From c57a877e24c8b49d28302915cd0e8a694e2124ba Mon Sep 17 00:00:00 2001 From: Robin VAN DE MERGHEL <robin.van_de_merghel@etu.uca.fr> Date: Thu, 23 Mar 2023 11:59:15 +0100 Subject: [PATCH] New courses of Algorithmic and MRP --- Algorithmique/TD/TD4.md | 107 +++++ Algorithmique/TD/td-complexite.pdf | Bin 0 -> 216817 bytes Modeliser Resolution/Exo_Assignment.py | 13 +- .../Exo_GaleShapleyAlgorithm-OO-Objets.py | 368 ++++++++++++++++++ 4 files changed, 480 insertions(+), 8 deletions(-) create mode 100644 Algorithmique/TD/TD4.md create mode 100644 Algorithmique/TD/td-complexite.pdf create mode 100644 Modeliser Resolution/Exo_GaleShapleyAlgorithm-OO-Objets.py diff --git a/Algorithmique/TD/TD4.md b/Algorithmique/TD/TD4.md new file mode 100644 index 0000000..d4dc80a --- /dev/null +++ b/Algorithmique/TD/TD4.md @@ -0,0 +1,107 @@ +# Exercice 4.1 + +> **On a les codes suivants :** +> **Quelle est la complexité de chacun de ces codes ?** + + +## Algorithme 1 + +```js +s = 0; +for (i=0;i<=n;i++) { + s = s + i; +} +``` + + +On a $n+1$ itérations, dont chaque itération prend $3$ unités de temps (incrémentation, addition, et comparaison). De plus il y a une affectation à l'initialisation qui coûte $3$ unités de temps. Donc la complexité est $O(3(n + 1) + 3) = O(n)$. + +## Algorithme 2 + +```js +s = 0; +for (i=0;i<=n;i++) { + for (j=i;j<=n;j++) { + s = s + i; + } +} +``` + +On a $n+1$ itérations, dont chaque itération prend $3$ unités de temps (incrémentation, addition, et comparaison). De plus, la boucle `for` interne a $n - i + 1$ itérations, donc $n(n+1)/2$ itérations au total. Donc la complexité est $O(3(n + 1) + 3(n(n+1)/2)) = O(n^2)$. + + +## Algorithme 3 + +```js +for (k=2; k<=n;k++) { + x = a[k]; + j = k - 1; + while (j!=0) { + if (x < a[j]) { + j = j - 1; + } else { + j = 0; + } + } +} +``` + + +On a une boucle `for` qui fait $n-2$ itérations. La boucle while elle a un nombre d'itérations qui dépend de la valeur de $j$. + +Dans le pire cas, on a $j=0$ à la fin de la boucle `while` grâce à la condition `j!=0`. Donc on a $n-2$ itérations dans la boucle `for` et $k$ itérations dans la boucle `while` pour $k=2,\dots,n$. Donc la complexité est $\sum_{k=2}^n O(k) = O(n^2)$. + + +## Algorithme 4 + +```js +c = n; +while (c > 1) { + c = c / 5; +} +``` + + + +On peut traduire cet algorithme par une suite récurrente : + +$$\begin{cases} c_0 = n \\ c_{k+1} = \frac{c_k}{5} \end{cases}$$ + +C'est à dire : + +$$c_k = \frac{n}{5^k}$$ + +On a donc $c_k \leq 1$ quand $n \geq \log_5(n)$. Donc la complexité est au pire $O(\log_5(n)) = O(\log(n))$. + + +# Exercice 4.2 + +> **Notations $O$** + +## Proposition 1 + +> **Montrez que $(n+a)^b = O(n^b)$ pour tout $a$ et $b$** + +On a : + +$$\begin{align} + (n+a)^b &= \sum_{k=0}^b \binom{b}{k} a^{b-k} n^k \\ + &= n^k + \alpha_1 n^{k-1} + \alpha_2 n^{k-2} + \dots + \alpha_{k-1} n + \alpha_k \\ +\end{align}$$ + +Avec $\alpha_i = \binom{b}{i}$. + +On a donc : + +$$\begin{align} + O((n+a)^b) &= O(n^b + \alpha_1 n^{b-1} + \alpha_2 n^{b-2} + \dots + \alpha_{b-1} n + \alpha_b) \\ + &= O(n^b) + O(\alpha_1 n^{b-1}) + O(\alpha_2 n^{b-2}) + \dots + O(\alpha_{b-1} n) + O(\alpha_b) \\ + &= O(n^b) + O(n^{b-1}) + O(n^{b-2}) + \dots + O(n) + O(1) \\ +\end{align}$$ + +On a $O(1)$ négligeable par rapport à $O(n)$, puis $O(n)$ négligeable par rapport à $O(n^2)$, etc. Jusqu'à $O(n^{b-1})$ qui est négligeable par rapport à $O(n^b)$. Donc $O((n+a)^b) = O(n^b)$. + +## Proposition 2 + +> **Est-ce que la phrase suivante a un sens : "Le temps d'exécution de l'algorithme est $O(n^2)$" ? 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candidates_choices): agencies = list(agencies_choices.keys()) candidates = list(candidates_choices.keys()) + n = len(agencies) def h(i): if i == n: yield resultat else: - resultat[agencies[i]] = candidates[0] - yield from h(i+1) - resultat[agencies[i]] = candidates[1] - yield from h(i+1) - resultat[agencies[i]] = candidates[2] - yield from h(i+1) - resultat[agencies[i]] = candidates[3] - yield from h(i+1) + for j in range(n): + if j not in resultat.values(): + resultat[agencies[i]] = candidates[j] + yield from h(i+1) resultat = {agency: 0 for agency in agencies} yield from h(0) diff --git a/Modeliser Resolution/Exo_GaleShapleyAlgorithm-OO-Objets.py b/Modeliser Resolution/Exo_GaleShapleyAlgorithm-OO-Objets.py new file mode 100644 index 0000000..dd81d70 --- /dev/null +++ b/Modeliser Resolution/Exo_GaleShapleyAlgorithm-OO-Objets.py @@ -0,0 +1,368 @@ +import random + +""" Un element est soit une agence, soit un candidat : les classes Agence et +Candidat héritent de la classe Element. +La classe Element a deux attributs : ++ nom : chaine de caractères. ++ tuple_choix : tuple (pas une liste) des objets constituant ses choix. +La classe Element a deux méthodes : ++ __init__ pour initialiser. ++ choix() qui retourne le tuple de ses choix. +Les deux classes filles, Agence et Candidat ont une méthode spéciale __repr__ +pour un affichage confortable de l'élément. +""" + +# Ecrivez ces trois classes ici + + +class Element: + def __init__(self, nom, tuple_choix): + self.nom = nom + self.tuple_choix = tuple_choix + + def choix(self): + return self.tuple_choix + + +class Agence(Element): + def __repr__(self): + return f"Agence {self.nom}" + + +class Candidat(Element): + def __repr__(self): + return f"Candidat {self.nom}" + + +# ------------------------------------------------------------------------------ +"""La classe Instance a deux attributs : ++ ens_agences : l'ensemble des agences de l'instance ++ ens_candidats : l'ensemble des candidats de l'instance. +Elle contient les méthodes : ++ __init__ pour initialiser. ++ ajouter_agence qui prend en entrée une agence et l'ajoute dans l'ensemble +des agences de l'instance. ++ ajouter_candidat qui prend en entrée un candidat et l'ajoute dans l'ensemble +des candidats de l'instance. ++ __repr__ pour un affichage confortable de l'instance. ++ __len__ pour retourner le nombre d'agences (ou de candidats) de l'instance. ++ affiche_choix qui affiche proprement les choix de chaque agence et de chaque +candidat.""" + +# Ecrivez cette classe ici + + +class Instance: + def __init__(self, ens_agences, ens_candidats): + self.ens_agences = ens_agences + self.ens_candidats = ens_candidats + + def ajouter_agence(self, agence): + self.ens_agences.append(agence) + + def ajouter_candidat(self, candidat): + self.ens_candidats.append(candidat) + + def __repr__(self): + return f"Instance avec {len(self.ens_agences)} agences et {len(self.ens_candidats)} candidats" + + def __len__(self): + return len(self.ens_agences) + + def affiche_choix(self): + for agence in self.ens_agences: + print(f"{agence} a pour choix : {agence.choix()}") + for candidat in self.ens_candidats: + print(f"{candidat} a pour choix : {candidat.choix()}") + + def __contains__(self, element): + return element in self.ens_agences or element in self.ens_candidats + + def agences(self): + return self.ens_agences + + def candidats(self): + return self.ens_candidats + + +# ------------------------------------------------------------------------------ +"""La classe Affectation a un attribut : ++ dico_agence_candidat : chaque clé est une agence et la valeur correspondante +est le candidat auquel il est affecté. +Cette classe contient les méthodes : ++ __init__ pour initialiser. ++ candidat_affecté_à : prend en entrée une agence et retourne le candidat qui +lui est affecté (None s'il n'y en a pas). ++ agence_affectée_à : prend en entrée un candidat et retourne l'agence qui +lui est affectée (None s'il n'y en a pas). ++ affecter : prend en entrée une agence, un candidat et affecte l'un à l'autre. ++ changer_affectation_candidat : prend en entrée agence, candidat, +nouvelle_agence et affecte candidat à nouvelle_agence et "dé-affecte" agence. +""" + +# Ecrivez cette classe ici + + +class Affectation: + def __init__(self, dico_agence_candidat): + self.dico_agence_candidat = dico_agence_candidat + + def candidat_affecté_à(self, agence): + return self.dico_agence_candidat[agence] + + def agence_affectée_à(self, candidat): + for agence, candidat_affecté in self.dico_agence_candidat.items(): + if candidat_affecté == candidat: + return agence + return None + + def agence_est_affectée(self, agence): + return agence in self.dico_agence_candidat + + def candidat_est_affecté(self, candidat): + for agence, candidat_affecté in self.dico_agence_candidat.items(): + if candidat_affecté == candidat: + return True + return False + + def affecter(self, agence, candidat): + self.dico_agence_candidat[agence] = candidat + + def changer_affectation_candidat(self, agence, candidat, nouvelle_agence): + # On affecte le candidat à la nouvelle agence + self.dico_agence_candidat[nouvelle_agence] = None + # On dé-affecte l'ancienne agence + del self.dico_agence_candidat[agence] + + def __repr__(self) -> str: + return f"Affectation : {self.dico_agence_candidat}" + + def __contains__(self, element): + return element in self.dico_agence_candidat or element in self.dico_agence_candidat.values() + + def __len__(self): + return len({candidat for candidat in self.dico_agence_candidat.values() if candidat is not None}) + +# ------------------------------------------------------------------------------ + + +def extraire_instance_du_fichier(file): + """Prend en entrée un nom de fichier (chaine de caractères) et retourne + l'instance stockée dans ce fichier. Le format du fichier est : + n (seul sur la première ligne) + suivi de n lignes, chacune étant sous le format : + X:Y1:Y2:...:Yn ou X est le nom d'une agence et les n Yi suivants sont les + noms des candidats, dans l'ordre des choix de X. + suivi de n lignes, chacune étant sous le format : + X:Y1:Y2:...:Yn ou X est le nom d'un candidat et les n Yi suivants sont les + noms des agences, dans l'ordre des choix de X. + Le séparateur est ':' sans espace entre les éléments. + Les noms des agences et candidats ne doivent pas contenir le caractère ":" + """ + with open(file, "r") as buffer: + + # Différence avec le code précédent : on utilise les classes + + n = int(buffer.readline()) + + # On va stocker les agences et les candidats dans des dictionnaires ("nom":"objet en classe") + dico_agences = {} + dico_candidats = {} + + choix_agences = {} + choix_candidats = {} + + for _ in range(n): + line = buffer.readline() + line = line.split(":") + agence = line[0] + candidats = line[1:] + candidats = [candidat.strip() for candidat in candidats] + choix_agences[agence] = candidats + dico_agences[agence] = Agence(agence, None) + + for _ in range(n): + line = buffer.readline() + line = line.split(":") + candidat = line[0] + agences = line[1:] + agences = [agence.strip() for agence in agences] + choix_candidats[candidat] = agences + dico_candidats[candidat] = Candidat(candidat, None) + + # Les agences ont comme liste de candidats des strings, on va les remplacer par des objets Candidat + for agence in dico_agences.values(): + agence.tuple_choix = tuple( + [dico_candidats[candidat] for candidat in choix_agences[agence.nom]]) + + # Les candidats ont comme liste d'agences des strings, on va les remplacer par des objets Agence + for candidat in dico_candidats.values(): + candidat.tuple_choix = tuple( + [dico_agences[agence] for agence in choix_candidats[candidat.nom]]) + + return Instance(dico_agences.values(), dico_candidats.values()) + + +# ------------------------------------------------------------------------------ + + +def génère_instance_aléatoire(n, version_number=1): + """Generate a random instance with n agencies, n candidates and put the + result in a file that is named GSEntries_Rand_{n}_{version_number} + (for example GSEntries_Rand_10_3) to distinguish different random files.""" + pass + +# ------------------------------------------------------------------------------ + + +def gale_shapley_algorithm2(instance): + """Exécute l'algorithme de Gale-Shapley sur l'instance et retourne + l'affectation construite.""" + agenciesAssign = Affectation({}) + candidatesAssign = Affectation({}) + + agenciesIndex = {agency: 0 for agency in instance.agences()} + + n = len(instance.agences()) + + while not len(agenciesAssign) == n: + for agency in instance.agences(): + if agency in agenciesAssign: + continue + + candidate = agency.tuple_choix[agenciesIndex[agency]] + + if candidate not in candidatesAssign: + agenciesAssign.affecter(agency, candidate) + candidatesAssign.affecter(candidate, agency) + else: + currentAgency = candidatesAssign.dico_agence_candidat[candidate] + candidateAgencyRank = candidate.tuple_choix.index(agency) + + candidateAgencyIndex = candidate.tuple_choix.index( + currentAgency) + + if candidateAgencyRank < candidateAgencyIndex: + agenciesAssign.changer_affectation_candidat( + currentAgency, candidate, agency) + candidatesAssign.changer_affectation_candidat( + candidate, currentAgency, agency) + + agenciesIndex[agency] += 1 + + return agenciesAssign + + +def gale_shapley_algorithm(instance): + + affectation = Affectation({}) + + agences_iter = {agence: iter(agence.choix()) + for agence in instance.ens_agences} + + liste_agences_non_affectees = list(instance.ens_agences) + + while len(liste_agences_non_affectees) > 0: + + a = random.choice(liste_agences_non_affectees) + c = next(agences_iter[a]) + + if affectation.agence_affectée_à(c) is None: + affectation.affecter(a, c) + liste_agences_non_affectees.remove(a) + else: + k = affectation.agence_affectée_à(c) + + if c.choix().index(a) < c.choix().index(k): + affectation.changer_affectation_candidat(k, c, a) + liste_agences_non_affectees.remove(a) + liste_agences_non_affectees.append(k) + + return affectation + + +# ------------------------------------------------------------------------------ + + +def nombre_de_couples_non_stables(instance, affectation): + """Retourne le nombre de couples non stables de l'affectation liée à + l'instance.""" + nombre_de_couples_non_stables = 0 + + for a in instance.ens_agences: + c = affectation.candidat_affecté_à(a) + rang_de_c = a.choix().index(c) + + if rang_de_c > 0: + + liste_cand_potentiels = a.choix()[:rang_de_c] + + for cand in liste_cand_potentiels: + agence_de_cand = affectation.agence_affectée_à(cand) + if cand.choix().index(a) < cand.choix().index(agence_de_cand): + nombre_de_couples_non_stables += 1 + + return nombre_de_couples_non_stables + +# ------------------------------------------------------------------------------ + + +def génère_affectation_aléatoire(instance): + """Retourne une affectation aléatoire de l'instance.""" + number = len(instance.agences) + import random + + # On créer deux listes de candidats et d'agences + candidats = list(instance.candidats) + agences = list(instance.agences) + + # On mélange les listes + random.shuffle(candidats) + random.shuffle(agences) + + # On créer un dictionnaire qui va contenir les affectations + + return Affectation({agence: candidat for agence, candidat in zip(agences, candidats)}) + + +# ------------------------------------------------------------------------------ + + +def iter_toutes_les_affectations(instance): + """Un itérateur de toutes les affectations possibles (stables ou non + stables). Utilise la bibliothèque itertools. """ + import itertools + + it = itertools.permutations(instance.ens_candidats) + tuple_des_agences = tuple(instance.ens_agences) + for perm in it: + yield Affectation({agence: candidat for agence, candidat in zip(tuple_des_agences, perm)}) + +# ------------------------------------------------------------------------------ +# Tests +# ------------------------------------------------------------------------------ + + +fichier = "couple.txt" +inst = extraire_instance_du_fichier(fichier) +# inst.affiche_choix() + + +# affect = gale_shapley_algorithm(inst) +# print("L'affectation produite par l'algorithme de Gayle-Shapley est :") +# print(affect) +# if nombre_de_couples_non_stables(inst, affect) != 0: +# raise Exception("Erreur : le nbre de couples non stables doit être 0 !!") + +# print(nombre_de_couples_non_stables(inst, affect)) + +# affect = génère_affectation_aléatoire(inst) +# print(f"Le nbre de couples non stables dans l'affectation aléatoire") +# print(affect) +# print(f"est de {nombre_de_couples_non_stables(inst, affect)}") + +it = iter_toutes_les_affectations(inst) +print([el for el in it]) +# liste = [nombre_de_couples_non_stables(inst, affect) for affect in it] +# liste.sort() +# print(liste) -- GitLab