316 lines
8.0 KiB
Python
316 lines
8.0 KiB
Python
|
"""
|
||
|
|
==========================
|
||
|
|
Bipartite Graph Algorithms
|
||
|
|
==========================
|
||
|
|
"""
|
||
|
|
import networkx as nx
|
||
|
|
from networkx.algorithms.components import connected_components
|
||
|
|
from networkx.exception import AmbiguousSolution
|
||
|
|
|
||
|
|
__all__ = [
|
||
|
|
"is_bipartite",
|
||
|
|
"is_bipartite_node_set",
|
||
|
|
"color",
|
||
|
|
"sets",
|
||
|
|
"density",
|
||
|
|
"degrees",
|
||
|
|
]
|
||
|
|
|
||
|
|
|
||
|
|
def color(G):
|
||
|
|
"""Returns a two-coloring of the graph.
|
||
|
|
|
||
|
|
Raises an exception if the graph is not bipartite.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
color : dictionary
|
||
|
|
A dictionary keyed by node with a 1 or 0 as data for each node color.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
NetworkXError
|
||
|
|
If the graph is not two-colorable.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms import bipartite
|
||
|
|
>>> G = nx.path_graph(4)
|
||
|
|
>>> c = bipartite.color(G)
|
||
|
|
>>> print(c)
|
||
|
|
{0: 1, 1: 0, 2: 1, 3: 0}
|
||
|
|
|
||
|
|
You can use this to set a node attribute indicating the biparite set:
|
||
|
|
|
||
|
|
>>> nx.set_node_attributes(G, c, "bipartite")
|
||
|
|
>>> print(G.nodes[0]["bipartite"])
|
||
|
|
1
|
||
|
|
>>> print(G.nodes[1]["bipartite"])
|
||
|
|
0
|
||
|
|
"""
|
||
|
|
if G.is_directed():
|
||
|
|
import itertools
|
||
|
|
|
||
|
|
def neighbors(v):
|
||
|
|
return itertools.chain.from_iterable([G.predecessors(v), G.successors(v)])
|
||
|
|
|
||
|
|
else:
|
||
|
|
neighbors = G.neighbors
|
||
|
|
|
||
|
|
color = {}
|
||
|
|
for n in G: # handle disconnected graphs
|
||
|
|
if n in color or len(G[n]) == 0: # skip isolates
|
||
|
|
continue
|
||
|
|
queue = [n]
|
||
|
|
color[n] = 1 # nodes seen with color (1 or 0)
|
||
|
|
while queue:
|
||
|
|
v = queue.pop()
|
||
|
|
c = 1 - color[v] # opposite color of node v
|
||
|
|
for w in neighbors(v):
|
||
|
|
if w in color:
|
||
|
|
if color[w] == color[v]:
|
||
|
|
raise nx.NetworkXError("Graph is not bipartite.")
|
||
|
|
else:
|
||
|
|
color[w] = c
|
||
|
|
queue.append(w)
|
||
|
|
# color isolates with 0
|
||
|
|
color.update(dict.fromkeys(nx.isolates(G), 0))
|
||
|
|
return color
|
||
|
|
|
||
|
|
|
||
|
|
def is_bipartite(G):
|
||
|
|
"""Returns True if graph G is bipartite, False if not.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms import bipartite
|
||
|
|
>>> G = nx.path_graph(4)
|
||
|
|
>>> print(bipartite.is_bipartite(G))
|
||
|
|
True
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
color, is_bipartite_node_set
|
||
|
|
"""
|
||
|
|
try:
|
||
|
|
color(G)
|
||
|
|
return True
|
||
|
|
except nx.NetworkXError:
|
||
|
|
return False
|
||
|
|
|
||
|
|
|
||
|
|
def is_bipartite_node_set(G, nodes):
|
||
|
|
"""Returns True if nodes and G/nodes are a bipartition of G.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
|
||
|
|
nodes: list or container
|
||
|
|
Check if nodes are a one of a bipartite set.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms import bipartite
|
||
|
|
>>> G = nx.path_graph(4)
|
||
|
|
>>> X = set([1, 3])
|
||
|
|
>>> bipartite.is_bipartite_node_set(G, X)
|
||
|
|
True
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
An exception is raised if the input nodes are not distinct, because in this
|
||
|
|
case some bipartite algorithms will yield incorrect results.
|
||
|
|
For connected graphs the bipartite sets are unique. This function handles
|
||
|
|
disconnected graphs.
|
||
|
|
"""
|
||
|
|
S = set(nodes)
|
||
|
|
|
||
|
|
if len(S) < len(nodes):
|
||
|
|
# this should maybe just return False?
|
||
|
|
raise AmbiguousSolution(
|
||
|
|
"The input node set contains duplicates.\n"
|
||
|
|
"This may lead to incorrect results when using it in bipartite algorithms.\n"
|
||
|
|
"Consider using set(nodes) as the input"
|
||
|
|
)
|
||
|
|
|
||
|
|
for CC in (G.subgraph(c).copy() for c in connected_components(G)):
|
||
|
|
X, Y = sets(CC)
|
||
|
|
if not (
|
||
|
|
(X.issubset(S) and Y.isdisjoint(S)) or (Y.issubset(S) and X.isdisjoint(S))
|
||
|
|
):
|
||
|
|
return False
|
||
|
|
return True
|
||
|
|
|
||
|
|
|
||
|
|
def sets(G, top_nodes=None):
|
||
|
|
"""Returns bipartite node sets of graph G.
|
||
|
|
|
||
|
|
Raises an exception if the graph is not bipartite or if the input
|
||
|
|
graph is disconnected and thus more than one valid solution exists.
|
||
|
|
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
|
||
|
|
for further details on how bipartite graphs are handled in NetworkX.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
G : NetworkX graph
|
||
|
|
|
||
|
|
top_nodes : container, optional
|
||
|
|
Container with all nodes in one bipartite node set. If not supplied
|
||
|
|
it will be computed. But if more than one solution exists an exception
|
||
|
|
will be raised.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
X : set
|
||
|
|
Nodes from one side of the bipartite graph.
|
||
|
|
Y : set
|
||
|
|
Nodes from the other side.
|
||
|
|
|
||
|
|
Raises
|
||
|
|
------
|
||
|
|
AmbiguousSolution
|
||
|
|
Raised if the input bipartite graph is disconnected and no container
|
||
|
|
with all nodes in one bipartite set is provided. When determining
|
||
|
|
the nodes in each bipartite set more than one valid solution is
|
||
|
|
possible if the input graph is disconnected.
|
||
|
|
NetworkXError
|
||
|
|
Raised if the input graph is not bipartite.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms import bipartite
|
||
|
|
>>> G = nx.path_graph(4)
|
||
|
|
>>> X, Y = bipartite.sets(G)
|
||
|
|
>>> list(X)
|
||
|
|
[0, 2]
|
||
|
|
>>> list(Y)
|
||
|
|
[1, 3]
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
color
|
||
|
|
|
||
|
|
"""
|
||
|
|
if G.is_directed():
|
||
|
|
is_connected = nx.is_weakly_connected
|
||
|
|
else:
|
||
|
|
is_connected = nx.is_connected
|
||
|
|
if top_nodes is not None:
|
||
|
|
X = set(top_nodes)
|
||
|
|
Y = set(G) - X
|
||
|
|
else:
|
||
|
|
if not is_connected(G):
|
||
|
|
msg = "Disconnected graph: Ambiguous solution for bipartite sets."
|
||
|
|
raise nx.AmbiguousSolution(msg)
|
||
|
|
c = color(G)
|
||
|
|
X = {n for n, is_top in c.items() if is_top}
|
||
|
|
Y = {n for n, is_top in c.items() if not is_top}
|
||
|
|
return (X, Y)
|
||
|
|
|
||
|
|
|
||
|
|
def density(B, nodes):
|
||
|
|
"""Returns density of bipartite graph B.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
B : NetworkX graph
|
||
|
|
|
||
|
|
nodes: list or container
|
||
|
|
Nodes in one node set of the bipartite graph.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
d : float
|
||
|
|
The bipartite density
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms import bipartite
|
||
|
|
>>> G = nx.complete_bipartite_graph(3, 2)
|
||
|
|
>>> X = set([0, 1, 2])
|
||
|
|
>>> bipartite.density(G, X)
|
||
|
|
1.0
|
||
|
|
>>> Y = set([3, 4])
|
||
|
|
>>> bipartite.density(G, Y)
|
||
|
|
1.0
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The container of nodes passed as argument must contain all nodes
|
||
|
|
in one of the two bipartite node sets to avoid ambiguity in the
|
||
|
|
case of disconnected graphs.
|
||
|
|
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
|
||
|
|
for further details on how bipartite graphs are handled in NetworkX.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
color
|
||
|
|
"""
|
||
|
|
n = len(B)
|
||
|
|
m = nx.number_of_edges(B)
|
||
|
|
nb = len(nodes)
|
||
|
|
nt = n - nb
|
||
|
|
if m == 0: # includes cases n==0 and n==1
|
||
|
|
d = 0.0
|
||
|
|
else:
|
||
|
|
if B.is_directed():
|
||
|
|
d = m / (2 * nb * nt)
|
||
|
|
else:
|
||
|
|
d = m / (nb * nt)
|
||
|
|
return d
|
||
|
|
|
||
|
|
|
||
|
|
def degrees(B, nodes, weight=None):
|
||
|
|
"""Returns the degrees of the two node sets in the bipartite graph B.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
B : NetworkX graph
|
||
|
|
|
||
|
|
nodes: list or container
|
||
|
|
Nodes in one node set of the bipartite graph.
|
||
|
|
|
||
|
|
weight : string or None, optional (default=None)
|
||
|
|
The edge attribute that holds the numerical value used as a weight.
|
||
|
|
If None, then each edge has weight 1.
|
||
|
|
The degree is the sum of the edge weights adjacent to the node.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
(degX,degY) : tuple of dictionaries
|
||
|
|
The degrees of the two bipartite sets as dictionaries keyed by node.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from networkx.algorithms import bipartite
|
||
|
|
>>> G = nx.complete_bipartite_graph(3, 2)
|
||
|
|
>>> Y = set([3, 4])
|
||
|
|
>>> degX, degY = bipartite.degrees(G, Y)
|
||
|
|
>>> dict(degX)
|
||
|
|
{0: 2, 1: 2, 2: 2}
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
The container of nodes passed as argument must contain all nodes
|
||
|
|
in one of the two bipartite node sets to avoid ambiguity in the
|
||
|
|
case of disconnected graphs.
|
||
|
|
See :mod:`bipartite documentation <networkx.algorithms.bipartite>`
|
||
|
|
for further details on how bipartite graphs are handled in NetworkX.
|
||
|
|
|
||
|
|
See Also
|
||
|
|
--------
|
||
|
|
color, density
|
||
|
|
"""
|
||
|
|
bottom = set(nodes)
|
||
|
|
top = set(B) - bottom
|
||
|
|
return (B.degree(top, weight), B.degree(bottom, weight))
|