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# ecole polytechnique - c.durr - 2009
# Kuhn-Munkres, The hungarian algorithm. Complexity O(n^3)
# Computes a max weight perfect matching in a bipartite graph
# for min weight matching, simply negate the weights.
""" Global variables:
n = number of vertices on each side
U,V vertex sets
lu,lv are the labels of U and V resp.
the matching is encoded as
- a mapping Mu from U to V,
- and Mv from V to U.
The algorithm repeatedly builds an alternating tree, rooted in a
free vertex u0. S is the set of vertices in U covered by the tree.
For every vertex v, T[v] is the parent in the tree and Mv[v] the
The algorithm maintains minSlack, s.t. for every vertex v not in
T, minSlack[v]=(val,u1), where val is the minimum slack
lu[u]+lv[v]-w[u][v] over u in S, and u1 is the vertex that
realizes this minimum.
Complexity is O(n^3), because there are n iterations in
maxWeightMatching, and each call to augment costs O(n^2). This is
because augment() makes at most n iterations itself, and each
updating of minSlack costs O(n).
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
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:
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
assert val>=0
if val>0:
# 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]
improveMatching(v) # v is a free vertex
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]
# 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]
if __name__=='__main__':
# n = int(raw_input())
# w = [[0 for v in range(n)] for u in range(n)]
# for _ in range(n*n):
# u,v,w[u][v] = map(int, raw_input().split())
# print maxWeightMatching(w)
print maxWeightMatching([[-31, -3, -2, -31, -31], [-31, -31, -2, -4, -31], [-31, -31, -31, -1, -5], [-31, -31, -31, -31, -6], [-31, -31, -31, -31, -31]])
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