From 571050fa4c8c889abbd5fcf975da8a59dae20920 Mon Sep 17 00:00:00 2001 From: Florian Forster Date: Mon, 21 Feb 2011 21:27:10 +0100 Subject: [PATCH] Diverses Neues zu SN-Evolution-Cut. Eigentlich will ich ja fertig werden :( --- diplomarbeit.tex | 203 ++++++++- images/11-ec-from-bs22-fast.tex | 161 +++++++ images/12-ec-from-bs24-fast.tex | 183 ++++++++ images/12-ec-from-oes24-efficient.tex | 178 ++++++++ images/12-ec-from-oes24-fast.tex | 186 ++++++++ images/23-ec-from-bs46-fast.tex | 521 ++++++++++++++++++++++ images/23-ec-from-oes46-fast.tex | 525 ++++++++++++++++++++++ images/32-ec-from-oes64.tex | 807 ++++++++++++++++++++++++++++++++++ 8 files changed, 2740 insertions(+), 24 deletions(-) create mode 100644 images/11-ec-from-bs22-fast.tex create mode 100644 images/12-ec-from-bs24-fast.tex create mode 100644 images/12-ec-from-oes24-efficient.tex create mode 100644 images/12-ec-from-oes24-fast.tex create mode 100644 images/23-ec-from-bs46-fast.tex create mode 100644 images/23-ec-from-oes46-fast.tex create mode 100644 images/32-ec-from-oes64.tex diff --git a/diplomarbeit.tex b/diplomarbeit.tex index d4b14e7..e0a902c 100644 --- a/diplomarbeit.tex +++ b/diplomarbeit.tex @@ -1382,8 +1382,6 @@ Abbildung~\ref{fig:16-e1-oddeven-1296543330} zu sehen. Ein Netzwerk, das $\operatorname{OES}(n)$ in mindestens einem Merkmal übertrifft, konnte jedoch nicht beobachtet werden. -\todo{Ggf. Abschnitt „Shmoo-Äquivalenz“ kürzen und hier einbauen.} - %\begin{figure} %\begin{center} %\input{images/08-e2-1237993371.tex} @@ -1530,8 +1528,8 @@ Leider sind die Schnittmuster, die \textsc{SN-Evolution-Cut} ausgibt, sehr unregelmäßig. Bisher ist es nicht gelungen eine Konstruktionsanweisung für gute Schnittmuster anzugeben. -Entscheidend für das Ergebnis eines Schnittmusters scheint beim bitonen -Mergesort-Netzwerk die Aufteilung der Minimum- und Maximumschnitte zu sein. +Entscheidend für das Ergebnis eines Schnittmusters scheint beim \emph{bitonen +Mergesort}-Netzwerk die Aufteilung der Minimum- und Maximumschnitte zu sein. Von Hundert 16-Schnittmustern für $\operatorname{BS}(32)$, die in Sortiernetzwerken mit 68~Komparatoren in 10~Schichten resultieren, hatten 73 ein Verhältnis von $5/11$, 13 hatten ein Verhältnis von $4/12$ und 14 hatten @@ -1540,6 +1538,57 @@ die Schnittmuster aufgrund der Symmetrie des bitonen Mergesort-Netzwerks leicht invertieren lassen, werden der Fall, dass es mehr Minimumschnitte, und der Fall, dass es mehr Maximumschnitte gibt, nicht unterschieden. +\begin{figure} + \centering + \subfigure[11-Sortiernetzwerk aus 37~Komparatoren in 9~Schichten. Das + Netzwerk wurde von \textsc{SN-Evolution-Cut} aus \bs{22} erzeugt.]{\input{images/11-ec-from-bs22-fast.tex}\label{fig:11-ec-from-bs22-fast}} + \subfigure[12-Sortiernetzwerk aus 42~Komparatoren in 9~Schichten. Das + Netzwerk wurde von \textsc{SN-Evolution-Cut} aus \bs{24} erzeugt.]{\input{images/12-ec-from-bs24-fast.tex}\label{fig:12-ec-from-bs24-fast}} + \caption{Startet man \textsc{SN-Evolution-Cut} mit \bs{22} und \bs{24}, kann + der Algorithmus schnelle Sortiernetzwerke ausgeben.} + \label{fig:11-12-ec-from-bs22-bs24} +\end{figure} + +Verwendet man als Eingabe für \textsc{SN-Evolution-Cut} Instanzen des +\emph{bitonen Mergesort}-Netzwerks, deren Leitungszahl keine Zweierpotenz ist, +können Sortiernetzwerke zurückgegeben werden, die sowohl schneller als auch +effizienter als das entsprechende \emph{bitone Mergesort}-Netzwerk sind. Die +folgende Tabelle listet einige interessante Fälle auf. Die Eingabe für +\textsc{SN-Evolution-Cut} war jeweils das \emph{bitone Mergesort}-Netzwerk mit +der doppelten Leitungszahl. +Abbildung~\ref{fig:23-ec-from-bs46} zeigt beispielhaft ein +23-Sortiernetzwerk, das aus \bs{46} generiert wurde. +\begin{center} +\begin{tabular}{|r|r|r|r|r|} +\hline +Leitungen & Komparatoren & Schichten & Komparatoren & Schichten \\ + & \textsc{SN-EC} & \textsc{SN-EC} & \bs{n} & + \bs{n} \\ +\hline +11 & 37 & 9 & 39 & 10 \\ +12 & 42 & 9 & 46 & 10 \\ +19 & 93 & 13 & 98 & 14 \\ +20 & 102 & 13 & 106 & 14 \\ +% 20: # sn-cut 2:MAX 3:MIN 4:MIN 9:MIN 10:MIN 13:MIN 14:MIN 15:MIN 19:MIN 20:MAX 24:MAX 26:MIN 27:MAX 29:MIN 31:MAX 33:MIN 34:MAX 35:MIN 37:MIN 39:MAX +21 & 109 & 14 & 114 & 15 \\ +22 & 116 & 14 & 123 & 15 \\ +23 & 124 & 14 & 133 & 15 \\ +\hline +\end{tabular} +\end{center} + +\begin{figure} + \begin{center} + \input{images/23-ec-from-bs46-fast.tex} + \end{center} + \caption{23-Sortiernetzwerk mit 124~Komparatoren in 14~Schichten. Das + Netzwerk wurde von \textsc{SN-Evolution-Cut} aus \bs{46} mit dem + Schnittmuster $\operatorname{MIN}(2, 4, 9, 12, 20, 22, 28, 30, 32, 33, 37, + 38, 41)$, $\operatorname{MAX}(1, 5, 16, 19, 21, 24, 25, 35, 36, 43)$ + erzeugt.} + \label{fig:23-ec-from-bs46} +\end{figure} + Dass die Ergebnisse von \textsc{SN-Evolution-Cut} keine erkennbare Struktur haben, ist jedoch kein Eigenschaft des Algorithmus, sondern hängt insbesondere von der Eingabe ab. Wird \textsc{SN-Evolution-Cut} beispielsweise mit dem @@ -1547,17 +1596,6 @@ von der Eingabe ab. Wird \textsc{SN-Evolution-Cut} beispielsweise mit dem $m$~Schnitten gestartet, so ist das beste Ergebnis immer das $\operatorname{OET}(n-m)$-Netzwerk. -\begin{figure} - \begin{center} - \input{images/16-ec-from-ps32.tex} - \end{center} - \caption{Sortiernetzwerk mit 16~Leitungen und 63~Komparatoren in - 10~Schichten. Das Netzwerk wurde von dem Algorithmus - \textsc{SN-Evolution-Cut} aus dem \emph{Pairwise-Sorting-Netzwerk} - $\operatorname{PS}(32)$ durch 16~Schnitte erzeugt.} - \label{fig:16-ec-from-ps32} -\end{figure} - \subsection{Versuche mit dem Pairwise-Sorting-Netzwerk} Anders verhält sich das \emph{Pairwise-Sorting-Netzwerk} @@ -1569,6 +1607,17 @@ Anzahl an Komparatoren und Schichten hat wie $\operatorname{PS}(16)$ und $\operatorname{OES}(16)$. Eines dieser Sortiernetzwerke ist in Abbildung~\ref{fig:16-ec-from-ps32} dargestellt. +\begin{figure} + \begin{center} + \input{images/16-ec-from-ps32.tex} + \end{center} + \caption{Sortiernetzwerk mit 16~Leitungen und 63~Komparatoren in + 10~Schichten. Das Netzwerk wurde von dem Algorithmus + \textsc{SN-Evolution-Cut} aus dem \emph{Pairwise-Sorting-Netzwerk} + $\operatorname{PS}(32)$ durch 16~Schnitte erzeugt.} + \label{fig:16-ec-from-ps32} +\end{figure} + Obwohl das \emph{Pairwise-Sorting-Netzwerk} den \emph{Odd-Even}-Mischer nicht einsetzt und auch nicht auf einem Mischer basiert, ist der $\operatorname{OEM}(8,8)$ im Sortiernetzwerk in @@ -1634,12 +1683,17 @@ besitzen. Eines der Ergebnisse war, dass von diesen Sortiernetzwerken das \emph{Odd-Even-Mergesort}-Netzwerk die wenigsten unterschiedlichen 16-Schnittmuster besitzt -- nur etwa $5,2$~Millionen. Entsprechend ist es wenig verwunderlich, dass \textsc{SN-Evolution-Cut} gestartet mit -$\operatorname{OES}(32)$ sehr schnell ein gutes 16-Schnittmuster findet. - -Eines der eher zufälligen Schnittmuster ist $\operatorname{MIN}(1, 6, 11, 14, -17, 23, 26, 29)$, $\operatorname{MAX}(2, 7, 8,$ $13, 18, 21, 27, 31)$. Das -Schnittmuster ist in Abbildung~\ref{fig:16-ec-from-oes32-cut} veranschaulicht, -das resultierende Netzwerk ist in Abbildung~\ref{fig:16-ec-from-oes32} zu sehen. +$\operatorname{OES}(32)$ sehr schnell\footnote{Auf dem Computer, auf dem diese +Arbeit geschrieben wurde, dauerte es in den meisten Fällen weniger als eine +Sekunde bis ein entsprechendes Schnittmuster gefunden wurde.} ein gutes +16-Schnittmuster findet. + +Eines der 16-Schnittmuster für \oes{32}, die ein Sortiernetzwerk erzeugen, das +bezüglich Effizienz und Geschwindigkeit identisch ist zu \oes{16}, ist +$\operatorname{MIN}(1, 6, 11, 14, 17, 23, 26, 29)$, $\operatorname{MAX}(2, 7, +8,$ $13, 18, 21, 27, 31)$. Das Schnittmuster ist in +Abbildung~\ref{fig:16-ec-from-oes32-cut} veranschaulicht, das resultierende +Netzwerk ist in Abbildung~\ref{fig:16-ec-from-oes32} zu sehen. \begin{figure} \begin{center} @@ -1662,6 +1716,109 @@ das resultierende Netzwerk ist in Abbildung~\ref{fig:16-ec-from-oes32} zu sehen. \label{fig:16-ec-from-oes32} \end{figure} +Bei diesem Schnittmuster fällt auf, dass es für jeweils vier Eingänge (0--3, +4--7, \dots, 28--31) einen Minimum- und einen Maximumschnitt gibt. Aus dieser +Beobachtung kann man das regelmäßige Schnittmuster +\begin{displaymath} +\textit{Eingang}_i = \left\{ \begin{array}{rl} + \infty & \quad \textrm{falls } i \bmod 4 = 0 \\ + -\infty & \quad \textrm{falls } i \bmod 4 = 3 \\ + ? & \quad \mathrm{sonst} + \end{array} \right. +\end{displaymath} +ableiten. Es entfernt die Hälfte der Leitungen, vorausgesetzt die Anzahl der +Leitungen ist durch Vier teilbar. Das Schnittmuster erzeugt effiziente +Netzwerke, wenn die Anzahl der Leitungen $n = 2^d$ eine Zweierpotenz ist. Ein +32-Sortiernetzwerk, das mit diesem Schnittmuster aus \oes{64} erzeugt wurde, +ist in Abbildung~\ref{fig:32-ec-from-oes64} zu sehen. + +\begin{figure} + \begin{center} + \input{images/32-ec-from-oes64.tex} + \end{center} + \caption{32-Sortiernetzwerk mit 191~Komparatoren in 15~Schichten. + Das Netzwerk wurde mit einem regelmäßigen Schnittmuster aus dem + \emph{Odd-Even-Mergesort}-Netzwerk \oes{64} erzeugt.} + \label{fig:32-ec-from-oes64} +\end{figure} + +Wenn die Anzahl der Leitungen keine Zweierpotenz ist, erreichen die so +erzeugten Sortiernetzwerke die Effizienz des +\emph{Odd-Even-Mergesort}-Netzwerks nicht. Wendet man das Schnittmuster +beispielsweise auf \oes{24} an, so erhält man ein Sortiernetzwerk mit +43~Komparatoren -- \oes{12} kommt mit 41~Komparatoren aus. Die Geschwindigkeit +beider Sortiernetzwerke ist mit 10~Schichten identisch. + +Startet man hingegen den \textsc{SN-Evolution-Cut}-Algorithmus mit \oes{24} +und dem Ziel, ein gutes 12-Schnittmuster zu finden, hängt die Ausgabe von der +verwendeten Gütefunktion ab. Werden effiziente Netzwerke bevorzugt, findet der +Algorithmus Schnittmuster wie $\operatorname{MIN}(6, 7, 8, 9, 16, 17, 20, +22)$, $\operatorname{MAX}(2, 4, 12, 14)$, dessen Ergebnis in +Abbildung~\ref{12-ec-from-oes24-efficient} zu sehen ist. Das resultierende +Sortiernetzwerk besteht aus 41~Komparatoren, die in 10~Schichten angeordnet +werden können. Damit ist das Netzwerk bezüglich Effizienz und Geschwindigkeit +gleichauf mit \oes{12}. Werden hingegen schnelle Sortiernetzwerke bevorzugt, +werden stattdessen Schnittmuster wie $\operatorname{MIN}(6, 7, 11, 12, 15, +16)$, $\operatorname{MAX}(1, 3, 10, 17, 20, 23)$ ausgegeben. Das Ergebnis +dieses Schnittmusters ist in Abbildung~\ref{fig:12-ec-from-oes24-fast} zu +sehen. Das Sortiernetzwerk besteht aus 43~Komparatoren, die in 9~Schichten +angeordnet sind. Das heißt, dass das resultierende Netzwerk zwar nicht so +effizient wie \oes{12} ist, dafür aber schneller als \oes{12} und \bs{12}. + +\begin{figure} + \centering + \subfigure[Schnelles 12-Sortiernetzwerk aus 43~Komparatoren in 9~Schichten, + das von \textsc{SN-Evolution-Cut} aus dem \emph{Odd-Even-Mergesort}-Netzwerk + generiert + wurde.]{\input{images/12-ec-from-oes24-fast.tex}\label{fig:12-ec-from-oes24-fast}} + \subfigure[Effizientes 12-Sortiernetzwerk aus 41~Komparatoren in + 10~Schichten, das von \textsc{SN-Evolution-Cut} aus dem + \emph{Odd-Even-Mergesort}-Netzwerk generiert + wurde.]{\input{images/12-ec-from-oes24-efficient.tex}\label{fig:12-ec-from-oes24-efficient}} + \caption{Startet man \textsc{SN-Evolution-Cut} mit \oes{24}, hängt das + Ergebnis von der Bewertungsfunktion ab.} + \label{fig:12-ec-from-oes24} +\end{figure} + +Das \oes{24}-Sortiernetzwerk ist kein Einzelfall: \textsc{SN-Evolution-Cut} +findet Sortiernetzwerke, die schneller sind als das entsprechende +\emph{Odd-Even-Mergesort}-Netzwerk, unter anderem für das +\emph{Odd-Even-Mergesort}-Netzwerk mit 22, 24, 38, 40, 42, 44 und 46 +Eingängen. In der folgenden Tabelle sind einige schnelle Netzwerke, die von +\textsc{SN-Evolution-Cut} generiert werden können, charakterisiert. Die +Eingabe für \textsc{SN-Evolution-Cut} war jeweils das +\emph{Odd-Even-Mergesort}-Netzwerk mit der doppelten Leitungszahl. +Abbildung~\ref{fig:23-ec-from-oes46} zeigt beispielhaft ein +23-Sortiernetzwerk, das aus \oes{46} generiert wurde. +\begin{center} +\begin{tabular}{|r|r|r|r|r|} +\hline +Leitungen & Komparatoren & Schichten & Komparatoren & Schichten \\ +(Resultat) & \textsc{SN-EC} & \textsc{SN-EC} & \oes{n} & \oes{n} \\ +\hline +11 & 38 & 9 & 37 & 10 \\ +12 & 43 & 9 & 41 & 10 \\ +19 & 93 & 13 & 91 & 14 \\ +20 & 101 & 13 & 97 & 14 \\ +21 & 108 & 14 & 107 & 15 \\ +22 & 116 & 14 & 114 & 15 \\ +23 & 125 & 14 & 122 & 15 \\ +\hline +\end{tabular} +\end{center} + +\begin{figure} + \begin{center} + \input{images/23-ec-from-oes46-fast.tex} + \end{center} + \caption{23-Sortiernetzwerk mit 125~Komparatoren in 14~Schichten. + Das Netzwerk wurde von \textsc{SN-Evolution-Cut} aus \oes{46} mit dem + Schnittmuster $\operatorname{MIN}(6, 7, 9, 17, 19, 22, 29, 30, 32, 34, 38, + 44)$, $\operatorname{MAX}(4, 5, 11, 16, 18, 25, 31, 36, 39, 42, 45)$ + erzeugt.} + \label{fig:23-ec-from-oes46} +\end{figure} + \newpage \section{Der \textsc{SN-Markov}-Algorithmus} \label{sect:markov} @@ -1836,8 +1993,6 @@ ein Phänomen, das mit der Initialisierung mit dem \begin{itemize} \item Beste erreichte Netzwerke (gleich zu \emph{OE-Mergesort}). \item Anzahl der erreichbaren Sortiernetzwerke. - \item Anzahl der Komparatoren und Anzahl der Schichten der durchlaufenen - Netzwerke. (Abbildung~\ref{fig:markov-comparators-16}) \item \textsc{SN-Count-Markov} (ggf) \end{itemize} @@ -1960,4 +2115,4 @@ Komparatornetzwerks auf eine Eingabe-Permutation. \end{document} -% vim: set shiftwidth=2 softtabstop=2 tabstop=8 fdm=marker tw=78 : +% vim: set shiftwidth=2 softtabstop=2 tabstop=8 fdm=marker tw=78 spelllang=de : diff --git a/images/11-ec-from-bs22-fast.tex b/images/11-ec-from-bs22-fast.tex new file mode 100644 index 0000000..418aaff --- /dev/null +++ b/images/11-ec-from-bs22-fast.tex @@ -0,0 +1,161 @@ +\begin{tikzpicture}[auto] +\node[vertex] (v0) at (0.49,0.00) {}; +\node[vertex] (v1) at (0.49,1.58) {}; +\path[comp] (v0) -- (v1); + +\node[vertex] (v2) at (0.64,0.39) {}; +\node[vertex] (v3) at (0.64,0.79) {}; +\path[comp] (v2) -- (v3); + +\node[vertex] (v4) at (0.64,1.18) {}; +\node[vertex] (v5) at (0.64,3.15) {}; +\path[comp] (v4) -- (v5); + +\node[vertex] (v6) at (0.49,1.97) {}; +\node[vertex] (v7) at (0.49,2.37) {}; +\path[comp] (v6) -- (v7); + +\node[vertex] (v8) at (0.49,3.55) {}; 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+ +\node[vertex] (v246) at (14.39,9.27) {}; +\node[vertex] (v247) at (14.39,9.76) {}; +\path[comp] (v246) -- (v247); + +\node[vertex] (v248) at (14.39,10.24) {}; +\node[vertex] (v249) at (14.39,10.73) {}; +\path[comp] (v248) -- (v249); + +\path[edge] (0,0.00) -- (15.00,0.00); +\path[edge] (0,0.49) -- (15.00,0.49); +\path[edge] (0,0.98) -- (15.00,0.98); +\path[edge] (0,1.46) -- (15.00,1.46); +\path[edge] (0,1.95) -- (15.00,1.95); +\path[edge] (0,2.44) -- (15.00,2.44); +\path[edge] (0,2.93) -- (15.00,2.93); +\path[edge] (0,3.41) -- (15.00,3.41); +\path[edge] (0,3.90) -- (15.00,3.90); +\path[edge] (0,4.39) -- (15.00,4.39); +\path[edge] (0,4.88) -- (15.00,4.88); +\path[edge] (0,5.37) -- (15.00,5.37); +\path[edge] (0,5.85) -- (15.00,5.85); +\path[edge] (0,6.34) -- (15.00,6.34); +\path[edge] (0,6.83) -- (15.00,6.83); +\path[edge] (0,7.32) -- (15.00,7.32); +\path[edge] (0,7.80) -- (15.00,7.80); +\path[edge] (0,8.29) -- (15.00,8.29); +\path[edge] (0,8.78) -- (15.00,8.78); +\path[edge] (0,9.27) -- (15.00,9.27); 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+\path[comp] (v84) -- (v85); + +\node[vertex] (v86) at (4.13,5.16) {}; +\node[vertex] (v87) at (4.13,10.32) {}; +\path[comp] (v86) -- (v87); + +\node[vertex] (v88) at (3.22,5.85) {}; +\node[vertex] (v89) at (3.22,6.53) {}; +\path[comp] (v88) -- (v89); + +\node[vertex] (v90) at (3.22,7.22) {}; +\node[vertex] (v91) at (3.22,7.91) {}; +\path[comp] (v90) -- (v91); + +\node[vertex] (v92) at (3.22,8.60) {}; +\node[vertex] (v93) at (3.22,9.28) {}; +\path[comp] (v92) -- (v93); + +\node[vertex] (v94) at (3.22,9.97) {}; +\node[vertex] (v95) at (3.22,10.66) {}; +\path[comp] (v94) -- (v95); + +% Schicht 3 +\node[vertex] (v96) at (4.56,0.00) {}; +\node[vertex] (v97) at (4.56,1.38) {}; +\path[comp] (v96) -- (v97); + +\node[vertex] (v98) at (4.68,0.34) {}; +\node[vertex] (v99) at (4.68,1.03) {}; +\path[comp] (v98) -- (v99); + +\node[vertex] (v100) at (4.56,1.72) {}; +\node[vertex] (v101) at (4.56,2.41) {}; +\path[comp] (v100) -- (v101); + +\node[vertex] (v102) at (4.56,2.75) {}; +\node[vertex] (v103) at (4.56,4.13) {}; +\path[comp] (v102) -- (v103); + +\node[vertex] (v104) at (4.68,3.09) {}; +\node[vertex] (v105) at (4.68,3.78) {}; +\path[comp] (v104) -- (v105); + +\node[vertex] (v106) at (4.56,4.47) {}; +\node[vertex] (v107) at (4.56,5.16) {}; +\path[comp] (v106) -- (v107); + +\node[vertex] (v108) at (4.56,5.50) {}; +\node[vertex] (v109) at (4.56,6.19) {}; +\path[comp] (v108) -- (v109); + +\node[vertex] (v110) at (4.56,6.53) {}; +\node[vertex] (v111) at (4.56,7.91) {}; +\path[comp] (v110) -- (v111); + +\node[vertex] (v112) at (4.68,6.88) {}; +\node[vertex] (v113) at (4.68,7.56) {}; +\path[comp] (v112) -- (v113); + +\node[vertex] (v114) at (4.56,8.25) {}; +\node[vertex] (v115) at (4.56,8.94) {}; +\path[comp] (v114) -- (v115); + +\node[vertex] (v116) at (4.56,9.28) {}; +\node[vertex] (v117) at (4.56,10.66) {}; +\path[comp] (v116) -- (v117); + +\node[vertex] (v118) at (4.68,9.63) {}; +\node[vertex] (v119) at (4.68,10.32) {}; +\path[comp] (v118) -- (v119); + +% Schicht 4 +\node[vertex] (v120) at (5.11,0.00) {}; 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+\path[comp] (v172) -- (v173); + +\node[vertex] (v174) at (7.52,3.78) {}; +\node[vertex] (v175) at (7.52,5.16) {}; +\path[comp] (v174) -- (v175); + +\node[vertex] (v176) at (7.26,5.50) {}; +\node[vertex] (v177) at (7.26,6.88) {}; +\path[comp] (v176) -- (v177); + +\node[vertex] (v178) at (7.39,5.85) {}; +\node[vertex] (v179) at (7.39,7.22) {}; +\path[comp] (v178) -- (v179); + +\node[vertex] (v180) at (7.52,6.19) {}; +\node[vertex] (v181) at (7.52,7.56) {}; +\path[comp] (v180) -- (v181); + +\node[vertex] (v182) at (7.26,8.25) {}; +\node[vertex] (v183) at (7.26,9.63) {}; +\path[comp] (v182) -- (v183); + +\node[vertex] (v184) at (7.39,8.60) {}; +\node[vertex] (v185) at (7.39,9.97) {}; +\path[comp] (v184) -- (v185); + +\node[vertex] (v186) at (7.52,8.94) {}; +\node[vertex] (v187) at (7.52,10.32) {}; +\path[comp] (v186) -- (v187); + +% Schicht 7 +\node[vertex] (v188) at (7.95,1.38) {}; +\node[vertex] (v189) at (7.95,5.50) {}; +\path[comp] (v188) -- (v189); + +\node[vertex] (v190) at (8.08,1.72) {}; +\node[vertex] (v191) at (8.08,5.85) {}; +\path[comp] (v190) -- (v191); + +\node[vertex] (v192) at (8.21,2.06) {}; +\node[vertex] (v193) at (8.21,6.19) {}; +\path[comp] (v192) -- (v193); + +\node[vertex] (v194) at (8.34,2.41) {}; +\node[vertex] (v195) at (8.34,6.53) {}; +\path[comp] (v194) -- (v195); + +\node[vertex] (v196) at (8.47,4.13) {}; +\node[vertex] (v197) at (8.47,8.25) {}; +\path[comp] (v196) -- (v197); + +\node[vertex] (v198) at (8.60,4.47) {}; +\node[vertex] (v199) at (8.60,8.60) {}; +\path[comp] (v198) -- (v199); + +\node[vertex] (v200) at (8.72,4.81) {}; +\node[vertex] (v201) at (8.72,8.94) {}; +\path[comp] (v200) -- (v201); + +\node[vertex] (v202) at (8.85,5.16) {}; +\node[vertex] (v203) at (8.85,9.28) {}; +\path[comp] (v202) -- (v203); + +% Schicht 8 +\node[vertex] (v204) at (9.28,0.69) {}; +\node[vertex] (v205) at (9.28,1.38) {}; +\path[comp] (v204) -- (v205); + +\node[vertex] (v206) at (9.41,1.03) {}; +\node[vertex] (v207) at (9.41,1.72) {}; +\path[comp] (v206) -- (v207); 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+\path[comp] (v224) -- (v225); + +\node[vertex] (v226) at (9.67,9.28) {}; +\node[vertex] (v227) at (9.67,9.97) {}; +\path[comp] (v226) -- (v227); + +% Schicht 9 +\node[vertex] (v228) at (10.10,0.34) {}; +\node[vertex] (v229) at (10.10,0.69) {}; +\path[comp] (v228) -- (v229); + +\node[vertex] (v230) at (10.10,1.03) {}; +\node[vertex] (v231) at (10.10,1.38) {}; +\path[comp] (v230) -- (v231); + +\node[vertex] (v232) at (10.10,1.72) {}; +\node[vertex] (v233) at (10.10,2.06) {}; +\path[comp] (v232) -- (v233); + +\node[vertex] (v234) at (10.10,2.41) {}; +\node[vertex] (v235) at (10.10,5.50) {}; +\path[comp] (v234) -- (v235); + +\node[vertex] (v236) at (10.23,3.09) {}; +\node[vertex] (v237) at (10.23,3.44) {}; +\path[comp] (v236) -- (v237); + +\node[vertex] (v238) at (10.23,3.78) {}; +\node[vertex] (v239) at (10.23,4.13) {}; +\path[comp] (v238) -- (v239); + +\node[vertex] (v240) at (10.23,4.47) {}; +\node[vertex] (v241) at (10.23,4.81) {}; +\path[comp] (v240) -- (v241); + +\node[vertex] (v242) at (10.23,5.16) {}; +\node[vertex] (v243) at (10.23,8.25) {}; +\path[comp] (v242) -- (v243); + +\node[vertex] (v244) at (10.10,5.85) {}; +\node[vertex] (v245) at (10.10,6.19) {}; +\path[comp] (v244) -- (v245); + +\node[vertex] (v246) at (10.10,6.53) {}; +\node[vertex] (v247) at (10.10,6.88) {}; +\path[comp] (v246) -- (v247); + +\node[vertex] (v248) at (10.10,7.22) {}; +\node[vertex] (v249) at (10.10,7.56) {}; +\path[comp] (v248) -- (v249); + +\node[vertex] (v250) at (10.23,8.60) {}; +\node[vertex] (v251) at (10.23,8.94) {}; +\path[comp] (v250) -- (v251); + +\node[vertex] (v252) at (10.23,9.28) {}; +\node[vertex] (v253) at (10.23,9.63) {}; +\path[comp] (v252) -- (v253); + +\node[vertex] (v254) at (10.23,9.97) {}; +\node[vertex] (v255) at (10.23,10.32) {}; +\path[comp] (v254) -- (v255); + +% Schicht 10 +\node[vertex] (v256) at (10.66,0.34) {}; +\node[vertex] (v257) at (10.66,3.09) {}; +\path[comp] (v256) -- (v257); + +\node[vertex] (v258) at (10.79,0.69) {}; +\node[vertex] (v259) at (10.79,3.44) {}; +\path[comp] (v258) -- (v259); + +\node[vertex] (v260) at (10.92,1.03) {}; +\node[vertex] (v261) at (10.92,3.78) {}; +\path[comp] (v260) -- (v261); + +\node[vertex] (v262) at (11.05,1.38) {}; +\node[vertex] (v263) at (11.05,4.13) {}; +\path[comp] (v262) -- (v263); + +\node[vertex] (v264) at (11.17,1.72) {}; +\node[vertex] (v265) at (11.17,4.47) {}; +\path[comp] (v264) -- (v265); + +\node[vertex] (v266) at (11.30,2.06) {}; +\node[vertex] (v267) at (11.30,4.81) {}; +\path[comp] (v266) -- (v267); + +\node[vertex] (v268) at (11.43,2.41) {}; +\node[vertex] (v269) at (11.43,5.16) {}; +\path[comp] (v268) -- (v269); + +\node[vertex] (v270) at (10.66,5.50) {}; +\node[vertex] (v271) at (10.66,8.25) {}; +\path[comp] (v270) -- (v271); + +\node[vertex] (v272) at (10.79,5.85) {}; +\node[vertex] (v273) at (10.79,8.60) {}; +\path[comp] (v272) -- (v273); + +\node[vertex] (v274) at (10.92,6.19) {}; +\node[vertex] (v275) at (10.92,8.94) {}; +\path[comp] (v274) -- (v275); + +\node[vertex] (v276) at (11.05,6.53) {}; +\node[vertex] (v277) at (11.05,9.28) {}; +\path[comp] (v276) -- (v277); + +\node[vertex] (v278) at (11.17,6.88) {}; +\node[vertex] (v279) at (11.17,9.63) {}; +\path[comp] (v278) -- (v279); + +\node[vertex] (v280) at (11.30,7.22) {}; +\node[vertex] (v281) at (11.30,9.97) {}; +\path[comp] (v280) -- (v281); + +\node[vertex] (v282) at (11.43,7.56) {}; +\node[vertex] (v283) at (11.43,10.32) {}; +\path[comp] (v282) -- (v283); + +\node[vertex] (v284) at (11.86,2.75) {}; +\node[vertex] (v285) at (11.86,5.50) {}; +\path[comp] (v284) -- (v285); + +\node[vertex] (v286) at (11.99,3.09) {}; +\node[vertex] (v287) at (11.99,5.85) {}; +\path[comp] (v286) -- (v287); + +\node[vertex] (v288) at (12.12,3.44) {}; +\node[vertex] (v289) at (12.12,6.19) {}; +\path[comp] (v288) -- (v289); + +\node[vertex] (v290) at (12.25,3.78) {}; +\node[vertex] (v291) at (12.25,6.53) {}; +\path[comp] (v290) -- (v291); + +\node[vertex] (v292) at (12.38,4.13) {}; +\node[vertex] (v293) at (12.38,6.88) {}; +\path[comp] (v292) -- (v293); + +\node[vertex] (v294) at (12.51,4.47) {}; +\node[vertex] (v295) at (12.51,7.22) {}; +\path[comp] (v294) -- (v295); + +\node[vertex] (v296) at (12.64,4.81) {}; +\node[vertex] (v297) at (12.64,7.56) {}; +\path[comp] (v296) -- (v297); + +\node[vertex] (v298) at (12.77,5.16) {}; +\node[vertex] (v299) at (12.77,7.91) {}; +\path[comp] (v298) -- (v299); + +\node[vertex] (v300) at (13.19,1.38) {}; +\node[vertex] (v301) at (13.19,2.75) {}; +\path[comp] (v300) -- (v301); + +\node[vertex] (v302) at (13.32,1.72) {}; +\node[vertex] (v303) at (13.32,3.09) {}; +\path[comp] (v302) -- (v303); + +\node[vertex] (v304) at (13.45,2.06) {}; +\node[vertex] (v305) at (13.45,3.44) {}; +\path[comp] (v304) -- (v305); + +\node[vertex] (v306) at (13.58,2.41) {}; +\node[vertex] (v307) at (13.58,3.78) {}; +\path[comp] (v306) -- (v307); + +\node[vertex] (v308) at (13.19,4.13) {}; +\node[vertex] (v309) at (13.19,5.50) {}; +\path[comp] (v308) -- (v309); + +\node[vertex] (v310) at (13.32,4.47) {}; +\node[vertex] (v311) at (13.32,5.85) {}; +\path[comp] (v310) -- (v311); + +\node[vertex] (v312) at (13.45,4.81) {}; +\node[vertex] (v313) at (13.45,6.19) {}; +\path[comp] (v312) -- (v313); + +\node[vertex] (v314) at (13.58,5.16) {}; +\node[vertex] (v315) at (13.58,6.53) {}; +\path[comp] (v314) -- (v315); + +\node[vertex] (v316) at (13.19,6.88) {}; +\node[vertex] (v317) at (13.19,8.25) {}; +\path[comp] (v316) -- (v317); + +\node[vertex] (v318) at (13.32,7.22) {}; +\node[vertex] (v319) at (13.32,8.60) {}; +\path[comp] (v318) -- (v319); + +\node[vertex] (v320) at (13.45,7.56) {}; +\node[vertex] (v321) at (13.45,8.94) {}; +\path[comp] (v320) -- (v321); + +\node[vertex] (v322) at (13.58,7.91) {}; +\node[vertex] (v323) at (13.58,9.28) {}; +\path[comp] (v322) -- (v323); + +\node[vertex] (v324) at (14.01,0.69) {}; +\node[vertex] (v325) at (14.01,1.38) {}; +\path[comp] (v324) -- (v325); + +\node[vertex] (v326) at (14.14,1.03) {}; +\node[vertex] (v327) at (14.14,1.72) {}; +\path[comp] (v326) -- (v327); + +\node[vertex] (v328) at (14.01,2.06) {}; +\node[vertex] (v329) at (14.01,2.75) {}; +\path[comp] (v328) -- (v329); + +\node[vertex] (v330) at (14.14,2.41) {}; +\node[vertex] (v331) at (14.14,3.09) {}; +\path[comp] (v330) -- (v331); + +\node[vertex] (v332) at (14.01,3.44) {}; +\node[vertex] (v333) at (14.01,4.13) {}; +\path[comp] (v332) -- (v333); + +\node[vertex] (v334) at (14.14,3.78) {}; +\node[vertex] (v335) at (14.14,4.47) {}; +\path[comp] (v334) -- (v335); + +\node[vertex] (v336) at (14.01,4.81) {}; +\node[vertex] (v337) at (14.01,5.50) {}; +\path[comp] (v336) -- (v337); + +\node[vertex] (v338) at (14.14,5.16) {}; +\node[vertex] (v339) at (14.14,5.85) {}; +\path[comp] (v338) -- (v339); + +\node[vertex] (v340) at (14.01,6.19) {}; +\node[vertex] (v341) at (14.01,6.88) {}; +\path[comp] (v340) -- (v341); + +\node[vertex] (v342) at (14.14,6.53) {}; +\node[vertex] (v343) at (14.14,7.22) {}; +\path[comp] (v342) -- (v343); + +\node[vertex] (v344) at (14.01,7.56) {}; +\node[vertex] (v345) at (14.01,8.25) {}; +\path[comp] (v344) -- (v345); + +\node[vertex] (v346) at (14.14,7.91) {}; +\node[vertex] (v347) at (14.14,8.60) {}; +\path[comp] (v346) -- (v347); + +\node[vertex] (v348) at (14.01,8.94) {}; +\node[vertex] (v349) at (14.01,9.63) {}; +\path[comp] (v348) -- (v349); + +\node[vertex] (v350) at (14.14,9.28) {}; +\node[vertex] (v351) at (14.14,9.97) {}; +\path[comp] (v350) -- (v351); + +\node[vertex] (v352) at (14.57,0.34) {}; +\node[vertex] (v353) at (14.57,0.69) {}; +\path[comp] (v352) -- (v353); + +\node[vertex] (v354) at (14.57,1.03) {}; +\node[vertex] (v355) at (14.57,1.38) {}; +\path[comp] (v354) -- (v355); + +\node[vertex] (v356) at (14.57,1.72) {}; +\node[vertex] (v357) at (14.57,2.06) {}; +\path[comp] (v356) -- (v357); + +\node[vertex] (v358) at (14.57,2.41) {}; +\node[vertex] (v359) at (14.57,2.75) {}; +\path[comp] (v358) -- (v359); + +\node[vertex] (v360) at (14.57,3.09) {}; +\node[vertex] (v361) at (14.57,3.44) {}; +\path[comp] (v360) -- (v361); + +\node[vertex] (v362) at (14.57,3.78) {}; +\node[vertex] (v363) at (14.57,4.13) {}; +\path[comp] (v362) -- (v363); + +\node[vertex] (v364) at (14.57,4.47) {}; +\node[vertex] (v365) at (14.57,4.81) {}; +\path[comp] (v364) -- (v365); + +\node[vertex] (v366) at (14.57,5.16) {}; +\node[vertex] (v367) at (14.57,5.50) {}; +\path[comp] (v366) -- (v367); + +\node[vertex] (v368) at (14.57,5.85) {}; +\node[vertex] (v369) at (14.57,6.19) {}; +\path[comp] (v368) -- (v369); + +\node[vertex] (v370) at (14.57,6.53) {}; +\node[vertex] (v371) at (14.57,6.88) {}; +\path[comp] (v370) -- (v371); + +\node[vertex] (v372) at (14.57,7.22) {}; +\node[vertex] (v373) at (14.57,7.56) {}; +\path[comp] (v372) -- (v373); + +\node[vertex] (v374) at (14.57,7.91) {}; +\node[vertex] (v375) at (14.57,8.25) {}; +\path[comp] (v374) -- (v375); + +\node[vertex] (v376) at (14.57,8.60) {}; +\node[vertex] (v377) at (14.57,8.94) {}; +\path[comp] (v376) -- (v377); + +\node[vertex] (v378) at (14.57,9.28) {}; +\node[vertex] (v379) at (14.57,9.63) {}; +\path[comp] (v378) -- (v379); + +\node[vertex] (v380) at (14.57,9.97) {}; +\node[vertex] (v381) at (14.57,10.32) {}; +\path[comp] (v380) -- (v381); + +\path[edge] (0,0.00) -- (15.00,0.00); +\path[edge] (0,0.34) -- (15.00,0.34); +\path[edge] (0,0.69) -- (15.00,0.69); +\path[edge] (0,1.03) -- (15.00,1.03); +\path[edge] (0,1.38) -- (15.00,1.38); +\path[edge] (0,1.72) -- (15.00,1.72); +\path[edge] (0,2.06) -- (15.00,2.06); +\path[edge] (0,2.41) -- (15.00,2.41); +\path[edge] (0,2.75) -- (15.00,2.75); +\path[edge] (0,3.09) -- (15.00,3.09); +\path[edge] (0,3.44) -- (15.00,3.44); +\path[edge] (0,3.78) -- (15.00,3.78); +\path[edge] (0,4.13) -- (15.00,4.13); +\path[edge] (0,4.47) -- (15.00,4.47); +\path[edge] (0,4.81) -- (15.00,4.81); +\path[edge] (0,5.16) -- (15.00,5.16); +\path[edge] (0,5.50) -- (15.00,5.50); +\path[edge] (0,5.85) -- (15.00,5.85); +\path[edge] (0,6.19) -- (15.00,6.19); +\path[edge] (0,6.53) -- (15.00,6.53); +\path[edge] (0,6.88) -- (15.00,6.88); +\path[edge] (0,7.22) -- (15.00,7.22); +\path[edge] (0,7.56) -- (15.00,7.56); +\path[edge] (0,7.91) -- (15.00,7.91); +\path[edge] (0,8.25) -- (15.00,8.25); +\path[edge] (0,8.60) -- (15.00,8.60); +\path[edge] (0,8.94) -- (15.00,8.94); +\path[edge] (0,9.28) -- (15.00,9.28); +\path[edge] (0,9.63) -- (15.00,9.63); +\path[edge] (0,9.97) -- (15.00,9.97); +\path[edge] (0,10.32) -- (15.00,10.32); +\path[edge] (0,10.66) -- (15.00,10.66); +\end{tikzpicture} -- 2.11.0