From c9e5fa2946f611a4b1ec032b3f0a2eb8d737c7cc Mon Sep 17 00:00:00 2001 From: Orestis <orestis.malaspinas@pm.me> Date: Thu, 25 May 2023 11:38:13 +0200 Subject: [PATCH] updated for next part --- slides/cours_20.md | 20 +- slides/cours_22.md | 127 -- slides/cours_23.md | 1013 +++++++++ slides/figs/Konigsberg_bridges.png | Bin 0 -> 32956 bytes slides/figs/PageRanks-Example.svg | 142 ++ slides/figs/Social_Network.svg | 2022 +++++++++++++++++ slides/figs/composantes_connexes.svg | 41 + .../figs/composantes_fortement_connexes.svg | 14 + slides/figs/ex_adj_non_or.svg | 156 ++ slides/figs/ex_adj_or.svg | 171 ++ slides/figs/ex_graphe_non_oriente.svg | 255 +++ slides/figs/ex_graphe_oriente.svg | 113 + slides/figs/ex_graphes.png | Bin 0 -> 26337 bytes slides/figs/graphe_connexe.svg | 39 + 14 files changed, 3976 insertions(+), 137 deletions(-) create mode 100644 slides/cours_23.md create mode 100644 slides/figs/Konigsberg_bridges.png create mode 100644 slides/figs/PageRanks-Example.svg create mode 100644 slides/figs/Social_Network.svg create mode 100644 slides/figs/composantes_connexes.svg create mode 100644 slides/figs/composantes_fortement_connexes.svg create mode 100644 slides/figs/ex_adj_non_or.svg create mode 100644 slides/figs/ex_adj_or.svg create mode 100644 slides/figs/ex_graphe_non_oriente.svg create mode 100644 slides/figs/ex_graphe_oriente.svg create mode 100644 slides/figs/ex_graphes.png create mode 100644 slides/figs/graphe_connexe.svg diff --git a/slides/cours_20.md b/slides/cours_20.md index a2b3129..5fbbc84 100644 --- a/slides/cours_20.md +++ b/slides/cours_20.md @@ -198,7 +198,7 @@ void rotate(node *qt) { } ``` -# Compression sans perte (1/N) +# Compression sans perte (1/5) ## Idée générale @@ -216,7 +216,7 @@ void rotate(node *qt) { * Comment faire? -# Compression sans perte (2/N) +# Compression sans perte (2/5) ## Que devient l'arbre suivant? @@ -228,7 +228,7 @@ void rotate(node *qt) {  -# Compression sans perte (3/N) +# Compression sans perte (3/5) * Si un nœud a tous ses enfants égaux: * Donner la valeur au nœud, @@ -251,7 +251,7 @@ rien compression_sans_pertes(arbre) detruire_enfants(arbre) ``` -# Compression sans perte (4/N) +# Compression sans perte (4/5) \footnotesize @@ -279,7 +279,7 @@ void lossless_compression(node *qt) { } ``` -# Compression sans perte (5/N) +# Compression sans perte (5/5) \footnotesize @@ -305,7 +305,7 @@ bool last_value(node *qt, int *val) { ``` -# Compression avec perte (1/N) +# Compression avec perte (1/5) ## Idée générale @@ -323,7 +323,7 @@ bool last_value(node *qt, int *val) { * On enlève si l'écart à la moyenne est "petit"? -# Compression avec perte (2/N) +# Compression avec perte (2/5) ## Que devient l'arbre suivant si l'écart est petit? @@ -335,7 +335,7 @@ bool last_value(node *qt, int *val) {  -# Compression avec perte (3/N) +# Compression avec perte (3/5) ## Comment mesurer l'écart à la moyenne? @@ -362,7 +362,7 @@ bool last_value(node *qt, int *val) { * Plus $\theta$ est grand, plus l'image sera compressée. -# Compression avec perte (4/N) +# Compression avec perte (4/5) ## Que devient l'arbre avec $\theta=0.5$? @@ -372,7 +372,7 @@ bool last_value(node *qt, int *val) {  -# Compression avec perte (6/N) +# Compression avec perte (5/5) ## Modifications sur la structure de données? diff --git a/slides/cours_22.md b/slides/cours_22.md index a3d629a..3940c6d 100644 --- a/slides/cours_22.md +++ b/slides/cours_22.md @@ -325,134 +325,7 @@ page ajouter_niveau(page, element) retourne tmp ``` -# Les B-arbres: suppression -## Cas simplissime - -{width=80%} - -. . . - -{width=80%} - -# Les B-arbres: suppression - -\footnotesize - -## Cas simple - - -{width=60%} - -. . . - -* On retire 27, mais.... - * Chaque page doit avoir au moins 2 éléments. - * On doit déplacer des éléments dans une autre feuille! Mais comment? - -. . . - -{width=60%} - -# Les B-arbres: suppression - -## Cas moins simple - -{width=60%} - -. . . - -* Un élément à droite, comment on fait? - * Remonter `7`, serait ok si racine, mais... c'est pas forcément. - * On redistribue les feuilles. - -. . . - -{width=60%} - -# Les B-arbres: suppression - -\footnotesize - -## Cas ultra moins simple - -{width=60%} - -. . . - -* `7` seul: - * Fusionner les feuilles et redistribuer, comment? - -. . . - -{width=60%} - -# Les B-arbres: suppression - -## Cas ultra moins simple - -{width=60%} - -. . . - -* `8` est seul, c'est plus un B-arbre : - * Fusionner le niveau 2 et redistribuer, comment? - -. . . - -{width=40%} - -. . . - -* La profondeur a diminué de 1. - -# Les B-arbres: suppression - -## Algorithme pour les feuilles! - -* Si la clé est supprimée d'une feuille: - * Si on a toujours `n` (ordre de l'arbre) clés dans la feuille on décale simplement les clés. - * Sinon on combine (récursivement) avec le nœud voisin et on descend la clé médiane. - -# Les B-arbres: suppression - -## Cas non-feuille! - -{width=60%} - -. . . - -* On sait comment effacer une valeur d'une feuille, donc? - -. . . - -{width=60%} - -* Ensuite? - -# Les B-arbres: suppression - -## Cas non-feuille! - -{width=60%} - -. . . - -* On sait comment effacer une valeur d'une feuille! - -. . . - -{width=60%} - -# Les B-arbres: suppression - -## Algorithme pour les non-feuilles! - -* Si la clé est supprimée d'une page qui n'est pas une feuille: - * On échange la valeur avec la valeur de droite de la page de gauche - * On supprime comme pour une feuille! - -## Et maintenant des exercices par millions! diff --git a/slides/cours_23.md b/slides/cours_23.md new file mode 100644 index 0000000..b834b94 --- /dev/null +++ b/slides/cours_23.md @@ -0,0 +1,1013 @@ +--- +title: "Graphes - Généralités" +date: "2023-05-24" +patat: + eval: + tai: + command: fish + fragment: false + replace: true + ccc: + command: fish + fragment: false + replace: true + images: + backend: auto +--- + + +# Les B-arbres: suppression + +## Cas simplissime + +{width=80%} + +. . . + +{width=80%} + +# Les B-arbres: suppression + +\footnotesize + +## Cas simple + + +{width=60%} + +. . . + +* On retire 27, mais.... + * Chaque page doit avoir au moins 2 éléments. + * On doit déplacer des éléments dans une autre feuille! Mais comment? + +. . . + +{width=60%} + +# Les B-arbres: suppression + +## Cas moins simple + +{width=60%} + +. . . + +* Un élément à droite, comment on fait? + * Remonter `7`, serait ok si racine, mais... c'est pas forcément. + * On redistribue les feuilles. + +. . . + +{width=60%} + +# Les B-arbres: suppression + +\footnotesize + +## Cas ultra moins simple + +{width=60%} + +. . . + +* `7` seul: + * Fusionner les feuilles et redistribuer, comment? + +. . . + +{width=60%} + +# Les B-arbres: suppression + +## Cas ultra moins simple + +{width=60%} + +. . . + +* `8` est seul, c'est plus un B-arbre : + * Fusionner le niveau 2 et redistribuer, comment? + +. . . + +{width=40%} + +. . . + +* La profondeur a diminué de 1. + +# Les B-arbres: suppression + +## Algorithme pour les feuilles! + +* Si la clé est supprimée d'une feuille: + * Si on a toujours `n` (ordre de l'arbre) clés dans la feuille on décale simplement les clés. + * Sinon on combine (récursivement) avec le nœud voisin et on descend la clé médiane. + +# Les B-arbres: suppression + +## Cas non-feuille! + +{width=60%} + +. . . + +* On sait comment effacer une valeur d'une feuille, donc? + +. . . + +{width=60%} + +* Ensuite? + +# Les B-arbres: suppression + +## Cas non-feuille! + +{width=60%} + +. . . + +* On sait comment effacer une valeur d'une feuille! + +. . . + +{width=60%} + +# Les B-arbres: suppression + +## Algorithme pour les non-feuilles! + +* Si la clé est supprimée d'une page qui n'est pas une feuille: + * On échange la valeur avec la valeur de droite de la page de gauche + * On supprime comme pour une feuille! + +## Et maintenant des exercices par millions! + +# Les graphes! Historique + +**Un mini-peu d'histoire...** + +## L. Euler et les 7 ponts de Koenigsberg: + +* Existe-t-il une promenade sympa, passant **une seule fois** par les 7 ponts et revenant au point de départ? + +{width=50%} + +. . . + +* Réponse: ben non! + +# Utilisation quotidienne + +## Réseau social + +{width=40%} + +* Chaque sommet est un individu. +* Chaque trait une relation d'amitié. +* Miam, Miam, Facebook. + +# Utilisation quotidienne + +## Moteurs de recherche + +{width=40%} + +* Sommet est un site. +* Liens sortants; +* Liens entrants; +* Notion d'importance d'un site: combien de liens entrants, pondérés par l'importance du site. +* Miam, Miam, Google (PageRank). + +# Introduction + +## Définition, plus ou moins + +* Un graphe est un ensemble de sommets, reliés par des lignes ou des flèches. + + + +* Des sommets (numérotés 1 à 6); +* Connectés ou pas par des traits ou des flèches! + +# Généralités + +## Définitions + +* Un **graphe** $G(V, E)$ est constitué + * $V$: un ensemble de sommets; + * $E$: un ensemble d'arêtes. +* Une **arête** relie une **paire** de sommets de $V$. + +## Remarques + +* Il y a **au plus** une arête $E$ par paire de sommets de $V$. +* La **complexité** d'un algorithme dans un graphe se mesure en terme de $|E|$ et $|V|$, le nombre d'éléments de $E$ et $V$ respectivement. + +# Généralités + +## Notations + +* Une arête d'un graphe **non-orienté** est représentée par une paire **non-ordonnée** $(u,v)=(v,u)$, avec $u,v\in V$. +* Les arêtes ne sont pas orientées dans un graphe non-orienté (elles sont bi-directionnelles, peuvent être parcourues dans n'importe quel ordre). + +## Exemple + + +::: columns + +:::: column + + + + +:::: + +:::: column + +## Que valent $V$, $|V|$, $E$, et $|E|$? + +. . . + +\begin{align*} +V&=\{1, 2, 3, 4\},\\ +|V|&=4,\\ +E&=\{(1,2),(2,3),(2,4),(4,1)\},\\ +|E|&=4. +\end{align*} + +:::: + +::: + +# Généralités + +## Notations + +* Une arête d'un graphe **orienté** est représentée par une paire **ordonnée** $(u,v)\neq(v,u)$, avec $u,v\in V$. +* Les arêtes sont orientées dans un graphe orienté (étonnant non?). + +## Exemple + + +::: columns + +:::: column + + + + +:::: + +:::: column + +## Que valent $V$, $|V|$, $E$, et $|E|$? + +. . . + +\begin{align*} +V&=\{1, 2, 3, 4\},\\ +|V|&=4,\\ +E&=\{(1,2),(2,3),(2,4),(4,1),(4,2)\},\\ +|E|&=5. +\end{align*} + +:::: + +::: + +# Généralités + +## Définition + +* Le somme $v$ est **adjacent** au sommet $u$, si et seulement si $(u,v)\in E$; +* Si un graphe non-orienté contient une arête $(u,v)$, $v$ est adjacent à $u$ et $u$ et adjacent à $v$. + +## Exemple + +::: columns + +:::: column + +{width=80%} + +:::: + +:::: column + +{width=80%} + +:::: + +::: + +# Généralités + +## Définition + +* Un **graphe pondéré** ou **valué** est un graphe dont chaque arête a un + poids associé, habituellement donné par une fonction de pondération $w:E\rightarrow\mathbb{R}$. + +## Exemples + +{width=80%} + + +# Généralités + +## Définition + +* Dans un graphe $G(V,E)$, une **chaîne** reliant un sommet $u$ à un sommet $v$ est une suite d'arêtes entre les sommets, $w_0$, $w_1$, ..., $w_k$, telles que +$$ +(w_i, w_{i+1})\in E,\quad u=w_0,\quad v=w_k,\quad \mbox{pour }0\leq i< k, +$$ +avec $k$ la longueur de la chaîne (le nombre d'arêtes du chemin). + +## Exemples + +{width=80%} + +# Généralités + +## Définition + +* Une **chaîne élémentaire** est une chaîne dont tous les sommets sont distincts, sauf les extrémités qui peuvent être égales + +## Exemples + +{width=80%} + +# Généralités + +## Définition + +* Une **boucle** est une arête $(v,v)$ d'un sommet vers lui-même. + +## Exemples + +{width=40%} + +# Généralités + +## Définition + +* Un graphe non-orienté est dit **connexe**, s'il existe un chemin reliant n'importe quelle paire de sommets distincts. + +## Exemples + +\ + +::: columns + +:::: column + +{width=80%} + +:::: + +:::: column +{width=60%} + +:::: + +::: + +# Généralités + +## Définition + +* Un graphe orienté est dit **fortement connexe**, s'il existe un chemin reliant n'importe quelle paire de sommets distincts. + +## Exemples + +\ + +::: columns + +:::: column + +{width=60%} + +:::: + +:::: column + +{width=100%} + +:::: + +::: + +# Généralités + +## Définition + +* Un **cycle** dans un graphe *non-orienté* est une chaîne de longueur $\leq 3$ telle que le 1er sommet de la chaîne est le même que le dernier, et dont les arêtes sont distinctes. +* Pour un graphe *orienté* on parle de **circuit**. +* Un graphe sans cycles est dit **acyclique**. + +## Exemples + +{width=100%} + +# Question de la mort + +* Qu'est-ce qu'un graphe connexe acyclique? + +. . . + +* Un arbre! + +# Représentations + +* La complexité des algorithmes sur les graphes s'expriment en fonction du nombre de sommets $V$, et du nombre d'arêtes $E$: + * Si $|E|\sim |V|^2$, on dit que le graphe est **dense**. + * Si $|E|\sim |V|$, on dit que le graphe est **peu dense**. +* Selon qu'on considère des graphes denses ou peu denses, différentes structure de données peuvent être envisagées. + +## Question + +* Comment peut-on représenter un graphe informatiquement? Des idées (réflexion de quelques minutes)? + +. . . + +* Matrice/liste d'adjacence. + +# Matrice d'adjacence + +* Soit le graphe $G(V,E)$, avec $V=\{1, 2, 3, ..., n\}$; +* On peut représenter un graphe par une **matrice d'adjacence**, $A$, de dimension $n\times n$ définie par +$$ +A_{ij}=\left\{ \begin{array}{ll} + 1 & \mbox{si } i,j\in E,\\ + 0 & \mbox{sinon}. + \end{array} \right. +$$ + + +::: columns + +:::: column + +## Exemple + +```{.mermaid format=pdf width=400 loc=figs/} +graph LR; + 1---2; + 1---4; + 2---5; + 4---5; + 5---3; +``` + +:::: + +:::: column + +\footnotesize + +## Quelle matrice d'adjacence? + +. . . + +``` + || 1 | 2 | 3 | 4 | 5 +===||===|===|===|===|=== + 1 || 0 | 1 | 0 | 1 | 0 +---||---|---|---|---|--- + 2 || 1 | 0 | 0 | 0 | 1 +---||---|---|---|---|--- + 3 || 0 | 0 | 0 | 0 | 1 +---||---|---|---|---|--- + 4 || 1 | 0 | 0 | 0 | 1 +---||---|---|---|---|--- + 5 || 0 | 1 | 1 | 1 | 0 +``` + +:::: + +::: + +# Matrice d'adjacence + +## Remarques + +* Zéro sur la diagonale. +* La matrice d'un graphe non-orienté est symétrique + +$$ +A_{ij}=A_{ji}, \forall i,j\in[1,n] +$$. + +::: columns + +:::: column + +```{.mermaid format=pdf width=400 loc=figs/} +graph LR; + 1---2; + 1---4; + 2---5; + 4---5; + 5---3; +``` + +:::: + +:::: column + +\footnotesize + +``` + || 1 | 2 | 3 | 4 | 5 +===||===|===|===|===|=== + 1 || 0 | 1 | 0 | 1 | 0 +---||---|---|---|---|--- + 2 || 1 | 0 | 0 | 0 | 1 +---||---|---|---|---|--- + 3 || 0 | 0 | 0 | 0 | 1 +---||---|---|---|---|--- + 4 || 1 | 0 | 0 | 0 | 1 +---||---|---|---|---|--- + 5 || 0 | 1 | 1 | 1 | 0 +``` + +:::: + +::: + +# Matrice d'adjacence + +* Pour un graphe orienté (digraphe) + +::: columns + +:::: column + +## Exemple + +```{.mermaid format=pdf width=400 loc=figs/} +graph LR; + 2-->1; + 1-->4; + 2-->5; + 5-->2; + 4-->5; + 5-->3; +``` + +:::: + +:::: column + +\footnotesize + +## Quelle matrice d'adjacence? + +. . . + +``` + || 1 | 2 | 3 | 4 | 5 +===||===|===|===|===|=== + 1 || 0 | 0 | 0 | 1 | 0 +---||---|---|---|---|--- + 2 || 1 | 0 | 0 | 0 | 1 +---||---|---|---|---|--- + 3 || 0 | 0 | 0 | 0 | 0 +---||---|---|---|---|--- + 4 || 0 | 0 | 0 | 0 | 1 +---||---|---|---|---|--- + 5 || 0 | 1 | 1 | 0 | 0 +``` + +:::: + +::: + +* La matrice d'adjacence n'est plus forcément symétrique +$$ +A_{ij}\neq A_{ji}. +$$ + +# Stockage + +* Quel est l'espace nécessaire pour stocker une matrice d'adjacence pour un graphe orienté? + +. . . + +* $\mathcal{O}(|V|^2)$. +* Quel est l'espace nécessaire pour stocker une matrice d'adjacence pour un graphe non-orienté? + +. . . + +* $\mathcal{O}(|V|-1)|V|/2$. + +# Considérations d'efficacité + +* Dans quel type de graphes la matrice d'adjacence est utile? + +. . . + +* Dans les graphes denses. +* Pourquoi? + +. . . + +* Dans les graphes peu denses, la matrice d'adjacence est essentiellement composée de `0`. + +## Remarque + +* Dans la majorité des cas, les grands graphes sont peu denses. +* Comment représenter un graphe autrement? + +# La liste d'adjacence (non-orienté) + +* Pour chaque sommet $v\in V$, stocker les sommets adjacents à $v$- +* Quelle structure de données pour la liste d'adjacence? + +. . . + +* Tableau de liste chaînée, vecteur (tableau dynamique), etc. + + +::: columns + +:::: column + +## Exemple + +{width=80%} + +:::: + +:::: column + + +## Quelle liste d'adjacence? + +. . . + + + + +:::: + +::: + +# La liste d'adjacence (orienté) + + +::: columns + +:::: column + +## Quelle liste d'adjacence pour... + +* Matrix (2min) + +```{.mermaid format=pdf width=400 loc=figs/} +graph LR; + 0-->1; + 0-->2; + 1-->2; + 3-->0; + 3-->1; + 3-->2; +``` + +:::: + +:::: column + +``` + + + + + + + + + + + +``` + + +:::: + +::: + +# Complexité + +## Stockage + +* Quelle espace est nécessaire pour stocker une liste d'adjacence (en fonction de $|E|$ et $|V|$)? + +. . . + +$$ +\mathcal{O}(|E|) +$$ + +* Pour les graphes *non-orientés*: $\mathcal{O}2|E|$. +* Pour les graphes *orientés*: $\mathcal{O}|E|$. + +## Définition + +* Le **degré** d'un sommet $v$, est le nombre d'arêtes incidentes du sommet (pour les graphes orientés on a un degré entrant ou sortant). +* Comment on retrouve le degré de chaque sommet avec la liste d'adjacence? + +. . . + +* C'est la longueur de la liste chaînée. + + +# Parcours + +* Beaucoup d'applications nécessitent de parcourir des graphes: + * Trouver un chemin d'un sommet à un autre; + * Trouver si le graphe est connexe; +* Il existe *deux* parcours principaux: + * en largeur (Breadth-First Search); + * en profondeur (Depth-First Search). +* Ces parcours créent *un arbre* au fil de l'exploration (si le graphe est non-connexe cela crée une *forêt*, un ensemble d'arbres). + +# Illustration: parcours en largeur + +{width=80%} + +# Exemple + +## Étape par étape (blanc non-visité) + +{width=50%} + +## Étape par étape (gris visité) + +{width=50%} + +# Exemple + +## Étape par étape + +{width=50%} + +## Étape par étape (vert à visiter) + +{width=50%} + +# Exemple + +## Étape par étape + +{width=50%} + +## Étape par étape + +{width=50%} + +# Exemple + +## Étape par étape + +{width=50%} + +## Étape par étape + +{width=50%} + +# Exemple + +## Étape par étape + +{width=50%} + +## Étape par étape + +{width=50%} + +# Exemple + +## Étape par étape + +{width=50%} + +## Étape par étape + +{width=50%} + +# En faisant ce parcours... + + +::: columns + +:::: column + +## Du parcours de l'arbre + +{width=100%} + +:::: + +:::: column + +## Quel arbre est créé par le parcours (2min)? + +. . . + +```{.mermaid format=pdf width=400 loc=figs/} +graph LR; + 0[x]-->1[w]; + 0-->2[t]; + 0-->3[y]; + 2-->9[u]; + 1-->4[s]; + 4-->5[r]; + 5-->6[v]; +``` + +:::: + +::: + +## Remarques + +* Le parcours dépend du point de départ dans le graphe. +* L'arbre sera différent en fonction du noeud de départ, et de l'ordre de parcours des voisins d'un noeud. + +# Le parcours en largeur + +## L'algorithme, idée générale (3min, matrix)? + +. . . + +```C +v = un sommet du graphe +i = 1 +pour sommet dans graphe et sommet non-visité + visiter(v, sommet, i) // marquer sommet à distance i visité + i += 1 +``` + +## Remarque + +* `i` est la distance de plus cours chemin entre `v` et les sommets en cours de visite. + + +# Le parcours en largeur + +## L'algorithme, pseudo-code (3min, matrix)? + +* Comment garder la trace de la distance? + +. . . + +* Utilisation d'une **file** + +. . . + +```C +initialiser(graphe) // tous sommets sont non-visités +file = visiter(sommet, vide) // sommet est un sommet du graphe au hasard +tant que !est_vide(file) + v = défiler(file) + file = visiter(v, file) +``` + +## Que fait visiter? + +``` +file visiter(sommet, file) + sommet = visité + pour w = chaque arête de sommet + si w != visité + file = enfiler(file, w) + retourne file +``` + +# Exercice (5min) + +## Appliquer l'algorithme sur le graphe + +{width=50%} + +* En partant de `v`, `s`, ou `u` (par colonne de classe). +* Bien mettre à chaque étape l'état de la file. + +# Complexité du parcours en largeur + +## Étape 1 + +* Extraire un sommet de la file; + +## Étape 2 + +* Traîter tous les sommets adjacents. + +## Quelle est la complexité? + +. . . + +* Étape 1: $\mathcal{O}(|V|)$, +* Étape 2: $\mathcal{O}(2|E|)$, +* Total: $\mathcal{O}(|V| + |2|E|)$. + +# Exercice + +* Établir la liste d'adjacence et appliquer l'algorithme de parcours en largeur au graphe + +```{.mermaid format=pdf width=400 loc=figs/} +graph LR; + 1---2; + 1---3; + 1---4; + 2---3; + 2---6; + 3---6; + 3---4; + 3---5; + 4---5; +``` + + +# Illustration: parcours en profondeur + +{width=80%} + +# Parcours en profondeur + +## Idée générale + +* Initialiser les sommets comme non-lus +* Visiter un sommet +* Pour chaque sommet visité, on visite un sommet adjacent s'il est pas encore visité récursivement. + +## Remarque + +* La récursivité est équivalent à l'utilisation d'une **pile**. + +# Parcours en profondeur + +## Pseudo-code (5min) + +. . . + +```C +initialiser(graphe) // tous sommets sont non-visités +pile = visiter(sommet, vide) // sommet est un sommet du graphe au hasard +tant que !est_vide(pile) + v = dépiler(pile) + pile = visiter(v, pile) +``` + +## Que fait visiter? + +. . . + +```C +pile visiter(sommet, pile) + sommet = visité + pour w = chaque arête de sommet + si w != visité + pile = empiler(pile, w) + retourne pile +``` + + +# Exercice + +* Établir la liste d'adjacence et appliquer l'algorithme de parcours en profondeur au graphe + +```{.mermaid format=pdf width=400 loc=figs/} +graph LR; + 1---2; + 1---3; + 1---4; + 2---3; + 2---6; + 3---6; + 3---4; + 3---5; + 4---5; +``` + +# Interprétation des parcours + +* Un graphe vu comme espace d'états (sommet: état, arête: action); + * Labyrinthe; + * Arbre des coups d'un jeu. + +. . . + +* BFS (Breadth-First) ou DFS (Depth-First) parcourent l'espace des états à la recherche du meilleur mouvement. + * Les deux parcourent *tout* l'espace; + * Mais si l'arbre est grand, l'espace est gigantesque! + +. . . + +* Quand on a un temps limité + * BFS explore beaucoup de 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