"""
********
Matching
********
The algorithm is taken from "Efficient Algorithms for Finding Maximum
Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
It is based on the "blossom" method for finding augmenting paths and
the "primal-dual" method for finding a matching of maximum weight, both
methods invented by Jack Edmonds.
"""
__author__ = """Joris van Rantwijk"""
# Copyright (C) 2004-2008 by
# Aric Hagberg
# Dan Schult
# Pieter Swart
# Distributed under the terms of the GNU Lesser General Public License
# http://www.gnu.org/copyleft/lesser.html
#
from itertools import repeat
_all__=['max_weight_matching']
def max_weight_matching(G, maxcardinality=False):
"""Compute a maximum-weighted matching in the undirected, weighted graph G.
If maxcardinality is True, compute the maximum-cardinality matching
with maximum weight among all maximum-cardinality matchings.
The matching is returned as a dictionary, mate, such that
mate[v] == w if node v is matched to node w. Unmatched nodes do not
occur as a key in mate.
A matching is a subset of edges in which no node occurs more than once.
The cardinality of a matching is the number of matched edges.
The weight of a matching is the sum of the weights of its edges.
If G is a weighted Graph, the edge data are used as weight values;
if G.weighted == False, all edge weights are taken to be 1.
Directed graphs and multi-edge graphs are not supported.
This function takes time O(number_of_nodes ** 3).
If all edge weights are integers, the algorithm uses only integer
computations. If floating point weights are used, the algorithm could
return a slightly suboptimal matching due to numeric precision errors.
"""
#
# The algorithm is taken from "Efficient Algorithms for Finding Maximum
# Matching in Graphs" by Zvi Galil, ACM Computing Surveys, 1986.
# It is based on the "blossom" method for finding augmenting paths and
# the "primal-dual" method for finding a matching of maximum weight, both
# methods invented by Jack Edmonds.
#
# A C program for maximum weight matching by Ed Rothberg was used
# extensively to validate this new code.
#
# Many terms used in the code comments are explained in the paper
# by Galil. You will probably need the paper to make sense of this code.
#
class NoNode:
"""Dummy value which is different from any node."""
pass
class Blossom:
"""Representation of a non-trivial blossom or sub-blossom."""
__slots__ = [ 'childs', 'edges', 'mybestedges' ]
# b.childs is an ordered list of b's sub-blossoms, starting with
# the base and going round the blossom.
# b.edges is the list of b's connecting edges, such that
# b.edges[i] = (v, w) where v is a vertex in b.childs[i]
# and w is a vertex in b.childs[wrap(i+1)].
# If b is a top-level S-blossom,
# b.mybestedges is a list of least-slack edges to neighbouring
# S-blossoms, or None if no such list has been computed yet.
# This is used for efficient computation of delta3.
# Generate the blossom's leaf vertices.
def leaves(self):
for t in self.childs:
if isinstance(t, Blossom):
for v in t.leaves():
yield v
else:
yield t
# Get a list of vertices.
gnodes = G.nodes()
if not gnodes:
return { } # don't bother with empty graphs
# Find the maximum edge weight.
maxweight = 0
allinteger = True
for (i, j, wt) in G.edges_iter(data=True):
if not G.weighted:
wt = 1
if i != j and wt > maxweight:
maxweight = wt
allinteger = allinteger and (type(wt) in (int, long))
# If v is a matched vertex, mate[v] is its partner vertex.
# If v is a single vertex, v does not occur as a key in mate.
# Initially all vertices are single; updated during augmentation.
mate = { }
# If b is a top-level blossom,
# label.get(b) is None if b is unlabeled (free),
# 1 if b is an S-blossom,
# 2 if b is a T-blossom.
# The label of a vertex is found by looking at the label of its top-level
# containing blossom.
# If v is a vertex inside a T-blossom, label[v] is 2 iff v is reachable
# from an S-vertex outside the blossom.
# Labels are assigned during a stage and reset after each augmentation.
label = { }
# If b is a labeled top-level blossom,
# labeledge[b] = (v, w) is the edge through which b obtained its label
# such that w is a vertex in b, or None if b's base vertex is single.
# If w is a vertex inside a T-blossom and label[w] == 2,
# labeledge[w] = (v, w) is an edge through which w is reachable from
# outside the blossom.
labeledge = { }
# If v is a vertex, inblossom[v] is the top-level blossom to which v
# belongs.
# If v is a top-level vertex, inblossom[v] == v since v is itself
# a (trivial) top-level blossom.
# Initially all vertices are top-level trivial blossoms.
inblossom = dict(zip(gnodes, gnodes))
# If b is a sub-blossom,
# blossomparent[b] is its immediate parent (sub-)blossom.
# If b is a top-level blossom, blossomparent[b] is None.
blossomparent = dict(zip(gnodes, repeat(None)))
# If b is a (sub-)blossom,
# blossombase[b] is its base VERTEX (i.e. recursive sub-blossom).
blossombase = dict(zip(gnodes, gnodes))
# If w is a free vertex (or an unreached vertex inside a T-blossom),
# bestedge[w] = (v, w) is the least-slack edge from an S-vertex,
# or None if there is no such edge.
# If b is a (possibly trivial) top-level S-blossom,
# bestedge[b] = (v, w) is the least-slack edge to a different S-blossom
# (v inside b), or None if there is no such edge.
# This is used for efficient computation of delta2 and delta3.
bestedge = { }
# If v is a vertex,
# dualvar[v] = 2 * u(v) where u(v) is the v's variable in the dual
# optimization problem (if all edge weights are integers, multiplication
# by two ensures that all values remain integers throughout the algorithm).
# Initially, u(v) = maxweight / 2.
dualvar = dict(zip(gnodes, repeat(maxweight)))
# If b is a non-trivial blossom,
# blossomdual[b] = z(b) where z(b) is b's variable in the dual
# optimization problem.
blossomdual = { }
# If (v, w) in allowedge or (w, v) in allowedge, then the edge
# (v, w) is known to have zero slack in the optimization problem;
# otherwise the edge may or may not have zero slack.
allowedge = { }
# Queue of newly discovered S-vertices.
queue = [ ]
# Return 2 * slack of edge (v, w) (does not work inside blossoms).
if G.weighted:
def slack(v, w):
return dualvar[v] + dualvar[w] - 2 * G[v][w]
else:
def slack(v, w):
return dualvar[v] + dualvar[w] - 2
# Assign label t to the top-level blossom containing vertex w,
# coming through an edge from vertex v.
def assignLabel(w, t, v):
b = inblossom[w]
assert label.get(w) is None and label.get(b) is None
label[w] = label[b] = t
if v is not None:
labeledge[w] = labeledge[b] = (v, w)
else:
labeledge[w] = labeledge[b] = None
bestedge[w] = bestedge[b] = None
if t == 1:
# b became an S-vertex/blossom; add it(s vertices) to the queue.
if isinstance(b, Blossom):
queue.extend(b.leaves())
else:
queue.append(b)
elif t == 2:
# b became a T-vertex/blossom; assign label S to its mate.
# (If b is a non-trivial blossom, its base is the only vertex
# with an external mate.)
base = blossombase[b]
assignLabel(mate[base], 1, base)
# Trace back from vertices v and w to discover either a new blossom
# or an augmenting path. Return the base vertex of the new blossom,
# or NoNode if an augmenting path was found.
def scanBlossom(v, w):
# Trace back from v and w, placing breadcrumbs as we go.
path = [ ]
base = NoNode
while v is not NoNode:
# Look for a breadcrumb in v's blossom or put a new breadcrumb.
b = inblossom[v]
if label[b] & 4:
base = blossombase[b]
break
assert label[b] == 1
path.append(b)
label[b] = 5
# Trace one step back.
if labeledge[b] is None:
# The base of blossom b is single; stop tracing this path.
assert blossombase[b] not in mate
v = NoNode
else:
assert labeledge[b][0] == mate[blossombase[b]]
v = labeledge[b][0]
b = inblossom[v]
assert label[b] == 2
# b is a T-blossom; trace one more step back.
v = labeledge[b][0]
# Swap v and w so that we alternate between both paths.
if w is not NoNode:
v, w = w, v
# Remove breadcrumbs.
for b in path:
label[b] = 1
# Return base vertex, if we found one.
return base
# Construct a new blossom with given base, through S-vertices v and w.
# Label the new blossom as S; set its dual variable to zero;
# relabel its T-vertices to S and add them to the queue.
def addBlossom(base, v, w):
bb = inblossom[base]
bv = inblossom[v]
bw = inblossom[w]
# Create blossom.
b = Blossom()
blossombase[b] = base
blossomparent[b] = None
blossomparent[bb] = b
# Make list of sub-blossoms and their interconnecting edge endpoints.
b.childs = path = [ ]
b.edges = edgs = [ (v, w) ]
# Trace back from v to base.
while bv != bb:
# Add bv to the new blossom.
blossomparent[bv] = b
path.append(bv)
edgs.append(labeledge[bv])
assert label[bv] == 2 or (label[bv] == 1 and labeledge[bv][0] == mate[blossombase[bv]])
# Trace one step back.
v = labeledge[bv][0]
bv = inblossom[v]
# Add base sub-blossom; reverse lists.
path.append(bb)
path.reverse()
edgs.reverse()
# Trace back from w to base.
while bw != bb:
# Add bw to the new blossom.
blossomparent[bw] = b
path.append(bw)
edgs.append((labeledge[bw][1], labeledge[bw][0]))
assert label[bw] == 2 or (label[bw] == 1 and labeledge[bw][0] == mate[blossombase[bw]])
# Trace one step back.
w = labeledge[bw][0]
bw = inblossom[w]
# Set label to S.
assert label[bb] == 1
label[b] = 1
labeledge[b] = labeledge[bb]
# Set dual variable to zero.
blossomdual[b] = 0
# Relabel vertices.
for v in b.leaves():
if label[inblossom[v]] == 2:
# This T-vertex now turns into an S-vertex because it becomes
# part of an S-blossom; add it to the queue.
queue.append(v)
inblossom[v] = b
# Compute b.mybestedges.
bestedgeto = { }
for bv in path:
if isinstance(bv, Blossom):
if bv.mybestedges is not None:
# Walk this subblossom's least-slack edges.
nblist = bv.mybestedges
# The sub-blossom won't need this data again.
bv.mybestedges = None
else:
# This subblossom does not have a list of least-slack
# edges; get the information from the vertices.
nblist = [ (v, w)
for v in bv.leaves()
for w in G.neighbors_iter(v)
if v != w ]
else:
nblist = [ (bv, w)
for w in G.neighbors_iter(bv)
if bv != w ]
for k in nblist:
(i, j) = k
if inblossom[j] == b:
i, j = j, i
bj = inblossom[j]
if (bj != b and label.get(bj) == 1 and
((bj not in bestedgeto) or
slack(i, j) < slack(*bestedgeto[bj]))):
bestedgeto[bj] = k
# Forget about least-slack edge of the subblossom.
bestedge[bv] = None
b.mybestedges = bestedgeto.values()
# Select bestedge[b].
mybestedge = None
bestedge[b] = None
for k in b.mybestedges:
kslack = slack(*k)
if mybestedge is None or kslack < mybestslack:
mybestedge = k
mybestslack = kslack
bestedge[b] = mybestedge
# Expand the given top-level blossom.
def expandBlossom(b, endstage):
# Convert sub-blossoms into top-level blossoms.
for s in b.childs:
blossomparent[s] = None
if isinstance(s, Blossom):
if endstage and blossomdual[s] == 0:
# Recursively expand this sub-blossom.
expandBlossom(s, endstage)
else:
for v in s.leaves():
inblossom[v] = s
else:
inblossom[s] = s
# If we expand a T-blossom during a stage, its sub-blossoms must be
# relabeled.
if (not endstage) and label.get(b) == 2:
# Start at the sub-blossom through which the expanding
# blossom obtained its label, and relabel sub-blossoms untili
# we reach the base.
# Figure out through which sub-blossom the expanding blossom
# obtained its label initially.
entrychild = inblossom[labeledge[b][1]]
# Decide in which direction we will go round the blossom.
j = b.childs.index(entrychild)
if j & 1:
# Start index is odd; go forward and wrap.
j -= len(b.childs)
jstep = 1
else:
# Start index is even; go backward.
jstep = -1
# Move along the blossom until we get to the base.
v, w = labeledge[b]
while j != 0:
# Relabel the T-sub-blossom.
if jstep == 1:
p, q = b.edges[j]
else:
q, p = b.edges[j-1]
label[w] = None
label[q] = None
assignLabel(w, 2, v)
# Step to the next S-sub-blossom and note its forward edge.
allowedge[(p, q)] = allowedge[(q, p)] = True
j += jstep
if jstep == 1:
v, w = b.edges[j]
else:
w, v = b.edges[j-1]
# Step to the next T-sub-blossom.
allowedge[(v, w)] = allowedge[(w, v)] = True
j += jstep
# Relabel the base T-sub-blossom WITHOUT stepping through to
# its mate (so don't call assignLabel).
bw = b.childs[j]
label[w] = label[bw] = 2
labeledge[w] = labeledge[bw] = (v, w)
bestedge[bw] = None
# Continue along the blossom until we get back to entrychild.
j += jstep
while b.childs[j] != entrychild:
# Examine the vertices of the sub-blossom to see whether
# it is reachable from a neighbouring S-vertex outside the
# expanding blossom.
bv = b.childs[j]
if label.get(bv) == 1:
# This sub-blossom just got label S through one of its
# neighbours; leave it be.
j += jstep
continue
if isinstance(bv, Blossom):
for v in bv.leaves():
if label.get(v):
break
else:
v = bv
# If the sub-blossom contains a reachable vertex, assign
# label T to the sub-blossom.
if label.get(v):
assert label[v] == 2
assert inblossom[v] == bv
label[v] = None
label[mate[blossombase[bv]]] = None
assignLabel(v, 2, labeledge[v][0])
j += jstep
# Remove the expanded blossom entirely.
label.pop(b, None)
labeledge.pop(b, None)
bestedge.pop(b, None)
del blossomparent[b]
del blossombase[b]
del blossomdual[b]
# Swap matched/unmatched edges over an alternating path through blossom b
# between vertex v and the base vertex. Keep blossom bookkeeping consistent.
def augmentBlossom(b, v):
# Bubble up through the blossom tree from vertex v to an immediate
# sub-blossom of b.
t = v
while blossomparent[t] != b:
t = blossomparent[t]
# Recursively deal with the first sub-blossom.
if isinstance(t, Blossom):
augmentBlossom(t, v)
# Decide in which direction we will go round the blossom.
i = j = b.childs.index(t)
if i & 1:
# Start index is odd; go forward and wrap.
j -= len(b.childs)
jstep = 1
else:
# Start index is even; go backward.
jstep = -1
# Move along the blossom until we get to the base.
while j != 0:
# Step to the next sub-blossom and augment it recursively.
j += jstep
t = b.childs[j]
if jstep == 1:
w, x = b.edges[j]
else:
x, w = b.edges[j-1]
if isinstance(t, Blossom):
augmentBlossom(t, w)
# Step to the next sub-blossom and augment it recursively.
j += jstep
t = b.childs[j]
if isinstance(t, Blossom):
augmentBlossom(t, x)
# Match the edge connecting those sub-blossoms.
mate[w] = x
mate[x] = w
# Rotate the list of sub-blossoms to put the new base at the front.
b.childs = b.childs[i:] + b.childs[:i]
b.edges = b.edges[i:] + b.edges[:i]
blossombase[b] = blossombase[b.childs[0]]
assert blossombase[b] == v
# Swap matched/unmatched edges over an alternating path between two
# single vertices. The augmenting path runs through S-vertices v and w.
def augmentMatching(v, w):
for (s, j) in ((v, w), (w, v)):
# Match vertex s to vertex j. Then trace back from s
# until we find a single vertex, swapping matched and unmatched
# edges as we go.
while 1:
bs = inblossom[s]
assert label[bs] == 1
assert (labeledge[bs] is None and blossombase[bs] not in mate) or (labeledge[bs][0] == mate[blossombase[bs]])
# Augment through the S-blossom from s to base.
if isinstance(bs, Blossom):
augmentBlossom(bs, s)
# Update mate[s]
mate[s] = j
# Trace one step back.
if labeledge[bs] is None:
# Reached single vertex; stop.
break
t = labeledge[bs][0]
bt = inblossom[t]
assert label[bt] == 2
# Trace one more step back.
s, j = labeledge[bt]
# Augment through the T-blossom from j to base.
assert blossombase[bt] == t
if isinstance(bt, Blossom):
augmentBlossom(bt, j)
# Update mate[j]
mate[j] = s
# Verify that the optimum solution has been reached.
def verifyOptimum():
if maxcardinality:
# Vertices may have negative dual;
# find a constant non-negative number to add to all vertex duals.
vdualoffset = max(0, -min(dualvar.values()))
else:
vdualoffset = 0
# 0. all dual variables are non-negative
assert min(dualvar.values()) + vdualoffset >= 0
assert len(blossomdual) == 0 or min(blossomdual.values()) >= 0
# 0. all edges have non-negative slack and
# 1. all matched edges have zero slack;
for (i, j, wt) in G.edges_iter(data=True):
if not G.weighted:
wt = 1
if i == j:
continue # ignore self-loops
s = dualvar[i] + dualvar[j] - 2 * wt
iblossoms = [ i ]
jblossoms = [ j ]
while blossomparent[iblossoms[-1]] is not None:
iblossoms.append(blossomparent[iblossoms[-1]])
while blossomparent[jblossoms[-1]] is not None:
jblossoms.append(blossomparent[jblossoms[-1]])
iblossoms.reverse()
jblossoms.reverse()
for (bi, bj) in zip(iblossoms, jblossoms):
if bi != bj:
break
s += 2 * blossomdual[bi]
assert s >= 0
if mate.get(i) == j or mate.get(j) == i:
assert mate[i] == j and mate[j] == i
assert s == 0
# 2. all single vertices have zero dual value;
for v in gnodes:
assert (v in mate) or dualvar[v] + vdualoffset == 0
# 3. all blossoms with positive dual value are full.
for b in blossomdual:
if blossomdual[b] > 0:
assert len(b.edges) % 2 == 1
for (i, j) in b.edges[1::2]:
assert mate[i] == j and mate[j] == i
# Ok.
# Main loop: continue until no further improvement is possible.
while 1:
# Each iteration of this loop is a "stage".
# A stage finds an augmenting path and uses that to improve
# the matching.
# Remove labels from top-level blossoms/vertices.
label.clear()
labeledge.clear()
# Forget all about least-slack edges.
bestedge.clear()
for b in blossomdual:
b.mybestedges = None
# Loss of labeling means that we can not be sure that currently
# allowable edges remain allowable througout this stage.
allowedge.clear()
# Make queue empty.
queue[:] = [ ]
# Label single blossoms/vertices with S and put them in the queue.
for v in gnodes:
if (v not in mate) and label.get(inblossom[v]) is None:
assignLabel(v, 1, None)
# Loop until we succeed in augmenting the matching.
augmented = 0
while 1:
# Each iteration of this loop is a "substage".
# A substage tries to find an augmenting path;
# if found, the path is used to improve the matching and
# the stage ends. If there is no augmenting path, the
# primal-dual method is used to pump some slack out of
# the dual variables.
# Continue labeling until all vertices which are reachable
# through an alternating path have got a label.
while queue and not augmented:
# Take an S vertex from the queue.
v = queue.pop()
assert label[inblossom[v]] == 1
# Scan its neighbours:
for w in G.neighbors_iter(v):
if w == v:
continue # ignore self-loops
# w is a neighbour to v
bv = inblossom[v]
bw = inblossom[w]
if bv == bw:
# this edge is internal to a blossom; ignore it
continue
if (v, w) not in allowedge:
kslack = slack(v, w)
if kslack <= 0:
# edge k has zero slack => it is allowable
allowedge[(v, w)] = allowedge[(w, v)] = True
if (v, w) in allowedge:
if label.get(bw) is None:
# (C1) w is a free vertex;
# label w with T and label its mate with S (R12).
assignLabel(w, 2, v)
elif label.get(bw) == 1:
# (C2) w is an S-vertex (not in the same blossom);
# follow back-links to discover either an
# augmenting path or a new blossom.
base = scanBlossom(v, w)
if base is not NoNode:
# Found a new blossom; add it to the blossom
# bookkeeping and turn it into an S-blossom.
addBlossom(base, v, w)
else:
# Found an augmenting path; augment the
# matching and end this stage.
augmentMatching(v, w)
augmented = 1
break
elif label.get(w) is None:
# w is inside a T-blossom, but w itself has not
# yet been reached from outside the blossom;
# mark it as reached (we need this to relabel
# during T-blossom expansion).
assert label[bw] == 2
label[w] = 2
labeledge[w] = (v, w)
elif label.get(bw) == 1:
# keep track of the least-slack non-allowable edge to
# a different S-blossom.
if bestedge.get(bv) is None or kslack < slack(*bestedge[bv]):
bestedge[bv] = (v, w)
elif label.get(w) is None:
# w is a free vertex (or an unreached vertex inside
# a T-blossom) but we can not reach it yet;
# keep track of the least-slack edge that reaches w.
if bestedge.get(w) is None or kslack < slack(*bestedge[w]):
bestedge[w] = (v, w)
if augmented:
break
# There is no augmenting path under these constraints;
# compute delta and reduce slack in the optimization problem.
# (Note that our vertex dual variables, edge slacks and delta's
# are pre-multiplied by two.)
deltatype = -1
delta = deltaedge = deltablossom = None
# Compute delta1: the minumum value of any vertex dual.
if not maxcardinality:
deltatype = 1
delta = min(dualvar.values())
# Compute delta2: the minimum slack on any edge between
# an S-vertex and a free vertex.
for v in G.nodes_iter():
if label.get(inblossom[v]) is None and bestedge.get(v) is not None:
d = slack(*bestedge[v])
if deltatype == -1 or d < delta:
delta = d
deltatype = 2
deltaedge = bestedge[v]
# Compute delta3: half the minimum slack on any edge between
# a pair of S-blossoms.
for b in blossomparent:
if ( blossomparent[b] is None and label.get(b) == 1 and
bestedge.get(b) is not None ):
kslack = slack(*bestedge[b])
if allinteger:
assert (kslack % 2) == 0
d = kslack // 2
else:
d = kslack / 2.0
if deltatype == -1 or d < delta:
delta = d
deltatype = 3
deltaedge = bestedge[b]
# Compute delta4: minimum z variable of any T-blossom.
for b in blossomdual:
if ( blossomparent[b] is None and label.get(b) == 2 and
(deltatype == -1 or blossomdual[b] < delta) ):
delta = blossomdual[b]
deltatype = 4
deltablossom = b
if deltatype == -1:
# No further improvement possible; max-cardinality optimum
# reached. Do a final delta update to make the optimum
# verifyable.
assert maxcardinality
deltatype = 1
delta = max(0, min(dualvar.values()))
# Update dual variables according to delta.
for v in gnodes:
if label.get(inblossom[v]) == 1:
# S-vertex: 2*u = 2*u - 2*delta
dualvar[v] -= delta
elif label.get(inblossom[v]) == 2:
# T-vertex: 2*u = 2*u + 2*delta
dualvar[v] += delta
for b in blossomdual:
if blossomparent[b] is None:
if label.get(b) == 1:
# top-level S-blossom: z = z + 2*delta
blossomdual[b] += delta
elif label.get(b) == 2:
# top-level T-blossom: z = z - 2*delta
blossomdual[b] -= delta
# Take action at the point where minimum delta occurred.
if deltatype == 1:
# No further improvement possible; optimum reached.
break
elif deltatype == 2:
# Use the least-slack edge to continue the search.
(v, w) = deltaedge
assert label[inblossom[v]] == 1
allowedge[(v, w)] = allowedge[(w, v)] = True
queue.append(v)
elif deltatype == 3:
# Use the least-slack edge to continue the search.
(v, w) = deltaedge
allowedge[(v, w)] = allowedge[(w, v)] = True
assert label[inblossom[v]] == 1
queue.append(v)
elif deltatype == 4:
# Expand the least-z blossom.
expandBlossom(deltablossom, False)
# End of a this substage.
# Paranoia check that the matching is symmetric.
for v in mate:
assert mate[mate[v]] == v
# Stop when no more augmenting path can be found.
if not augmented:
break
# End of a stage; expand all S-blossoms which have zero dual.
for b in blossomdual.keys():
if b not in blossomdual:
continue # already expanded
if ( blossomparent[b] is None and label.get(b) == 1 and
blossomdual[b] == 0 ):
expandBlossom(b, True)
# Verify that we reached the optimum solution (only for integer weights).
if allinteger:
verifyOptimum()
return mate