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\documentclass{llncs}
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\input{prelude}
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\begin{document}
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-\title{Star Discrepancies for generalized Halton points}
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+\title{$\mbox{\EightStarBold}$ Discrepancies for generalized Halton points\\
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+Comparison of three heuristics for generating points set}
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% \titlerunning{} % abbreviated title (for running head)
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% also used for the TOC unless
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% \toctitle is used
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-\author{Thomas Espitau\inst{1} \and Olivier Marty\inst{1}}
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+\author{Thomas Espitau$\mbox{}^{\mbox{\SnowflakeChevron}}$ \and Olivier Marty$\mbox{}^{\mbox{\SnowflakeChevron}}$}
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%
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% \authorrunning{} % abbreviated author list (for running head)
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%
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%%%% list of authors for the TOC (use if author list has to be modified)
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% \tocauthor{}
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%
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-\institute{
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- $\mbox{}^1$ ENS Cachan \qquad
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-}
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+\institute{$\mbox{}^{\mbox{\SnowflakeChevron}}$ ENS Cachan}
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\maketitle
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\makeatletter
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-
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+\renewcommand\bibsection%
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+{
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+ \section*{\refname
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+ \@mkboth{\MakeUppercase{\refname}}{\MakeUppercase{\refname}}}
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+ }
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\makeatother
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\begin{abstract}
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+ Geometric discrepancies are standard measures to quantify the irregularity of
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+ distributions. They are an important notion in numerical integration.
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+ One of the most important discrepancy notions is the so-called star
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+ discrepancy. Roughly speaking, a point set of low star discrepancy value
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+ allows for a small approximation error in quasi-Monte Carlo integration.
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+ In this work we present a tool realizing the implantation of three
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+ basics heuristics for construction low discrepancy points sets
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+ in the generalized Halton model: fully random search, local search with
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+ simmulated annealing and genetic $(5+5)$ search with a ad-hoc
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+ crossover function.
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\end{abstract}
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-\section{Introduction}
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-
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\section{General architecture of the tool}
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The testing tool is aimed to be modular: it is made of independents blocks that
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@@ -41,6 +52,8 @@ layer is written in C++ for performances. The given discrepancy algorithm
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The wrapper dispatch the computations on the multi-core architecture of
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modern computers\footnote{for us, between 2 and 4 physical cores and 4 or 8
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virtual cores}. This basic architecture is described in figure~\ref{main_flow}.
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+More precisely the class diagram for the unitary test layer is
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+presented in figure~\ref{class_flow}.
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Experiments were conducted on two machines:
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\begin{itemize}
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\item 2.4 GHz Intel Dual Core i5 hyper-threaded to 2.8GHz, 8 Go 1600 MHz DDR3.
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@@ -53,6 +66,12 @@ Experiments were conducted on two machines:
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\label{main_flow}
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\end{figure}
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+\begin{figure}
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+\includegraphics[scale=1]{graph_classes.pdf}
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+\caption{Class dependencies}
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+ \label{class_flow}
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+\end{figure}
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+
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On these machines, some basic profiling has make clear that
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the main bottleneck of the computations is hiding in the \emph{computation
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of the discrepancy}. The chosen algorithm and implantation of this
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@@ -79,7 +98,7 @@ extremal values are also given in order to construct error bands graphs.
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A flowchart of the conduct of one experiment is described in the
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flowchart~\ref{insight_flow}. The number of iteration of the heuristic is
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-I and the number of full restart is N. Th function Heuristic() correspond to
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+I and the number of full restart is N. The function Heuristic() correspond to
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a single step of the chosen heuristic. We now present an in-depth view of
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the implemented heuristics.
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@@ -91,7 +110,7 @@ the implemented heuristics.
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\end{mdframed}
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\end{figure}
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-Graph are presented not with the usual "mustache boxes" to show the
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+Graph are presented not with the usual box plot to show the
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error bounds, but in a more graphical way with error bands. The graph
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of the mean result is included inside a band of the same color which
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represents the incertitude with regards to the values obtained.
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@@ -99,11 +118,12 @@ represents the incertitude with regards to the values obtained.
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\section{Heuristics developed}
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\subsection{Fully random search (Test case)}
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- The first heuristic implemented is the random search. We generates
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- random sets of Halton points and select the best set with regard to its
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- discrepancy iteratively. The process is wrapped up in the
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- flowchart~\ref{random_flow}. In order to generate at each step a random
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- permutation, we transform it directly from the previous one.
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+ The first heuristic implemented is the random search. We generate
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+ random permutations, compute the corresponging sets of Halton points
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+ and select the best set with regard to its discrepancy.
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+ The process is wrapped up in the
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+ flowchart~\ref{random_flow}. In order to generate at each step random
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+ permutations, we transform them directly from the previous ones.
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More precisely the permutation is a singleton object which have method
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random, built on the Knuth Fisher Yates shuffle. This algorithm allows
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us to generate an uniformly chosen permutation at each step. We recall
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@@ -172,11 +192,11 @@ The final wrap up graphs are much more lighter and only presents the best
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results obtained.
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As expected from a fully random search, the error bands are very large for
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low number of iterations ($15\%$ of the value for 400 iterations) and tends
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-to shrink with a bigger number of iterations (around $5\%$ for 1600 iterations).
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+to shrink with a bigger number of iterations (around $5\%$ for 1500 iterations).
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This shrinkage is a direct consequence of well known concentrations bounds
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(Chernoff and Asuma-Hoeffding).
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The average results are quite stable, they decrease progressively with
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-the growing number of iterations, but seems to get to a limits after 1000
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+the growing number of iterations, but seem to get to a limits after 1000
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iterations. This value acts as a threshold for the interesting number of iterations.
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As such interesting results can be conducted with \emph{only} 1000 iterations,
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without altering too much the quality of the set with regards to its
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@@ -193,6 +213,7 @@ discrepancy and this heuristic.
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\label{rand_iter3}
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\end{figure}
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+%# TODO sa n'est pas evolutionnaire
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\subsection{Evolutionary heuristic: Simulated annealing and local search}
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The second heuristic implemented is a randomized local search with
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simulated annealing. This heuristic is inspired by the physical
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@@ -200,15 +221,20 @@ process of annealing in metallurgy.
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Simulated annealing interprets the physical slow cooling as a
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slow decrease in the probability of accepting worse solutions as it
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explores the solution space.
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-More precisely the neighbours are here the permutations which can be obtained
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-by application of exactly one transposition of the current permutation.
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+More precisely a state is a $d$-tuple of permutations, one for each dimension,
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+and the neighbourhood is the set of $d$-tuple of permutations which can be obtained
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+by application of exactly one transposition of one of the permutation of
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+the current state.
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The selection phase is dependant on the current temperature:
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-after applying a random transposition on the current permutation, either
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+after applying a random transposition on one of the current permutations, either
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the discrepancy of the corresponding Halton set is decreased and the
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evolution is kept, either it does not but is still kept with
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a probability $e^{\frac{\delta}{T}}$ where $\delta$ is the difference
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between the old and new discrepancy, and $T$ the current temperature.
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-The all algorithm is described in the flowchart~\ref{flow_rec}.
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+If the de discrepancy has decreased, the temperature $T$ is multiplied
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+by a factor $\lambda$ (fixed to $0.992$ in all our simultations), hence
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+is decreased.
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+The whole algorithm is described in the flowchart~\ref{flow_rec}.
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\begin{figure}
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\begin{mdframed}
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@@ -221,7 +247,7 @@ The all algorithm is described in the flowchart~\ref{flow_rec}.
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\subsubsection{Dependence on the temperature}
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First experiments were made to select the best initial temperature.
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Results are compiled in graphs~\ref{temp_2},~\ref{temp3},\ref{temp3_z}.
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-Graphs~\ref{temp_2},~\ref{temp3} represents the results obtained respectively
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+Graphs~\ref{temp_2},~\ref{temp3} represent the results obtained respectively
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in dimension 2 and 3 between 10 and 500 points. The curve obtained is
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characteristic of the average evolution of the discrepancy optimization
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algorithms for Halton points sets: a very fast decrease for low number of
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@@ -230,7 +256,7 @@ after~\cite{Doerr}.
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The most interesting part of these results are concentrated between 80 and 160
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points were the different curves splits. The graph~\ref{temp3_z} is a zoom
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of~\ref{temp3} in this window. We remark on that graph that the lower the
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-temperature is, the best the results are.
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+temperature is, the best the results are, with a threshold at $10^{-3}$.
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\begin{figure}
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\includegraphics[scale=0.3]{Results/resu_2_temp.png}
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@@ -258,7 +284,7 @@ of the algorithm with regards to the number of iterations. We present here
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the result in dimension 3 in the graph~\ref{iter_sa}. Once again we
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restricted the window between 80 and 180 points were curves are split.
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An interesting phenomena can be observed: the error rates are somehow
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-invariant w.r.t.\ the number of iteration and once again the 1000 iterations
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+invariant w.r.t.\ the number of iterations and once again the 1000 iterations
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threshold seems to appear --- point 145 is a light split between iteration
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1600 and the others, but excepted for that point, getting more than 1000
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iterations tends be be a waste of time. The error rate is for 80 points the
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@@ -276,47 +302,68 @@ rates for fully random search with 400 iterations.
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\begin{figure}
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\begin{mdframed}
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- \label{rand_flow}
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\includegraphics[scale=0.4]{crossover_flow.pdf}
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\caption{Flowchart of the crossover algorithm.}
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+ \label{cross_flow}
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\end{mdframed}
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\end{figure}
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+\subsubsection{Dependence on the parameter p}
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+First experiments were made to select the value for the crossover parameter
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+p. Results are compiled in graphs~\ref{res_gen2},~\ref{res_gen2z},\ref{res_gen3}
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+and~\ref{res_gen4}.
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+Graph~\ref{res_gen2}, represents the results obtained
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+in dimension 2 between 10 and 500 points. The curve obtained is, with no
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+surprise again,
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+the characteristic curve of the average evolution of the discrepancy we already
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+saw with the previous experiments.
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+The most interesting part of these results are concentrated --- once again ---
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+between 80 and 160 points were the different curves splits.
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+The graph~\ref{res_gen2z} is a zoom of~\ref{res_gen2} in this window, and
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+graphs~\ref{res_gen3} and~\ref{res_gen4} are focused directly into it too.
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+We remark that in dimension 2, the results are better for $p$ close to $0.5$
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+whereas for dimension 3 and 4 the best results are obtained for $p$ closer to
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+$0.1$.
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+
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+
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\begin{figure}
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- \label{rand_flow}
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\includegraphics[scale=0.3]{Results/res_gen_2.png}
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-\caption{Dependence on iterations number: D=3}
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+\caption{Dependence on parameter p: D=2}
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+ \label{res_gen2}
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\end{figure}
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\begin{figure}
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- \label{rand_flow}
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\includegraphics[scale=0.3]{Results/res_gen_2_zoom.png}
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-\caption{Dependence on iterations number: D=3}
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+\caption{Dependence on parameter p (zoom): D=2}
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+\label{res_gen2z}
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\end{figure}
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\begin{figure}
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- \label{rand_flow}
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\includegraphics[scale=0.3]{Results/res_gen_3_zoom.png}
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-\caption{Dependence on iterations number: D=3}
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+\caption{Dependence on parameter p: D=3}
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+ \label{res_gen3}
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\end{figure}
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\begin{figure}
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- \label{rand_flow}
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\includegraphics[scale=0.3]{Results/res_gen_4_zoom.png}
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-\caption{Dependence on iterations number: D=3}
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+\caption{Dependence on parameter p: D=4}
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+ \label{res_gen4}
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\end{figure}
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-As prev we investigated the stability
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-of the algorithm with regards to the number of iterations. We present here
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-the result in dimension 3 in the graph~\ref{iter_sa}. Once again we
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+Once again we investigated the stability
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+of the algorithm with regards to the number of iterations. Once again we
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restricted the window between 80 and 180 points were curves are split.
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-An interesting phenomena can be observed: the error rates are somehow
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-invariant w.r.t.\ the number of iteration and once again the 1000 iterations
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-threshold seems to appear --- point 145 is a light split between iteration
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-1600 and the others, but excepted for that point, getting more than 1000
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-iterations tends be be a waste of time. The error rate is for 80 points the
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-biggest and is about $15\%$ of the value, which is similar to the error
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-rates for fully random search with 400 iterations.
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+The results are compiled in grah~\ref{gen_iter}.
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+An interesting phenomena can be observed: the error rates are getting really
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+big for 1400 iterations at very low points (up to 120),
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+even if the average results are stables after the threshold 1000 iterations,
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+like we get before.
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+
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+\begin{figure}
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+\includegraphics[scale=0.3]{Results/gen_iter.png}
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+\caption{Stability w.r.t.\ number of iterations: D=2}
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+ \label{gen_iter}
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+\end{figure}
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\section{Results}
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Eventually we made extensive experiments to compare the three previously
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@@ -333,9 +380,9 @@ annealing and genetic search --- is clear over fully random search.
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Both curves for these heuristics are way below the error band of random
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search. As a result \emph{worse average results of non trivial heuristics are
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better than best average results when sampling points at random}.
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-In dimension 2~\ref{wrap2z}, the best results are given by the gentic search,
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-wheras in dimension 3 and 4~\ref{wrap3z},~\ref{wrap4z}, best results are
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-given by simmulated annealing. It is also noticable that in that range
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+In dimension 2~\ref{wrap2z}, the best results are given by the simulated annealing,
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+whereas in dimension 3 and 4~\ref{wrap3z},~\ref{wrap4z}, best results are
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+given by genetic search. It is also noticeable that in that range
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of points the error rates are roughly the same for all heuristics:
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\emph{for 1000 iteration, the stability of the results is globally the
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same for each heuristic}.
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@@ -366,6 +413,13 @@ same for each heuristic}.
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\section{Conclusion}
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+\section*{Acknoledgments}
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+We would like to thank Magnus Wahlstrom from the Max Planck Institute for Informatics
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+for providing an implementation of the DEM algorithm [DEM96].
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+We would also like to thank Christoff Durr and Carola Doerr
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+for several very helpful talks on the topic of this work.
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+Both Thomas Espitau and Olivier Marty supported by the French Ministry for
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+Research and Higher Education, trough the Ecole Normale Supérieure.
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\bibliographystyle{alpha}
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\bibliography{bi}
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\end{document}
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espitau