From 286797268fd68ce6e03d99d3a39816f6dcacc992 Mon Sep 17 00:00:00 2001 From: couchot Date: Mon, 5 Mar 2012 22:52:53 +0100 Subject: [PATCH 1/1] mixage synchrone asynchrone --- chao_iterate_excerpt.pdf | Bin 0 -> 5000 bytes chao_iterate_excerpt.ps | 509 ++++++++++++++ combMExp.tex | 11 + combMExp.tex~ | 13 + combMXpl.tex | 9 + combMXpl.tex~ | 9 + combMixed.tex | 27 + combMixed.tex~ | 28 + combMixedTheo.tex | 6 + combMixedTheo.tex~ | 15 + introModes.tex | 26 + introModes.tex~ | 32 + introMotivations.tex | 5 + introMotivations.tex~ | 13 + introRIter.tex | 55 ++ introRIter.tex~ | 49 ++ introRunning.tex | 17 + introRunning.tex~ | 18 + para_iterate_dec.pdf | Bin 0 -> 12920 bytes para_iterate_dec.ps | 1032 ++++++++++++++++++++++++++++ slides_and.aux | 78 +++ slides_and.log | 1376 ++++++++++++++++++++++++++++++++++++++ slides_and.nav | 47 ++ slides_and.out | 9 + slides_and.pdf | Bin 0 -> 402850 bytes slides_and.snm | 0 slides_and.tex | 33 + slides_and.toc | 11 + xpl_iteration_final.pdf | Bin 0 -> 17306 bytes xpl_iteration_final.ps | 641 ++++++++++++++++++ xplgraph.pdf | Bin 0 -> 3491 bytes xplgraph.ps | 426 ++++++++++++ 32 files changed, 4495 insertions(+) create mode 100644 chao_iterate_excerpt.pdf create mode 100644 chao_iterate_excerpt.ps create mode 100755 combMExp.tex create mode 100755 combMExp.tex~ create mode 100755 combMXpl.tex create mode 100755 combMXpl.tex~ create mode 100755 combMixed.tex create mode 100755 combMixed.tex~ create mode 100755 combMixedTheo.tex create mode 100755 combMixedTheo.tex~ create mode 100755 introModes.tex create mode 100755 introModes.tex~ create mode 100755 introMotivations.tex create mode 100755 introMotivations.tex~ create mode 100755 introRIter.tex create mode 100755 introRIter.tex~ create mode 100755 introRunning.tex create mode 100755 introRunning.tex~ create mode 100644 para_iterate_dec.pdf create mode 100644 para_iterate_dec.ps create mode 100644 slides_and.aux create mode 100644 slides_and.log create mode 100644 slides_and.nav 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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} diff --git a/combMXpl.tex b/combMXpl.tex new file mode 100755 index 0000000..fd2c583 --- /dev/null +++ b/combMXpl.tex @@ -0,0 +1,9 @@ + +\begin{exampleblock}{} +\begin{itemize} +\item Graphe de connection: +\begin{center} +\includegraphics[width=3cm]{xplgraph.pdf} +\end{center} +\end{itemize} +\end{exampleblock} diff --git a/combMXpl.tex~ b/combMXpl.tex~ new file mode 100755 index 0000000..58d7514 --- /dev/null +++ b/combMXpl.tex~ @@ -0,0 +1,9 @@ + +\begin{exampleblock}{} +\begin{itemize} +\item Connection graph: +\begin{center} +\includegraphics[width=3cm]{../xplgraph.ps} +\end{center} +\end{itemize} +\end{exampleblock} diff --git a/combMixed.tex b/combMixed.tex new file mode 100755 index 0000000..29e122b --- /dev/null +++ b/combMixed.tex @@ -0,0 +1,27 @@ +\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} diff --git a/combMixed.tex~ b/combMixed.tex~ new file mode 100755 index 0000000..a22a809 --- /dev/null +++ b/combMixed.tex~ @@ -0,0 +1,28 @@ +\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} diff --git a/combMixedTheo.tex b/combMixedTheo.tex new file mode 100755 index 0000000..68af77f --- /dev/null +++ b/combMixedTheo.tex @@ -0,0 +1,6 @@ +\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} diff --git a/combMixedTheo.tex~ b/combMixedTheo.tex~ new file mode 100755 index 0000000..f686b77 --- /dev/null +++ b/combMixedTheo.tex~ @@ -0,0 +1,15 @@ +\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} diff --git a/introModes.tex b/introModes.tex new file mode 100755 index 0000000..edbe918 --- /dev/null +++ b/introModes.tex @@ -0,0 +1,26 @@ +\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} + diff --git a/introModes.tex~ b/introModes.tex~ new file mode 100755 index 0000000..ab0ce9e --- /dev/null +++ b/introModes.tex~ @@ -0,0 +1,32 @@ +\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} + diff --git a/introMotivations.tex b/introMotivations.tex new file mode 100755 index 0000000..2426495 --- /dev/null +++ b/introMotivations.tex @@ -0,0 +1,5 @@ +\begin{itemize} +\item + + + diff --git a/introMotivations.tex~ b/introMotivations.tex~ new file mode 100755 index 0000000..db5c024 --- /dev/null +++ b/introMotivations.tex~ @@ -0,0 +1,13 @@ +\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} + + + diff --git a/introRIter.tex b/introRIter.tex new file mode 100755 index 0000000..8aa4f23 --- /dev/null +++ b/introRIter.tex @@ -0,0 +1,55 @@ +\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} + diff --git a/introRIter.tex~ b/introRIter.tex~ new file mode 100755 index 0000000..601d302 --- /dev/null +++ b/introRIter.tex~ @@ -0,0 +1,49 @@ +\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} diff --git a/introRunning.tex b/introRunning.tex new file mode 100755 index 0000000..2db7ac2 --- /dev/null +++ b/introRunning.tex @@ -0,0 +1,17 @@ +\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} diff --git a/introRunning.tex~ b/introRunning.tex~ new file mode 100755 index 0000000..a212770 --- /dev/null +++ b/introRunning.tex~ @@ -0,0 +1,18 @@ +\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 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+%\begin{small} +\input{#2} +%\end{small} +}} @@ -141,4 +149,29 @@ \end{myitemize} \end{frame} + +\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} +\end{frame} + + +\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} + + + + \end{document} diff --git a/slides_and.toc b/slides_and.toc new file mode 100644 index 0000000..ef002dc --- /dev/null +++ b/slides_and.toc @@ -0,0 +1,11 @@ +\beamer@endinputifotherversion {3.10pt} +\select@language {french} +\beamer@sectionintoc {1}{Combinaison synchrone/asynchrone}{8}{0}{1} +\beamer@subsectionintoc {1}{1}{Plan}{8}{0}{1} +\beamer@subsectionintoc {1}{2}{Exemple jouet}{9}{0}{1} +\beamer@subsectionintoc {1}{3}{Du mode parall\IeC {\`e}le au mode asynchrone}{10}{0}{1} +\beamer@subsectionintoc {1}{4}{Iterations de l'exemple jouet}{11}{0}{1} +\beamer@subsectionintoc {1}{5}{Mode mixe}{15}{0}{1} +\beamer@subsectionintoc {1}{6}{Composantes Connexes de l'exemple jouet}{16}{0}{1} +\beamer@subsectionintoc {1}{7}{R\IeC {\'e}sultats th\IeC {\'e}oriques du mode mixe}{17}{0}{1} +\beamer@subsectionintoc {1}{8}{Exp\IeC {\'e}riences}{18}{0}{1} diff --git a/xpl_iteration_final.pdf b/xpl_iteration_final.pdf new file mode 100644 index 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