def bipartite_vertex_cover(bigraph):
"""Bipartite minimum vertex cover by Koenig's theorem
:param bigraph: adjacency list, index = vertex in U,
value = neighbor list in V
:assumption: U = V = {0, 1, 2, ..., n - 1} for n = len(bigraph)
:returns: boolean table for U, boolean table for V
:comment: selected vertices form a minimum vertex cover,
i.e. every edge is adjacent to at least one selected vertex
and number of selected vertices is minimum
:complexity: `O(|V|*|E|)`
"""
V = range(len(bigraph))
matchV = max_bipartite_matching(bigraph)
matchU = [None for u in V]
for v in V: # -- build the mapping from U to V
if matchV[v] is not None:
matchU[matchV[v]] = v
visitU = [False for u in V] # -- build max alternating forest
visitV = [False for v in V]
for u in V:
if matchU[u] is None: # -- starting with free vertices in U
_alternate(u, bigraph, visitU, visitV, matchV)
inverse = [not b for b in visitU]
return (inverse, visitV)
def bipartite_vertex_cover(bigraph):
"""Bipartie minimum vertex cover by Koenig's theorem
:param bigraph: adjacency list, index = vertex in U,
value = neighbor list in V
:assumption: U = V = {0, 1, 2, ..., n - 1} for n = len(bigraph)
:returns: boolean table for U, boolean table for V
:comment: selected vertices form a minimum vertex cover,
i.e. every edge is adjacent to at least one selected vertex
and number of selected vertices is minimum
:complexity: `O(|V|*|E|)`
"""
V = range(len(bigraph))
matchV = max_bipartite_matching(bigraph)
matchU = [None for u in V]
for v in V: # -- build the mapping from U to V
if matchV[v] is not None:
matchU[matchV[v]] = v
visitU = [False for u in V] # -- build max alternating forest
visitV = [False for v in V]
for u in V:
if matchU[u] is None: # -- starting with free vertices in U
_alternate(u, bigraph, visitU, visitV, matchV)
inverse = [not b for b in visitU]
return (inverse, visitV)
def dilworth(graph):
"""Decompose a DAG into a minimum number of chains by Dilworth
:param graph: directed graph in listlist or listdict format
:assumes: graph is acyclic
:returns: table giving for each vertex the number of its chains
:complexity: same as matching
"""
n = len(graph)
match = max_bipartite_matching(graph) # couplage maximum
part = [None] * n # partition en chaînes
nb_chains = 0
for v in range(n - 1, -1, -1): # dans l'ordre topologique inverse
if part[v] is None: # début d'une chaîne
u = v
while u is not None: # suivre la chaîne
part[u] = nb_chains # marquer
u = match[u]
nb_chains += 1
return part
def dilworth(graph):
"""Decompose a DAG into a minimum number of chains by Dilworth
:param graph: directed graph in listlist or listdict format
:assumes: graph is acyclic
:returns: table giving for each vertex the number of its chains
:complexity: same as matching
"""
n = len(graph)
match = max_bipartite_matching(graph) # maximum matching
part = [None] * n # partition into chains
nb_chains = 0
for v in range(n - 1, -1, -1): # in inverse topological order
if part[v] is None: # start of chain
u = v
while u is not None: # follow the chain
part[u] = nb_chains # mark
u = match[u]
nb_chains += 1
return part
