diff --git a/slides/cours_20.md b/slides/cours_20.md
index a2b3129ab104e5bbead7dab763a46d64595ea9cc..5fbbc844e15c8fe6607c6442350e374f6d462e47 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) {
 
 ![](figs/quad_img_simple_comp.svg)
 
-# 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) {
 
 ![](figs/quad_img_simple_comp_loss.svg)
 
-# 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) {
 
 ![Arbre compressé.](figs/quad_img_simple_comp_avg.svg)
 
-# 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 a3d629a0a1d0d19afe5ef7e16f1b9b3184490ccd..3940c6d559336882800283d028a1be66cc483cce 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
-
-![Suppression de 25.](figs/barbres_ordre2_supp1.svg){width=80%}
-
-. . .
-
-![25 supprimé, on décale juste 27.](figs/barbres_ordre2_supp2.svg){width=80%}
-
-# Les B-arbres: suppression
-
-\footnotesize
-
-## Cas simple
-
-
-![Suppression de 27.](figs/barbres_ordre2_supp2.svg){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?
-
-. . .
-
-![La médiane de la racine descend, fusion de 20 à gauche, et suppression à droite.](figs/barbres_ordre2_supp3.svg){width=60%}
-
-# Les B-arbres: suppression
-
-## Cas moins simple
-
-![Suppression de 5.](figs/barbres_ordre2_supp4.svg){width=60%}
-
-. . .
-
-* Un élément à droite, comment on fait?
-    * Remonter `7`, serait ok si racine, mais... c'est pas forcément.
-    * On redistribue les feuilles.
-
-. . .
-
-![Descente de `3`, remontée médiane des feuilles `2`.](figs/barbres_ordre2_supp5.svg){width=60%}
-
-# Les B-arbres: suppression
-
-\footnotesize
-
-## Cas ultra moins simple
-
-![Suppression de 3.](figs/barbres_ordre2_supp6.svg){width=60%}
-
-. . .
-
-* `7` seul:
-    * Fusionner les feuilles et redistribuer, comment?
-
-. . .
-
-![Descendre `-1`, déplacer `7` à gauche, et décaler les éléments de droite au milieu.](figs/barbres_ordre2_supp7.svg){width=60%}
-
-# Les B-arbres: suppression
-
-## Cas ultra moins simple
-
-![On a pas fini...](figs/barbres_ordre2_supp7.svg){width=60%}
-
-. . .
-
-* `8` est seul, c'est plus un B-arbre :
-    * Fusionner le niveau 2 et redistribuer, comment?
-
-. . .
-
-![Fusionner `8`, `17`, `22` et descendre `12`.](figs/barbres_ordre2_supp8.svg){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!
-
-![Suppression de 8.](figs/barbres_ordre2_supp9.svg){width=60%}
-
-. . .
-
-* On sait comment effacer une valeur d'une feuille, donc?
-
-. . .
-
-![Échanger le `8` avec le plus grand du sous-arbre de gauche.](figs/barbres_ordre2_supp10.svg){width=60%}
-
-* Ensuite?
-
-# Les B-arbres: suppression
-
-## Cas non-feuille!
-
-![Suppression de 8.](figs/barbres_ordre2_supp10.svg){width=60%}
-
-. . .
-
-* On sait comment effacer une valeur d'une feuille!
-
-. . .
-
-![Yaka enlever le 8 de la feuille comme avant!](figs/barbres_ordre2_supp11.svg){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 0000000000000000000000000000000000000000..b834b9496c3d3c5dfb6b6b118596bcef34f68211
--- /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
+
+![Suppression de 25.](figs/barbres_ordre2_supp1.svg){width=80%}
+
+. . .
+
+![25 supprimé, on décale juste 27.](figs/barbres_ordre2_supp2.svg){width=80%}
+
+# Les B-arbres: suppression
+
+\footnotesize
+
+## Cas simple
+
+
+![Suppression de 27.](figs/barbres_ordre2_supp2.svg){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?
+
+. . .
+
+![La médiane de la racine descend, fusion de 20 à gauche, et suppression à droite.](figs/barbres_ordre2_supp3.svg){width=60%}
+
+# Les B-arbres: suppression
+
+## Cas moins simple
+
+![Suppression de 5.](figs/barbres_ordre2_supp4.svg){width=60%}
+
+. . .
+
+* Un élément à droite, comment on fait?
+    * Remonter `7`, serait ok si racine, mais... c'est pas forcément.
+    * On redistribue les feuilles.
+
+. . .
+
+![Descente de `3`, remontée médiane des feuilles `2`.](figs/barbres_ordre2_supp5.svg){width=60%}
+
+# Les B-arbres: suppression
+
+\footnotesize
+
+## Cas ultra moins simple
+
+![Suppression de 3.](figs/barbres_ordre2_supp6.svg){width=60%}
+
+. . .
+
+* `7` seul:
+    * Fusionner les feuilles et redistribuer, comment?
+
+. . .
+
+![Descendre `-1`, déplacer `7` à gauche, et décaler les éléments de droite au milieu.](figs/barbres_ordre2_supp7.svg){width=60%}
+
+# Les B-arbres: suppression
+
+## Cas ultra moins simple
+
+![On a pas fini...](figs/barbres_ordre2_supp7.svg){width=60%}
+
+. . .
+
+* `8` est seul, c'est plus un B-arbre :
+    * Fusionner le niveau 2 et redistribuer, comment?
+
+. . .
+
+![Fusionner `8`, `17`, `22` et descendre `12`.](figs/barbres_ordre2_supp8.svg){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!
+
+![Suppression de 8.](figs/barbres_ordre2_supp9.svg){width=60%}
+
+. . .
+
+* On sait comment effacer une valeur d'une feuille, donc?
+
+. . .
+
+![Échanger le `8` avec le plus grand du sous-arbre de gauche.](figs/barbres_ordre2_supp10.svg){width=60%}
+
+* Ensuite?
+
+# Les B-arbres: suppression
+
+## Cas non-feuille!
+
+![Suppression de 8.](figs/barbres_ordre2_supp10.svg){width=60%}
+
+. . .
+
+* On sait comment effacer une valeur d'une feuille!
+
+. . .
+
+![Yaka enlever le 8 de la feuille comme avant!](figs/barbres_ordre2_supp11.svg){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?
+
+![Les ponts c'est beau. Source: Wikipédia, <https://bit.ly/37h0yOG>](figs/Konigsberg_bridges.png){width=50%}
+
+. . .
+
+* Réponse: ben non!
+
+# Utilisation quotidienne
+
+## Réseau social
+
+![Source, Wikipedia: <https://bit.ly/3kG6cgo>](figs/Social_Network.svg){width=40%}
+
+* Chaque sommet est un individu.
+* Chaque trait une relation d'amitié.
+* Miam, Miam, Facebook.
+
+# Utilisation quotidienne
+
+## Moteurs de recherche
+
+![Source, Wikipedia: <https://bit.ly/3kG6cgo>](figs/PageRanks-Example.svg){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.
+
+![Deux exemples de graphes.](figs/ex_graphes.png)
+
+* 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
+
+![Un graphe non-orienté.](figs/ex_graphe_non_oriente.svg)
+
+
+::::
+
+:::: 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
+
+![Un graphe non-orienté.](figs/ex_graphe_oriente.svg)
+
+
+::::
+
+:::: 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
+
+![Sommet $a$  adjacent à $c$, $c$ adjacent à $a$.](figs/ex_adj_non_or.svg){width=80%}
+
+::::
+
+:::: column
+
+![Sommet $a$  adjacent à $c$.](figs/ex_adj_or.svg){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
+
+![Graphe pondéré orienté (gauche) et non-orienté (droite).](figs/ex_graph_pond.pdf){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
+
+![Illustration d'une chaîne, ou pas chaîne dans un graphe.](figs/ex_graphe_chaine.pdf){width=80%}
+
+# Généralités
+
+## Définition
+
+* Une **chaîne élémentaire** est une chaîne dont tous les sommets sont distincts, sauf les extrémités qui peuvent être égales
+
+## Exemples
+
+![Illustration d'une chaîne élémentaire.](figs/ex_graphe_chaine_elem.pdf){width=80%}
+
+# Généralités
+
+## Définition
+
+* Une **boucle** est une arête $(v,v)$ d'un sommet vers lui-même.
+
+## Exemples
+
+![Illustration d'une boucle.](figs/ex_graphe_boucle.pdf){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
+
+![Graphe connexe. Source, Wikipédia: <https://bit.ly/3yiUzUv>](figs/graphe_connexe.svg){width=80%}
+
+::::
+
+:::: column
+![Graphe non-connexe avec composantes connexes. Source, Wikipédia: <https://bit.ly/3KJB76d>](figs/composantes_connexes.svg){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
+
+![Graphe fortement connexe.](figs/ex_graph_fort_connexe.pdf){width=60%}
+
+::::
+
+:::: column
+
+![Composantes fortement connexes. Source, Wikipédia: <https://bit.ly/3w5PL2l>](figs/composantes_fortement_connexes.svg){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
+
+![Illustration de cycles, ou pas.](figs/ex_graphe_cycle.pdf){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
+
+![Un graphe non-orienté.](figs/ex_graph_adj.pdf){width=80%}
+
+::::
+
+:::: column
+
+
+## Quelle liste d'adjacence? 
+
+. . .
+
+![La liste d'adjacence.](figs/ex_graph_list_adj.pdf)
+
+
+::::
+
+:::
+
+# 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
+
+![Le parcours en largeur.](figs/parcours_larg.pdf){width=80%}
+
+# Exemple
+
+## Étape par étape (blanc non-visité)
+
+![Initialisation.](figs/parcours_larg_0.pdf){width=50%}
+
+## Étape par étape (gris visité)
+
+![On commence en `x`.](figs/parcours_larg_1.pdf){width=50%}
+
+# Exemple
+
+## Étape par étape
+
+![On commence en `x`.](figs/parcours_larg_1.pdf){width=50%}
+
+## Étape par étape (vert à visiter)
+
+![Vister `w`, `t`, `y`.](figs/parcours_larg_2.pdf){width=50%}
+
+# Exemple
+
+## Étape par étape
+
+![Vister `w`, `t`, `y`.](figs/parcours_larg_2.pdf){width=50%}
+
+## Étape par étape
+
+![`w`, `t`, `y` visités. `u`, `s` à visiter.](figs/parcours_larg_3.pdf){width=50%}
+
+# Exemple
+
+## Étape par étape
+
+![`w`, `t`, `y` visités. `u`, `s` à visiter.](figs/parcours_larg_3.pdf){width=50%}
+
+## Étape par étape
+
+![`u`, `s`, visités. `r` à visiter.](figs/parcours_larg_4.pdf){width=50%}
+
+# Exemple
+
+## Étape par étape
+
+![`u`, `s`, visités. `r` à visiter.](figs/parcours_larg_4.pdf){width=50%}
+
+## Étape par étape
+
+![`r` visité. `v` à visiter.](figs/parcours_larg_5.pdf){width=50%}
+
+# Exemple
+
+## Étape par étape
+
+![`r` visité. `v` à visiter.](figs/parcours_larg_5.pdf){width=50%}
+
+## Étape par étape
+
+![The end. Plus rien à visiter!](figs/parcours_larg_6.pdf){width=50%}
+
+# En faisant ce parcours...
+
+
+::: columns
+
+:::: column
+
+## Du parcours de l'arbre
+
+![](figs/parcours_larg_6.pdf){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
+
+![](figs/parcours_larg_0.pdf){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
+
+![Le parcours en profondeur. À quel parcours d'arbre cela ressemble-t-il?](figs/parcours_prof.pdf){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.
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