From f8d2de514e1304f6e41bb7f73b2f623abcb764ee Mon Sep 17 00:00:00 2001 From: Hugo LEVY-FALK Date: Mon, 5 Feb 2018 11:46:13 +0100 Subject: [PATCH] =?UTF-8?q?R=C3=A9daction=20finale.?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- compte_rendu/compte_rendu.aux | 76 ++++++++++++----- compte_rendu/compte_rendu.lof | 24 ++++-- compte_rendu/compte_rendu.pdf | Bin 343617 -> 618625 bytes compte_rendu/compte_rendu.synctex.gz | Bin 39864 -> 62892 bytes compte_rendu/compte_rendu.tex | 121 +++++++++++++++++++++++++-- compte_rendu/compte_rendu.toc | 26 ++++-- compte_rendu/images/3_1.png | Bin 0 -> 31393 bytes compte_rendu/images/3_3.png | Bin 0 -> 29364 bytes compte_rendu/images/3_9.png | Bin 0 -> 51633 bytes compte_rendu/images/4_2.png | Bin 0 -> 29516 bytes compte_rendu/images/4_3.png | Bin 0 -> 25287 bytes compte_rendu/images/4_8_approx.png | Bin 0 -> 31207 bytes compte_rendu/images/4_8_naif.png | Bin 0 -> 30870 bytes graphe.py | 57 +++++++++---- 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self.arrivee = None def renomme(self, nom): @@ -292,6 +292,7 @@ class Graphe: """Calcule la matrice de cout d'une tournée. :param tournee:La liste des étapes avec en première position le départ. + :return: la matrice des couts. """ n = len(tournee) matrice = np.zeros((n, n)) @@ -302,6 +303,11 @@ class Graphe: return matrice def voyageurDeCommerceNaif(self, tournee): + """Résout le problème du voyageur de commerce par backtracking. + + :param tournee: La liste des sommets à visiter. + :return: le cout minimal et l'itinéraire associé. + """ meilleurItineraire = [] minCout = sys.float_info.max matrice = self.matriceCout(tournee) @@ -323,15 +329,20 @@ class Graphe: visite[i] = True indicePrec = itineraire[-1] itineraire.append(i) - cout += matrice[i, indicePrec] - backtrack() + if cout + matrice[i, indicePrec] < minCout: + cout += matrice[i, indicePrec] + backtrack() + cout -= matrice[i, indicePrec] visite[i] = False - cout -= matrice[i, indicePrec] itineraire.pop() backtrack() return minCout, meilleurItineraire def traceItineraire(self, itineraire): + """Colorie les arêtes d'un itinéraire. + + :param itineraire: les points de la tournée ordonnés. + """ for i, x in enumerate(itineraire): x.nom = "s" + str(i) suivant = itineraire[(i + 1) % len(itineraire)] @@ -339,6 +350,12 @@ class Graphe: self.colorieChemin(c, (0.8, 0.1, 0.8)) def Prim(self): + """Applique l'algorithme de Prim sur le graphe. + Les sommets et arêtes sont ajoutés à l'arbre en passant leur flag + selection à True. + + :return: le poids de l'arbre minimal + """ for s in self.sommets: s.selection = False for a in self.aretes: @@ -347,7 +364,6 @@ class Graphe: for a in self.sommets[0].aretes: aretes.ajoute(a) self.sommets[0].selection = True - arbre = [] ptot = 0 while not aretes.empty(): a = aretes.pop() @@ -375,6 +391,11 @@ class Graphe: s.couleur = couleur def grapheDeCout(self, tournee): + """Génère le graphe de coût pour la tournée. + + :param tournee: la tournée. + :return: le graphe de coûts. + """ g = Graphe("Graphe de cout de " + self.nom) for s in tournee: s_a = g.ajouteSommet(s.x(), s.y()) @@ -389,6 +410,11 @@ class Graphe: return g def tourneeApproximative(self, tournee): + """Résout le problème du voyageur de commerce de manière approximative. + + :param tournee: les sommets à visiter + :return: Le chemin optimal approché + """ g_cout = self.grapheDeCout(tournee) g_cout.Prim() pile = stack.Stack() @@ -431,6 +457,8 @@ class Sommet: @property def minCumulRestant(self): + """Minore le cumul restant pour le sommet considéré. + """ x, y = self.graphe.arrivee.pos d = ((self.pos[0] - x)**2 + (self.pos[1] - y)**2)**(1 / 2) if self.graphe.cout is Graphe.Cout.TEMPS: @@ -466,8 +494,7 @@ class Sommet: class Arete: """Implémente une arête de graphe.""" - def __init__( - self, s1, s2, longueur, v_moyenne, graph, couleur=(0., 0., 0.)): + def __init__(self, **kwargs): """Initialise une arête. :param s1: sommet 1. @@ -477,14 +504,14 @@ class Arete: :param couleur: couleur de l'arête. :param graph: le graphe de l'arête. """ - self.s1 = s1 - self.s2 = s2 - s1.aretes.add(self) - s2.aretes.add(self) - self.longueur = longueur - self.v_moyenne = v_moyenne - self.couleur = couleur - self.graph = graph + self.s1 = kwargs.get('s1') + self.s2 = kwargs.get('s2') + self.s1.aretes.add(self) + self.s2.aretes.add(self) + self.longueur = kwargs.get('longueur') + self.v_moyenne = kwargs.get('v_moyenne') + self.couleur = kwargs.get('couleur', (0., 0., 0.)) + self.graph = kwargs.get('graph') self.selection = False def voisin(self, s): diff --git a/test.py b/test.py index ce81b39..e266b1d 100644 --- a/test.py +++ b/test.py @@ -15,6 +15,7 @@ import numpy as np import matplotlib.pyplot as pl import time + def creerGrapheFigure1(): """ Crée le graphe de la figure 1 """ g = graphe.Graphe("Graphe de la figure 1") @@ -86,22 +87,25 @@ def chronometre(fonctionTest, fonctionPreparation, parametres): temps = [] # Pour chaque valeur de paramètre for p in parametres: - # Génère les entrées du test pour la valeur p - entrees = fonctionPreparation(p) # Lance le test pour ces entrées print("t({}) = ".format(p), end="", flush=True) - debut = time.time() - fonctionTest(entrees) - fin = time.time() - # Mesure le temps d ' exécution - t = (fin - debut) + t = [] + for i in range(10): + entrees = fonctionPreparation(p) + debut = time.time() + fonctionTest(entrees) + fin = time.time() + # Mesure le temps d ' exécution + t.append(fin - debut) + t = sum(t)/10 print("{:.2f} s".format(t)) temps.append(t) return temps + def testQuestion2_6(): """Mesure la performance de graphe.reseau""" - prepare = lambda p : graphe.pointsAleatoires(p, 10) + def prepare(p): return graphe.pointsAleatoires(p, 10) valeurs_n = list(map(int, np.logspace(1, 2.7, 100))) temps = chronometre(graphe.reseau, prepare, valeurs_n) pl.close('all') @@ -112,15 +116,16 @@ def testQuestion2_6(): pl.show() pl.title("Mesure du temps d'exécution de `reseau`.") pl.loglog(valeurs_n, temps, label='temps de calcul') - pl.loglog(valeurs_n, (lambda x : x**3)(valeurs_n), label='$x\mapsto x^3$') + pl.loglog(valeurs_n, (lambda x: x**3)(valeurs_n), label='$x\mapsto x^3$') pl.legend(loc='best') pl.xlabel("n") pl.ylabel("temps") pl.show() + def testQuestion2_7(): """Compare la création de graphe de gabriel et ed delaunay.""" - prepare = lambda p : graphe.pointsAleatoires(p, 10) + def prepare(p): return graphe.pointsAleatoires(p, 10) valeurs_n = np.arange(1, 200) temps1 = chronometre(graphe.gabriel, prepare, valeurs_n) temps2 = chronometre(graphe.delaunay, prepare, valeurs_n) @@ -133,6 +138,7 @@ def testQuestion2_7(): pl.ylabel('temps') pl.show() + def testQuestion2_8(): """Compare le reseau naif et non naif.""" g = graphe.pointsAleatoires(30, 30) @@ -145,9 +151,10 @@ def testQuestion2_8(): graphique.affiche(g, (0, 0), 10.) graphique.affiche(g1, (0, 0), 10.) + def testQuestion2_9(): """Compare le temps de création des deux méthodes de création de réseau""" - prepare = lambda p : graphe.pointsAleatoires(p, 10) + def prepare(p): return graphe.pointsAleatoires(p, 10) valeurs_n = list(map(lambda x: int(x), np.logspace(1, 3, 30))) temps1 = chronometre(graphe.reseau, prepare, valeurs_n) temps2 = chronometre(graphe.reseauRapide, prepare, valeurs_n) @@ -199,19 +206,24 @@ def testQuestion3_2(): def testQuestion3_3(): """Mesure le temps d'exécution de Dijkstra.""" - prepare = lambda p : graphe.reseauRapide(graphe.pointsAleatoires(p, 100)) - valeurs_n = list(map(lambda x: int(x), np.logspace(1, 4, 10))) + def prepare(p): return graphe.reseauRapide(graphe.pointsAleatoires(p, 100)) + valeurs_n = list(map(lambda x: int(x), np.logspace(1, 3, 100))) temps = chronometre(graphe.Graphe.dijkstra, prepare, valeurs_n) + + def f(x): return x*np.log(x) + theorie = [f(x) for x in valeurs_n] pl.close('all') pl.title("Temps d'exécution en fonction de la taille du graphe.") - pl.plot(valeurs_n, temps, label='reseau') - pl.legend(loc='best') + pl.plot(valeurs_n, temps, label="Dijkstra") + pl.plot(valeurs_n, theorie, label="$x \mapsto x\log x$") + pl.legend(loc="best") pl.xlabel("n") pl.ylabel("temps") pl.show() pl.title("Temps d'exécution en fonction de la taille du graphe.") - pl.loglog(valeurs_n, temps, label='reseau') - pl.legend(loc='best') + pl.loglog(valeurs_n, temps) + pl.loglog(valeurs_n, theorie, label="$x \mapsto x\log x$") + pl.legend(loc="best") pl.xlabel("n") pl.ylabel("temps") pl.show() @@ -226,7 +238,7 @@ def testQuestion3_4(): g.fixeTempsCommeCout() g.dijkstra(g.sommets[0]) c = g.cheminOptimal(g.sommets[-1]) - g.colorieChemin(c, (0.,1.,0.)) + g.colorieChemin(c, (0., 1., 0.)) g.renomme("Temps") graphique.affiche(g, (0, 0), 10.) @@ -249,7 +261,7 @@ def testQuestion3_5(): def testQuestion3_6(): """Compare Dijkstra avec et sans tas.""" - prepare = lambda p : graphe.reseauRapide(graphe.pointsAleatoires(p, 10)) + def prepare(p): return graphe.reseauRapide(graphe.pointsAleatoires(p, 10)) valeurs_n = list(map(lambda x: int(x), np.logspace(1, 3, 10))) temps1 = chronometre(graphe.Graphe.dijkstra, prepare, valeurs_n) temps2 = chronometre(graphe.Graphe.dijkstraAvecTas, prepare, valeurs_n) @@ -300,13 +312,14 @@ def testQuestion3_8(): def testQuestion3_9(): """Compare Dijkstra avec et sans tas et A*.""" - prepare = lambda p : graphe.reseauRapide(graphe.pointsAleatoires(p, 10)) - valeurs_n = list(map(lambda x: int(x), np.logspace(1, 3, 50))) + def prepare(p): return graphe.reseauRapide(graphe.pointsAleatoires(p, 10)) + valeurs_n = list(map(lambda x: int(x), np.logspace(1, 2.5, 30))) temps1 = chronometre(graphe.Graphe.dijkstra, prepare, valeurs_n) temps2 = chronometre(graphe.Graphe.dijkstraPartiel, prepare, valeurs_n) temps3 = chronometre(graphe.Graphe.astar, prepare, valeurs_n) pl.close('all') - pl.title("Comparaison du temps d'exécution de `dijkstra`, `dijkstraPartiel` et `astar`.") + pl.title( + "Comparaison du temps d'exécution de `dijkstra`, `dijkstraPartiel` et `astar`.") pl.plot(valeurs_n, temps1, label='Dijkstra') pl.plot(valeurs_n, temps2, label='Dijkstra partiel avec tas') pl.plot(valeurs_n, temps3, label='A*') @@ -314,7 +327,8 @@ def testQuestion3_9(): pl.xlabel("n") pl.ylabel("temps") pl.show() - pl.title("Comparaison du temps d'exécution de `dijkstra`, `dijkstraPartiel` et `astar`.") + pl.title( + "Comparaison du temps d'exécution de `dijkstra`, `dijkstraPartiel` et `astar`.") pl.loglog(valeurs_n, temps1, label='Dijkstra') pl.loglog(valeurs_n, temps2, label='Dijkstra partiel avec tas') pl.loglog(valeurs_n, temps3, label='A*') @@ -343,7 +357,7 @@ def testQuestion4_2(): g.renomme("graphe") print("Points 0,1,2 et 7") tournee = [g.sommets[i] for i in (0, 1, 2, 7)] - cout,iti = g.voyageurDeCommerceNaif(tournee) + cout, iti = g.voyageurDeCommerceNaif(tournee) g.traceItineraire(iti) graphique.affiche(g, (0, 0), 10.) @@ -351,8 +365,9 @@ def testQuestion4_2(): def testQuestion4_3(): """Mesure le temps de calcul.""" g = graphe.reseauRapide(graphe.pointsAleatoires(1000, 1000)) - prepare = lambda p : random.sample(g.sommets, p) - valeurs_n = list(map(int, np.linspace(1, 15, 10))) + + def prepare(p): return random.sample(g.sommets, p) + valeurs_n = list(map(int, np.linspace(1, 11, 10))) temps = chronometre(g.voyageurDeCommerceNaif, prepare, valeurs_n) pl.close('all') pl.title("Temps d'exécution pour le voyageur de commerce.") @@ -396,14 +411,15 @@ def testQuestion4_8(): graphe.reseauRapide(g) g.fixeTempsCommeCout() - g.renomme("Graphe de test (cout=temps)") + g.renomme("Graphe de test (cout=temps) tournée approximative") g1 = copy.deepcopy(g) + g1.renomme("Graphe de test (cout=temps) tournée naïve") print("Tournée 0, 1, 2, 7") tournee = [g.sommets[i] for i in (0, 1, 2, 7)] iti = g.tourneeApproximative(tournee) g.traceItineraire(iti) tournee = [g1.sommets[i] for i in (0, 1, 2, 7)] - cout,iti = g1.voyageurDeCommerceNaif(tournee) + cout, iti = g1.voyageurDeCommerceNaif(tournee) g1.traceItineraire(iti) graphique.affiche(g, (0, 0), 10., blocage=False) graphique.affiche(g1, (0, 0), 10.) @@ -412,8 +428,9 @@ def testQuestion4_8(): def testQuestion4_9(): """Mesure le temps de calcul.""" g = graphe.reseauRapide(graphe.pointsAleatoires(1000, 1000)) - prepare = lambda p : random.sample(g.sommets, p) - valeurs_n = list(map(int, np.logspace(1, 2, 30))) + + def prepare(p): return random.sample(g.sommets, p) + valeurs_n = list(map(int, np.logspace(1, 3, 30))) temps = chronometre(g.tourneeApproximative, prepare, valeurs_n) pl.close('all') pl.title("Temps d'exécution pour le voyageur de commerce approximatif.") @@ -427,6 +444,7 @@ def testQuestion4_9(): pl.ylabel("temps") pl.show() + if __name__ == '__main__': n = sys.argv[1] if n in ('--help', '-h'):