# 1A.algo - Parcours de graphe - correction¶

Algorithme assez simple sur les graphes (correction).

```from jyquickhelper import 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)
```
```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
```