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--- /dev/null
+\begin{exampleblock}{}
+
+\begin{itemize}
+\item Mode mixe avec
+ $SCC(1)=\{1,2\}$,
+ $SCC(3)=\{3\}$ and $SCC(4)=\{4,5\}$
+\begin{center}
+ \includegraphics[width=10cm]{xpl_iteration_final.pdf}
+\end{center}
+\end{itemize}
+\end{exampleblock}
--- /dev/null
+\begin{exampleblock}{}
+
+\begin{itemize}
+\item Our Prototype: confirmation of convergence
+of mixed asynchronous iterations with uniform delays
+\item Mixed iterations with
+ $SCC(1)=\{1,2\}$,
+ $SCC(3)=\{3\}$ and $SCC(4)=\{4,5\}$
+\begin{center}
+ \includegraphics[width=2.5cm]{../xpl_iteration_final.ps}
+\end{center}
+\end{itemize}
+\end{exampleblock}
--- /dev/null
+
+\begin{exampleblock}{}
+\begin{itemize}
+\item Graphe de connection:
+\begin{center}
+\includegraphics[width=3cm]{xplgraph.pdf}
+\end{center}
+\end{itemize}
+\end{exampleblock}
--- /dev/null
+
+\begin{exampleblock}{}
+\begin{itemize}
+\item Connection graph:
+\begin{center}
+\includegraphics[width=3cm]{../xplgraph.ps}
+\end{center}
+\end{itemize}
+\end{exampleblock}
--- /dev/null
+\begin{itemize}
+\item Intuition:
+ \begin{itemize}
+ \item N{\oe}uds problématiques: regroupés et synchronisés
+ \item N{\oe}uds entre les groupes distincts: laisser l'asynchronisme
+ \end{itemize}
+\item Formalisation:
+ \begin{itemize}
+ \item Regroupement selon les composantes connexes (SCC) du graphe de connection
+ \item Mode mixe:
+ \begin{itemize}
+ \item $S_{ij}^t = t$ si $i \in SCC(j)$.
+ \item $
+ \begin{array}{l}
+ \forall p_0,p_1,q_0,q_1,t \, .\,
+ \quad (
+ p_1 \in SCC(p_0) \land
+ q_1 \in SCC(q_0))\\
+ \qquad \Rightarrow
+ S_{p_0q_0}^{t} =
+ S_{p_1q_1}^{t}
+ \end{array}
+ $
+ \end{itemize}
+ \end{itemize}
+\item Elements d'un même groupe: équivalents de l'extérieur.
+\end{itemize}
--- /dev/null
+\begin{itemize}
+\item Intuition:
+ \begin{itemize}
+ \item Nodes that may introduce cyclic iterations: grouped
+ \item Nodes inside each group: synchronized at each other
+ \item Nodes of distinct groups: keeping asynchronism
+ \end{itemize}
+\item Formalization:
+ \begin{itemize}
+ \item Group: SCC of the connection graph
+ \item Mixed iterations of DN with Uniform Delays:
+ \begin{itemize}
+ \item $S_{ij}^t$ is $t$ if $i \in SCC(j)$.
+ \item $
+ \begin{array}{l}
+ \forall p_0,p_1,q_0,q_1,t \, .\,
+ \quad (
+ p_1 \in SCC(p_0) \land
+ q_1 \in SCC(q_0))\\
+ \qquad \Rightarrow
+ S_{p_0q_0}^{t} =
+ S_{p_1q_1}^{t}
+ \end{array}
+ $
+ \end{itemize}
+ \end{itemize}
+\item Elements in the same SCC: cannot be differentiated
+\end{itemize}
--- /dev/null
+\begin{Theorem}{}
+ Soit une fonction $F$ contenant un unique point fixe $X^*$.
+ Si les itérations chaotiques convergent vers $X^*$,
+ alors les itérations mixes convergent aussi $X^*$
+ pour n'imoporte quelle stratégie pseudo périodique.
+\end{Theorem}
--- /dev/null
+\newcommand{\Nats}[0]{\ensuremath{\mathbb{N}}}
+
+\begin{Theorem}{}
+ Given an initial configuration $X^0 = (X_1^0, \ldots, X_n^0)$ and a
+ strategy $(J^t)^{t \in \Nats}$. If chaotic iterations with strategy
+ $(J^t)^{t \in \Nats}$ converge to some fixed-point, then mixed iterations with
+ uniform delays with the same strategy $(J^t)^{t \in \Nats}$ converge to the
+ same fixed-point.
+\end{Theorem}
+\begin{Proof}
+\begin{itemize}
+\item Element renaming wrt. a partial order
+\item Induction on the SCC index
+\end{itemize}
+\end{Proof}
--- /dev/null
+\begin{itemize}
+ \item \emph{Strategie}: les éléments $J^t$ modifiés au temps $t$;
+ \item \emph{Date de Visibilité}: suite de la date la plus récente où un
+ composant connaît la valeur d'un autre.
+ \begin{itemize}
+ \item $n*n$ matrice $S_{ij}^t$: le $t'$ plus élevé
+ t.-q. $t' \le t$ et $X_j^{t'}$ est accessible à $i$
+ \item Délai de communication entre $j$ et $i$: $t - D_{ij}^t$
+ \end{itemize}
+ \item \emph{Formalisation des modes}:
+\[
+X^{t+1}_i= \left\{
+\begin{array}{l}
+X^{t}_i \textrm{ si $i \not \in J^t$} \\
+F_i
+\left( X_1^{S_{i1}^t},\ldots, X_{n}^{S_{i{n}}^t} \right)
+\end{array}
+\right.
+\]
+\begin{itemize}
+\item Parallèle: $J^t=\{1,\ldots,n\}$ and $S^t=(t)$;
+\item Chaotique: $S^t=(t)$;
+\item Asynchrone: aucune constrainte.
+\end{itemize}
+\end{itemize}
+
--- /dev/null
+\begin{itemize}
+ \item \emph{Strategy}: sequence of elements updated at time $t$;
+ \begin{itemize}
+ \item $n*n$ diagonal matrix: $J_{ii}^t = 1$ iff $X_i \in E_i$
+ is updated at time $t$
+ \end{itemize}
+ \item \emph{Visibility Dates}: sequence of the more recent dates where
+ components know values of other ones
+ \begin{itemize}
+ \item $n*n$ matrix $S_{ij}^t$: highest $t'$
+ s.t. $t' \le t$ and $X_j^{t'}$ is accessible for $i$
+ \item Communcation delays from $j$ to $i$: $t - S_{ij}^t$
+ \end{itemize}
+ \item \emph{Iterations modes}:
+\[
+X^{t+1}= (I -J^t)X^t + J^t
+\left(
+\begin{array}{c}
+F_1
+\left( X_1^{S_{11}^t},\ldots, X_{n}^{S_{1{n}}^t} \right) \\
+\vdots \\
+F_{n} \left( X_1^{S_{{n}1}^t},\ldots, X_{n}^{S_{{n}{n}}^t} \right)
+\end{array}
+\right)
+\]
+\begin{itemize}
+\item Parallel mode: $J^t=I$ and $S^t=(t)$
+\item Chaotic mode: $S^t=(t)$
+\item Asynchronous: no constraint
+\end{itemize}
+\end{itemize}
+
--- /dev/null
+\begin{itemize}
+\item
+
+
+
--- /dev/null
+\begin{itemize}
+\item<1-3> Objectives: ``similar behaviors'' in asynchronous iterations
+ as in some chaotic iterations
+\item<2-3> Issues:
+\begin{itemize}
+\item Strengthening the convergence conditions of chaotic mode
+\item Lightly introduce synchronism into asynchronous modes [ABCVS05]
+\end{itemize}
+\item<3> \alert{Proof of convergence conjecture}
+\end{itemize}
+
+
+
--- /dev/null
+\begin{exampleblock}{}
+\begin{itemize}
+\only<1>{\item Itérations Parallèles : convergence
+\begin{center}
+\includegraphics[width=8cm]{para_iterate_dec.pdf}
+\end{center}}
+\only<2>{\item Itérations chaotiques: divergence avec $J^t=\{24;15;14;15;\ldots\}$
+\begin{center}
+\includegraphics[width=4cm]{chao_iterate_excerpt.pdf}
+\end{center}}
+\only<3>{\item Iterations asynchrones : divergence même avec $J^t=\{1,\ldots,n\}$
+\begin{itemize}
+\item $S^t = \left(
+\begin{array}{lllll}
+t & t' & t & t & t \\
+t & t & t & t & t \\
+t & t & t & t & t \\
+t & t & t & t & t\\
+t & t & t & t & t \\
+\end{array}
+\right)
+\textrm{ where }
+t' = \left\{
+\begin{array}{l}
+t \textrm{si $t$ est pair;} \\
+t-1 \textrm{sinon.}
+\end{array}
+\right.
+$
+\item
+$
+\begin{array}{rcl}
+X^{2k+1} & =& F(X^{2k}) \\
+X^{2k} & =& \left(
+\begin{array}{l}
+F_1(X_1^{2k-1},X_2^{2k-2},X_3^{2k-1},X_4^{2k-1},X_5^{2k-1}) \\
+F_2(X^{2k-1})\\
+\vdots \\
+F_5(X^{2k-1})
+\end{array}
+\right).
+\end{array}
+$
+
+$\rightarrow$ cycle entre 3 et 11.
+\end{itemize}
+}
+
+\only<4>{
+\item Objectif: obtenir des comportements similaires en asynchrone et en
+ chaotique.
+}
+\end{itemize}
+\end{exampleblock}
+
--- /dev/null
+\begin{exampleblock}{}
+\begin{itemize}
+\only<1>{\item Parallel iterations: convergence
+\begin{center}
+\includegraphics[width=8cm]{../implementation/para_iterate_dec.ps}
+\end{center}}
+\only<2>{\item Chaotic iterations: divergence with $J^t=\{24;15;14;15;\ldots\}$
+\begin{center}
+\includegraphics[width=4cm]{../chao_iterate_excerpt.ps}
+\end{center}}
+\only<3>{\item Asynchronous iterations: divergence even with $J^t=I$
+\begin{itemize}
+\item $S^t = \left(
+\begin{array}{lllll}
+t & t' & t & t & t \\
+t & t & t & t & t \\
+t & t & t & t & t \\
+t & t & t & t & t\\
+t & t & t & t & t \\
+\end{array}
+\right)
+\textrm{ where }
+t' = \left\{
+\begin{array}{l}
+t \textrm{ if t is even;} \\
+t-1 \textrm{ otherwise.}
+\end{array}
+\right.
+$
+\item
+$
+\begin{array}{rcl}
+X^{2k+1} & =& F(X^{2k}) \\
+X^{2k} & =& \left(
+\begin{array}{l}
+F_1(X_1^{2k-1},X_2^{2k-2},X_3^{2k-1},X_4^{2k-1},X_5^{2k-1}) \\
+F_2(X^{2k-1})\\
+\vdots \\
+F_5(X^{2k-1})
+\end{array}
+\right).
+\end{array}
+$
+
+$\rightarrow$ cycle between 3 and 11.
+\end{itemize}
+}
+\end{itemize}
+\end{exampleblock}
--- /dev/null
+\begin{exampleblock}{}
+\begin{itemize}
+\item Cinq éléments $1 \le i \le 5$ dont les valeurs $X_i$ dans $\{0,1\}$
+\item Dynamique:
+\[
+F(X)= \left \{
+\begin{array}{lll}
+f_1(X_1,X_2,X_3,X_4,X_5) & = & X_1 \overline{X_2} + \overline{X_1} X_2 \\
+f_2(X_1,X_2,X_3,X_4,X_5) & = & \overline{X_1 + X_2} \\
+f_3(X_1,X_2,X_3,X_4,X_5) & = & X_3 \overline{X_1} \\
+f_4(X_1,X_2,X_3,X_4,X_5) & = & X_5 \\
+f_5(X_1,X_2,X_3,X_4,X_5) & = & \overline{X_3} + X_4
+\end{array}
+\right.
+\]
+\end{itemize}
+\end{exampleblock}
--- /dev/null
+\begin{exampleblock}{}
+\begin{itemize}
+\item Five elements $1 \le i \le 5$ taking their value $X_i$ in $\{0,1\}$
+\item Configuration: a binary number in 0,1,\ldots,31
+\item Dynamic: a map
+\[
+F(X)= \left \{
+\begin{array}{lll}
+f_1(X_1,X_2,X_3,X_4,X_5) & = & X_1 \overline{X_2} + \overline{X_1} X_2 \\
+f_2(X_1,X_2,X_3,X_4,X_5) & = & \overline{X_1 + X_2} \\
+f_3(X_1,X_2,X_3,X_4,X_5) & = & X_3 \overline{X_1} \\
+f_4(X_1,X_2,X_3,X_4,X_5) & = & X_5 \\
+f_5(X_1,X_2,X_3,X_4,X_5) & = & \overline{X_3} + X_4
+\end{array}
+\right.
+\]
+\end{itemize}
+\end{exampleblock}
--- /dev/null
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+\BOOKMARK [3][]{Outline0.1.7.17}{R\351sultats th\351oriques du mode mixe}{Outline0.1}
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+
+\begin{frame} %-------------------------------------------------------
+ \frametitle{Equilibrage de charge}
+ \begin{myitemize}
+ \item Conception de nombreux algorithmes d'équilibrage de charge
+ \item Particularités : contexte distribué, preuve de convergence, support de pertes de liens, conception de stratégie d'équilibrage
+ \end{myitemize}
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+
+
+\section{Combinaison synchrone/asynchrone}
+\frame{\subsection{Plan}\tableofcontents[currentsection,hideallsubsections]}
+\inputFrame{Exemple jouet}{introRunning}
+\inputFrame{Du mode parallèle au mode asynchrone}{introModes}
+\inputFrame{Iterations de l'exemple jouet}{introRIter}
+\inputFrame{Mode mixe}{combMixed}
+\inputFrame{Composantes Connexes de l'exemple jouet}{combMXpl}
+\inputFrame{Résultats théoriques du mode mixe}{combMixedTheo}
+\inputFrame{Expériences}{combMExp}
+
+\section{Avancées autours du chaos}
+
+
+
+
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