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@@ -46,6 +46,46 @@
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\subsection{Fully random search (Test case)}
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Blabla+pipeline
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+ \subsection{The Knuth-Fisher-Yates shuffle}
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+
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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 Computrer programming}.
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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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+
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+\begin{algorithm}[H]
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+ \SetAlgoLined
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+ \SetKwFunction{Rand}{Rand}
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+ \SetKwFunction{Swap}{Swap}
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+ \KwData{A table T[1..n]}
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+ \KwResult{Same table T, shuffled}
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+ \For{$i\leftarrow 1$ \KwTo $n-1$}{
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+ $j \leftarrow$ \Rand{$[1,n-i]$}\;
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+ \Swap{$T[i], T[i+j]$}\;
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+ }
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+ \caption{KFY algorithm}
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+\end{algorithm}
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+
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+
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+\begin{lemma}
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+ The resulting permutation of KFY is unbiaised.
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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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+ 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 connex
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+ --- 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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+ 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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+ algorithm is thus unbiaised.
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+
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+\end{proof}
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\subsection{Local Search}
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Blabla+Pipeline
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\subsection{Evolutionary heuristics}
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espitau