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% ------------------------------------------------------------------------
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\documentclass{llncs}
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\input{prelude}
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-\begin{document}
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+\begin{document}
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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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@@ -29,16 +29,16 @@ Comparison of three heuristics for generating points set}
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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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+ 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 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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+ simulated 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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@@ -47,14 +47,14 @@ Comparison of three heuristics for generating points set}
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The testing tool is aimed to be modular: it is made of independents blocks that
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are interfaced trough a scheduler. More precisely a master wrapper is written
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in Python that calls a first layer which performs the chosen heuristic. This
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-layer is written in C++ for performances. The given discrepancy algorithm
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+layer is written in C++ for performances. The given discrepancy algorithm
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--- written in C --- is called when evaluations of a state is needed.
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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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+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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+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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+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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\item 2.8 GHz Intel Quad Core i7 hyper-threaded to 3.1GHz, 8 Go 1600 MHz DDR3.
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@@ -72,32 +72,32 @@ Experiments were conducted on two machines:
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\label{class_flow}
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\end{figure}
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-On these machines, some basic profiling has make clear that
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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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-cost function is the DEM-algorithm~\cite{Dobkin} of
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+of the discrepancy}. The chosen algorithm and implantation of this
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+cost function is the DEM-algorithm~\cite{Dobkin} of
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\emph{Magnus Wahlstr\o m}~\cite{Magnus}.\medskip
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-All the experiments has been conducted on dimension 2,3,4
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+All the experiments has been conducted on dimension 2,3,4
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--- with a fixed Halton basis 7, 13, 29, 3 ---. Some minor tests have
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-been made in order to discuss the dependency of the discrepancy and
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+been made in order to discuss the dependency of the discrepancy and
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efficiency of the heuristics with regards to the values chosen for the
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-prime base. The average results remains roughly identical when taking
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+prime base. The average results remains roughly identical when taking
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changing these primes and taking them in the range [2, 100]. For such
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-a reason we decided to pursue the full computations with a fixed
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+a reason we decided to pursue the full computations with a fixed
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basis.
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\subsection{Algorithmic insights}
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-To perform an experiment we made up a
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-loop above the main algorithm which calls the chosen heuristic multiple
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+To perform an experiment we made up a
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+loop above the main algorithm which calls the chosen heuristic multiple
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times in order to smooth up the results and obtain more exploitable datas.
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Then an arithmetic mean of the result is performed on the values. In addition
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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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+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. 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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@@ -110,7 +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 box plot 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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@@ -119,14 +119,14 @@ represents the incertitude with regards to the values obtained.
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\subsection{Fully random search (Test case)}
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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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+ random permutations, compute the corresponding 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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+ 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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+ 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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+ us to generate an uniformly chosen permutation at each step. We recall
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this fact and detail the algorithm in the following section.
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\begin{figure}
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\begin{mdframed}
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@@ -139,12 +139,12 @@ represents the incertitude with regards to the values obtained.
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\subsubsection{The Knuth-Fisher-Yates shuffle}
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-The Fisher–Yates shuffle is an algorithm for generating a random permutation
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-of a finite sets. The Fisher–Yates shuffle is unbiased, so that every
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-permutation is equally likely. We present here the Durstenfeld variant of
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+The Fisher–Yates shuffle is an algorithm for generating a random permutation
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+of a finite sets. The Fisher–Yates shuffle is unbiased, so that every
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+permutation is equally likely. We present here the Durstenfeld variant of
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the algorithm, presented by Knuth in \emph{The Art of Computer programming}
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vol. 2~\cite{Knuth}.
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-The algorithm's time complexity is here $O(n)$, compared to $O(n^2)$ of
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+The algorithm's time complexity is here $O(n)$, compared to $O(n^2)$ of
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the naive implementation.
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\begin{algorithm}[H]
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@@ -165,40 +165,40 @@ the naive implementation.
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The resulting permutation of KFY is unbiased.
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\end{lemma}
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\begin{proof}
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- Let consider the set $[1,\ldots n]$ as the vertices of a random graph
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- constructed as the trace of the execution of the algorithm:
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+ Let consider the set $[1,\ldots n]$ as the vertices of a random graph
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+ constructed as the trace of the execution of the algorithm:
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an edge $(i,j)$ exists in the graph if and only if the swap of $T[i]$ and
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$T[j]$ had been executed. This graph encodes the permutation represented by
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- $T$. To be able to encode any permutation the considered graph must be
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+ $T$. To be able to encode any permutation the considered graph must be
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connected --- in order to allow any pairs of points to be swapped ---.
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- Since by construction every points is reached by an edge, and that there
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+ Since by construction every points is reached by an edge, and that there
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exists exactly $n-1$ edges, we can conclude directly that any permutation can
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- be reached by the algorithm. Since the probability of getting a fixed graph
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- of $n-1$ edges with every edges of degree at least one is $n!^{-1}$, the
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+ be reached by the algorithm. Since the probability of getting a fixed graph
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+ of $n-1$ edges with every edges of degree at least one is $n!^{-1}$, the
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algorithm is thus unbiased.
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\end{proof}
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\subsubsection{Results and stability}
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-We first want to analyze the dependence of the results on the number of
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-iterations of the heuristic, in order to discuss its stability.
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+We first want to analyze the dependence of the results on the number of
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+iterations of the heuristic, in order to discuss its stability.
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The results are compiled in the figures~\ref{rand_iter2},~\ref{rand_iter3},
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restricted to a number of points between 80 and 180.
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-We emphasize on the fact the lots of datas appears on the graphs,
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+We emphasize on the fact the lots of datas appears on the graphs,
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and the error bands representation make them a bit messy. These graphs
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were made for extensive internal experiments and parameters researches.
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-The final wrap up graphs are much more lighter and only presents the best
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+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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+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 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 seem to get to a limits after 1000
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+The average results are quite stable, they decrease progressively with
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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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+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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discrepancy and this heuristic.
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@@ -213,33 +213,32 @@ 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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+\subsection{Local search with simulated annealing}
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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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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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+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 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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+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 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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+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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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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+by a factor $\lambda$ (fixed to $0.992$ in all our simulations), 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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\includegraphics[scale=0.4]{flow_recuit.pdf}
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-\caption{Flowchart of the simulated annealing local search heuristic}
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+\caption{Flowchart of the local search with simulated annealing heuristic}
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\label{flow_rec}
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\end{mdframed}
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\end{figure}
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@@ -248,14 +247,14 @@ The whole algorithm is described in the flowchart~\ref{flow_rec}.
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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} 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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-points --- roughly up to 80 points --- and then a very slow one
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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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+points --- roughly up to 80 points --- and then a very slow one
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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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+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, with a threshold at $10^{-3}$.
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\begin{figure}
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@@ -283,9 +282,9 @@ As for the fully random search heuristic 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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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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+An interesting phenomena can be observed: the error rates are somehow
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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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+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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@@ -297,7 +296,26 @@ rates for fully random search with 400 iterations.
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\label{iter_sa}
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\end{figure}
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-\subsection{Genetic (5+5) search}
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+\subsection{Genetic $(\mu+\lambda)$ search}
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+
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+The third heuristic implemented is the $(\mu+\lambda)$ genectic search.
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+This heuristic is inspired from the evolution of species: a family
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+of $\mu$ genes is known (they are generated randomly at the beginning),
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+from which $\lambda$ new genes are derived. A gene is the set of parameters
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+we are optimising, i.e. the permutations.
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+Each one is derived either from one gene applying a mutation
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+(here a transposition of one of the permutations), or from two
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+genes applying a crossover : a blending of both genes (the
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+algorithm is described in details further). The probability of
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+making a mutation is $c$, the third parameter of the algorithm,
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+among $\mu$ and $\lambda$. After that, only the $\mu$ best
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+genes are kept, according to their fitness, and the process
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+can start again.
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+
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+Because variating over $\mu$ or $\lambda$ does not change fundamentaly
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+the algorithm, we have chosen to fix $\mu=\lambda=5$ once and for all,
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+which seemed to be a good trade-off between the running time of
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+each iteration and the size of the family.
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\begin{figure}
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@@ -308,54 +326,110 @@ rates for fully random search with 400 iterations.
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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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+\subsubsection{Algorithm of crossover}
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+
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+We designed a crossover for permutations. The idea is simple: given two
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+permutations $A$ and $B$ of $\{1..n\}$, it constructs a new permutations
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+$C$ value after value, in a random order (we use our class permutation
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+for it). For each indice $i$, we take either $A_i$ or $B_i$. If exactly
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+one of those values is available (understand it was not already chosen)
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+we choose it. If both are available, we choose randomly and we remember the
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+second. If both are unavailable, we choose a remembered value.
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+
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+The benefits of this method are that it keeps common values of $A$ and $B$,
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+the values $C_i$ are often among $\{A_i, B_i\}$ (hence $C$ is close to $A$
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+and $B$), and it does not favor either the begining or the ending of
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+permutations.
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+
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+\begin{algorithm}[H]
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+ \SetAlgoLined
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+ \SetKwFunction{Rand}{Rand}
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+ \SetKwFunction{RPO}{a random permutation of }
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+ \SetKwFunction{RVI}{a random value in }
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+ \KwData{Two permutations A[1..n], B[1..n]}
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+ \KwResult{A permutation C[1..n]}
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+ $pi \leftarrow \RPO \{1,\dots,n\}$\;
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+ $available \leftarrow \{\}$\;
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+ $got \leftarrow \{\}$\;
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+ \For{$i\leftarrow 1$ \KwTo $n$}{
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+ $j \leftarrow pi_i$\;
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+ $a \leftarrow A_j$\;
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+ $b \leftarrow B_j$\;
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+ \If{$a \in got \land b \in got$}{
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+ $v \leftarrow \RVI available$\;
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+ }
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+ \ElseIf{$a \in got$}{
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+ $v \leftarrow b$\;
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+ }
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+ \ElseIf{$b \in got$}{
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+ $v \leftarrow a$\;
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+ }
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+ \Else{
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+ \If{$\Rand(\{0,1\}) = 1$}{
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+ $v \leftarrow a$\;
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+ $available \leftarrow available \cup \{b\}$\;
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+ }
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+ \Else{
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+ $v \leftarrow b$\;
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+ $available \leftarrow available \cup \{a\}$\;
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+ }
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+ }
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+ $C_j \leftarrow v$\;
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+ $got \leftarrow got \cup \{v\}$\;
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+ $available \leftarrow available \setminus \{v\}$\;
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+ }
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+ \caption{Permutations crossover}
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+\end{algorithm}
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+
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+
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+\subsubsection{Dependence on the parameter $c$}
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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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+$c$. 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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+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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+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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+We remark that in dimension 2, the results are better for $c$ close to $0.5$
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+whereas for dimension 3 and 4 the best results are obtained for $c$ closer to
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$0.1$.
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\begin{figure}
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\includegraphics[scale=0.3]{Results/res_gen_2.png}
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-\caption{Dependence on parameter p: D=2}
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+\caption{Dependence on parameter $c$: 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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\includegraphics[scale=0.3]{Results/res_gen_2_zoom.png}
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-\caption{Dependence on parameter p (zoom): D=2}
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+\caption{Dependence on parameter $c$ (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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\includegraphics[scale=0.3]{Results/res_gen_3_zoom.png}
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-\caption{Dependence on parameter p: D=3}
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+\caption{Dependence on parameter $c$: 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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|
\includegraphics[scale=0.3]{Results/res_gen_4_zoom.png}
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-\caption{Dependence on parameter p: D=4}
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+\caption{Dependence on parameter $c$: D=4}
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\label{res_gen4}
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\end{figure}
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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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+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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-The results are compiled in grah~\ref{gen_iter}.
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|
+The results are compiled in graph~\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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+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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|
@@ -367,23 +441,23 @@ like we get before.
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|
\section{Results}
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|
Eventually we made extensive experiments to compare the three previously
|
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|
-presented heuristics. The parameters chosen for the heuristics have been
|
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|
+presented heuristics. The parameters chosen for the heuristics have been
|
|
|
chosen using the experiments conducted in the previous sections
|
|
|
-Results are compiled in the last
|
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|
-figures~\ref{wrap2},~\ref{wrap2z},~\ref{wrap3z},~\ref{wrap4z}. The
|
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|
+Results are compiled in the last
|
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|
+figures~\ref{wrap2},~\ref{wrap2z},~\ref{wrap3z},~\ref{wrap4z}. The
|
|
|
recognizable curve of decrease
|
|
|
-of the discrepancy is still clearly recognizable in the graph~\ref{wrap2},
|
|
|
-made for points ranged between 10 and 600. We then present the result
|
|
|
-in the --- now classic --- window 80 points - 180 points ---.
|
|
|
+of the discrepancy is still clearly recognizable in the graph~\ref{wrap2},
|
|
|
+made for points ranged between 10 and 600. We then present the result
|
|
|
+in the --- now classic --- window 80 points - 180 points ---.
|
|
|
For all dimensions, the superiority of non-trivial algorithms --- simulated
|
|
|
-annealing and genetic search --- is clear over fully random search.
|
|
|
-Both curves for these heuristics are way below the error band of random
|
|
|
+annealing and genetic search --- is clear over fully random search.
|
|
|
+Both curves for these heuristics are way below the error band of random
|
|
|
search. As a result \emph{worse average results of non trivial heuristics are
|
|
|
better than best average results when sampling points at random}.
|
|
|
In dimension 2~\ref{wrap2z}, the best results are given by the simulated annealing,
|
|
|
whereas in dimension 3 and 4~\ref{wrap3z},~\ref{wrap4z}, best results are
|
|
|
given by genetic search. It is also noticeable that in that range
|
|
|
-of points the error rates are roughly the same for all heuristics:
|
|
|
+of points the error rates are roughly the same for all heuristics:
|
|
|
\emph{for 1000 iteration, the stability of the results is globally the
|
|
|
same for each heuristic}.
|
|
|
|
|
@@ -413,10 +487,10 @@ same for each heuristic}.
|
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|
\section{Conclusion}
|
|
|
|
|
|
-\section*{Acknoledgments}
|
|
|
-We would like to thank Magnus Wahlstrom from the Max Planck Institute for Informatics
|
|
|
-for providing an implementation of the DEM algorithm [DEM96].
|
|
|
-We would also like to thank Christoff Durr and Carola Doerr
|
|
|
+\section*{Acknowledgments}
|
|
|
+We would like to thank Magnus Wahlstrom from the Max Planck Institute for Informatics
|
|
|
+for providing an implementation of the DEM algorithm [DEM96].
|
|
|
+We would also like to thank Christoff Durr and Carola Doerr
|
|
|
for several very helpful talks on the topic of this work.
|
|
|
Both Thomas Espitau and Olivier Marty supported by the French Ministry for
|
|
|
Research and Higher Education, trough the Ecole Normale Supérieure.
|