Реализация на python.

В листинге 3 приводится пример реализации решения примера, описанного выше, на языке программирования python.

Листинг 3.

def improveLabels(val):

""" change the labels, and maintain minSlack.

"""

for u in S:

lu[u] -= val

for v in V:

if v in T:

lv[v] += val

else:

minSlack[v][0] -= val

def improveMatching(v):

""" apply the alternating path from v to the root in the tree.

"""

u = T[v]

if u in Mu:

improveMatching(Mu[u])

Mu[u] = v

Mv[v] = u

def slack(u,v): return lu[u]+lv[v]-w[u][v]

def augment():

""" augment the matching, possibly improving the lablels on the way.

"""

while True:

# select edge (u,v) with u in S, v not in T and min slack

((val, u), v) = min([(minSlack[v], v) for v in V if v not in T])

assert u in S

if val>0:

improveLabels(val)

# now we are sure that (u,v) is saturated

assert slack(u,v)==0

T[v] = u # add (u,v) to the tree

if v in Mv:

u1 = Mv[v] # matched edge,

assert not u1 in S

S[u1] = True # ... add endpoint to tree

for v in V: # maintain minSlack

if not v in T and minSlack[v][0] > slack(u1,v):

minSlack[v] = [slack(u1,v), u1]

else:

improveMatching(v) # v is a free vertex

return

def maxWeightMatching(weights):

""" given w, the weight matrix of a complete bipartite graph,

returns the mappings Mu : U->V ,Mv : V->U encoding the matching

as well as the value of it.

"""

global U,V,S,T,Mu,Mv,lu,lv, minSlack, w

w = weights

n = len(w)

U = V = range(n)

lu = [ max([w[u][v] for v in V]) for u in U] # start with trivial labels

lv = [ 0 for v in V]

Mu = {} # start with empty matching

Mv = {}

while len(Mu)<n:

free = [u for u in V if u not in Mu] # choose free vertex u0

u0 = free[0]

S = {u0: True} # grow tree from u0 on

T = {}

minSlack = [[slack(u0,v), u0] for v in V]

augment()

# val. of matching is total edge weight

val = sum(lu)+sum(lv)

return (Mu, Mv, val)

# a small example

#print maxWeightMatching([[1,2,3,4],[2,4,6,8],[3,6,9,12],[4,8,12,16]])

# read from standard input a line with n

# then n*n lines with u,v,w[u][v]

n = 3 #Размер матрицы

w = [[1, 2, 4], #Матрица весов

[2, 5, 3],

[6, 7, 8]]

print(maxWeightMatching(w))

Входные данные:

n = 3 #Размер матрицы

w = [[1, 2, 4], #Матрица весов

[2, 5, 3],

[6, 7, 8]]

Выходные данные:

({0: 2, 1: 1, 2: 0}, {0: 2, 1: 1, 2: 0}, 15)

Вывод.

Матричное представление графов – наиболее универсальное представление графов в памяти эвм, для дальнейшей их обработки и решения разного рода задач.