From c7c7f9b7e2309f3817f329f4c1cd8d1588055370 Mon Sep 17 00:00:00 2001 From: couchot <couchot@localhost.localdomain> Date: Fri, 24 Jun 2016 14:46:52 +0200 Subject: [PATCH] modification section stopping time --- complexity.eps | 1324 ++++++++++++++++++++++++++++++++++++++++++ complexityDisplay.py | 40 ++ main.pdf | Bin 627209 -> 508770 bytes stopping.tex | 66 ++- 4 files changed, 1404 insertions(+), 26 deletions(-) create mode 100644 complexity.eps create mode 100644 complexityDisplay.py diff --git a/complexity.eps b/complexity.eps new file mode 100644 index 0000000..41e0c2b --- /dev/null +++ b/complexity.eps @@ -0,0 +1,1324 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Title: complexity.eps +%%Creator: matplotlib version 1.5.1, http://matplotlib.org/ +%%CreationDate: Fri Jun 24 14:36:55 2016 +%%Orientation: portrait +%%BoundingBox: 18 180 594 612 +%%EndComments +%%BeginProlog +/mpldict 8 dict def +mpldict begin +/m { moveto } bind def +/l { lineto } bind def +/r { rlineto } bind def +/c { curveto } bind def +/cl { closepath } bind def +/box { +m +1 index 0 r +0 exch r +neg 0 r +cl +} bind def +/clipbox { +box +clip +newpath +} bind def +%!PS-Adobe-3.0 Resource-Font +%%Title: Bitstream Vera Sans +%%Copyright: Copyright (c) 2003 by Bitstream, Inc. 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plt.plot(x,yp,marker=None,color='r',label="L1") +l1 = plt.legend([pexp,pd], ["experimental evaluation","2xln(2x+8)"], loc=0) + +#plt.show() +plt.savefig("complexity.eps") + +""" +n=np.linspace(512,4096,10) +phugo, = plt.plot(n,2*pow(n,2)*(343**2+np.log(n)),marker='o',label="L1") +pwow, = plt.plot(n,6*pow(n,4)*np.log(n)+pow(n,2),marker='v',label="L2") +pcanny, = plt.plot(n,pow(n,2)*(5**3+4*T+1),marker='x',label="L3") +plt.yscale('log') +plt.ylabel('nb. of primary operations') +plt.xlabel("nb of lines and of rows in each image") +#plt.legend(handles=[phugo], loc=1) +#plt.legend([phugo, pwow], ["Hugo", "Wow/Uniward"]) +l1 = plt.legend([pwow,phugo,pcanny], ["Wow/Uniward","Hugo","Stabylo"], loc=0) +plt.gca().add_artist(l1) +#plt.gca().add_artist(l2) +#plt.gca().add_artist(l3) +plt.savefig("complexity.eps") +#plt.show() +""" diff --git a/main.pdf b/main.pdf index cba5430ad797da97fb199017b0be4f066bf8a3a1..4f393a062c5fbaec199b8b1b95bb334f62f5d7e1 100644 GIT binary patch delta 245774 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functions have been generaed according to method presented in section~\ref{sec:hamilton}. For each of them, the calculus of the approximation of $E[\ts]$ -is executed 10000 times with a random seed. The table~\ref{table:stopping:moy} -summarizes results. It can be observed that the approximation is largely -wÅsmaller than the upper bound given in theorem~\ref{prop:stop}. - \begin{algorithm}[ht] %\begin{scriptsize} \KwIn{a function $f$, an initial configuration $x^0$ ($\mathsf{N}$ bits)} @@ -389,39 +383,59 @@ wÅsmaller than the upper bound given in theorem~\ref{prop:stop}. $\textit{nbit} \leftarrow 0$\; $x\leftarrow x^0$\; -$\textit{visited}\leftarrow\emptyset$\; - -\While{$\left\vert{\textit{visited}}\right\vert < \mathsf{N} $} +$\textit{fair}\leftarrow\emptyset$\; +\While{$\left\vert{\textit{fair}}\right\vert < \mathsf{N} $} { - $ s \leftarrow \textit{Random}(n)$ \; + $ s \leftarrow \textit{Random}(\mathsf{N})$ \; $\textit{image} \leftarrow f(x) $\; - \If{$x[s] \neq \textit{image}[s]$}{ - $\textit{visited} \leftarrow \textit{visited} \cup \{s\}$ + \If{$\textit{Random}(1) \neq 0$ and $x[s] \neq \textit{image}[s]$}{ + $\textit{fair} \leftarrow \textit{fair} \cup \{s\}$\; + $x[s] \leftarrow \textit{image}[s]$\; } - $x[s] \leftarrow \textit{image}[s]$\; $\textit{nbit} \leftarrow \textit{nbit}+1$\; } \Return{$\textit{nbit}$}\; %\end{scriptsize} -\caption{Pseudo Code of the stoping time calculus} +\caption{Pseudo Code of stoping time calculus } \label{algo:stop} \end{algorithm} +Practically speaking, for each number $\mathsf{N}$, $ 3 \le \mathsf{N} \le 16$, +10 functions have been generaed according to method presented in section~\ref{sec:hamilton}. For each of them, the calculus of the approximation of $E[\ts]$ +is executed 10000 times with a random seed. The Figure~\ref{fig:stopping:moy} +summarizes these results. In this one, a circle represents the +approximation of $E[\ts]$ for a given $\mathsf{N}$. +The line is the graph of the function $x \mapsto 2x\ln(2x+8)$. +It can firstly +be observed that the approximation is largely +smaller than the upper bound given in theorem~\ref{prop:stop}. +It can be further deduced that the conjecture of the previous section +is realistic according the graph of $x \mapsto 2x\ln(2x+8)$. -\begin{table} -$$ -\begin{array}{|*{15}{l|}} -\hline -\mathsf{N} & 3 & 4 & 5 & 6 & 7& 8 & 9 & 10& 11 & 12 & 13 & 14 & 15 & 16 \\ -\hline -\mathsf{N} & 3 & 10.9 & 5 & 17.7 & 7& 25 & 9 & 32.7& 11 & 40.8 & 13 & 49.2 & 15 & 16 \\ -\hline -\end{array} -$$ -\caption{Average Stopping Time}\label{table:stopping:moy} -\end{table} + + +% \begin{table} +% $$ +% \begin{array}{|*{14}{l|}} +% \hline +% \mathsf{N} & 4 & 5 & 6 & 7& 8 & 9 & 10& 11 & 12 & 13 & 14 & 15 & 16 \\ +% \hline +% \mathsf{N} & 21.8 & 28.4 & 35.4 & 42.5 & 50 & 57.7 & 65.6& 73.5 & 81.6 & 90 & 98.3 & 107.1 & 16 \\ +% \hline +% \end{array} +% $$ +% \caption{Average Stopping Time}\label{table:stopping:moy} +% \end{table} + +\begin{figure} +\centering +\includegraphics[scale=0.5]{complexity} +\caption{Average Stopping Time Approximation}\label{fig:stopping:moy} +\end{figure} + + %%% Local Variables: %%% mode: latex -- 2.39.5