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-% showpage\r
-%%EOF\r
\:$\Theta(1)$ 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
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projects / ads2.git / commitdiff
