1A.algo - Parcours de graphe - correction

Links: notebook, html, PDF, python, slides, GitHub

Algorithme assez simple sur les graphes (correction).

from jyquickhelper import add_notebook_menu
add_notebook_menu()
# tutoriel_graphe
noeuds = {0: 'le', 1: 'silences', 2: 'quelques', 3: '\xe9crit', 4: 'non-dits.', 5: 'Et', 6: 'risque', 7: '\xe0', 8: "qu'elle,", 9: 'parfois', 10: 'aim\xe9', 11: 'lorsque', 12: 'que', 13: 'plus', 14: 'les', 15: 'Minelli,', 16: "n'oublierai", 17: 'je', 18: 'prises', 19: 'sa', 20: 'la', 21: 'jeune,', 22: "qu'elle,", 23: '\xe0', 24: 'ont', 25: "j'ai", 26: 'chemin', 27: '\xe9tranger', 28: 'lente', 29: 'de', 30: 'voir', 31: 'quand', 32: 'la', 33: 'recul,', 34: 'de', 35: 'trop', 36: 'ce', 37: 'Je', 38: 'Il', 39: "l'extr\xeame", 40: "J'ai", 41: 'silences,', 42: "qu'elle,", 43: 'le', 44: 'trace,', 45: 'avec', 46: 'seras', 47: 'dire,', 48: 'femme', 49: 'soit'}
arcs   = {(3, 15): None, (46, 47): None, (42, 33): None, (35, 45): None, (1, 14): None, (22, 26): None, (26, 28): None, (43, 29): None, (40, 41): None, (29, 44): None, (17, 3): None, (32, 37): None, (24, 19): None, (46, 34): None, (11, 19): None, (34, 49): None, (22, 2): None, (37, 48): None, (14, 12): None, (3, 10): None, (5, 18): None, (12, 24): None, (34, 32): None, (45, 39): None, (37, 26): None, (33, 45): None, (34, 47): None, (36, 31): None, (29, 47): None, (13, 11): None, (12, 21): None, (2, 16): None, (5, 4): None, (33, 35): None, (28, 49): None, (25, 49): None, (21, 0): None, (3, 13): None, (18, 24): None, (12, 7): None, (13, 15): None, (11, 1): None, (16, 23): None, (37, 45): None, (27, 32): None, (32, 41): None, (8, 24): None, (10, 1): None, (2, 24): None, (24, 11): None, (2, 14): None, (47, 36): None, (48, 39): None, (30, 25): None, (30, 43): None, (15, 14): None, (26, 27): None, (6, 8): None, (20, 10): None, (19, 17): None, (5, 7): None, (44, 25): None, (27, 38): None, (2, 0): None, (3, 18): None, (3, 9): None, (25, 33): None, (42, 48): None, (2, 15): None, (26, 48): None, (26, 38): None, (7, 8): None, (8, 4): None}
points = [(0.84737386691659533, 0.95848816613228727), (0.28893525107454354, 0.66073249195336492), (0.60382037086559148, 0.13747945088383384), (0.21951613156582261, 0.040905525433785228), (0.21613062123493632, 0.096875623632852625), (0.99787588721497178, 0.79337171783327132), (0.18576957348508683, 0.78396225027633837), (0.23875443625588322, 0.35497638429086975), (0.8713637939628045, 0.22983756618811024), (0.28301724069085921, 0.99408996134013161), (0.39792684083973429, 0.77105362865540716), (0.75452041353842147, 0.330325155167562), (0.24824845436118537, 0.95998690078041737), (0.92318434139996397, 0.38115765401571988), (0.54660304309415886, 0.62093667623480242), (0.58899996464290505, 0.9017292023892568), (0.60541336358687847, 0.28929082523865812), (0.87925379747840293, 0.94834058131858756), (0.61449632813730748, 0.94264237081849722), (0.13119804743502139, 0.44158556198130949), (0.20660796942108339, 0.915599021810789), (0.3097131996826511, 0.81979953110332837), (0.89711055197298928, 0.7298496710091944), (0.22499060312661545, 0.072786594549671291), (0.012604758185058018, 0.36199484670070914), (0.92050750708863993, 0.91447248587261709), (0.26304069827339327, 0.026058147250910935), (0.59289937178711172, 0.86673111722782969), (0.70640070176443837, 0.64096733852134291), (0.049399266565914535, 0.54027723332288746), (0.26450585597978316, 0.50883097182669357), (0.91987410679455195, 0.97753050553942622), (0.5618293073273094, 0.27688371997865069), (0.91241761244784847, 0.090310675429991605), (0.90925789663628509, 0.40628594240956295), (0.3832814495252409, 0.66221025722485627), (0.74928785967005418, 0.32840192750838815), (0.25478832731446643, 0.70269825611412617), (0.54293534537395793, 0.87800254191632932), (0.89603330911109724, 0.77106655965183546), (0.29830084404349644, 0.97117954065316903), (0.075137754060910056, 0.086473140735377596), (0.120307047737505, 0.073651360408690802), (0.87835916829742444, 0.34622147871872355), (0.20567119579830373, 0.42658381934346423), (0.27715586337053655, 0.87999487046170488), (0.16364186693234739, 0.98604111274325335), (0.31830209002283116, 0.36372930495109934), (0.73434680601907532, 0.65926820980026724), (0.9830474686174655, 0.12246834322318068), (4.0293130665095358, -3.0529459366329164), (-3.7755737603387041, 2.2685053357046323), (-2.1926920625846602, 2.4857321786911326), (5.1445647965531025, 4.8943143876324848), (0.87403644639763023, 5.6464000746270226), (-3.5545355219233219, 3.8988261206085766), (2.0785612031685732, -1.2948920530351256), (-3.4682717483474708, 2.2364561845005868), (2.0695530720860349, -2.9439062757612424), (3.9563571060210054, -2.0678946581365616), (3.2485209278176157, -2.6386418932454814), (-3.4800728241977779, 0.72646452125011518), (-1.8341241854718167, 3.3482541467971951), (-4.5558692651012178, 3.5624030818263908), (-4.6768285328272157, -1.0106699901361971), (3.9175303893386597, 0.1087117017596031), (-3.9111941479785823, 2.70001353796486), (-5.5501953466420737, -3.8544512068951891), (1.9246058344257151, 4.123740240481137), (-4.110657752575519, -2.0774760107085393), (2.6547967574269418, 4.6868873425221045), (-1.9308254017076039, -2.9448006865754279), (-3.0788555249744247, 3.396205767032443), (-4.0516249434348621, 0.42035392996461629), (-2.2989465364173602, -3.2706795830191275), (4.651698949077459, 1.1364194264447973), (3.3637257964296152, -2.5082040184760555), (-3.2502121678035314, 4.5383631321594571), (4.5274668721202556, 4.473426056956777), (3.400114365788911, -3.0434200740148363), (3.513062501300436, 2.718209259961025), (-2.3986743034356737, -4.0590996420222467), (2.6632346815268289, 4.8894243587379433), (4.2802341564965607, -3.4921791441653762), (-1.5297912885016269, 5.5780900056883569), (4.0634598983096293, -2.2904478604819776), (1.0857595813036722, 5.6366192967000295), (-4.5596385297232223, -1.3177709282351766), (-2.1361714943468244, -3.9107871995830976), (3.7240531749202161, -4.8033709892679886), (-4.1017624989859351, -0.54374796617700816), (2.3715344477591818, -3.2387553898801391), (3.8187172884547076, -5.1522284671097314), (1.0454193728074506, 3.1688190599740418), (-3.9848808505730315, -3.5176013894081675), (-4.1965918931505275, 2.0248869962483522), (3.4535361867324776, 3.4437155145638751), (-3.2171776428648808, -2.0867326734388021), (-3.5763512667620065, -3.785293447306691), (-3.2489915323631275, -4.6589505137265448), (1.2817385669950028, -4.0553290947191964), (-5.5481507299407191, 5.2080477057573553), (-2.2817876881965624, 0.12512408298772948), (-3.4831125975271719, 1.7834950195462245), (-4.2064606598908139, -3.2421411165648886), (-5.3461204499811092, 0.65966593807378215), (-0.36559473517464181, -3.9248327086099932), (-4.4223418217602317, 4.790875007038224), (-3.9026572243192548, -0.21621909226838504), (0.16100173690141428, -4.8875278273011942), (-4.2792213808538602, 1.9041297697847308), (-4.4298318748123444, -3.8717874765920124), (3.2660121035644738, 3.8922848961161609), (4.4724681658043082, -1.8875314666371643), (-3.1337207059785208, 7.2290596706950154), (5.0970619686963916, 2.4188864705446997), (-1.824501293502089, 0.87811217547665232), (2.6141377553638456, 2.4736768016729647), (-3.9646033676482686, 1.7291507868196327), (-5.6494860793108481, -1.1744278681124489), (3.3291564189715617, 3.1892910878432268), (2.6260111359196396, -4.8029748349762125), (-4.1110554486386404, 0.0087017311510849682), (-3.812034605848817, 1.8310006567642712), (-4.0643824785110239, 4.7806635726760689), (-3.8724397920934015, 0.65927045141188367), (-3.6202135060380289, -0.18281430910806151), (1.8134764145891591, -4.0328054369849538), (4.0315824591034124, -3.5339867923196042), (-3.0906912982614791, -3.8390710019489158), (0.77019164393866146, 4.0099320163703895), (-3.2239134319849398, 2.5227757084315567), (-2.5342615497190861, -4.5402720724503229), (0.52313297572359074, 5.8268409663350287), (-2.0896974241486603, -0.83931337455192145), (5.9824769771009292, 1.8062615072223389), (-1.7151819974072808, -4.6553638508191835), (-0.94296691141453703, -4.3332773280899097), (-2.9080659785364102, 3.8017876981653527), (-4.146797854411842, -2.4943345068020939), (-1.6135304662636716, -4.5968234340599352), (-5.2240732422979015, -0.40050907128273239), (3.0003615064702411, 4.3564534485947091), (1.5251603471425388, 5.3602495377614252), (0.70829180528117897, 4.8705912438690024), (-1.9857439387875215, 4.3495410597763557), (-1.7415118623160484, -2.8482449535792851), (3.1227029816875906, -3.943690794192229), (2.5533372938495322, 0.23654193364300019), (4.9320538122814632, 0.27398085527961841), (3.5379571426787906, 3.5479478416595258), (-3.9952197756192462, 0.9519866242123729), (-0.63418929807710789, 4.9714021509147459), (3.7514419719026835, -3.7952656655539831), (5.8168652955867248, -5.8059389896821614), (-3.86083201462211, 1.6763339473293351), (5.2346287443442741, -2.0049022214331869), (3.0159172780756807, -4.6747832401686313), (1.9625789720275502, 0.21332969214064601), (-5.4459656516053521, 1.8490131071943328), (5.4887755131556295, 1.0537691340713213), (4.1214658457920255, 1.8180419262808878), (1.0417225435808637, 6.4876076903545457), (5.2056831059665383, 3.4403227294912879), (-3.29183542445509, 1.1299087065549616), (-4.6894950904308068, 0.67877427899602139), (-4.2334935303450196, 0.66692066781151726), (6.918359229911677, -0.43825691963852248), (5.0912552685819197, 5.9256467457380193), (3.9995400634925016, -4.2633779062253305), (-1.3270510253578853, -2.8998811026998816), (-3.4372749748248483, -2.800876689538256), (2.5720483206059228, -4.5479241832525954), (3.5107697954439923, -5.6063323885377114), (3.45355690226015, -1.3924594206301864), (4.8170391803389006, -1.3343907023480963), (1.1592191821861308, 4.551692003143347), (-2.2147820707711716, 0.55930561729387951), (-3.2364813901253862, -1.7059292544869302), (3.5980046177747229, 3.0606302788023871), (3.0235041652892747, -0.27015781708378661), (2.4303330714757383, 3.3989583334332432), (2.4649562148782955, -4.3524552397826168), (-3.3322237797463616, 1.6813558717119386), (4.3359544685337736, 2.7104894884469877), (3.350410042767797, -3.8412188670946792), (-2.8993273426849919, 5.5101185505218293), (3.3563537615645282, 5.0439247587050282), (3.3738404946436238, -0.43277784903448813), (1.6236691719193734, -4.8192122194763103), (-4.3000303214498619, 2.7045156595962521), (3.2036876689968699, -0.22379027409222038), (5.0078193725337679, -0.33061456656172339), (1.3173753727230917, 2.3292728936983247), (0.17305051546078376, -2.3708524146324814), (0.18920570140751003, 2.7288547711089577), (4.5559793038807355, 2.4460955268542377), (-0.65537111745445098, 4.3024274811626642), (0.32733974310015845, -2.6653194005399481), (-4.3495524342659682, 0.50620561077402126), (3.6859406925109957, 1.0042337939426813), (-5.4168309661540643, -2.3784247121303279), (2.0873449293614152, 3.8206900404120345), (3.3397623772131446, -2.2347446764630474), (2.8720948774765485, 2.6955132035521556), (5.9472576652843694, -3.3542922693748149), (1.030233796538444, 1.6199282129862145), (-1.7351581782776853, -5.5709314373179808), (0.14607908112131446, 2.79251837326064), (0.37002429983216167, 4.3653059393186942), (3.8616789948811956, 3.6100436336617339), (-4.8019087210485418, -3.5911421188072357), (1.6953052292111459, -4.3928959775316905), (-1.049532260408768, -2.9169000088107522), (-4.8042700374731648, -2.6636201843555991), (2.2856117402115821, -4.497386564362329), (-1.1085015582769402, -4.1635806015318408), (-0.51764720743541925, 3.3207617687324866), (2.6552485122750968, 1.9457154950840061), (4.4574030967957459, 0.13220998701481373), (4.1064026703010086, -4.6992062016898437), (3.6218017958370492, 2.4171784152426357), (2.1893570148164336, -0.53987360896641756), (-0.62289304323418893, 5.6377915319211773), (0.95656595366184183, -3.5482370903224183), (4.6552715153624238, -0.42419842122106877), (3.9138981541477369, 1.5211086418661788), (-5.7643908686171743, 3.3462875243179644), (4.4001664954474204, 1.8715548148469952), (3.7209034976257116, -4.3132712976844925), (2.0077653108424371, -3.8044349295045858), (-2.7004396541700451, 3.6313151291578776), (2.7805282578575432, -1.3496033840422226), (2.5149407509344646, -4.4491799573779538), (-3.4969549443875327, 0.59052341158001964), (2.5871839418980924, -2.8626995345211439), (4.530084220131168, 0.73947783901217035), (-4.2278934560638541, -1.4480933790189707), (-3.6638968948801822, -1.8603129450393652), (1.0034748779660814, 4.3783603559660618), (-0.24711046251746965, 5.0245225170472958), (-0.75233017871629115, -3.4003624728787472), (-5.3204808270534789, 0.8530050107548528), (-0.66555456366565435, -3.210607962975542), (4.4312598575388913, -1.8510534338146063), (-1.0579141292803367, -3.8599892658343156), (5.1580465239922022, -1.6376354853614972), (-2.6525127599513731, 2.9406618825179196), (3.3353268107001339, 4.5193520805659642), (4.9838132614191322, -4.5937246171656669)]
from mlstatpy.graph.graphviz_helper import draw_graph_graphviz
draw_graph_graphviz(noeuds, arcs, "image.png")
from IPython.display import Image
Image("image.png", width=400)
../_images/graph1exo_parcours_correction_3_0.png
class MonGraphe :
    def __init__ (self, num, valeur):
        self.valeur = valeur
        self.num = num
        self.arcs = [] # liste d’éléments MonGraphe

Q1

def ConstruireMonGraphe(noeuds, arcs) :
    nodes = {}
    for i,v in noeuds.items():
        nodes [i] = MonGraphe(i,v)
    for k in arcs:
        i,j = k
        nodes[i].arcs.append(nodes[j])
        nodes[j].arcs.append(nodes[i])
    return nodes[0]
ConstruireMonGraphe(noeuds, arcs)
<__main__.MonGraphe at 0x180b6285518>

Q2

La fonction est récursive, elle parcourt le graphe en profondeur et peut repasser plusieurs fois par le même noeud.

# déclenche une exception
class MonGraphe:
    def __init__ (self, num, valeur):
        self.valeur = valeur
        self.num = num
        self.arcs = [] # liste d’éléments MonGraphe
        self.passage = 0
    def NombreTotalNoeud(self):
        self.passage += 1
        if self.passage > 3:
            if self.passage == 10:
                print("Trop de passages, on s'arrête.")
            self.passage += 1
            return 0
        else:
            r = 1
            for a in self.arcs:
                r += a.NombreTotalNoeud()
            return r

g = ConstruireMonGraphe(noeuds, arcs)
g.NombreTotalNoeud()
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
Trop de passages, on s'arrête.
150

Q3

La fonction retourne le bon résultat pour les graphes orientés et acyclique. C’est la seule façon de ne jamais repasser deux fois par le même noeud.

Q4

Version non récursive.

def NombreArcsRecursive (r, edges=None, done=None):
    if edges is None:
        edges = {}
    if done is None:
        done = {}
    if r.num in done:
        return 0
    done[r.num] = None
    for node in r.arcs:
        edges[r.num, node.num] = None
        NombreArcsRecursive(node, edges, done)

    final = { }
    for k in arcs:
        i,j = k
        if i != j and (j,i) not in final:
            final[i,j] = None

    return len(final)

g = ConstruireMonGraphe(noeuds, arcs)
NombreArcsRecursive(g)
73

Version non récursive

def NombreArcs(racine) :
    edges = {}
    pile = [ ]
    done = { }
    pile.append (racine)
    while len(pile) > 0 :
        r               = pile.pop ()
        if r.num in done : continue
        done [r.num]    = None
        for node in r.arcs :
            if node.num in done : continue
            edges [ r.num, node.num ] = None
            pile.append (node)

    final = { }
    for k in arcs :
        i,j = k
        if i != j and (j,i) not in final:
            final [i,j] = None

    return len(final)

g = ConstruireMonGraphe(noeuds, arcs)
NombreArcs(g)
73