From c5afaf6d902cd9097415ecf4de5144991005a2e3 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Fran=C3=A7ois=20Pelletier?= Date: Tue, 27 Feb 2018 23:42:59 -0500 Subject: [PATCH] =?UTF-8?q?notes=20mi-session=20termin=C3=A9es?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- algo_trick.png | Bin 0 -> 34602 bytes ch5hypergraph.png | Bin 0 -> 25305 bytes ch5hypergraph.svg | 161 ++++++++++++++++++++++++++++++++++++++++++++ chapitre4.tex | 135 ++++++++++++++++++++++++++++++++++++- chapitre5.tex | 120 ++++++++++++++++++++++++++++++++- notes_de_cours.tex | 1 + tableau_regular.png | Bin 0 -> 51973 bytes 7 files changed, 415 insertions(+), 2 deletions(-) create mode 100644 algo_trick.png create mode 100644 ch5hypergraph.png create mode 100644 ch5hypergraph.svg create mode 100644 tableau_regular.png diff --git a/algo_trick.png b/algo_trick.png new file mode 100644 index 0000000000000000000000000000000000000000..a32e732c599c3259d26b4fe67309cc81a1b3b686 GIT 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-\section{Chapitre 4} +\section{Chapitre 4: Algorithmes de filtrage pour les contraintes arithmétiques} \label{sec:ch4} +\subsection{Contrainte d'égalité} +\label{sec:ch4egal} + +\begin{itemize} +\item $X=Y, \texttt{dom}(X) = [a,b], \mathtt{dom}(Y) = [c,d]$ +\item Cohérence de domaine: $\texttt{dom}(X) \cap \mathtt{dom}(Y)$ +\item Cohérence de bornes: $[\max(a,c),\min(b,d)]$ +\end{itemize} + +\subsection{Arithmétique des intervalles} +\label{sec:ch4arith} + +\paragraph{Addition} + +\begin{itemize} +\item $X+Y=Z, \texttt{dom}(X) = [a,b], \mathtt{dom}(Y) = [c,d], \mathtt{dom}(Z) = [e,f]$ +\item $\mathtt{dom}(X+Y) = [a,b]+[c,d]=[a+c,b+d]$ +\item $\mathtt{dom}(Z)=[a+c,b+d] \cap [e,f] = [\max(e,a+c),\min(f,b+d)]$ +\item Addition d'une constante: $[a,b]+k = [a+k,b+k]$ +\end{itemize} + +\paragraph{Multiplication par une constante} + +\begin{align} + k[a,b] &= \begin{cases} + [ka,kb], k \geq 0\\ + [kb,ka], k < 0 + \end{cases} +\end{align} + +\paragraph{Soustraction} + +\begin{align} + [a,b]-[c,d] = [a-d,b-c] +\end{align} + +\paragraph{Multiplication et division} + +La multiplication par l'inverse équivaut à une division. + +\begin{align} + [a,b] \times [c,d] &= [\min(ac, ad, bc, bd), \max(ac, ad, bc, bd)]\\ + \frac{1}{[a,b]} &= \begin{cases} + [\frac{1}{b},\frac{1}{a} ], & 0 \notin [a,b]\\ + [\frac{1}{b},\infty ), & a=0 0$ + \item $X_i \leftarrow a_i, \frac{df}{dX_i} < 0$ + \end{itemize} + \item Borne inférieure: + \begin{itemize} + \item $X_i \leftarrow a_i, \frac{df}{dX_i} > 0$ + \item $X_i \leftarrow b_i, \frac{df}{dX_i} < 0$ + \end{itemize} + \end{itemize} +\item On peut utiliser des résultats où la fonction est monotone seulement par rapport à quelques variables. On fixe les bornes inférieures et supérieures et on applique l'arithmétique des intervalles ensuite. On obtient un surensemble davantage restreint. +\end{itemize} + +\paragraph{Filtrer une contrainte arithmétique} + +Exprimer une variable en fonction des autres variables, puis utiliser l'arithmétique des intervalles. $X_i = f(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_n)$. + +Si une variable ne peut être isolée: on crée des contraintes plus petites en utilisant des variables additionnelles. + +Variables entières: \begin{itemize} +\item Calcul des intervalles continus +\item Borne supérieure: fonction plancher +\item Borne inférieure: fonction plafond +\item \textbf{Attention}: Pas de cohérence de bornes. NP-Difficile +\end{itemize} + +\paragraph{Réduction} + +$S=\lbrace a_1,\ldots,a_n \rbrace$ un ensemble d'entiers. On pose $\sum_{a_i\in A}a_i = \sum_{a_i\in S \setminus A}a_i$ Est-ce que $A$ existe ? + +On peut réduire le problème à $a_1X_1+a_2X_2+\ldots+a_nX_n = \frac{1}{2} \sum_{i=1}^{n}a_i, X_i \in \lbrace 0,1 \rbrace$. + +\subsection{Cohérence de domaine} +\label{sec:ch4cohdom} + +La cohérence de domaine est aussi NP-Difficile. +Algorithme pseudo-polynomial (dépend de la taille des coefficients et des cartinalité des domaines): Algorithme de Trick + +\paragraph{Algorithme de Trick} + +\begin{itemize} +\item Filtre $< \leq X_1 + \ldots + X_n \leq u$. Égalité si $l=u$. +\item Graphe de n+1 niveaux. Niveau 0: Un noeud étiqueté 0. +\item On crée un noeud $v+u$ au niveau $i$ pour chaque valeur $u$ au niveau $i-1$ et chaque valeur $u$ du domaine de $X_i$. +\item L'arête prend la valeur $u$. +\item Illustration: \ref{fig:algotrick} +\end{itemize} + +\begin{figure}[H] + \centering + \includegraphics[width=10cm]{algo_trick} + \caption{Algorithme de Trick} + \label{fig:algotrick} +\end{figure} + + %%% Local Variables: %%% mode: latex %%% TeX-master: "notes_de_cours" diff --git a/chapitre5.tex b/chapitre5.tex index 44ce209..857152a 100644 --- a/chapitre5.tex +++ b/chapitre5.tex @@ -1,7 +1,125 @@ -\section{Chapitre 5} +\section{Chapitre 5: Structures traitables} \label{sec:ch5} +\textbf{Structure traitable}: propriétés d'un problème qui garantissent une résolution en temps polynomial. + +\paragraph{Éviter un retour arrière} + +\begin{itemize} +\item Heuristique qui ne fait que de bons choix +\item Filtrage assez fort pour éliminer tous les mauvais choix +\end{itemize} + +\subsection{Graphe de contraintes} + +\begin{itemize} +\item Problème de satisfaction binaire +\item Variable:noeud, Contrainte:arête +\item \textbf{Arbre}: graphe acyclique +\item \textbf{Arbre orienté}: possède une racine, chaque noeud a des enfants et un parent (sauf le noeud racine) +\item Hauteur: + \begin{itemize} + \item feuille: 0 + \item autres: maximum des hauteurs des enfants + 1 + \end{itemize} +\item Théorème: Arbre + Cohérence de domaine = Aucun retour arrière +\item Sous-ensemble de contraintes forment un arbre = filtrage fort. Dans le cas des horaires de travail, la structure en arbre donne des horaires valides. Il reste à valider les autres contraintes. +\end{itemize} + +Modèle avec deux séquences de variables: + +\begin{align} + \mathtt{dom}(X_i) &= \lbrace 0,1 \rbrace \forall X_1, \ldots, X_n\\ + \mathtt{dom}(Y_i) &= \lbrace 0,1,2 \rbrace \forall Y_1, \ldots, Y_n\\ + X_i = 1 &\Leftrightarrow Y_i=1 et Y_i \leq Y_{i+1} +\end{align} + +\paragraph{Hypergraphes de contraintes} + +Tuple $(V,E)$ +\begin{itemize} +\item $V$: ensemble de noeuds: variables du problème +\item $E$: ensemble d'hyperarêtes: portées des contraintes +\item Décomposition de contrainte: arbre arithmétique +\end{itemize} + +\begin{figure}[H] + \centering + \includegraphics[width=6cm]{ch5hypergraph} + \caption{Hypergraphe} + \label{fig:ch5hypergraphe} +\end{figure} + +\subsection{Automate} + +Tuple $(\Sigma,Q,q_0,F,T)$ +\begin{itemize} +\item $\Sigma$ : alphabet +\item $Q$: Ensemble d'états +\item $q_0$: État initial +\item $F \subset Q$: États finaux +\item $T \subset Q \times \Sigma \times Q$: Un ensemble de transitions +\end{itemize} + +Une séquence de caractères $c_1, \ldots, c_n$ est reconnue par l'automate si $\exists q_1, \ldots, q_n t.q. (q_{i-1},c_i,q_i) \in T \wedge q_n \in F$. + +\paragraph{Contrainte \textsc{Regular}} + +\begin{itemize} +\item Comprend une séquence de variable et un automate +\item Un algorithme peut appliquer la cohérence de domaine +\item Peut aussi être encodée avec une structure en forme d'arbre \ref{fig:arbreregular} +\end{itemize} + +\paragraph{Encoder avec des contraintes} + +\begin{figure}[H] + \centering + \includegraphics[width=14cm]{tableau_regular} + \caption{Structure d'arbre pour \textsc{Regular}} + \label{fig:arbreregular} +\end{figure} + +\begin{itemize} +\item $dom(x_i) = \Sigma$ +\item $n+1$ variables $Q_0,\ldots,Q_n$ +\item \begin{itemize} + \item $dom(Q_0)=\lbrace q_o \rbrace$ + \item $dom(Q_i)=Q, 0gMo5=uySHxdHUNHv|h2L=%Rh{7v zG<%6Bnpjv^GfOIq@J}KKXymyyzTA3<+qgAKzR`>oHcCp0K_;yk5r*@4OZVQ8-@LJOjalwS;7MIBK z|NcspJo!iN(evtme>vs+zpsl3s(;+0_}^cs5lFn5YAYQrt%-bNyEkvPCyI4MGoS0J 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