From 46edfe5592d85122354f2f39674d0e28986c856d Mon Sep 17 00:00:00 2001 From: Florian Forster Date: Sat, 26 Feb 2011 18:57:38 +0100 Subject: [PATCH] SN-Evolution-Cut: Mehr zu den Versuchen mit OES(n). --- diplomarbeit.tex | 106 +++++++++++++++++++++- images/11-ec-from-oes17-fast.tex | 165 ++++++++++++++++++++++++++++++++++ images/12-ec-from-oes18-fast.tex | 186 +++++++++++++++++++++++++++++++++++++++ 3 files changed, 456 insertions(+), 1 deletion(-) create mode 100644 images/11-ec-from-oes17-fast.tex create mode 100644 images/12-ec-from-oes18-fast.tex diff --git a/diplomarbeit.tex b/diplomarbeit.tex index 3caa170..0358d88 100644 --- a/diplomarbeit.tex +++ b/diplomarbeit.tex @@ -2165,6 +2165,110 @@ $\operatorname{OET}(n-k)$-Netzwerk. \subsection[Odd-Even-Mergesort-Netzwerk]{Versuche mit dem Odd-Even-Mergesort-Netzwerk} \label{sect:sn-evolution-cut:oes} +Wird \textsc{SN-Evolution-Cut} mit dem \emph{Odd-Even-Mergesort}-Netzwerk +\oes{n} gestartet, gibt der Algorithmus meist Sortiernetzwerke zurück, die +genauso effizient und schnell wie das entsprechende +\emph{Odd-Even-Mergesort}-Netzwerk \oes{m} sind. Die Effizienz der +Sortiernetzwerke, die mit Schnittmustern von \textsc{SN-Evolution-Cut} aus +\oes{n} entstehen können, zeigt Tabelle~\ref{tbl:ec-oes-efficiency} +tabellarisch. + +\begin{table} + \begin{center} + \rowcolors{2}{black!5}{} +\begin{tabular}{|r|rrrrrrrrrrrrrrrr|} +\hline + & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ +\hline + 9 & 19 & & & & & & & & & & & & & & & \\ + 10 & 19 & 26 & & & & & & & & & & & & & & \\ + 11 & 19 & 26 & 31 & & & & & & & & & & & & & \\ + 12 & 19 & 26 & 31 & 37 & & & & & & & & & & & & \\ + 13 & 19 & 26 & 31 & 37 & 41 & & & & & & & & & & & \\ + 14 & 19 & 26 & 31 & 37 & 41 & 48 & & & & & & & & & & \\ + 15 & 19 & 26 & 31 & 37 & 41 & 48 & 53 & & & & & & & & & \\ + 16 & 19 & 26 & 31 & 37 & 41 & 48 & 53 & 59 & & & & & & & & \\ + 17 & 19 & 26 & 31 & 38 & 41 & 48 & 53 & 59 & 63 & & & & & & & \\ + 18 & 19 & 26 & 31 & 38 & 43 & 48 & 53 & 59 & 63 & 74 & & & & & & \\ + 19 & 19 & 26 & 31 & 38 & 43 & 48 & 53 & 59 & 63 & 74 & 82 & & & & & \\ + 20 & 19 & 26 & 31 & 38 & 43 & 48 & 53 & 59 & 63 & 74 & 82 & 91 & & & & \\ + 21 & 19 & 26 & 31 & 38 & 43 & 48 & 53 & 59 & 63 & 74 & 82 & 91 & 97 & & & \\ + 22 & 19 & 26 & 31 & 38 & 43 & 48 & 53 & 59 & 63 & 74 & 82 & 91 & 97 & 107 & & \\ + 23 & 19 & 26 & 31 & 38 & 43 & 48 & 53 & 59 & 63 & 74 & 82 & 91 & 97 & 107 & 114 & \\ + 24 & 19 & 26 & 31 & 38 & 43 & 48 & 53 & 59 & 63 & 74 & 82 & 91 & 97 & 107 & 114 & 122 \\ +\hline +\end{tabular} + \end{center} + \caption{Anzahl der Komparatoren der Ergebnisse von + \textsc{SN-Evolution-Cut} mit verschiedenen Größen des + \emph{Odd-Even-Mergesort}-Netzwerks und unterschiedlichen Werten für~$k$. + Jede Zeile gibt die Ergebnisse für ein Eingabenetzwerk \oes{n} an, jede + Spalte enthält die Ergebnisse für $m=n-k$, die Anzahl der Leitungen des + Ausgabenetzwerks.} + \label{tbl:ec-oes-efficiency} +\end{table} + +\begin{figure} + \centering + \subfigure[11-Sortiernetzwerk aus 38~Komparatoren in 9~Schichten. Das + Netzwerk wurde von \textsc{SN-Evolution-Cut} aus \oes{17} erzeugt.]{\input{images/11-ec-from-oes17-fast.tex}\label{fig:11-ec-from-oes17-fast}} + \subfigure[12-Sortiernetzwerk aus 43~Komparatoren in 9~Schichten. Das + Netzwerk wurde von \textsc{SN-Evolution-Cut} aus \oes{18} erzeugt.]{\input{images/12-ec-from-oes18-fast.tex}\label{fig:12-ec-from-oes18-fast}} + \caption{Für einige Ziel-Leitungszahlen, unter anderem $m = 10$ und $m = + 11$, kann der \textsc{SN-Evolution-Cut}-Algorithmus Sortiernetzwerke + erzeugen, die \emph{schneller} aber weniger \emph{effizient} als \oes{m} + sind.} + \label{fig:ec-oes-fast_networks} +\end{figure} + +Die Bewertungsfunktion, die \textsc{SN-Evolution-Cut} verwendet, bevorzugt +schnelle Sortiernetzwerke. Dadurch kann es vorkommen, dass ein +$m$-Sortiernetzwerk, das durch ein von \textsc{SN-Evolution-Cut} ausgegebenes +Schnittmuster entsteht, schneller als \oes{m} ist. Diese Geschwindigkeit +war allerdings in allen beobachteten Fällen nur dann möglich, wenn +zusätzliche Komparatoren in Kauf genommen wurden. In den +Tabellen~\ref{tbl:ec-oes-efficiency} und~\ref{tbl:ec-oes-speed} ist dieser +Fall für $m = 11$ und $k \geqq 6$, beziehungsweise $m = 12$ und $k \geqq 6$ zu +beobachten. Die entsprechenden schnellen Sortiernetzwerke sind in +Abbildung~\ref{fig:ec-oes-fast_networks} dargestellt. + +\begin{table} + \begin{center} + \rowcolors{2}{black!5}{} +\begin{tabular}{|r|rrrrrrrrrrrrrrrr|} +\hline + & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 \\ +\hline + 9 & 6 & & & & & & & & & & & & & & & \\ + 10 & 6 & 8 & & & & & & & & & & & & & & \\ + 11 & 6 & 8 & 9 & & & & & & & & & & & & & \\ + 12 & 6 & 8 & 9 & 10 & & & & & & & & & & & & \\ + 13 & 6 & 8 & 9 & 10 & 10 & & & & & & & & & & & \\ + 14 & 6 & 8 & 9 & 10 & 10 & 10 & & & & & & & & & & \\ + 15 & 6 & 8 & 9 & 10 & 10 & 10 & 10 & & & & & & & & & \\ + 16 & 6 & 8 & 9 & 10 & 10 & 10 & 10 & 10 & & & & & & & & \\ + 17 & 6 & 8 & 9 & 9 & 10 & 10 & 10 & 10 & 10 & & & & & & & \\ + 18 & 6 & 8 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 12 & & & & & & \\ + 19 & 6 & 8 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 12 & 13 & & & & & \\ + 20 & 6 & 8 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 12 & 13 & 14 & & & & \\ + 21 & 6 & 8 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 12 & 13 & 14 & 14 & & & \\ + 22 & 6 & 8 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 12 & 13 & 14 & 14 & 15 & & \\ + 23 & 6 & 8 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 12 & 13 & 14 & 14 & 15 & 15 & \\ + 24 & 6 & 8 & 9 & 9 & 9 & 10 & 10 & 10 & 10 & 12 & 13 & 14 & 14 & 15 & 15 & 15 \\ +\hline +\oes{m}& 6 & 8 & 9 & 10 & 10 & 10 & 10 & 10 & 10 & 12 & 13 & 14 & 14 & 15 & 15 & 15 \\ +\hline +\end{tabular} + \end{center} + \caption{Anzahl der Schichten der Ergebnisse von + \textsc{SN-Evolution-Cut} mit verschiedenen Größen des + \emph{Odd-Even-Mergesort}-Netzwerks und unterschiedlichen Werten für~$k$. + Jede Zeile gibt die Ergebnisse für ein Eingabenetzwerk \oes{n} an, jede + Spalte enthält die Ergebnisse für $m=n-k$, die Anzahl der Leitungen des + Ausgabenetzwerks.} + \label{tbl:ec-oes-speed} +\end{table} + In Abschnitt~\ref{sect:anzahl_schnittmuster} wurde bereits untersucht, wie viele \emph{unterschiedliche} Schnittmuster die konstruktiven Sortiernetzwerke $\operatorname{OES}(32)$, $\operatorname{BS}(32)$ und $\operatorname{PS}(32)$ @@ -2353,7 +2457,7 @@ wie und warum es jede beliebige Eingabe sortiert. Jede Zeile gibt die Ergebnisse für ein Eingabenetzwerk \ps{n} an, jede Spalte enthält die Ergebnisse für $m=n-k$, die Anzahl der Leitungen des Ausgabenetzwerks.} - \label{tbl:ec-ps-fast} + \label{tbl:ec-ps-speed} \end{table} Das \emph{Pairwise-Sorting-Netzwerk} $\operatorname{PS}(n)$, das \textit{Ian diff --git a/images/11-ec-from-oes17-fast.tex b/images/11-ec-from-oes17-fast.tex new file mode 100644 index 0000000..c46afe8 --- /dev/null +++ b/images/11-ec-from-oes17-fast.tex @@ -0,0 +1,165 @@ +\begin{tikzpicture}[auto] +\node[vertex] (v0) at (0.56,0.45) {}; +\node[vertex] (v1) at (0.56,1.81) {}; +\path[comp] (v0) -- (v1); + +\node[vertex] (v2) at (0.73,0.90) {}; +\node[vertex] (v3) at (0.73,1.35) {}; +\path[comp] (v2) -- (v3); + +\node[vertex] (v4) at (0.56,2.26) {}; +\node[vertex] (v5) at (0.56,3.61) {}; +\path[comp] (v4) -- (v5); + +\node[vertex] (v6) at (0.73,2.71) {}; +\node[vertex] (v7) at (0.73,3.16) {}; +\path[comp] (v6) -- (v7); + +\node[vertex] (v8) at (0.56,4.06) {}; +\node[vertex] (v9) at (0.56,4.52) {}; 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