Représentation des graphes

Les différentes représentations des graphes

Afin de le représenter en Python, on peut utiliser une matrice d'adjacence ou de listes de successeurs.

La matrice d'adjacence est une matrice carrée dont chaque ligne et chaque colonne correspond à un sommet. Il y a un \(1\) sur la ligne du sommet \(i\) et de la colonne \(j\) si et seulement s'il y a un arc reliant \(i\) à \(j\).

Le dictionnaire de successeurs associe à chaque sommet \(i\) la liste des sommets \(j\) tels qu'il y a un arc entre \(i\) et \(j\). Voici les 3 représentations d'un même graphe.

En Python, on peut utiliser des tableaux à deux dimensions pour les matrices et des dictionnaires associant à chaque sommet des listes pour les listes de successeurs.

Matrice d'adjacence

graphe1_matrice = [[0, 1, 1, 0],
                   [0, 0, 1, 0],
                   [1, 1, 0, 0],
                   [0, 1, 1, 0]]

Dictionnaire de successeurs

graphe1_successeurs = {0: [1, 2],
                       1: [2],
                       2: [0, 1],
                       3: [1, 2]}

Exercice 1

On considère le graphe suivant, que l'on appellera graphe2 :

Écrire les représentations du graphe sous forme de matrice et de dictionnaire de successeurs :

MatriceDictionnaire

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###(Dés-)Active le code après la ligne # Tests (insensible à la casse)
(Ctrl+I)

Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)

Exécuter le code
(Ctrl+S)Valider
(Ctrl+Enter)TéléchargerTéléverserRéinitialiser l'éditeurSauvegarder dans le navigateurÉvaluations restantes : 5/5

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Exercice 2

Écrire le code de la fonction matrice_vers_dico qui permet de passer d'une matrice d'adjacence à un dictionnaire de successeurs.

Exemple

>>> matrice_vers_dico(graphe1_matrice)
{0: [1, 2], 1: [2], 2: [0, 1], 3: [1, 2]}

Version videVersion à compléter

###(Dés-)Active le code après la ligne # Tests (insensible à la casse)
(Ctrl+I)

Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)

Exécuter le code
(Ctrl+S)Valider
(Ctrl+Enter)TéléchargerTéléverserRéinitialiser l'éditeurSauvegarder dans le navigateurÉvaluations restantes : 5/5

43908.43996.43997.43980.43993.43985.43988.43979.43928.43995.43988.43993.43979.43979.43909.43930.43980.43985.43976.43930.43910.43954.43908.43979.43981.43989.43989.43993.43978.43969.43910.44011.43991.43988.43981.43980.43985.43991.43990.43908.43927.43979.43981.43989.43989.43993.43978.43969.43910.43954.43908.43996.43985.43982.43928.43989.43993.43978.43987.43996.43991.43983.43990.43909.43930.43913.43930.43928.43979.43980.43969.43988.43997.43909.43930.43989.43993.43978.43999.43985.43990.43906.43913.43926.43919.43997.43989.43928.43913.43997.43989.43930.43928.43910.43954.43908.43996.43985.43982.43928.43995.43988.43993.43979.43979.43909.43930.43984.43985.43999.43984.43988.43985.43999.43984.43980.43930.43910.43908.43980.43993.43994.43988.43997.43928.43995.43988.43993.43979.43979.43909.43930.43984.43985.43999.43984.43988.43985.43999.43984.43980.43980.43993.43994.43988.43997.43930.43910.43908.43980.43978.43910.43908.43980.43984.43928.43995.43991.43988.43979.43976.43993.43990.43909.43930.43914.43930.43928.43995.43988.43993.43979.43979.43909.43930.43998.43985.43988.43997.43990.43993.43989.43997.43930.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43998.43985.43988.43997.43990.43993.43989.43997.43930.43910.90037.43928.44008.43978.43991.43976.43991.43979.43985.43980.43985.43991.43990.43928.43996.43997.43928.43995.43991.43978.43978.43997.43995.43980.43985.43991.43990.43908.43927.43979.43976.43993.43990.43910.43908.43927.43980.43984.43910.43908.43927.43980.43978.43910.43908.43980.43978.43910.43908.43980.43996.43928.43995.43988.43993.43979.43979.43909.43930.43988.43985.43990.43997.43990.43991.43979.43930.43910.43908.43996.43985.43982.43928.43995.43988.43993.43979.43979.43909.43930.43988.43985.43990.43997.43990.43991.43996.43985.43982.43930.43910.43908.43976.43978.43997.43910.43908.43979.43976.43993.43990.43910.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43913.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43914.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43915.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43916.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43917.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43918.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43919.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43904.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43905.43908.43927.43979.43976.43993.43990.43910.43908.43927.43976.43978.43997.43910.43908.43927.43996.43985.43982.43910.43908.43927.43980.43996.43910.43908.43980.43996.43928.43995.43988.43993.43979.43979.43909.43930.43995.43991.43996.43997.43930.43910.43908.43996.43985.43982.43910.43908.43976.43978.43997.43910.43908.43979.43976.43993.43990.43910.43908.43927.43979.43976.43993.43990.43910.43908.43995.43991.43996.43997.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43996.43997.43998.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43998.43930.43910.43989.43993.43980.43978.43985.43995.43997.44007.43982.43997.43978.43979.44007.43996.43985.43995.43991.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43989.43993.43980.43978.43985.43995.43997.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43906.43908.43927.43979.43976.43993.43990.43910.43954.43928.43928.43928.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43979.43981.43995.43995.43997.43979.43979.43997.43981.43978.43979.43908.43927.43979.43976.43993.43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###(Dés-)Active le code après la ligne # Tests (insensible à la casse)
(Ctrl+I)

Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)

Exécuter le code
(Ctrl+S)Valider
(Ctrl+Enter)TéléchargerTéléverserRéinitialiser l'éditeurSauvegarder dans le navigateurÉvaluations restantes : 5/5

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Exercice 3

Écrire le code de la fonction dico_vers_matrice qui permet de passer d'un dictionnaire de successeurs à une matrice. On supposera que tous les sommets du graphe apparaissent comme clef du dictionnaire, même s'ils n'ont pas de successeurs.

Exemple

>>> dico_vers_matrice(graphe1_matrice)
[[0, 1, 1, 0], [0, 0, 1, 0], [1, 1, 0, 0], [0, 1, 1, 0]]

Explications de l'algorithme

Au début, on crée une matrice contenant uniquement des 0 puis on rajoute les 1 pour les cases correspondant à la ligne sommet et la colonne voisin pour tous les voisins de chaque sommet.

Version videVersion à compléter

###(Dés-)Active le code après la ligne # Tests (insensible à la casse)
(Ctrl+I)

Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)

Exécuter le code
(Ctrl+S)Valider
(Ctrl+Enter)TéléchargerTéléverserRéinitialiser l'éditeurSauvegarder dans le navigateurÉvaluations restantes : 5/5

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###(Dés-)Active le code après la ligne # Tests (insensible à la casse)
(Ctrl+I)

Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)

Exécuter le code
(Ctrl+S)Valider
(Ctrl+Enter)TéléchargerTéléverserRéinitialiser l'éditeurSauvegarder dans le navigateurÉvaluations restantes : 5/5

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Liste d'adjacence

Lorsque les sommets d'un graphe sont numérotés à partir de 0, on peut le représenter à l'aide d'une liste d'adjacence. Pour chaque indice, la liste d'adjacence contient la liste des successeurs. Si on a le dictionnaire de successeurs, le passage à la liste d'adjacence est très simple :

Exemple avec le graphe1

graphe1_successeurs = {0: [1, 2],
                       1: [2],
                       2: [0, 1],
                       3: [1, 2]}
graphe1_adjacence = [[1, 2], [2], [0, 1], [1, 2]]

Exercice 4

Écrire le code de la fonction dico_vers_adjacence qui prend en paramètre un dictionnaire de successeurs et qui renvoie la liste d'adjacence. On supposera que tous les sommets du graphe apparaissent comme clef du dictionnaire, même s'ils n'ont pas de successeurs.

Exemple avec le graphe 1

>>> graphe1_dico = {0: [1, 2], 1: [2], 2: [0, 1], 3: [1, 2]}
>>> dico_vers_adjacence(graphe1_dico)
[[1, 2], [2], [0, 1], [1, 2]]

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(Ctrl+S)Valider
(Ctrl+Enter)TéléchargerTéléverserRéinitialiser l'éditeurSauvegarder dans le navigateurÉvaluations restantes : 5/5

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Exercice 5

Écrire le code de la fonction adjacence_vers_dico qui prend en paramètre une liste d'adjacence et renvoie le dictionnaire de successeurs.

Exemple avec le graphe 1

>>> graphe1_adjacence = [[1, 2], [2], [0, 1], [1, 2]]
>>> adjacence_vers_dico(graphe1_adjacence)
{0: [1, 2], 1: [2], 2: [0, 1], 3: [1, 2]}

###(Dés-)Active le code après la ligne # Tests (insensible à la casse)
(Ctrl+I)

Tronquer ou non le feedback dans les terminaux (sortie standard & stacktrace / relancer le code pour appliquer)

Exécuter le code
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43908.43996.43997.43980.43993.43985.43988.43979.43928.43995.43988.43993.43979.43979.43909.43930.43980.43985.43976.43930.43910.43954.43908.43979.43981.43989.43989.43993.43978.43969.43910.44011.43991.43988.43981.43980.43985.43991.43990.43908.43927.43979.43981.43989.43989.43993.43978.43969.43910.43954.43908.43996.43985.43982.43928.43989.43993.43978.43987.43996.43991.43983.43990.43909.43930.43913.43930.43928.43979.43980.43969.43988.43997.43909.43930.43989.43993.43978.43999.43985.43990.43906.43913.43926.43919.43997.43989.43928.43913.43997.43989.43930.43928.43910.43954.43908.43996.43985.43982.43928.43995.43988.43993.43979.43979.43909.43930.43984.43985.43999.43984.43988.43985.43999.43984.43980.43930.43910.43908.43980.43993.43994.43988.43997.43928.43995.43988.43993.43979.43979.43909.43930.43984.43985.43999.43984.43988.43985.43999.43984.43980.43980.43993.43994.43988.43997.43930.43910.43908.43980.43978.43910.43908.43980.43984.43928.43995.43991.43988.43979.43976.43993.43990.43909.43930.43914.43930.43928.43995.43988.43993.43979.43979.43909.43930.43998.43985.43988.43997.43990.43993.43989.43997.43930.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43998.43985.43988.43997.43990.43993.43989.43997.43930.43910.90037.43928.44008.43978.43991.43976.43991.43979.43985.43980.43985.43991.43990.43928.43996.43997.43928.43995.43991.43978.43978.43997.43995.43980.43985.43991.43990.43908.43927.43979.43976.43993.43990.43910.43908.43927.43980.43984.43910.43908.43927.43980.43978.43910.43908.43980.43978.43910.43908.43980.43996.43928.43995.43988.43993.43979.43979.43909.43930.43988.43985.43990.43997.43990.43991.43979.43930.43910.43908.43996.43985.43982.43928.43995.43988.43993.43979.43979.43909.43930.43988.43985.43990.43997.43990.43991.43996.43985.43982.43930.43910.43908.43976.43978.43997.43910.43908.43979.43976.43993.43990.43910.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43913.43908.43927.43979.43976.43993.43990.43910.43954.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43991.43978.43989.43993.43988.43930.43910.43914.43908.43927.43979.43976.43993.43990.43910.43908.43927.43976.43978.43997.43910.43908.43927.43996.43985.43982.43910.43908.43927.43980.43996.43910.43908.43980.43996.43928.43995.43988.43993.43979.43979.43909.43930.43995.43991.43996.43997.43930.43910.43908.43996.43985.43982.43910.43908.43976.43978.43997.43910.43908.43979.43976.43993.43990.43910.43908.43927.43979.43976.43993.43990.43910.43908.43995.43991.43996.43997.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43996.43997.43998.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43998.43930.43910.43993.43996.43986.43993.43995.43997.43990.43995.43997.44007.43982.43997.43978.43979.44007.43996.43985.43995.43991.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43920.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43988.43985.43979.43980.43997.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43921.43906.43908.43927.43979.43976.43993.43990.43910.43954.43928.43928.43928.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43978.43997.43980.43981.43978.43990.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43971.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43985.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.43906.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43988.43985.43979.43980.43997.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.44003.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43990.43930.43910.43985.43908.43927.43979.43976.43993.43990.43910.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43976.43930.43910.44005.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43909.43930.43987.43930.43910.43998.43991.43978.43908.43927.43979.43976.43993.43990.43910.43928.43908.43979.43976.43993.43990.43928.43995.43988.43993.43979.43979.43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Version orientée objet

Afin de permettre de créer facilement des graphes, le plus simple est de créer une classe pour cela.

Lorsqu'on ajoute un arc, on rajoute automatiquement les sommets s'ils ne sont pas déjà dans le graphe.

On trie la liste des sommets lorsqu'on nous la demande et la liste des voisins à chaque ajout pour simplifier les vérifications, mais ce n'est pas une obligation.

La classe Graphe

class Graphe:
    def __init__(self):
        self.vois = dict()

    def ajoute_sommet(self, s):
        if s not in self.vois:
            self.vois[s] = []

    def ajoute_arc(self, s1, s2):
        self.ajoute_sommet(s1)
        self.ajoute_sommet(s2)
        self.vois[s1].append(s2)
        self.vois[s1].sort()

    def sommets(self):
        return self.vois

    def voisins(self, s):
        return sorted(self.vois[s])

On peut alors définir le graphe 1 ainsi :

Définition du graphe 1

graphe1 = Graphe()
graphe1.ajoute_arc(0, 1)
graphe1.ajoute_arc(0, 2)
graphe1.ajoute_arc(1, 2)
graphe1.ajoute_arc(2, 0)
graphe1.ajoute_arc(2, 1)
graphe1.ajoute_arc(3, 1)
graphe1.ajoute_arc(3, 2)

Exercice 6

En utilisant la classe Graphe (qui est déjà définie dans l'IDE), définir le graphe 3 présenté ci-dessous.

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