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- % ------------------------------------------------------------------------
- \documentclass{llncs}
- \input{prelude}
- \begin{document}
- \title{Star Discrepancies for generalized Halton points}
- % \titlerunning{} % abbreviated title (for running head)
- % also used for the TOC unless
- % \toctitle is used
- \author{Thomas Espitau\inst{1} \and Olivier Marty\inst{1}}
- %
- % \authorrunning{} % abbreviated author list (for running head)
- %
- %%%% list of authors for the TOC (use if author list has to be modified)
- % \tocauthor{}
- %
- \institute{
- $\mbox{}^1$ ENS Cachan \qquad
- }
- \maketitle
- \makeatletter
- \makeatother
- \begin{abstract}
- \end{abstract}
- \section{Introduction}
- \section{General architecture of the toolset}
- Big pipeline + pool + blabla\medskip
- Un truc bien ca serait d'avoir un truc modulaire (genre les heuristiques
- sont des "blocs" et l'outil appelle un bloc quand il veut faire son test
- en l'appelant en multicore"
- (c'est stupide mais ca fera un beau schéma!)
- \section{Heuristics devlopped}
- \subsection{Fully random search (Test case)}
- The first heuristic implemented is rge random search. We generates
- random sets of Halton points and select the best set with regard to its
- discrepancy iteratively. The process is wraped up in the following
- flowchart~\ref{rand_flow}. In order to generate at each step a random
- permutation, we transform it directly from the previous one.
- More precisely the permutation is a singleton object which have method
- random, built on the Knuth Fisher Yates shuffle. This algorithm allows
- us to generate an uniformly chosen permutation at each step. We recall
- this fact and detail the algorithm in the following section.
- \begin{figure}
- \begin{mdframed}
- \label{rand_flow}
- \includegraphics[scale=0.4]{flow_rand.pdf}
- \caption{Flowchart of the random search}
- \end{mdframed}
- \end{figure}
- \subsubsection{The Knuth-Fisher-Yates shuffle}
- The Fisher–Yates shuffle is an algorithm for generating a random permutation
- of a finite sets. The Fisher–Yates shuffle is unbiased, so that every
- permutation is equally likely. We present here the Durstenfeld variant of
- the algorithm, presented by Knuth in \emph{The Art of Computrer programming}.
- The algorithm's time complexity is here $O(n)$, compared to $O(n^2)$ of
- the naive implementation.
- \begin{algorithm}[H]
- \SetAlgoLined
- \SetKwFunction{Rand}{Rand}
- \SetKwFunction{Swap}{Swap}
- \KwData{A table T[1..n]}
- \KwResult{Same table T, shuffled}
- \For{$i\leftarrow 1$ \KwTo $n-1$}{
- $j \leftarrow$ \Rand{$[1,n-i]$}\;
- \Swap{$T[i], T[i+j]$}\;
- }
- \caption{KFY algorithm}
- \end{algorithm}
- \begin{lemma}
- The resulting permutation of KFY is unbiaised.
- \end{lemma}
- \begin{proof}
- Let consider the set $[1,\ldots n]$ as the vertices of a random graph
- constructed as the trace of the execution of the algorithm:
- an edge $(i,j)$ exists in the graph if and only if the swap of $T[i]$ and
- $T[j]$ had been executed. This graph encodes the permutation represented by
- $T$. To be able to encode any permutation the considered graph must be connex
- --- in order to allow any pairs of points to be swapped ---.
- Since by construction every points is reached by an edge, and that there
- exists exactly $n-1$ edges, we can conclude directly that any permutation can
- be reached by the algorithm. Since the probability of getting a fixed graph
- of $n-1$ edges with every edges of degree at least one is $n!^{-1}$, the
- algorithm is thus unbiaised.
- \end{proof}
- \subsubsection{Results and stability}
- \subsection{Local Search}
- Blabla+Pipeline
- \subsection{Evolutionary heuristics}
- Blabla+Pipeline
- \subsection{Genetic heuristics}
- Blabla+Pipeline
- \section{Results}
- \section{Conclusion}
- \end{document}
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