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 coups dans un futur proche;
+ * DFS explore peu de coups dans un futur lointain.
diff --git a/slides/figs/Konigsberg_bridges.png b/slides/figs/Konigsberg_bridges.png
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