491 lines
14 KiB
Python
491 lines
14 KiB
Python
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"""
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Algorithms for chordal graphs.
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A graph is chordal if every cycle of length at least 4 has a chord
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(an edge joining two nodes not adjacent in the cycle).
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https://en.wikipedia.org/wiki/Chordal_graph
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"""
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import sys
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import warnings
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import networkx as nx
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from networkx.algorithms.components import connected_components
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from networkx.utils import arbitrary_element, not_implemented_for
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__all__ = [
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"is_chordal",
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"find_induced_nodes",
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"chordal_graph_cliques",
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"chordal_graph_treewidth",
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"NetworkXTreewidthBoundExceeded",
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"complete_to_chordal_graph",
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]
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class NetworkXTreewidthBoundExceeded(nx.NetworkXException):
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"""Exception raised when a treewidth bound has been provided and it has
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been exceeded"""
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def is_chordal(G):
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"""Checks whether G is a chordal graph.
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A graph is chordal if every cycle of length at least 4 has a chord
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(an edge joining two nodes not adjacent in the cycle).
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Parameters
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----------
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G : graph
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A NetworkX graph.
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Returns
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-------
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chordal : bool
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True if G is a chordal graph and False otherwise.
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Raises
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------
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NetworkXError
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The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
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If the input graph is an instance of one of these classes, a
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:exc:`NetworkXError` is raised.
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Examples
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--------
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>>> e = [
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... (1, 2),
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... (1, 3),
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... (2, 3),
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... (2, 4),
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... (3, 4),
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... (3, 5),
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... (3, 6),
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... (4, 5),
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... (4, 6),
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... (5, 6),
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... ]
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>>> G = nx.Graph(e)
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>>> nx.is_chordal(G)
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True
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Notes
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-----
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The routine tries to go through every node following maximum cardinality
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search. It returns False when it finds that the separator for any node
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is not a clique. Based on the algorithms in [1]_.
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References
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----------
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.. [1] R. E. Tarjan and M. Yannakakis, Simple linear-time algorithms
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to test chordality of graphs, test acyclicity of hypergraphs, and
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selectively reduce acyclic hypergraphs, SIAM J. Comput., 13 (1984),
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pp. 566–579.
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"""
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if G.is_directed():
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raise nx.NetworkXError("Directed graphs not supported")
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if G.is_multigraph():
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raise nx.NetworkXError("Multiply connected graphs not supported.")
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if len(_find_chordality_breaker(G)) == 0:
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return True
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else:
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return False
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def find_induced_nodes(G, s, t, treewidth_bound=sys.maxsize):
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"""Returns the set of induced nodes in the path from s to t.
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Parameters
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----------
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G : graph
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A chordal NetworkX graph
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s : node
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Source node to look for induced nodes
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t : node
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Destination node to look for induced nodes
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treewith_bound: float
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Maximum treewidth acceptable for the graph H. The search
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for induced nodes will end as soon as the treewidth_bound is exceeded.
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Returns
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-------
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Induced_nodes : Set of nodes
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The set of induced nodes in the path from s to t in G
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Raises
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------
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NetworkXError
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The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
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If the input graph is an instance of one of these classes, a
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:exc:`NetworkXError` is raised.
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The algorithm can only be applied to chordal graphs. If the input
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graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
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Examples
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--------
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>>> G = nx.Graph()
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>>> G = nx.generators.classic.path_graph(10)
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>>> induced_nodes = nx.find_induced_nodes(G, 1, 9, 2)
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>>> sorted(induced_nodes)
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[1, 2, 3, 4, 5, 6, 7, 8, 9]
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Notes
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-----
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G must be a chordal graph and (s,t) an edge that is not in G.
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If a treewidth_bound is provided, the search for induced nodes will end
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as soon as the treewidth_bound is exceeded.
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The algorithm is inspired by Algorithm 4 in [1]_.
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A formal definition of induced node can also be found on that reference.
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References
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----------
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.. [1] Learning Bounded Treewidth Bayesian Networks.
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Gal Elidan, Stephen Gould; JMLR, 9(Dec):2699--2731, 2008.
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http://jmlr.csail.mit.edu/papers/volume9/elidan08a/elidan08a.pdf
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"""
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if not is_chordal(G):
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raise nx.NetworkXError("Input graph is not chordal.")
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H = nx.Graph(G)
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H.add_edge(s, t)
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Induced_nodes = set()
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triplet = _find_chordality_breaker(H, s, treewidth_bound)
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while triplet:
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(u, v, w) = triplet
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Induced_nodes.update(triplet)
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for n in triplet:
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if n != s:
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H.add_edge(s, n)
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triplet = _find_chordality_breaker(H, s, treewidth_bound)
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if Induced_nodes:
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# Add t and the second node in the induced path from s to t.
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Induced_nodes.add(t)
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for u in G[s]:
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if len(Induced_nodes & set(G[u])) == 2:
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Induced_nodes.add(u)
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break
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return Induced_nodes
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def chordal_graph_cliques(G):
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"""Returns the set of maximal cliques of a chordal graph.
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The algorithm breaks the graph in connected components and performs a
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maximum cardinality search in each component to get the cliques.
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Parameters
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----------
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G : graph
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A NetworkX graph
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Returns
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-------
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cliques : A set containing the maximal cliques in G.
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Raises
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------
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NetworkXError
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The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
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If the input graph is an instance of one of these classes, a
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:exc:`NetworkXError` is raised.
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The algorithm can only be applied to chordal graphs. If the input
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graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
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Examples
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--------
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>>> e = [
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... (1, 2),
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... (1, 3),
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... (2, 3),
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... (2, 4),
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... (3, 4),
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... (3, 5),
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... (3, 6),
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... (4, 5),
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... (4, 6),
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... (5, 6),
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... (7, 8),
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... ]
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>>> G = nx.Graph(e)
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>>> G.add_node(9)
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>>> setlist = nx.chordal_graph_cliques(G)
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"""
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msg = "This will return a generator in 3.0."
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warnings.warn(msg, DeprecationWarning)
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return {c for c in _chordal_graph_cliques(G)}
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def chordal_graph_treewidth(G):
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"""Returns the treewidth of the chordal graph G.
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Parameters
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----------
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G : graph
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A NetworkX graph
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Returns
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-------
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treewidth : int
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The size of the largest clique in the graph minus one.
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Raises
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------
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NetworkXError
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The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
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If the input graph is an instance of one of these classes, a
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:exc:`NetworkXError` is raised.
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The algorithm can only be applied to chordal graphs. If the input
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graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
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Examples
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--------
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>>> e = [
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... (1, 2),
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... (1, 3),
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... (2, 3),
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... (2, 4),
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... (3, 4),
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... (3, 5),
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... (3, 6),
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... (4, 5),
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... (4, 6),
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... (5, 6),
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... (7, 8),
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... ]
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>>> G = nx.Graph(e)
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>>> G.add_node(9)
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>>> nx.chordal_graph_treewidth(G)
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3
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References
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----------
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.. [1] https://en.wikipedia.org/wiki/Tree_decomposition#Treewidth
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"""
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if not is_chordal(G):
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raise nx.NetworkXError("Input graph is not chordal.")
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max_clique = -1
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for clique in nx.chordal_graph_cliques(G):
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max_clique = max(max_clique, len(clique))
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return max_clique - 1
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def _is_complete_graph(G):
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"""Returns True if G is a complete graph."""
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if nx.number_of_selfloops(G) > 0:
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raise nx.NetworkXError("Self loop found in _is_complete_graph()")
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n = G.number_of_nodes()
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if n < 2:
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return True
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e = G.number_of_edges()
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max_edges = (n * (n - 1)) / 2
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return e == max_edges
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def _find_missing_edge(G):
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""" Given a non-complete graph G, returns a missing edge."""
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nodes = set(G)
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for u in G:
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missing = nodes - set(list(G[u].keys()) + [u])
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if missing:
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return (u, missing.pop())
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def _max_cardinality_node(G, choices, wanna_connect):
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"""Returns a the node in choices that has more connections in G
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to nodes in wanna_connect.
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"""
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max_number = -1
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for x in choices:
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number = len([y for y in G[x] if y in wanna_connect])
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if number > max_number:
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max_number = number
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max_cardinality_node = x
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return max_cardinality_node
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def _find_chordality_breaker(G, s=None, treewidth_bound=sys.maxsize):
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""" Given a graph G, starts a max cardinality search
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(starting from s if s is given and from an arbitrary node otherwise)
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trying to find a non-chordal cycle.
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If it does find one, it returns (u,v,w) where u,v,w are the three
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nodes that together with s are involved in the cycle.
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"""
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unnumbered = set(G)
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if s is None:
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s = arbitrary_element(G)
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unnumbered.remove(s)
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numbered = {s}
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current_treewidth = -1
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while unnumbered: # and current_treewidth <= treewidth_bound:
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v = _max_cardinality_node(G, unnumbered, numbered)
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unnumbered.remove(v)
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numbered.add(v)
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clique_wanna_be = set(G[v]) & numbered
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sg = G.subgraph(clique_wanna_be)
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if _is_complete_graph(sg):
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# The graph seems to be chordal by now. We update the treewidth
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current_treewidth = max(current_treewidth, len(clique_wanna_be))
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if current_treewidth > treewidth_bound:
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raise nx.NetworkXTreewidthBoundExceeded(
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f"treewidth_bound exceeded: {current_treewidth}"
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)
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else:
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# sg is not a clique,
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# look for an edge that is not included in sg
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(u, w) = _find_missing_edge(sg)
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return (u, v, w)
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return ()
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def _chordal_graph_cliques(G):
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"""Returns all maximal cliques of a chordal graph.
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The algorithm breaks the graph in connected components and performs a
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maximum cardinality search in each component to get the cliques.
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Parameters
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----------
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G : graph
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A NetworkX graph
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Returns
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-------
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iterator
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An iterator over maximal cliques, each of which is a frozenset of
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nodes in `G`. The order of cliques is arbitrary.
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Raises
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------
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NetworkXError
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The algorithm does not support DiGraph, MultiGraph and MultiDiGraph.
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|||
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If the input graph is an instance of one of these classes, a
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:exc:`NetworkXError` is raised.
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The algorithm can only be applied to chordal graphs. If the input
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graph is found to be non-chordal, a :exc:`NetworkXError` is raised.
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|||
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Examples
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|||
|
|
--------
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|||
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>>> e = [
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... (1, 2),
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... (1, 3),
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|
|
... (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)
|
|||
|
|
>>> cliques = [c for c in _chordal_graph_cliques(G)]
|
|||
|
|
>>> cliques[0]
|
|||
|
|
frozenset({1, 2, 3})
|
|||
|
|
"""
|
|||
|
|
if not is_chordal(G):
|
|||
|
|
raise nx.NetworkXError("Input graph is not chordal.")
|
|||
|
|
|
|||
|
|
for C in (G.subgraph(c).copy() for c in connected_components(G)):
|
|||
|
|
if C.number_of_nodes() == 1:
|
|||
|
|
yield frozenset(C.nodes())
|
|||
|
|
else:
|
|||
|
|
unnumbered = set(C.nodes())
|
|||
|
|
v = arbitrary_element(C)
|
|||
|
|
unnumbered.remove(v)
|
|||
|
|
numbered = {v}
|
|||
|
|
clique_wanna_be = {v}
|
|||
|
|
while unnumbered:
|
|||
|
|
v = _max_cardinality_node(C, unnumbered, numbered)
|
|||
|
|
unnumbered.remove(v)
|
|||
|
|
numbered.add(v)
|
|||
|
|
new_clique_wanna_be = set(C.neighbors(v)) & numbered
|
|||
|
|
sg = C.subgraph(clique_wanna_be)
|
|||
|
|
if _is_complete_graph(sg):
|
|||
|
|
new_clique_wanna_be.add(v)
|
|||
|
|
if not new_clique_wanna_be >= clique_wanna_be:
|
|||
|
|
yield frozenset(clique_wanna_be)
|
|||
|
|
clique_wanna_be = new_clique_wanna_be
|
|||
|
|
else:
|
|||
|
|
raise nx.NetworkXError("Input graph is not chordal.")
|
|||
|
|
yield frozenset(clique_wanna_be)
|
|||
|
|
|
|||
|
|
|
|||
|
|
@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
|
|||
|
|
--------
|
|||
|
|
>>> 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
|