From: Martin Mares Date: Sat, 19 Nov 2011 19:13:10 +0000 (+0100) Subject: Hradla: Prepsana cast o nasobeni cisel X-Git-Url: http://mj.ucw.cz/gitweb/?a=commitdiff_plain;h=1f505896891873dcba3568faf24771aeee0101cc;p=ads2.git Hradla: Prepsana cast o nasobeni cisel --- diff --git a/5-hradla/1_9_deleni_bloku.eps b/5-hradla/1_9_deleni_bloku.eps deleted file mode 100644 index 2c96b1c..0000000 --- a/5-hradla/1_9_deleni_bloku.eps +++ /dev/null @@ -1,3225 +0,0 @@ -%!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 exch def - /proc exch cvx def - /mtx exch 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b/5-hradla/5-hradla.tex index 082c380..c47a8b3 100644 --- a/5-hradla/5-hradla.tex +++ b/5-hradla/5-hradla.tex @@ -317,45 +317,57 @@ Obecn \:$\Theta(1)$ hladin na samotné seètení: $\forall i: z_i = x_i \oplus y_i \oplus c_i$. \endalgo -\figure{1_9_deleni_bloku.eps}{Výpoèet pøenosu}{2.5in} +\figure{deleni_bloku.eps}{Výpoèet pøenosu}{2.5in} Algoritmus tedy pracuje v~èase $\Theta(\log n)$. Hradel je pou¾ito lineárnì: pøi výpoètu chování blokù na jednotlivých hladinách poèet hradel exponenciálnì klesá od~$n$ k~1, bìhem zahu¹»ování pøenosù naopak exponenciálnì stoupá od~1 k~$n$. -Obì geometrické øady mají souèet $\Theta(n)$. +Obì geometrické øady se seètou 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 paralelní a~zvládneme ho za~konstantnì mnoho hladin. Celé násobení tedy zvládneme v~logaritmickém èase. +Je¹tì si rozmysleme, jak rychle by bylo mo¾né èísla násobit. Opìt se inspirujeme +¹kolním algoritmem: pokud násobíme dvì $n$-ciferná èísla $x$ a~$y$, uvá¾íme +v¹ech~$n$ posunutí èísla~$y$, ka¾dé z~nich vynásobíme pøíslu¹nou èíslicí v~$y$ +a výsledky posèítáme. + +\figure{skolni_nasobeni.eps}{©kolní násobení}{2in} + +Ve~dvojkové soustavì je to je¹tì jednodu¹¹í: násobení jednou èíslicí je prostý +{\csc and.} Paralelnì tedy vytvoøíme v¹echna posunutí a spoèítáme v¹echny {\csc and}y. +To v¹e stihneme za 1~takt výpoètu. + +Zbývá seèíst $n$~èísel, z~nich¾ ka¾dé má $\Theta(n)$ bitù. Mohli bychom opìt +pou¾ít osvìdèený trik: sèítat dvojice èísel, pak dvojice souètù dvojic, atd. +Taková sí» by mìla tvar binárního stromu hloubky $\log n$, jeho¾ ka¾dý vrchol +obsahuje jednu sèítaèku a na tu, jak víme, postaèí $\Theta(\log n)$ hladin. +Celý výpoèet tedy bì¾í v~èase $\Theta(\log^2 n)$. + +Jde to ale rychleji, pou¾ijeme-li jednoduchý, témìø kouzelnický trik. +Sestrojíme {\I kompresor} -- to bude obvod konstantní hloubky, který na~vstupu +dostane tøi èísla a vypoète z~nich dvì èísla mající stejný souèet jako zadaná +trojice. + +K~èemu je to dobré? Máme-li seèíst $n$~èísel, v~konstantním èase doká¾eme tento +úkol pøevést na seètení $\lceil 2/3\cdot n\rceil$ èísel, to pak opìt v~konstantním èase +na seètení $\lceil (2/3)^2\cdot n\rceil$ èísel atd., a¾ nám po $\lceil\log_{3/2} n\rceil = \Theta(\log n)$ krocích +zbudou dvì èísla a ta seèteme klasickou sèítaèkou. Zbývá vymyslet kompresor. + +\s{Konstrukce kompresoru:} +Oznaème vstupy kompresoru $x$, $y$ a~$z$ a výstupy $p$ a~$q$. +Pro ka¾dý øád~$i$ spoèteme souèet $x_i + y_i + z_i$. To je nìjaké +dvoubitové èíslo, tak¾e mù¾eme jeho ní¾¹í bit prohlásit za~$p_i$ +a vy¹¹í za~$q_{i+1}$. + +Jinými slovy jsme v¹echna tøi èísla normálnì seèetli, ale místo abychom +pøenosy posílali do vy¹¹ího øádu, vytvoøili jsme z~nich dal¹í èíslo, +které má být k~výsledku èasem pøièteno. + +\figure{add_3to2.eps}{Kompresor}{0.9in} + +\s{Shrnutí:} Na¹e sí» pro paralelní násobení pracuje v~èase $\Theta(\log n)$ +-- nejdøíve v~konstantním èase vytváøíme mezivýsledky, pak pou¾ijeme $\Theta(\log n)$ +hladin kompresorù konstantní hloubky a nakonec jednu sèítaèku hloubky $\Theta(\log n)$. +Jistou vadou na kráse ov¹em je, ¾e na to potøebujeme $\Theta(n^2)$ hradel. \bye diff --git a/5-hradla/add_3to2.eps b/5-hradla/add_3to2.eps new file mode 100644 index 0000000..f468f45 --- /dev/null +++ b/5-hradla/add_3to2.eps @@ -0,0 +1,879 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: inkscape 0.46 +%%Pages: 1 +%%Orientation: Portrait +%%BoundingBox: 84 577 284 728 +%%HiResBoundingBox: 84.350568 577.24478 283.94358 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