Clique algorithms (taken from NetworkX)
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Dimitri Lozeve
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3 changed files with 269 additions and 70 deletions
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clique.py
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clique.py
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import graph_tool.all as gt
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def find_cliques(G):
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"""Returns all maximal cliques in an undirected graph.
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For each node *v*, a *maximal clique for v* is a largest complete
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subgraph containing *v*. The largest maximal clique is sometimes
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called the *maximum clique*.
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This function returns an iterator over cliques, each of which is a
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list of nodes. It is an iterative implementation, so should not
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suffer from recursion depth issues.
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Parameters
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----------
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G : graph-tool graph
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An undirected 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 list of
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nodes in `G`. The order of cliques is arbitrary.
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See Also
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--------
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find_cliques_recursive
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A recursive version of the same algorithm.
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Notes
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-----
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Taken from NetworkX.
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https://github.com/networkx/networkx/blob/master/networkx/algorithms/clique.py
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To obtain a list of all maximal cliques, use
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`list(find_cliques(G))`. However, be aware that in the worst-case,
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the length of this list can be exponential in the number of nodes in
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the graph (for example, when the graph is the complete graph). This
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function avoids storing all cliques in memory by only keeping
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current candidate node lists in memory during its search.
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This implementation is based on the algorithm published by Bron and
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Kerbosch (1973) [1]_, as adapted by Tomita, Tanaka and Takahashi
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(2006) [2]_ and discussed in Cazals and Karande (2008) [3]_. It
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essentially unrolls the recursion used in the references to avoid
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issues of recursion stack depth (for a recursive implementation, see
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:func:`find_cliques_recursive`).
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This algorithm ignores self-loops and parallel edges, since cliques
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are not conventionally defined with such edges.
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References
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----------
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.. [1] Bron, C. and Kerbosch, J.
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"Algorithm 457: finding all cliques of an undirected graph".
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*Communications of the ACM* 16, 9 (Sep. 1973), 575--577.
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<http://portal.acm.org/citation.cfm?doid=362342.362367>
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.. [2] Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi,
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"The worst-case time complexity for generating all maximal
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cliques and computational experiments",
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*Theoretical Computer Science*, Volume 363, Issue 1,
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Computing and Combinatorics,
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10th Annual International Conference on
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Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28--42
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<https://doi.org/10.1016/j.tcs.2006.06.015>
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.. [3] F. Cazals, C. Karande,
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"A note on the problem of reporting maximal cliques",
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*Theoretical Computer Science*,
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Volume 407, Issues 1--3, 6 November 2008, Pages 564--568,
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<https://doi.org/10.1016/j.tcs.2008.05.010>
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"""
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if len(G.get_vertices()) == 0:
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return
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adj = {u: {v for v in G.get_out_neighbors(u)} for u in G.vertices()}
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Q = [None]
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subg = set(G.get_vertices())
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cand = set(G.get_vertices())
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u = max(subg, key=lambda u: len(cand & adj[u]))
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ext_u = cand - adj[u]
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stack = []
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try:
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while True:
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if ext_u:
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q = ext_u.pop()
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cand.remove(q)
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Q[-1] = q
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adj_q = adj[q]
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subg_q = subg & adj_q
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if not subg_q:
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yield Q[:]
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else:
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cand_q = cand & adj_q
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if cand_q:
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stack.append((subg, cand, ext_u))
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Q.append(None)
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subg = subg_q
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cand = cand_q
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u = max(subg, key=lambda u: len(cand & adj[u]))
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ext_u = cand - adj[u]
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else:
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Q.pop()
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subg, cand, ext_u = stack.pop()
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except IndexError:
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pass
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def find_cliques_recursive(G):
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"""Returns all maximal cliques in a graph.
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For each node *v*, a *maximal clique for v* is a largest complete
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subgraph containing *v*. The largest maximal clique is sometimes
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called the *maximum clique*.
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This function returns an iterator over cliques, each of which is a
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list of nodes. It is a recursive implementation, so may suffer from
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recursion depth issues.
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Parameters
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----------
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G : graph-tool 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 list of
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nodes in `G`. The order of cliques is arbitrary.
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See Also
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--------
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find_cliques
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An iterative version of the same algorithm.
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Notes
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-----
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Taken from NetworkX.
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https://github.com/networkx/networkx/blob/master/networkx/algorithms/clique.py
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To obtain a list of all maximal cliques, use
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`list(find_cliques_recursive(G))`. However, be aware that in the
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worst-case, the length of this list can be exponential in the number
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of nodes in the graph (for example, when the graph is the complete
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graph). This function avoids storing all cliques in memory by only
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keeping current candidate node lists in memory during its search.
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This implementation is based on the algorithm published by Bron and
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Kerbosch (1973) [1]_, as adapted by Tomita, Tanaka and Takahashi
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(2006) [2]_ and discussed in Cazals and Karande (2008) [3]_. For a
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non-recursive implementation, see :func:`find_cliques`.
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This algorithm ignores self-loops and parallel edges, since cliques
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are not conventionally defined with such edges.
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References
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----------
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.. [1] Bron, C. and Kerbosch, J.
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"Algorithm 457: finding all cliques of an undirected graph".
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*Communications of the ACM* 16, 9 (Sep. 1973), 575--577.
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<http://portal.acm.org/citation.cfm?doid=362342.362367>
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.. [2] Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi,
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"The worst-case time complexity for generating all maximal
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cliques and computational experiments",
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*Theoretical Computer Science*, Volume 363, Issue 1,
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Computing and Combinatorics,
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10th Annual International Conference on
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Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28--42
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<https://doi.org/10.1016/j.tcs.2006.06.015>
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.. [3] F. Cazals, C. Karande,
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"A note on the problem of reporting maximal cliques",
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*Theoretical Computer Science*,
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Volume 407, Issues 1--3, 6 November 2008, Pages 564--568,
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<https://doi.org/10.1016/j.tcs.2008.05.010>
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"""
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if len(G.get_vertices()) == 0:
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return iter([])
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adj = {u: {v for v in G.get_out_neighbors(u)} for u in G.vertices()}
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Q = []
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def expand(subg, cand):
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u = max(subg, key=lambda u: len(cand & adj[u]))
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for q in cand - adj[u]:
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cand.remove(q)
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Q.append(q)
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adj_q = adj[q]
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subg_q = subg & adj_q
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if not subg_q:
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yield Q[:]
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else:
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cand_q = cand & adj_q
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if cand_q:
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for clique in expand(subg_q, cand_q):
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yield clique
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Q.pop()
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return expand(set(G.get_vertices()), set(G.get_vertices()))
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def cliques_containing_node(G, nodes=None, cliques=None):
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"""Returns a list of cliques containing the given node.
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Returns a single list or list of lists depending on input nodes.
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Optional list of cliques can be input if already computed.
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Taken from NetworkX.
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https://github.com/networkx/networkx/blob/master/networkx/algorithms/clique.py
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"""
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if cliques is None:
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cliques = list(find_cliques(G))
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if nodes is None:
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nodes = list(G.get_vertices()) # none, get entire graph
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if not isinstance(nodes, list): # check for a list
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v = nodes
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# assume it is a single value
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vcliques = [c for c in cliques if v in c]
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else:
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vcliques = {}
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for v in nodes:
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vcliques[v] = [c for c in cliques if v in c]
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return vcliques
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if __name__ == "__main__":
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g = gt.collection.data["karate"]
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cliques = list(find_cliques(g))
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print(cliques)
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print(cliques_containing_node(g, 1))
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