +_s("1",_n0,1.00000,18.1564,0.0000,); +setbounds _p to (0,-3.4869)--(23.1377,-3.4869)-- + (23.1377,8.4682)--(0,8.4682)--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; +_n0="cmr10"; +_s("p",_n0,1.00000,0.0000,0.0000,); +_s("osloupnost",_n0,1.00000,5.8116,0.0000,); +_s("prohozen"&char19,_n0,1.00000,55.1821,0.0000,); +_s("a",_n0,1.00000,94.5069,0.0000,); +_s("separ"&char19,_n0,1.00000,102.8091,0.0000,); +_s("atorem",_n0,1.00000,125.5849,0.0000,); +setbounds _p to (0,-1.9372)--(156.0540,-1.9372)-- + (156.0540,6.9185)--(0,6.9185)--cycle; +_p endgroup +mpxbreak diff --git a/old/4-hradla/4-hradla.tex b/old/4-hradla/4-hradla.tex deleted file mode 100644 index ccb0c50..0000000 --- a/old/4-hradla/4-hradla.tex +++ /dev/null @@ -1,139 +0,0 @@ -\input ../lecnotes.tex - -\prednaska{4}{Hradlové sítì}{(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~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. - -\h{Hradlové sítì} - -\s{Definice:} {\I Hradlo} je prvek, který umí vyhodnocovat nìjakou funkci -nad~koneènou abecedou $\Sigma$. - -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: - -\itemize\ibull -\:nulární: to jsou konstanty (FALSE=0, TRUE=1), -\: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 provádící 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 (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}) \hfil\break 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~$e$ konèí v~$h$ a~$z(e)=j$, - (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 -její výpoèet probíhá. - -\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{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, 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} - -\bye diff --git a/old/4-hradla/Makefile b/old/4-hradla/Makefile deleted file mode 100644 index 65a25ae..0000000 --- a/old/4-hradla/Makefile +++ /dev/null @@ -1,3 +0,0 @@ -P=4-hradla - -include ../Makerules diff --git a/old/4-hradla/chytry_or.eps b/old/4-hradla/chytry_or.eps deleted file mode 100644 index 3d88cff..0000000 --- a/old/4-hradla/chytry_or.eps +++ /dev/null @@ -1,2324 +0,0 @@ -%!PS-Adobe-3.0 EPSF-3.0 -%%Creator: inkscape 0.46 -%%Pages: 1 -%%Orientation: Portrait -%%BoundingBox: 2 325 592 773 -%%HiResBoundingBox: 2.9557909 325.79571 591.83322 772.94254 -%%EndComments -%%Page: 1 1 -0 842 translate -0.8 -0.8 scale -0 0 0 setrgbcolor -[] 0 setdash -1 setlinewidth -0 setlinejoin -0 setlinecap -gsave [1 0 0 1 0 0] concat -gsave [1.086765 0 0 1.086765 -114.17957 349.28806] concat 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bitonické tøídìní}{(zapsal: Petr Jankovský)} - -Minule jsme si zavedli paralelní výpoèetní model, ve kterém si nyní nìco naprogramujeme \dots - -\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. - -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: -$$ -z_i=x_i \oplus y_i \oplus c_i, -$$ -kde $z_i$ je $i$-tá èíslice souètu, $\oplus$ znaèí operaci {\sc 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): -$$ -\eqalign{ -c_{0} &= 0, \cr -c_{i+1} &= (x_i~\&~y_i)\lor(x_i~\&~c_i)\lor(y_i~\&~c_i).\cr -} -$$ - -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} - -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. - -Zamysleme se nad tím, jak by se proces sèítání mohl zrychlit. - -\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. - -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 $\left$. 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} - -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: - -\figure{bloky_1bit.eps}{Tabulka triviálních blokù.}{1.1in} - -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, -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: - -\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, -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 -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: - -\itemize\ibull -\:$(1,*) = <$, -\:$(0,0) = 0$, -\:$(0,1) = 1$ -\endlist - -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. - -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 - -\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. - -\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: $\forall i: z_i = x_i \oplus y_i \oplus c_i$. -\endalgo - -\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~$\Theta(n)$. - -\h{Tøídìní} - -Nyní se podíváme na~druhý problém, a~to na~problém tøídìní. Ji¾ známe pomìrnì efektivní sekvenèní algoritmy, které dovedou tøídit v~èase $\O(n\log n)$. Byli bychom jistì rádi, kdybychom to zvládli je¹tì rychleji. Pojïme se podívat, zda by nám v tom nepomohlo problém paralelizovat. - -Budeme pøi tom pracovat ve~výpoèetním modelu, kterému se øíká komparátorová sí». Ta je postavená z~hradel, kterým se øíká komparátory. - -\s{Definice:} {\I Komparátorová sí»} je kombinaèní obvod, jeho¾ hradla jsou -komparátory. - -\figure{sortnet.0}{Komparátor}{0.7in} - -Komparátor umí porovnat dvì hodnoty a~rozhodnout, která z~nich je vìt¹í -a~která men¹í. Nevrací v¹ak booleovský výsledek jako bì¾né hradlo, ale má dva -výstupy, pøièem¾ na~jednom vrátí men¹í ze~vstupních hodnot a~na~druhém vìt¹í. - -Výstupy komparátorù se nevìtví. Nemù¾eme tedy jeden výstup \uv{rozdvojit} a~pøipojit ho na~dva vstupy. (Vìtvení by dokonce ani nemìlo smysl, proto¾e zatímco rozdvojit bychom mohli, slouèit u¾~ne. Pokud tedy chceme, aby sí» mìla $n$~vstupù i~$n$~výstupù, rozdvojení stejnì nesmíme provést, i kdybychom jej mìli povolené.) - -\s{Pøíklad:} {\sl Bubble sort} - -Obrázek Bubble 1 ilustruje pou¾ití komparátorù pro tøídìní Bubble sortem. -©ipky pøedstavují jednotlivé komparátory. Výpoèet v¹ak je¹tì mù¾eme vylep¹it. - -\twofigures{sortnet.1}{Bubble 1}{143pt}{sortnet.2}{Bubble 2}{143pt} - -Sna¾íme se výpoèet co nejvíce paralelizovat (viz obrázek Bubble 2). -Jak je vidìt, komparátory na sebe nemusejí èekat. Tím mù¾eme výpoèet urychlit a místo èasu $\Theta{(n^2)}$ docílit èasové slo¾itosti $\Theta{(n)}$. V obou pøípadech je zachován kvadratický prostor. - -Nyní si uká¾eme je¹tì rychlej¹í tøídící algoritmus. Pùjdeme na nìj v¹ak trochu \uv{od lesa}. Nejdøíve vymyslíme sí», která bude umìt tøídit jenom nìco -- toti¾ bitonické posloupnosti. Bez újmy na obecnosti budeme pøedpokládat, ¾e ka¾dé dva prvky na vstupu jsou navzájem rùzné. - -\medskip -\s{Definice:} Øekneme, ¾e posloupnost $x_0,\dots,x_{n-1} $ je {\I èistì bitonická} právì tehdy, kdy¾ -pro nìjaké $x_k, k\in\{0, \dots, n-1\} $ platí, ¾e~v¹echny prvky pøed ním (vèetnì jeho samotného) -tvoøí rostoucí poslopnost, kde¾to prvky stojící za~ním tvoøí poslopnost klesající. -Formálnì zapsáno musí platit, ¾e: -$$x_0\leq x_1\leq \dots \leq x_k \geq x_{k+1}\geq\dots \geq x_{n-1}.$$ - -\s{Definice:} Posloupnost $x_0 \dots x_{n-1}$ je {\I bitonická}, právì kdy¾ $\exists~j\in \{0,\dots ,n-1\}$, pro -které je rotace pùvodní posloupnosti o $j$ prvkù, tedy posloupnost -$$x_j,x_{(j+1) \bmod n},\dots, x_{(j+n-1) \bmod n},$$ èistì bitonická. - -\s{Definice:} {\I Separátor $S_n$} je sí», ve které jsou v¾dy~$i$-tý a~$(i+{n/2})$-tý prvek vstupu -(pro $i=0,\dots, {n/2}-1$) propojeny komparátorem. Minimum se pak stane~$i$-tým, -maximum $(i+{n/2})$-tým prvkem výstupu. -\figure{sortnet.3}{$(y_i, y_{i+{n/2}}) = \(x_i, x_{i+{n/2}})$} {300pt} - -\s{Lemma:} Pokud separátor dostane na~vstupu bitonickou posloupnost, pak jeho výstup $y_0, \dots, y_{n-1}$ -splòuje: - -(i) $y_0,\dots, y_{n/2 -1}$ a~$y_{n/2},\dots, y_{n-1}$ jsou -bitonické posloupnosti, - -(ii) Pro v¹echna $i,j< {n/2}$ platí $y_i < y_{j + {n/2}}$. - -Separátor nám tedy jednu bitonickou posloupnost na~vstupu rozdìlí na~dvì -bitonické posloupnosti, pøièem¾ navíc ka¾dý prvek první posloupnosti ($y_0,\dots, y_{n/2 -1}$) -je men¹í nebo roven prvkùm druhé posloupnosti ($y_{n/2}, \dots, y_{n-1}$). - -\>Ne¾ pøistoupíme k dùkazu lemmatu, uka¾me si, k èemu se nám bude hodit. - - -\s{Definice:} {\I Bitonická tøídièka $B_n$} je obvod sestavený ze separátorù, který dostane-li na vstupu bitonickou posloupnost délky $n$ (BÚNO konstruujeme tøídièku pro $n=2^k$), vydá setøídìnou zadanou posloupnost délky $n$. - -Tøídièka dostane na~vstupu bitonickou posloupnost. Separátor~$S_n$ ji pak dle lemmatu rozdìlí na~dvì bitonické posloupnosti, kdy je ka¾dý prvek z~první posloupnosti men¹í ne¾ libovolný prvek z~druhé. Tyto poloviny pak dal¹í separátory rozdìlí na~ètvrtiny, ..., a¾~na~konci zbudou pouze jednoduché posloupnosti délky jedna (zjevnì setøídìné), které mezi sebou mají po¾adovanou nerovnost -- tedy ka¾dá posloupnost (nebo spí¹e prvek) nalevo je $\leq$ ne¾ prvek napravo od~nìj. - -\centerline{\epsfbox{sortnet.5}} - -Jak je vidìt, bitonická tøídièka nám libovolnou bitonickou posloupnost délky~$n$ -setøídí na~$\Theta(\log n)$ hladin. - -Nyní se dá odvodit, ¾e pokud umíme tøídit bitonické posloupnosti, umíme setøídit v¹echno. -Vzpomeòme si na~tøídìní sléváním -- Merge sort. To funguje tak, ¾e zaène s~jednoprvkovými posloupnostmi, které jsou evidentnì setøídìné, a~poté v¾dy dvì setøídìné posloupnosti slévá do~jedné. Kdybychom nyní umìli paralelnì slévat, mohli bychom vytvoøit i~paralelní Merge sort. Jinými slovy, potøebujeme dvì rostoucí posloupnosti nìjak efektivnì slít do~jedné setøídìné. Uvìdomme si, ¾e to zvládneme jednodu¹e -- staèí druhou z~posloupností obrátit a~\uv{pøilepit} za~první, èím¾ vznikne bitonická posloupnost, kterou poté mù¾eme setøídit na¹í bitonickou tøídièkou. - -\s{Pøíklad:} {\sl Merge sort} - -Bitonická tøídièka se tedy dá pou¾ít ke~slévání setøídìných posloupností. -Uka¾me si, jak s~její pomocí sestavíme souèástky {\I slévaèky} $M_n$: - -Setøídìné posloupnosti $x_0,\dots, x_{n-1}$ a~$y_0,\dots, y_{n-1}$ spojíme do jedné bitonické -posloupnosti $x_0,\dots, x_{n-1},y_{n-1},\dots, y_0$. Z~této posloupnosti vytvoøíme pomocí bitonické tøídièky -$B_{2n}$ setøídìnou posloupnost. Vytvoøíme tedy blok $M_{2n}$, který se ov¹em sestává de~facto pouze z~bloku $B_{2n}$, jeho¾ druhá polovina vstupù je zapojena v~obráceném poøadí. -\figure{sortnet.6}{Paralelní MergeSort.}{3in} - -Nyní se pokusme odhadnout èasovou slo¾itost. Ná¹ MergeSort bude mít øádovì hloubku blokù $\log n$. -V ka¾dém bloku $M_n$ je navíc ukryta bitonická tøídièka s takté¾ logaritmickou hloubkou. -Celková hloubka tedy bude $\log 2 + \log4 + \dots + \log 2^k + \dots + \log n$. -Po seètení nakonec dostáváme výslednou èasovou slo¾itost $\Theta (\log^2 n)$. - -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. - -Vra»me se nyní k dùkazu lemmatu, který jsme na pøedminulé stránce vynechali. - -\h{Dùkaz Lemmatu:} -(i) Nejprve nahlédneme, ¾e lemma platí, je-li vstupem èistì bitonická -posloupnost. Dále BÚNO pøedpokládejme, ¾e vrchol posloupnosti je v~první polovinì (kdyby -byl vrchol za~polovinou, staèilo by zrcadlovì obrátit posloupnost i~komparátory a~øe¹ili -bychom stejný problém). Nyní si definujme $k := \min j: x_j > x_{j+n/2}$. (Pokud -by~takové~$k$ neexistovalo, znamenalo by~to, ¾e vstupní posloupnost je monotónní. -Separátor by tedy nic nedìlal a~pouze zkopíroval vstup na~výstup, co¾ jistì lemma splòuje.) Nyní si v¹imìme, -¾e jakmile jednou zaène platit, ¾e prvek na~levé stranì je men¹í ne¾ na~pravé, bude -nám tato relace platit a¾~do~konce. Oznaème vrchol vstupní posloupnosti jako~$x_m$. -Pak~$k$ bude jistì men¹í ne¾~$m$ a~$k+{n/2}$ bude vìt¹í ne¾~$m$. Mezi~$k$ a~$m$ je tedy -vstupní posloupnost neklesající, mezi $k+{n/2}$ a~$n-1$ nerostoucí. - -Do pozice~$k$ tedy separátor bude pouze kopírovat vstup na~výstup, od~pozice~$k$ -dál u¾~jen prohazuje. Pro ka¾dé~$i$, $(k\leq i\leq {n/2}-1)$ se prvky -$x_i$ a~$x_{i+{n/2}}$ prohodí. Úsek mezi~$k$ a~${n/2}-1$ tedy -nahradíme nerostoucí posloupností, první polovina výstupu tedy bude -(dokonce èistì) bitonická. Podobnì úsek $k+{n/2}$ a¾~$n-1$ nahradíme èistì -bitonickou posloupností. Obì poloviny tedy budou bitonické. - -Dostaneme-li na vstupu obecnou bitonickou posloupnost, pøedstavíme si, -¾e je to èistì bitonická posloupnost zrotovaná o~$r$ prvkù (BÚNO -doprava). Zjistíme, ¾e v~komparátorech se porovnávají tyté¾ prvky, -jako kdyby zrotovaná nebyla. Výstup se od výstupu èistì bitonické -posloupnosti zrotovaného o~$r$ bude li¹it prohozením úsekù $x_0$ a¾ -$x_{r-1}$ a~$x_{n/2}$ a¾ $x_{{n/2}+r-1}$. Obì výstupní -posloupnosti tedy budou zrotované o~$r$ prvkù, ale na jejich -bitoniènosti se nic nezmìní. - -(ii) Z dùkazu (i) pro èistì bitonickou posloupnost víme, ¾e $y_0\dots y_{n/2-1}$ je èistì bitonická a bude rovna $x_0\dots x_{k-1},x_{k+n/2}\dots x_{n-1}$ pro vhodné $k$ a navíc bude mít maximum v $x_{k-1}$ nebo $x_k+{n/2}$. Mezi tìmito body ov¹em ve vstupní posloupnosti urèitì nele¾el ¾ádný $x_i$ men¹í ne¾ $x_k-1$ nebo $x_k+{n/2}$ (jak je vidìt z obrázku) a posloupnost $x_k \dots x_{k-1+{n/2}}$ je rovna $y_{n/2}\dots y_{n-1}$. Pro obecné bitonické posloupnosti uká¾eme stejnì jako v (i). -\qed - -\medskip - -\centerline{\epsfbox{sortnet.7}} - - -\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 -{\sc and} ) a~nakonec potøebujeme výsledných~$n$ èísel seèíst. -\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} - -Toto øe¹ení by v¹ak vedlo na~èasovou slo¾itost $\Theta (\log^2 n)$, nebo» sèítat umíme v~èase $\Theta (\log n)$. To je sice dle na¹ich mìøítek -docela efektivní, ale pøekvapivì to jde je¹tì lépe -- toti¾ na~$\Theta (\log n)$ hladin. Této -slo¾itosti dosáhneme malým trikem. - -Vymyslíme obvod konstantní hloubky, který na~vstupu dostane tøi èísla. Odpoví pak dvìma èísly -takovými, ¾e budou mít stejný souèet jako pùvodní tøi èísla. Jinými slovy pomocí -tohoto obvodu budeme umìt seètení tøí èísel pøevést na seètení dvou èísel. - -\figure{obvod.eps}{$x+y+z = p+q$}{0.3in} - -V¹imnìme si, ¾e~kdy¾ sèítáme tøi bity, mù¾e být pøenos do~vy¹¹ího øádu nula èi~jednièka. Vezmeme si tedy bity $x_i$, $y_i$, $z_i$ a~ty seèteme. To nám dá dvoubitový výsledek, pøièem¾ ni¾¹í bit z~tohoto výsledku po¹leme do~èísla~$p$, vy¹¹í do~èísla~$q$. - -\figure{obvod_real.eps}{Redukování sèítání}{1.2in} - -Toto zredukování sèítání nám nyní umo¾ní opìt stavit strom, by» o malièko slo¾itìj¹í. - -\figure{sl_stromecek.eps}{Slo¾itìj¹í stromeèek}{0.9in} - -Pokud jsme mìli na~zaèátku $n$ èísel, po~první redukci nám jich zbývá jen $2/3 \cdot n$ a~obecnì v~$k$-té hladiné $(2/3)^k \cdot n$. Znamená to, ¾e èísel nám ubývá exponenciálnì, tak¾e poèet hladin bude logaritmický. Redukující obvod je pøi tom jen konstantnì hluboký, tak¾e celé redukování zvládneme v~èase $\Theta (\log n)$. Na~konci Slo¾itìj¹ího stromeèku pak máme umístìnou jednu obyèejnou sèítaèku, která zbývající dvì èísla seète v~logaritmickém èase. - -Seètení v¹ech $n$ èísel tedy zabere $\Theta (\log n)$ hladin. - -Kdy¾ se nyní vrátíme k~násobení, zbývá nám vyøe¹it posouvání a~{\sc and}ování. Uvìdomme si, ¾e to je plnì paralelení a~zvládneme ho za~konstantnì mnoho hladin. 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-z34=(3v,4v); -z42=(1v,5v); -z43=(2v,5v); - -pickup pencircle scaled 0.4pt; -draw(z21--z22--z23--z24--z34--z33--z32--z31--z21); -drawarrow(z22--z12); -drawarrow(z23--z13); -drawarrow(z42--z32); -drawarrow(z43--z33); - -label.bot(btex \strut a etex,z32); -label.bot(btex \strut b etex,z33); -label.top(btex min etex,z22); -label.top(btex max etex,z23); - - -endfig; - -beginfig(1); -v:=u*5mm; -z10=(0v,0v); -z20=(2v,0v); -z30=(4v,0v); -z40=(6v,0v); -z50=(8v,0v); - -z11=(0v,1v); -z12=(0v,2v); -z13=(0v,3v); -z14=(0v,4v); -z15=(0v,5v); -z16=(0v,6v); -z17=(0v,7v); -z18=(0v,8v); -z19=(0v,9v); -z110=(0v,10v); - -z21=(2v,1v); -z22=(2v,2v); -z23=(2v,3v); -z24=(2v,4v); -z25=(2v,5v); -z26=(2v,6v); -z27=(2v,7v); -z28=(2v,8v); -z29=(2v,9v); -z210=(2v,10v); - -z31=(4v,1v); -z32=(4v,2v); -z33=(4v,3v); -z34=(4v,4v); -z35=(4v,5v); -z36=(4v,6v); -z37=(4v,7v); -z38=(4v,8v); -z39=(4v,9v); -z310=(4v,10v); - -z41=(6v,1v); -z42=(6v,2v); -z43=(6v,3v); -z44=(6v,4v); -z45=(6v,5v); -z46=(6v,6v); -z47=(6v,7v); -z48=(6v,8v); -z49=(6v,9v); -z410=(6v,10v); - -z51=(8v,1v); -z52=(8v,2v); -z53=(8v,3v); -z54=(8v,4v); -z55=(8v,5v); -z56=(8v,6v); -z57=(8v,7v); -z58=(8v,8v); -z59=(8v,9v); -z510=(8v,10v); - -z111=(0v,11v); -z211=(2v,11v); -z311=(4v,11v); -z411=(6v,11v); -z511=(8v,11v); - -z1015=(-1v,1.5v); -z1615=(9v,1.5v); -z1035=(-1v,3.5v); -z1635=(9v,3.5v); -z1065=(-1v,6.5v); -z1665=(9v,6.5v); - -pickup pencircle scaled 0.4pt; -draw(z10--z111); -draw(z20--z211); -draw(z30--z311); -draw(z40--z411); -draw(z50--z511); -drawarrow(z11--z21); -drawarrow(z13--z23); -drawarrow(z16--z26); -drawarrow(z110--z210); -drawarrow(z22--z32); -drawarrow(z25--z35); -drawarrow(z29--z39); -drawarrow(z34--z44); -drawarrow(z38--z48); -drawarrow(z47--z57); - -pickup pencircle scaled 0.7pt; -draw(z1015--z1615) dashed withdots scaled 0.7; -draw(z1035--z1635) dashed withdots scaled 0.7; -draw(z1065--z1665) dashed withdots scaled 0.7; - -label.top(btex x1 etex,z111); -label.top(btex x2 etex,z211); -label.top(btex x3 etex,z311); -label.top(btex x4 etex,z411); -label.top(btex x5 etex,z511); - -endfig; - -beginfig(2); -v:=u*5mm; -z13=(0v,3v); -z23=(2v,3v); -z33=(4v,3v); -z43=(6v,3v); -z53=(8v,3v); - -z14=(0v,4v); -z15=(0v,5v); -z16=(0v,6v); -z17=(0v,7v); -z18=(0v,8v); -z19=(0v,9v); -z110=(0v,10v); - -z24=(2v,4v); -z25=(2v,5v); -z26=(2v,6v); -z27=(2v,7v); -z28=(2v,8v); -z29=(2v,9v); -z210=(2v,10v); - -z34=(4v,4v); -z35=(4v,5v); -z36=(4v,6v); -z37=(4v,7v); -z38=(4v,8v); -z39=(4v,9v); -z310=(4v,10v); - -z44=(6v,4v); -z45=(6v,5v); -z46=(6v,6v); -z47=(6v,7v); -z48=(6v,8v); -z49=(6v,9v); -z410=(6v,10v); - -z54=(8v,4v); -z55=(8v,5v); -z56=(8v,6v); -z57=(8v,7v); -z58=(8v,8v); -z59=(8v,9v); -z510=(8v,10v); - -z111=(0v,11v); -z211=(2v,11v); -z311=(4v,11v); -z411=(6v,11v); -z511=(8v,11v); - -pickup pencircle scaled 0.4pt; -draw(z13--z111); -draw(z23--z211); -draw(z33--z311); -draw(z43--z411); -draw(z53--z511); -drawarrow(z14--z24); -drawarrow(z16--z26); -drawarrow(z18--z28); -drawarrow(z110--z210); -drawarrow(z25--z35); -drawarrow(z27--z37); -drawarrow(z29--z39); -drawarrow(z36--z46); -drawarrow(z38--z48); -drawarrow(z47--z57); - -label.top(btex x1 etex,z111); -label.top(btex x2 etex,z211); -label.top(btex x3 etex,z311); -label.top(btex x4 etex,z411); -label.top(btex x5 etex,z511); - -endfig; - - -beginfig(3); -v:=u*5mm; - -z10=(0v,0v); -z15=(0v,5v); -z16=(0v,6v); - -z20=(2v,0v); -z24=(2v,4v); -z26=(2v,6v); - -z30=(4v,0v); -z33=(4v,3v); -z36=(4v,6v); - -z40=(6v,0v); -z42=(6v,2v); -z46=(6v,6v); - -z50=(8v,0v); -z51=(8v,1v); -z56=(8v,6v); - -z60=(10v,0v); -z65=(10v,5v); -z66=(10v,6v); - -z70=(12v,0v); -z74=(12v,4v); -z76=(12v,6v); - -z80=(14v,0v); -z83=(14v,3v); -z86=(14v,6v); - -z90=(16v,0v); -z92=(16v,2v); -z96=(16v,6v); - -z100=(18v,0v); -z101=(18v,1v); -z106=(18v,6v); - -z110=(20v,0v); -z116=(20v,6v); - - -pickup pencircle scaled 0.4pt; -draw(z10--z16); -draw(z20--z26); -draw(z30--z36); -draw(z40--z46); -draw(z50--z56); -draw(z60--z66); -draw(z70--z76); -draw(z80--z86); -draw(z90--z96); -draw(z100--z106); -drawarrow(z15--z65); -drawarrow(z24--z74); -drawarrow(z33--z83); -drawarrow(z42--z92); -drawarrow(z51--z101); - -label.top(btex $x_0$ etex,z16); -label.top(btex $x_1$ etex,z26); -label.top(btex $x_2$ etex,z36); -label.top(btex \dots etex,z66); -label.top(btex $x_{n-2}$ etex,z96); -label.top(btex $x_{n-1}$ etex,z106); - -endfig; - - - -beginfig(4); -v:=u*5mm; - -z1075=(7.5v,0v); - -z10=(0v,1v); -z175=(7.5v,1v); -z115=(15v,1v); - -z20=(0v,2v); -z235=(3.5v,2v); -z211=(11.5v,2v); -z215=(15v,2v); - -z30=(0v,3v); -z335=(3.5v,3v); -z37=(7v,3v); -z38=(8v,3v); -z311=(11.5v,3v); -z315=(15v,3v); - -z40=(0v,4v); -z415=(1.5v,4v); -z455=(5.5v,4v); -z47=(7v,4v); -z48=(8v,4v); -z495=(9.5v,4v); -z413=(13.5v,4v); -z416=(15v,4v); - -z50=(0v,5v); -z515=(1.5v,5v); -z53=(3v,5v); -z54=(4v,5v); -z555=(5.5v,5v); -z57=(7v,5v); -z58=(8v,5v); -z595=(9.5v,5v); -z511=(11v,5v); -z512=(12v,5v); -z513=(13.5v,5v); -z516=(15v,5v); - -z60=(0v,6v); -z61=(1v,6v); -z62=(2v,6v); -z63=(3v,6v); -z64=(4v,6v); -z65=(5v,6v); -z66=(6v,6v); -z67=(7v,6v); -z68=(8v,6v); -z69=(9v,6v); -z610=(10v,6v); -z611=(11v,6v); -z612=(12v,6v); -z613=(13v,6v); -z614=(14v,6v); -z615=(15v,6v); - -z71=(1v,7v); -z72=(2v,7v); -z75=(5v,7v); -z76=(6v,7v); -z79=(9v,7v); -z710=(10v,7v); -z713=(13v,7v); -z714=(14v,7v); - - - - -pickup pencircle scaled 0.4pt; -draw(z10--z115--z215--z20--cycle); -draw(z30--z37--z47--z40--cycle); -draw(z38--z315--z416--z48--cycle); -draw(z50--z53--z63--z60--cycle); -draw(z54--z57--z67--z64--cycle); -draw(z58--z511--z611--z68--cycle); -draw(z512--z516--z615--z612--cycle); -drawarrow(z71--z61); -drawarrow(z72--z62); -drawarrow(z75--z65); -drawarrow(z76--z66); -drawarrow(z79--z69); -drawarrow(z710--z610); -drawarrow(z713--z613); -drawarrow(z714--z614); -drawarrow(z515--z415); -drawarrow(z555--z455); -drawarrow(z595--z495); -drawarrow(z513--z413); -drawarrow(z335--z235); -drawarrow(z311--z211); -drawarrow(z175--z1075); - -endfig; - - -beginfig(5); -v:=u*5mm; - -z075=(7.5v,8v); - -z10=(0v,7v); -z175=(7.5v,7v); -z115=(15v,7v); - -z20=(0v,6v); -z235=(3.5v,6v); -z211=(11.5v,6v); -z215=(15v,6v); - -z30=(0v,5v); -z335=(3.5v,5v); -z37=(7v,5v); -z38=(8v,5v); -z311=(11.5v,5v); -z315=(15v,5v); - -z40=(0v,4v); -z415=(1.5v,4v); -z455=(5.5v,4v); -z47=(7v,4v); -z48=(8v,4v); -z495=(9.5v,4v); -z413=(13.5v,4v); -z416=(15v,4v); - -z50=(0v,3v); -z515=(1.5v,3v); -z53=(3v,3v); -z54=(4v,3v); -z555=(5.5v,3v); -z57=(7v,3v); -z58=(8v,3v); -z595=(9.5v,3v); -z511=(11v,3v); -z512=(12v,3v); -z513=(13.5v,3v); -z516=(15v,3v); - -z60=(0v,2v); -z61=(1v,2v); -z62=(2v,2v); -z63=(3v,2v); -z64=(4v,2v); -z65=(5v,2v); -z66=(6v,2v); -z67=(7v,2v); -z68=(8v,2v); -z69=(9v,2v); -z610=(10v,2v); -z611=(11v,2v); -z612=(12v,2v); -z613=(13v,2v); -z614=(14v,2v); -z615=(15v,2v); - -% ve skutecnosti dle znaceni -% by melo byt z6* ale uz -% obsazeno -z815=(1.5v,2v); -z855=(5.5v,2v); -z895=(9.5v,2v); -z813=(13.5v,2v); - -z715=(1.5v,1v); -z755=(5.5v,1v); -z795=(9.5v,1v); -z713=(13.5v,1v); - -z9=(4v,0v); - -pickup pencircle scaled 0.4pt; -draw(z10--z115--z215--z20--cycle); -draw(z30--z37--z47--z40--cycle); -draw(z38--z315--z416--z48--cycle); -draw(z50--z53--z63--z60--cycle); -draw(z54--z57--z67--z64--cycle); -draw(z58--z511--z611--z68--cycle); -draw(z512--z516--z615--z612--cycle); -drawarrow(z075--z175); -drawarrow(z415--z515); -drawarrow(z455--z555); -drawarrow(z495--z595); -drawarrow(z413--z513); -drawarrow(z235--z335); -drawarrow(z211--z311); -drawarrow(z815--z715); -drawarrow(z855--z755); -drawarrow(z895--z795); -drawarrow(z813--z713); - -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 $S_{n\over 4}$ etex,z515); -label.bot(btex $S_{n\over 4}$ etex,z555); -label.bot(btex $S_{n\over 4}$ etex,z595); -label.bot(btex $S_{n\over 4}$ etex,z513); -label.rt(btex Bitonick\'a t\v r\'\i di\v cka $B_{n}$ etex,z9); - -endfig; - - - -beginfig(6); -v:=u*5mm; - -z1075=(7.5v,0v); - -z10=(0v,1v); -z175=(7.5v,1v); -z115=(15v,1v); - -z20=(0v,2v); -z235=(3.5v,2v); -z211=(11.5v,2v); -z215=(15v,2v); - -z30=(0v,3v); -z335=(3.5v,3v); -z37=(7v,3v); -z38=(8v,3v); -z311=(11.5v,3v); -z315=(15v,3v); - -z40=(0v,4v); -z415=(1.5v,4v); -z455=(5.5v,4v); -z47=(7v,4v); -z48=(8v,4v); -z495=(9.5v,4v); -z413=(13.5v,4v); -z416=(15v,4v); - -z50=(0v,5v); -z515=(1.5v,5v); -z53=(3v,5v); -z54=(4v,5v); -z555=(5.5v,5v); -z57=(7v,5v); -z58=(8v,5v); -z595=(9.5v,5v); -z511=(11v,5v); -z512=(12v,5v); -z513=(13.5v,5v); -z516=(15v,5v); - -z60=(0v,6v); -z61=(1v,6v); -z62=(2v,6v); -z63=(3v,6v); -z64=(4v,6v); -z65=(5v,6v); -z66=(6v,6v); -z67=(7v,6v); -z68=(8v,6v); -z69=(9v,6v); -z610=(10v,6v); -z611=(11v,6v); -z612=(12v,6v); -z613=(13v,6v); -z614=(14v,6v); -z615=(15v,6v); - -z71=(1v,7v); -z72=(2v,7v); -z75=(5v,7v); -z76=(6v,7v); -z79=(9v,7v); -z710=(10v,7v); -z713=(13v,7v); -z714=(14v,7v); - - - - -pickup pencircle scaled 0.4pt; -draw(z10--z115--z215--z20--cycle); -draw(z30--z37--z47--z40--cycle); -draw(z38--z315--z416--z48--cycle); -draw(z50--z53--z63--z60--cycle); -draw(z54--z57--z67--z64--cycle); -draw(z58--z511--z611--z68--cycle); -draw(z512--z516--z615--z612--cycle); -drawarrow(z71--z61); -drawarrow(z72--z62); -drawarrow(z75--z65); -drawarrow(z76--z66); -drawarrow(z79--z69); -drawarrow(z710--z610); -drawarrow(z713--z613); -drawarrow(z714--z614); -drawarrow(z515--z415); -drawarrow(z555--z455); -drawarrow(z595--z495); -drawarrow(z513--z413); -drawarrow(z335--z235); -drawarrow(z311--z211); -drawarrow(z175--z1075); - -label.top(btex $M_8$ etex,z175); -label.top(btex $M_4$ etex,z335); -label.top(btex $M_4$ etex,z311); -label.top(btex $M_2$ etex,z515); -label.top(btex $M_2$ etex,z555); -label.top(btex $M_2$ etex,z595); -label.top(btex $M_2$ etex,z513); - -endfig; - - -beginfig(7); -v:=u*7mm; - -z12=(1v,2v); -z13=(1v,3v); -z16=(1v,6v); -z356=(3.5v,6v); -z40=(5v,1v); -z42=(5v,2v); -z47=(5v,6.5v); -z72=(9v,2v); -z74=(9v,4v); -z76=(9v,6v); - -z100=(0.5v,0.5v); -z101=(1.5v,0.5v); - -z0=whatever[z13,z356]; -z1=whatever[z356,z74]; -z1=z0+4v*right; -z2=whatever[z12,z72]; -z2=z0+whatever*down; -z3=whatever[z12,z72]; -z3=z1+whatever*down; -z4=z356+4v*right; -z5=whatever[z72,z76]; -z5=whatever[z4,z1+4v*right]; -z6=whatever[z40,z47]; -z6=z74+4v*left; - - -pickup pencircle scaled 0.4pt; -draw(z16--z12--z72--z76); -draw(z13--z356--z74); -draw(z40--z47) dashed evenly; -draw(z1--z4) dashed withdots scaled 0.7; -draw(z4--z5) dashed withdots scaled 0.7; -draw(z0--z6) dashed withdots scaled 0.7; -draw(z0--z2) dashed evenly; -draw(z1--z3) dashed evenly; - -draw(z100--z101) dashed withdots scaled 0.7; - -pickup pencircle scaled 3pt; -drawdot(z0); -drawdot(z1); - -label.bot(btex \strut 0 etex,z12); -label.bot(btex $k$ etex,z2); -label.llft(btex \strut ${n\over 2} - 1$ etex,z42); -label.bot(btex \strut $k+{n\over 2}$ etex,z3); -label.bot(btex \strut $n-1$ etex,z72); -label.rt(btex posloupnost prohozen\'a separ\'atorem etex,z101); - -endfig; - - - - - - - -end; diff --git a/old/5-addsort/sortnet.mpx b/old/5-addsort/sortnet.mpx deleted file mode 100644 index 9814d6d..0000000 --- a/old/5-addsort/sortnet.mpx +++ /dev/null @@ -1,548 +0,0 @@ -% Written by DVItoMP, Version 0.64/color (Web2C 7.5.4) -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; -_n0="cmr10"; -_s("a",_n0,1.00000,0.0000,0.0000,); -setbounds _p to (0,-3.4869)--(4.9813,-3.4869)-- - (4.9813,8.4682)--(0,8.4682)--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; -_n0="cmr10"; -_s("b",_n0,1.00000,0.0000,0.0000,); -setbounds _p to (0,-3.4869)--(5.5348,-3.4869)-- - (5.5348,8.4682)--(0,8.4682)--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; -_n0="cmr10"; -_s("min",_n0,1.00000,0.0000,0.0000,); -setbounds _p to (0,0.0000)--(16.6045,0.0000)-- - (16.6045,6.6536)--(0,6.6536)--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; -_n0="cmr10"; -_s("max",_n0,1.00000,0.0000,0.0000,); -setbounds _p to (0,0.0000)--(18.5416,0.0000)-- - (18.5416,4.2895)--(0,4.2895)--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; -_n0="cmr10"; -_s("x1",_n0,1.00000,0.0000,0.0000,); -setbounds _p to (0,0.0000)--(10.2394,0.0000)-- - (10.2394,6.4204)--(0,6.4204)--cycle; -_p endgroup -mpxbreak -begingroup save 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(10.2394,6.4204)--(0,6.4204)--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; -_n0="cmr10"; -_s("x3",_n0,1.00000,0.0000,0.0000,); -setbounds _p to (0,0.0000)--(10.2394,0.0000)-- - (10.2394,6.4204)--(0,6.4204)--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; -_n0="cmr10"; -_s("x4",_n0,1.00000,0.0000,0.0000,); -setbounds _p to (0,0.0000)--(10.2394,0.0000)-- - (10.2394,6.4204)--(0,6.4204)--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; -_n0="cmr10"; -_s("x5",_n0,1.00000,0.0000,0.0000,); -setbounds _p to 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addto _p also _t infont _f scaled _m shifted (_x,_y) _c; enddef; -_n1="cmmi10"; -_s("x",_n1,1.00000,0.0000,0.0000,); -_n2="cmr7"; -_s("2",_n2,1.00000,5.6939,-1.4944,); -setbounds _p to (0,-1.4944)--(10.1633,-1.4944)-- - (10.1633,4.2895)--(0,4.2895)--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(":",_n1,1.00000,0.0000,0.0000,); -_s(":",_n1,1.00000,4.4278,0.0000,); -_s(":",_n1,1.00000,8.8556,0.0000,); -setbounds _p to (0,0.0000)--(13.2834,0.0000)-- - (13.2834,1.0516)--(0,1.0516)--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("x",_n1,1.00000,0.0000,0.0000,); -_n3="cmmi7"; -_s("n",_n3,1.00000,5.6939,-1.4944,); -_n4="cmsy7"; 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(5.9799,4.2895)--(0,4.2895)--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,); -_n3="cmmi7"; -_s("n",_n3,1.00000,6.1090,-1.4944,); -setbounds _p to (0,-1.4944)--(11.5320,-1.4944)-- - (11.5320,6.8078)--(0,6.8078)--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,); -interim linecap:=0; -vardef _r(expr _a,_w)(text _t) = - addto _p doublepath _a withpen pencircle scaled _w _t enddef; -_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); -_n6="cmr5"; -_s("2",_n6,1.00000,7.8033,-3.8949,); -setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- - (13.3857,6.8078)--(0,6.8078)--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,); -interim linecap:=0; -vardef _r(expr _a,_w)(text _t) = - addto _p doublepath _a withpen pencircle scaled _w _t enddef; -_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); -_n6="cmr5"; -_s("2",_n6,1.00000,7.8033,-3.8949,); -setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- - (13.3857,6.8078)--(0,6.8078)--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,); -interim linecap:=0; -vardef _r(expr _a,_w)(text _t) = - addto _p doublepath _a withpen pencircle scaled _w _t enddef; -_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); -_n6="cmr5"; -_s("4",_n6,1.00000,7.8033,-3.8949,); -setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- - (13.3857,6.8078)--(0,6.8078)--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,); -interim linecap:=0; -vardef _r(expr _a,_w)(text _t) = - addto _p doublepath _a withpen pencircle scaled _w _t enddef; -_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); -_n6="cmr5"; -_s("4",_n6,1.00000,7.8033,-3.8949,); -setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- - (13.3857,6.8078)--(0,6.8078)--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,); -interim linecap:=0; -vardef _r(expr _a,_w)(text _t) = - addto _p doublepath _a withpen pencircle scaled _w _t enddef; -_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); -_n6="cmr5"; -_s("4",_n6,1.00000,7.8033,-3.8949,); -setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- - (13.3857,6.8078)--(0,6.8078)--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,); -interim linecap:=0; -vardef _r(expr _a,_w)(text _t) = - addto _p doublepath _a withpen pencircle scaled _w _t enddef; -_r((7.3045,0.2491)..(11.6921,0.2491), 0.3985,); -_n6="cmr5"; -_s("4",_n6,1.00000,7.8033,-3.8949,); -setbounds _p to (0,-3.8949)--(13.3857,-3.8949)-- - (13.3857,6.8078)--(0,6.8078)--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; -_n0="cmr10"; -_s("Bitonic",_n0,1.00000,0.0000,0.0000,); -_s("k"&char19,_n0,1.00000,31.1333,0.0000,); -_s("a",_n0,1.00000,36.3914,0.0000,); -_s("t",_n0,1.00000,44.6936,0.0000,); -_s(char20,_n0,1.00000,48.0283,0.0000,); -_s("r",_n0,1.00000,48.5680,0.0000,); -_s(char19,_n0,1.00000,51.3631,0.0000,); -_s(char16&"di",_n0,1.00000,52.4700,0.0000,); -_s(char20,_n0,1.00000,63.2629,0.0000,); -_s("ck",_n0,1.00000,63.5397,0.0000,); -_s("a",_n0,1.00000,72.6721,0.0000,); -_n1="cmmi10"; -_s("B",_n1,1.00000,80.9743,0.0000,); -_n3="cmmi7"; -_s("n",_n3,1.00000,88.5311,-1.4944,); -setbounds _p to (0,-1.4944)--(93.9541,-1.4944)-- - (93.9541,6.9185)--(0,6.9185)--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("8",_n2,1.00000,9.6652,-1.4944,); -setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- - (14.1345,6.8078)--(0,6.8078)--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,); -setbounds _p to (0,-1.4944)--(14.1345,-1.4944)-- - (14.1345,6.8078)--(0,6.8078)--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)= 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