From 34201281c3a282836d85197ff5f2aeb30a51c50e Mon Sep 17 00:00:00 2001
From: Orestis <orestis.malaspinas@pm.me>
Date: Thu, 8 Jun 2023 20:12:20 +0200
Subject: [PATCH] added cours 25

---
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diff --git a/slides/cours_25.md b/slides/cours_25.md
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+---
+title: "Graphes - Plus court chemin (2)"
+date: "2023-06-23"
+---
+
+# Questions
+
+* A quoi sert l'algorithme de Dijkstra?
+
+. . .
+
+* A trouver le plus court chemin entre un sommet, $s$, d'un graphe pondéré et tous les autres sommets.
+* Quelle est la limitation de l'algorithme de Dijkstra?
+
+. . .
+
+* Les poids doivent être positifs.
+* Résumer les étapes de l'algorithme de Dijkstra.
+
+. . .
+
+* `distance[source] = 0`, `ditance[reste]=inf`;
+* enfiler tous les sommets, `distance <=> priorité`;
+* tant qu'il y a des sommets dans la file:
+    * u = défiler;
+    * pour tous les sommets `v` dans le voisinage de `u`;
+    * mettre à jour `distance[v]` (priorité et précédence) si `distance[v] > distance[u] + w(u,v)`.
+
+
+# Plus cours chemin pour toute paire de sommets
+
+## Comment faire pour avoir toutes les paires?
+
+. . .
+
+* Appliquer Dijkstra sur tous les sommets d'origine.
+* Complexité:
+    * Graphe dense: $\mathcal{O}(|V|)\mathcal{O}(|V|^2\log|V|)=\mathcal{O}(|V|^3\log|V|)$.
+    * Graphe peu dense: $\mathcal{O}(|V|)\mathcal{O}(|V|\log|V|)=\mathcal{O}(|V|^2\log|V|)$.
+
+. . .
+
+## Solution alternative: Floyd--Warshall
+
+* Pour toutes paires de sommets $u,v\in V$, trouver le chemin de poids minimal reliant $u$ à $v$.
+* Complexité $\mathcal{O}(|V|^3)$, indiqué pour graphes denses.
+* Fonctionne avec la matrice d'adjacence.
+
+# Algorithme de Floyd--Warshall
+
+## Idée générale
+
+* Soit l'ensemble de sommets $V=\{1, 2, 3, 4, ..., n\}$.
+* Pour toute paire de sommets, $i,j$, on considère tous les chemins passant par les sommets intermédiaires $\in\{1, 2, ..., k\}$ avec $k\leq n$.
+* On garde pour chaque $k$ la plus petite valeur.
+
+## Principe
+
+* A chaque étape, $k$, on vérifie s'il est plus court d'aller de $i$ à $j$ en passant par le sommet $k$.
+* Si à l'étape $k-1$, le coût du parcours est $p$, on vérifie si $p$ est plus petit que $p_1+p_2$, le chemin de $i$ à $k$, et $k$ à $j$ respectivement.
+
+# Algorithme de Floyd--Warshall
+
+## The algorithme
+
+Soit $d_{ij}(k)$ le plus court chemin de $i$ à $j$ passant par les sommets $\in\{1,2,...,k\}$
+
+$$
+d_{ij}(k)=\left\{
+\begin{array}{ll}
+         w(i,j), & \mbox{si } k=0,\\
+         \min(d_{ij}(k-1),d_{ik}(k-1)+d_{kj}(k-1)), & \mbox{sinon}.
+\end{array}
+\right.
+$$
+
+# Algorithme de Floyd--Warshall (exemple)
+
+
+::: columns
+
+:::: column
+
+![Le graphe, $D=w$.](figs/floyd_exemple.png)
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(0)}$ (3min)?
+
+. . .
+
+$$
+D^{(0)}=\begin{bmatrix}
+0      & 2      & 4      & \infty & 3 \\
+2      & 0      & 8      & \infty & 1 \\
+6      & 2      & 0      & 4      & 3 \\
+1      & \infty & \infty & 0      & 5 \\
+\infty & \infty & \infty & 1      & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+
+::: columns
+
+:::: column
+
+## On part de $D^{(0)}$?
+
+$$
+D^{(0)}=\begin{bmatrix}
+0      & 2      & 4      & \infty & 3 \\
+2      & 0      & 8      & \infty & 1 \\
+6      & 2      & 0      & 4      & 3 \\
+1      & \infty & \infty & 0      & 5 \\
+\infty & \infty & \infty & 1      & 0 \\
+\end{bmatrix}
+$$
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(1)}$ (3min)?
+
+. . .
+
+$$
+D^{(0)}=\begin{bmatrix}
+0      & 2          & 4               & \infty & 3 \\
+2      & 0          & \mathbf{6}      & \infty & 1 \\
+6      & 2          & 0               & 4      & 3 \\
+1      & \mathbf{3} & \mathbf{5}      & 0      & \mathbf{4} \\
+\infty & \infty     & \infty          & 1      & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+
+::: columns
+
+:::: column
+
+## On part de $D^{(0)}$
+
+$$
+D^{(0)}=\begin{bmatrix}
+0      & 2      & 4      & \infty & 3 \\
+2      & 0      & 8      & \infty & 1 \\
+6      & 2      & 0      & 4      & 3 \\
+1      & \infty & \infty & 0      & 5 \\
+\infty & \infty & \infty & 1      & 0 \\
+\end{bmatrix}
+$$
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(1)}$ (3min)?
+
+. . .
+
+$$
+D^{(1)}=\begin{bmatrix}
+0      & 2          & 4               & \infty & 3 \\
+2      & 0          & \mathbf{6}      & \infty & 1 \\
+6      & 2          & 0               & 4      & 3 \\
+1      & \mathbf{3} & \mathbf{5}      & 0      & \mathbf{4} \\
+\infty & \infty     & \infty          & 1      & 0 \\
+\end{bmatrix}
+$$
+
+## Exemple
+
+$$
+D_{42}^{(1)}=D_{41}^{(0)}+D_{12}^{(0)}=1+2<\infty.
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+::: columns
+
+:::: column
+
+## On part de $D^{(1)}$
+
+$$
+D^{(1)}=\begin{bmatrix}
+0      & 2          & 4      & \infty & 3 \\
+2      & 0          & 6      & \infty & 1 \\
+6      & 2          & 0      & 4      & 3 \\
+1      & 3          & 5      & 0      & 4 \\
+\infty & \infty     & \infty & 1      & 0 \\
+\end{bmatrix}
+$$
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(2)}$ (3min)?
+
+. . .
+
+$$
+D^{(2)}=\begin{bmatrix}
+0          & 2          & 4      & \infty & 3 \\
+2          & 0          & 6      & \infty & 1 \\
+\mathbf{4} & 2          & 0      & 4      & 3 \\
+1          & 3          & 5      & 0      & 4 \\
+\infty     & \infty     & \infty & 1      & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+::: columns
+
+:::: column
+
+## On part de $D^{(2)}$
+
+$$
+D^{(2)}=\begin{bmatrix}
+0          & 2          & 4      & \infty & 3 \\
+2          & 0          & 6      & \infty & 1 \\
+4          & 2          & 0      & 4      & 3 \\
+1          & 3          & 5      & 0      & 4 \\
+\infty     & \infty     & \infty & 1      & 0 \\
+\end{bmatrix}
+$$
+
+
+::::
+
+:::: column
+
+## Que vaut $D^{(3)}$ (3min)?
+
+. . .
+
+$$
+D^{(3)}=\begin{bmatrix}
+0          & 2          & 4      & \mathbf{8}  & 3 \\
+2          & 0          & 6      & \mathbf{10} & 1 \\
+4          & 2          & 0      & 4           & 3 \\
+1          & 3          & 5      & 0           & 4 \\
+\infty     & \infty     & \infty & 1           & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+::: columns
+
+:::: column
+
+## On part de $D^{(3)}$
+
+$$
+D^{(3)}=\begin{bmatrix}
+0          & 2          & 4      & 8  & 3 \\
+2          & 0          & 6      & 10 & 1 \\
+4          & 2          & 0      & 4  & 3 \\
+1          & 3          & 5      & 0  & 4 \\
+\infty     & \infty     & \infty & 1  & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::: column
+
+## Que vaut $D^{(4)}$ (3min)?
+
+. . .
+
+$$
+D^{(4)}=\begin{bmatrix}
+0          & 2          & 4         & 8  & 3 \\
+2          & 0          & 6         & 10 & 1 \\
+4          & 2          & 0         & 4  & 3 \\
+1          & 3          & 5         & 0  & 4 \\
+\mathbf{2} & \mathbf{4} & \mathbf{6} & 1  & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exemple)
+
+::: columns
+
+:::: column
+
+## On part de $D^{(4)}$
+
+$$
+D^{(4)}=\begin{bmatrix}
+0          & 2          & 4         & 8  & 3 \\
+2          & 0          & 6         & 10 & 1 \\
+4          & 2          & 0         & 4  & 3 \\
+1          & 3          & 5         & 0  & 4 \\
+2          & 4          & 6         & 1  & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::: column
+
+## Que vaut $D^{(5)}$ (3min)?
+
+. . .
+
+$$
+D^{(5)}=\begin{bmatrix}
+0          & 2          & 4         & \mathbf{4} & 3 \\
+2          & 0          & 6         & \mathbf{2} & 1 \\
+4          & 2          & 0         & 4          & 3 \\
+1          & 3          & 5         & 0          & 4 \\
+2          & 4          & 6         & 1          & 0 \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall
+
+## The pseudo-code (10min)
+
+* Quelle structure de données?
+* Quelle initialisation?
+* Quel est le code pour le calcul de la matrice $D$?
+
+# Algorithme de Floyd--Warshall
+
+## The pseudo-code
+
+* Quelle structure de données?
+
+```C
+int distance[n][n];
+```
+
+. . .
+
+* Quelle initialisation?
+
+```C
+matrice ini_floyd_warshall(distance, n, w)
+    pour i de 1 à n
+        pour j de 1 à n
+            distance[i][j] = w(i,j)
+    retourne distance
+```
+
+# Algorithme de Floyd--Warshall
+
+## The pseudo-code
+
+* Quel est le code pour le calcul de la matrice $D$?
+
+```C
+matrice floyd_warshall(distance, n, w)
+    pour k de 1 à n
+        pour i de 1 à n
+            pour j de 1 à n
+                distance[i][j] = min(distance[i][j], 
+                    distance[i][k] + distance[k][j])
+    retourne distance
+```
+
+# Algorithme de Floyd--Warshall
+
+## La matrice de précédence
+
+* On a pas encore vu comment reconstruire le plusc court chemin!
+* On définit, $P_{ij}^{(k)}$, qui est le prédécesseur du sommet $j$ depuis $i$ avec les sommets intermédiaires $\in\{1, 2, ..., k\}$.
+$$
+P^{(0)}_{ij}=\left\{
+\begin{array}{ll}
+         \mbox{vide}, & \mbox{si } i=j\mbox{, ou }w(i,j)=\infty\\
+         i, & \mbox{sinon}.
+\end{array}
+\right.
+$$
+
+* Mise à jour
+$$
+P^{(k)}_{ij}=\left\{
+\begin{array}{ll}
+         P^{(k-1)}_{\mathbf{i}j}, & \mbox{si } d_{ij}^{(k)}\leq d_{ik}^{(k-1)}+d_{kj}^{(k-1)}\\
+         P^{(k-1)}_{\mathbf{k}j}, & \mbox{sinon}.
+\end{array}
+\right.
+$$
+
+. . .
+
+* Moralité: si le chemin est plus court en passant par $k$, alors il faut utiliser son prédécesseur!
+
+# Algorithme de Floyd--Warshall
+
+## La matrice de précédence (pseudo-code, 3min)
+
+. . .
+
+```C
+matrice, matrice floyd_warshall(distance, n, w)
+    pour k de 1 à n
+        pour i de 1 à n
+            pour j de 1 à n
+                n_distance = distance[i][k] + distance[k][j]
+                if n_distance < distance[i][j]
+                    distance[i][j] = n_distance
+                    précédence[i][j] = précédence[k][j]
+    retourne distance, précédence
+```
+
+# Algorithme de Floyd--Warshall (exercice)
+
+
+::: columns
+
+:::: column
+
+![Le graphe, $D=w$.](figs/floyd_exemple.png)
+
+
+::::
+
+:::: column
+
+## Que vaut $P^{(0)}$ (3min)?
+
+. . .
+
+$$
+P^{(0)}=\begin{bmatrix}
+-          & 1          & 1         & -          & 1 \\
+2          & -          & 2         & -          & 2 \\
+3          & 3          & -         & 3          & 3 \\
+4          & -          & -         & -          & 4 \\
+-          & -          & -         & 5          & - \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Algorithme de Floyd--Warshall (exercice)
+
+
+::: columns
+
+:::: column
+
+![Le graphe, $D=w$.](figs/floyd_exemple.png)
+
+
+::::
+
+:::: column
+
+## Que vaut $P^{(5)}$ (10min)?
+
+. . .
+
+$$
+P^{(5)}=\begin{bmatrix}
+-          & 1          & 1         & 5          & 1 \\
+2          & -          & 1         & 5          & 2 \\
+2          & 3          & -         & 3          & 3 \\
+4          & 1          & 1         & -          & 1 \\
+4          & 1          & 1         & 5          & - \\
+\end{bmatrix}
+$$
+
+::::
+
+:::
+
+# Exercice: retrouver le chemin entre 1 et 4 (5min)
+
+$$
+P=\begin{bmatrix}
+-          & 1          & 1         & 5          & 1 \\
+2          & -          & 1         & 5          & 2 \\
+2          & 3          & -         & 3          & 3 \\
+4          & 1          & 1         & -          & 4 \\
+4          & 1          & 1         & 5          & - \\
+\end{bmatrix}
+$$
+
+. . .
+
+## Solution
+
+* Le sommet $5=P_{14}$, on a donc, $5\rightarrow 4$, on veut connaître le prédécesseur de 5.
+* Le sommet $1=P_{15}$, on a donc, $1\rightarrow 5\rightarrow 4$. The end.
+
+# Exercice complet
+
+## Appliquer l'algorithme de Floyd--Warshall au graphe suivant
+
+![The exorcist.](figs/floyd_exercice.png){width=50%}
+
+* Bien indiquer l'état de $D$ et $P$ à chaque étape!
+* Ne pas oublier de faire la matrice d'adjacence évidemment...
+
+# La suite
+
+* Sans transition.... la suite!
+
+# Trouver un réseau électrique pour
+
+![Ces maisons n'ont pas d'électricité.](figs/arbre_couvrant_vide.png)
+
+# Solution: pas optimale
+
+![Le réseau simple, mais nul.](figs/arbre_couvrant_mal.png)
+
+* La longueur totale des câbles est super longue!
+
+# Solution: optimale
+
+![Le meilleur réseau.](figs/arbre_couvrant_bien.png)
+
+# Formalisation: Les arbres couvrants
+
+## Application: minimisation des coûrs
+
+* Équipement d'un lotissement avec des lignes électriques/téléphoniques, des canalisations, ...
+
+. . .
+
+* Pour réduire les coûts, on cherche à minimiser la longueur totale des câbles/tuyaux.
+
+. . .
+
+* Les lignes/tuyaux forment un *arbre couvrant*.
+
+. . .
+
+* La meilleure option est un *arbre couvrant minimal*.
+
+
+# Formalisation: Les arbres couvrants
+
+* Qu'est-ce qu'un arbre couvrant? Des idées? De quel objet on part? Où va-t-on?
+
+. . .
+
+* Un arbre couvrant d'un graphe non-orienté et connexe est:
+    * un arbre inclu dans le graphe qui connecte tous les sommets du graphe.
+
+. . .
+
+![Exemple d'arbres couvrants d'un graphe connexe.](figs/arbre_couvrant_exemples.png)
+
+# Arbres couvrants
+
+* Quels algorithmes que nous avons déjà vus permettent de construire des arbres couvrants?
+
+. . .
+
+* Les parcours en largeur et en profondeur!
+
+. . .
+
+![Graphe, et parcours comme arbres couvrants.](figs/arbres_couvrants_parcours.png)
+
+# Arbres couvrants minimaux
+
+* Un *arbre couvrant minimal* est un sous-graphe d'un graphe non-orienté pondéré $G(V,E)$, tel quel:
+    * C'est un arbre (graphe acyclique);
+    * Il couvre tous les sommets de $G$ et contient $|V|-1$ arêtes;
+    * Le coût total associé aux arêtes de l'arbre est minimum parmi tous les arbres couvrants possibles.
+
+. . .
+
+* Est-il unique?
+
+. . .
+
+* Pas forcément.
+
+# Arbres couvrants minimaux
+
+* Comment générer un arbre couvrant minimal?
+
+![Un graphe, connexe, non-orienté, pondéré, et son arbre couvrant minimal.](figs/arbre_couvrant_minimal_exemple.png)
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## Un exemple
+
+![Le graphe de départ.](figs/prim_0.png)
+
+::::
+
+:::: column
+
+## On part de `e` (au hasard)
+
+![Le sommet `e` est couvert.](figs/prim_1.png)
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## On choisit comment? 
+
+![Quelle arête choisir?](figs/prim_1.png)
+
+. . .
+
+* L'arête la plus courte sortant d'un sommet déjà visité, et entrant dans un sommet non-visité.
+
+::::
+
+:::: column
+
+. . .
+
+## L'arête `e->d`
+
+![Le sommet `d` est couvert.](figs/prim_2.png)
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## On choisit comment? 
+
+![Quelle arête choisir?](figs/prim_2.png)
+
+. . .
+
+* L'arête la plus courte sortant d'un sommet déjà visité, et entrant dans un sommet non-visité.
+
+::::
+
+:::: column
+
+. . .
+
+## L'arête `d->a`
+
+![Le sommet `a` est couvert.](figs/prim_3.png)
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## On choisit comment? 
+
+![Quelle arête choisir?](figs/prim_3.png)
+
+. . .
+
+* L'arête la plus courte sortant d'un sommet déjà visité, et entrant dans un sommet non-visité.
+
+::::
+
+:::: column
+
+. . .
+
+## L'arête `d->c`
+
+![Le sommet `c` est couvert.](figs/prim_4.png)
+
+::::
+
+:::
+
+# Algorithme de Prim
+
+::: columns
+
+:::: column
+
+## On choisit comment? 
+
+![Quelle arête choisir?](figs/prim_4.png)
+
+. . .
+
+* L'arête la plus courte sortant d'un sommet déjà visité, et entrant dans un sommet non-visité.
+
+::::
+
+:::: column
+
+. . .
+
+## L'arête `e->b`
+
+![Le sommet `b` est couvert.](figs/prim_5.png)
+
+::::
+
+:::
+
+* Game over!
+
diff --git a/slides/figs/floyd_exemple.png b/slides/figs/floyd_exemple.png
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-- 
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