422 lines
13 KiB
Python
422 lines
13 KiB
Python
# -*- coding: utf-8 -*-
|
||
"""
|
||
Algorithms for chordal graphs.
|
||
|
||
A graph is chordal if every cycle of length at least 4 has a chord
|
||
(an edge joining two nodes not adjacent in the cycle).
|
||
https://en.wikipedia.org/wiki/Chordal_graph
|
||
"""
|
||
import sys
|
||
|
||
import networkx as nx
|
||
from networkx.algorithms.components import connected_components
|
||
from networkx.utils import arbitrary_element, not_implemented_for
|
||
|
||
__authors__ = "\n".join(['Jesus Cerquides <cerquide@iiia.csic.es>',
|
||
'Julien Klaus <julien.klaus@uni-jena.de'])
|
||
# Copyright (C) 2010 by
|
||
# Jesus Cerquides <cerquide@iiia.csic.es>
|
||
# All rights reserved.
|
||
# BSD license.
|
||
|
||
__all__ = ['is_chordal',
|
||
'find_induced_nodes',
|
||
'chordal_graph_cliques',
|
||
'chordal_graph_treewidth',
|
||
'NetworkXTreewidthBoundExceeded',
|
||
'complete_to_chordal_graph']
|
||
|
||
|
||
class NetworkXTreewidthBoundExceeded(nx.NetworkXException):
|
||
"""Exception raised when a treewidth bound has been provided and it has
|
||
been exceeded"""
|
||
|
||
|
||
def is_chordal(G):
|
||
"""Checks whether G is a chordal graph.
|
||
|
||
A graph is chordal if every cycle of length at least 4 has a chord
|
||
(an edge joining two nodes not adjacent in the cycle).
|
||
|
||
Parameters
|
||
----------
|
||
G : graph
|
||
A NetworkX graph.
|
||
|
||
Returns
|
||
-------
|
||
chordal : bool
|
||
True if G is a chordal graph and False otherwise.
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
|
||
If the input graph is an instance of one of these classes, a
|
||
:exc:`NetworkXError` is raised.
|
||
|
||
Examples
|
||
--------
|
||
>>> import networkx as nx
|
||
>>> e=[(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)]
|
||
>>> G=nx.Graph(e)
|
||
>>> nx.is_chordal(G)
|
||
True
|
||
|
||
Notes
|
||
-----
|
||
The routine tries to go through every node following maximum cardinality
|
||
search. It returns False when it finds that the separator for any node
|
||
is not a clique. Based on the algorithms in [1]_.
|
||
|
||
References
|
||
----------
|
||
.. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms
|
||
to test chordality of graphs, test acyclicity of hypergraphs, and
|
||
selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984),
|
||
pp. 566–579.
|
||
"""
|
||
if G.is_directed():
|
||
raise nx.NetworkXError('Directed graphs not supported')
|
||
if G.is_multigraph():
|
||
raise nx.NetworkXError('Multiply connected graphs not supported.')
|
||
if len(_find_chordality_breaker(G)) == 0:
|
||
return True
|
||
else:
|
||
return False
|
||
|
||
|
||
def find_induced_nodes(G, s, t, treewidth_bound=sys.maxsize):
|
||
"""Returns the set of induced nodes in the path from s to t.
|
||
|
||
Parameters
|
||
----------
|
||
G : graph
|
||
A chordal NetworkX graph
|
||
s : node
|
||
Source node to look for induced nodes
|
||
t : node
|
||
Destination node to look for induced nodes
|
||
treewith_bound: float
|
||
Maximum treewidth acceptable for the graph H. The search
|
||
for induced nodes will end as soon as the treewidth_bound is exceeded.
|
||
|
||
Returns
|
||
-------
|
||
Induced_nodes : Set of nodes
|
||
The set of induced nodes in the path from s to t in G
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
|
||
If the input graph is an instance of one of these classes, a
|
||
:exc:`NetworkXError` is raised.
|
||
The algorithm can only be applied to chordal graphs. If the input
|
||
graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
|
||
|
||
Examples
|
||
--------
|
||
>>> import networkx as nx
|
||
>>> G=nx.Graph()
|
||
>>> G = nx.generators.classic.path_graph(10)
|
||
>>> Induced_nodes = nx.find_induced_nodes(G,1,9,2)
|
||
>>> sorted(Induced_nodes)
|
||
[1, 2, 3, 4, 5, 6, 7, 8, 9]
|
||
|
||
Notes
|
||
-----
|
||
G must be a chordal graph and (s,t) an edge that is not in G.
|
||
|
||
If a treewidth_bound is provided, the search for induced nodes will end
|
||
as soon as the treewidth_bound is exceeded.
|
||
|
||
The algorithm is inspired by Algorithm 4 in [1]_.
|
||
A formal definition of induced node can also be found on that reference.
|
||
|
||
References
|
||
----------
|
||
.. [1] Learning Bounded Treewidth Bayesian Networks.
|
||
Gal Elidan, Stephen Gould; JMLR, 9(Dec):2699--2731, 2008.
|
||
http://jmlr.csail.mit.edu/papers/volume9/elidan08a/elidan08a.pdf
|
||
"""
|
||
if not is_chordal(G):
|
||
raise nx.NetworkXError("Input graph is not chordal.")
|
||
|
||
H = nx.Graph(G)
|
||
H.add_edge(s, t)
|
||
Induced_nodes = set()
|
||
triplet = _find_chordality_breaker(H, s, treewidth_bound)
|
||
while triplet:
|
||
(u, v, w) = triplet
|
||
Induced_nodes.update(triplet)
|
||
for n in triplet:
|
||
if n != s:
|
||
H.add_edge(s, n)
|
||
triplet = _find_chordality_breaker(H, s, treewidth_bound)
|
||
if Induced_nodes:
|
||
# Add t and the second node in the induced path from s to t.
|
||
Induced_nodes.add(t)
|
||
for u in G[s]:
|
||
if len(Induced_nodes & set(G[u])) == 2:
|
||
Induced_nodes.add(u)
|
||
break
|
||
return Induced_nodes
|
||
|
||
|
||
def chordal_graph_cliques(G):
|
||
"""Returns the set of maximal cliques of a chordal graph.
|
||
|
||
The algorithm breaks the graph in connected components and performs a
|
||
maximum cardinality search in each component to get the cliques.
|
||
|
||
Parameters
|
||
----------
|
||
G : graph
|
||
A NetworkX graph
|
||
|
||
Returns
|
||
-------
|
||
cliques : A set containing the maximal cliques in G.
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
|
||
If the input graph is an instance of one of these classes, a
|
||
:exc:`NetworkXError` is raised.
|
||
The algorithm can only be applied to chordal graphs. If the input
|
||
graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
|
||
|
||
Examples
|
||
--------
|
||
>>> import networkx as nx
|
||
>>> e= [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
|
||
>>> G = nx.Graph(e)
|
||
>>> G.add_node(9)
|
||
>>> setlist = nx.chordal_graph_cliques(G)
|
||
"""
|
||
if not is_chordal(G):
|
||
raise nx.NetworkXError("Input graph is not chordal.")
|
||
|
||
cliques = set()
|
||
for C in (G.subgraph(c).copy() for c in connected_components(G)):
|
||
cliques |= _connected_chordal_graph_cliques(C)
|
||
|
||
return cliques
|
||
|
||
|
||
def chordal_graph_treewidth(G):
|
||
"""Returns the treewidth of the chordal graph G.
|
||
|
||
Parameters
|
||
----------
|
||
G : graph
|
||
A NetworkX graph
|
||
|
||
Returns
|
||
-------
|
||
treewidth : int
|
||
The size of the largest clique in the graph minus one.
|
||
|
||
Raises
|
||
------
|
||
NetworkXError
|
||
The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
|
||
If the input graph is an instance of one of these classes, a
|
||
:exc:`NetworkXError` is raised.
|
||
The algorithm can only be applied to chordal graphs. If the input
|
||
graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
|
||
|
||
Examples
|
||
--------
|
||
>>> import networkx as nx
|
||
>>> e = [(1,2),(1,3),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6),(7,8)]
|
||
>>> G = nx.Graph(e)
|
||
>>> G.add_node(9)
|
||
>>> nx.chordal_graph_treewidth(G)
|
||
3
|
||
|
||
References
|
||
----------
|
||
.. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
|
||
"""
|
||
if not is_chordal(G):
|
||
raise nx.NetworkXError("Input graph is not chordal.")
|
||
|
||
max_clique = -1
|
||
for clique in nx.chordal_graph_cliques(G):
|
||
max_clique = max(max_clique, len(clique))
|
||
return max_clique - 1
|
||
|
||
|
||
def _is_complete_graph(G):
|
||
"""Returns True if G is a complete graph."""
|
||
if nx.number_of_selfloops(G) > 0:
|
||
raise nx.NetworkXError("Self loop found in _is_complete_graph()")
|
||
n = G.number_of_nodes()
|
||
if n < 2:
|
||
return True
|
||
e = G.number_of_edges()
|
||
max_edges = ((n * (n - 1)) / 2)
|
||
return e == max_edges
|
||
|
||
|
||
def _find_missing_edge(G):
|
||
""" Given a non-complete graph G, returns a missing edge."""
|
||
nodes = set(G)
|
||
for u in G:
|
||
missing = nodes - set(list(G[u].keys()) + [u])
|
||
if missing:
|
||
return (u, missing.pop())
|
||
|
||
|
||
def _max_cardinality_node(G, choices, wanna_connect):
|
||
"""Returns a the node in choices that has more connections in G
|
||
to nodes in wanna_connect.
|
||
"""
|
||
max_number = -1
|
||
for x in choices:
|
||
number = len([y for y in G[x] if y in wanna_connect])
|
||
if number > max_number:
|
||
max_number = number
|
||
max_cardinality_node = x
|
||
return max_cardinality_node
|
||
|
||
|
||
def _find_chordality_breaker(G, s=None, treewidth_bound=sys.maxsize):
|
||
""" Given a graph G, starts a max cardinality search
|
||
(starting from s if s is given and from an arbitrary node otherwise)
|
||
trying to find a non-chordal cycle.
|
||
|
||
If it does find one, it returns (u,v,w) where u,v,w are the three
|
||
nodes that together with s are involved in the cycle.
|
||
"""
|
||
|
||
unnumbered = set(G)
|
||
if s is None:
|
||
s = arbitrary_element(G)
|
||
unnumbered.remove(s)
|
||
numbered = set([s])
|
||
current_treewidth = -1
|
||
while unnumbered: # and current_treewidth <= treewidth_bound:
|
||
v = _max_cardinality_node(G, unnumbered, numbered)
|
||
unnumbered.remove(v)
|
||
numbered.add(v)
|
||
clique_wanna_be = set(G[v]) & numbered
|
||
sg = G.subgraph(clique_wanna_be)
|
||
if _is_complete_graph(sg):
|
||
# The graph seems to be chordal by now. We update the treewidth
|
||
current_treewidth = max(current_treewidth, len(clique_wanna_be))
|
||
if current_treewidth > treewidth_bound:
|
||
raise nx.NetworkXTreewidthBoundExceeded(
|
||
"treewidth_bound exceeded: %s" % current_treewidth)
|
||
else:
|
||
# sg is not a clique,
|
||
# look for an edge that is not included in sg
|
||
(u, w) = _find_missing_edge(sg)
|
||
return (u, v, w)
|
||
return ()
|
||
|
||
|
||
def _connected_chordal_graph_cliques(G):
|
||
"""Returns the set of maximal cliques of a connected chordal graph."""
|
||
if G.number_of_nodes() == 1:
|
||
x = frozenset(G.nodes())
|
||
return set([x])
|
||
else:
|
||
cliques = set()
|
||
unnumbered = set(G.nodes())
|
||
v = arbitrary_element(G)
|
||
unnumbered.remove(v)
|
||
numbered = set([v])
|
||
clique_wanna_be = set([v])
|
||
while unnumbered:
|
||
v = _max_cardinality_node(G, unnumbered, numbered)
|
||
unnumbered.remove(v)
|
||
numbered.add(v)
|
||
new_clique_wanna_be = set(G.neighbors(v)) & numbered
|
||
sg = G.subgraph(clique_wanna_be)
|
||
if _is_complete_graph(sg):
|
||
new_clique_wanna_be.add(v)
|
||
if not new_clique_wanna_be >= clique_wanna_be:
|
||
cliques.add(frozenset(clique_wanna_be))
|
||
clique_wanna_be = new_clique_wanna_be
|
||
else:
|
||
raise nx.NetworkXError("Input graph is not chordal.")
|
||
cliques.add(frozenset(clique_wanna_be))
|
||
return cliques
|
||
|
||
|
||
@not_implemented_for('directed')
|
||
def complete_to_chordal_graph(G):
|
||
"""Return a copy of G completed to a chordal graph
|
||
|
||
Adds edges to a copy of G to create a chordal graph. A graph G=(V,E) is
|
||
called chordal if for each cycle with length bigger than 3, there exist
|
||
two non-adjacent nodes connected by an edge (called a chord).
|
||
|
||
Parameters
|
||
----------
|
||
G : NetworkX graph
|
||
Undirected graph
|
||
|
||
Returns
|
||
-------
|
||
H : NetworkX graph
|
||
The chordal enhancement of G
|
||
alpha : Dictionary
|
||
The elimination ordering of nodes of G
|
||
|
||
Notes
|
||
------
|
||
There are different approaches to calculate the chordal
|
||
enhancement of a graph. The algorithm used here is called
|
||
MCS-M and gives at least minimal (local) triangulation of graph. Note
|
||
that this triangulation is not necessarily a global minimum.
|
||
|
||
https://en.wikipedia.org/wiki/Chordal_graph
|
||
|
||
References
|
||
----------
|
||
.. [1] Berry, Anne & Blair, Jean & Heggernes, Pinar & Peyton, Barry. (2004)
|
||
Maximum Cardinality Search for Computing Minimal Triangulations of
|
||
Graphs. Algorithmica. 39. 287-298. 10.1007/s00453-004-1084-3.
|
||
|
||
Examples
|
||
--------
|
||
>>> import networkx as nx
|
||
>>> from networkx.algorithms.chordal import complete_to_chordal_graph
|
||
>>> G = nx.wheel_graph(10)
|
||
>>> H,alpha = complete_to_chordal_graph(G)
|
||
"""
|
||
H = G.copy()
|
||
alpha = {node: 0 for node in H}
|
||
if nx.is_chordal(H):
|
||
return H, alpha
|
||
chords = set([])
|
||
weight = {node: 0 for node in H.nodes()}
|
||
unnumbered_nodes = list(H.nodes())
|
||
for i in range(len(H.nodes()), 0, -1):
|
||
# get the node in unnumbered_nodes with the maximum weight
|
||
z = max(unnumbered_nodes, key=lambda node: weight[node])
|
||
unnumbered_nodes.remove(z)
|
||
alpha[z] = i
|
||
update_nodes = []
|
||
for y in unnumbered_nodes:
|
||
if G.has_edge(y, z):
|
||
update_nodes.append(y)
|
||
else:
|
||
# y_weight will be bigger than node weights between y and z
|
||
y_weight = weight[y]
|
||
lower_nodes = [node for node in unnumbered_nodes
|
||
if weight[node] < y_weight]
|
||
if nx.has_path(H.subgraph(lower_nodes + [z, y]), y, z):
|
||
update_nodes.append(y)
|
||
chords.add((z, y))
|
||
# during calculation of paths the weights should not be updated
|
||
for node in update_nodes:
|
||
weight[node] += 1
|
||
H.add_edges_from(chords)
|
||
return H, alpha
|