313 lines
12 KiB
C++
313 lines
12 KiB
C++
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// (C) Copyright 2007-2009 Andrew Sutton
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//
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0 (See accompanying file
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// LICENSE_1_0.txt or http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_GRAPH_CLIQUE_HPP
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#define BOOST_GRAPH_CLIQUE_HPP
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#include <vector>
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#include <deque>
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#include <boost/config.hpp>
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#include <boost/concept/assert.hpp>
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#include <boost/graph/graph_concepts.hpp>
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#include <boost/graph/lookup_edge.hpp>
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#include <boost/concept/detail/concept_def.hpp>
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namespace boost
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{
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namespace concepts
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{
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BOOST_concept(CliqueVisitor, (Visitor)(Clique)(Graph))
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{
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BOOST_CONCEPT_USAGE(CliqueVisitor) { vis.clique(k, g); }
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private:
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Visitor vis;
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Graph g;
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Clique k;
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};
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} /* namespace concepts */
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using concepts::CliqueVisitorConcept;
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} /* namespace boost */
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#include <boost/concept/detail/concept_undef.hpp>
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namespace boost
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{
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// The algorithm implemented in this paper is based on the so-called
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// Algorithm 457, published as:
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//
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// @article{362367,
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// author = {Coen Bron and Joep Kerbosch},
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// title = {Algorithm 457: finding all cliques of an undirected graph},
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// journal = {Communications of the ACM},
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// volume = {16},
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// number = {9},
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// year = {1973},
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// issn = {0001-0782},
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// pages = {575--577},
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// doi = {http://doi.acm.org/10.1145/362342.362367},
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// publisher = {ACM Press},
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// address = {New York, NY, USA},
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// }
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//
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// Sort of. This implementation is adapted from the 1st version of the
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// algorithm and does not implement the candidate selection optimization
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// described as published - it could, it just doesn't yet.
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//
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// The algorithm is given as proportional to (3.14)^(n/3) power. This is
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// not the same as O(...), but based on time measures and approximation.
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//
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// Unfortunately, this implementation may be less efficient on non-
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// AdjacencyMatrix modeled graphs due to the non-constant implementation
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// of the edge(u,v,g) functions.
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//
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// TODO: It might be worthwhile to provide functionality for passing
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// a connectivity matrix to improve the efficiency of those lookups
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// when needed. This could simply be passed as a BooleanMatrix
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// s.t. edge(u,v,B) returns true or false. This could easily be
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// abstracted for adjacency matricies.
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//
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// The following paper is interesting for a number of reasons. First,
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// it lists a number of other such algorithms and second, it describes
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// a new algorithm (that does not appear to require the edge(u,v,g)
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// function and appears fairly efficient. It is probably worth investigating.
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//
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// @article{DBLP:journals/tcs/TomitaTT06,
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// author = {Etsuji Tomita and Akira Tanaka and Haruhisa Takahashi},
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// title = {The worst-case time complexity for generating all maximal
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// cliques and computational experiments}, journal = {Theor. Comput.
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// Sci.}, volume = {363}, number = {1}, year = {2006}, pages = {28-42}
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// ee = {https://doi.org/10.1016/j.tcs.2006.06.015}
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// }
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/**
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* The default clique_visitor supplies an empty visitation function.
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*/
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struct clique_visitor
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{
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template < typename VertexSet, typename Graph >
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void clique(const VertexSet&, Graph&)
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{
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}
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};
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/**
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* The max_clique_visitor records the size of the maximum clique (but not the
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* clique itself).
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*/
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struct max_clique_visitor
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{
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max_clique_visitor(std::size_t& max) : maximum(max) {}
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template < typename Clique, typename Graph >
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inline void clique(const Clique& p, const Graph& g)
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{
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BOOST_USING_STD_MAX();
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maximum = max BOOST_PREVENT_MACRO_SUBSTITUTION(maximum, p.size());
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}
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std::size_t& maximum;
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};
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inline max_clique_visitor find_max_clique(std::size_t& max)
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{
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return max_clique_visitor(max);
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}
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namespace detail
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{
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template < typename Graph >
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inline bool is_connected_to_clique(const Graph& g,
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typename graph_traits< Graph >::vertex_descriptor u,
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typename graph_traits< Graph >::vertex_descriptor v,
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typename graph_traits< Graph >::undirected_category)
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{
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return lookup_edge(u, v, g).second;
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}
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template < typename Graph >
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inline bool is_connected_to_clique(const Graph& g,
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typename graph_traits< Graph >::vertex_descriptor u,
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typename graph_traits< Graph >::vertex_descriptor v,
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typename graph_traits< Graph >::directed_category)
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{
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// Note that this could alternate between using an || to determine
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// full connectivity. I believe that this should produce strongly
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// connected components. Note that using && instead of || will
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// change the results to a fully connected subgraph (i.e., symmetric
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// edges between all vertices s.t., if a->b, then b->a.
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return lookup_edge(u, v, g).second && lookup_edge(v, u, g).second;
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}
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template < typename Graph, typename Container >
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inline void filter_unconnected_vertices(const Graph& g,
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typename graph_traits< Graph >::vertex_descriptor v,
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const Container& in, Container& out)
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{
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BOOST_CONCEPT_ASSERT((GraphConcept< Graph >));
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typename graph_traits< Graph >::directed_category cat;
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typename Container::const_iterator i, end = in.end();
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for (i = in.begin(); i != end; ++i)
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{
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if (is_connected_to_clique(g, v, *i, cat))
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{
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out.push_back(*i);
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}
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}
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}
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template < typename Graph,
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typename Clique, // compsub type
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typename Container, // candidates/not type
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typename Visitor >
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void extend_clique(const Graph& g, Clique& clique, Container& cands,
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Container& nots, Visitor vis, std::size_t min)
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{
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BOOST_CONCEPT_ASSERT((GraphConcept< Graph >));
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BOOST_CONCEPT_ASSERT((CliqueVisitorConcept< Visitor, Clique, Graph >));
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typedef typename graph_traits< Graph >::vertex_descriptor Vertex;
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// Is there vertex in nots that is connected to all vertices
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// in the candidate set? If so, no clique can ever be found.
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// This could be broken out into a separate function.
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{
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typename Container::iterator ni, nend = nots.end();
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typename Container::iterator ci, cend = cands.end();
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for (ni = nots.begin(); ni != nend; ++ni)
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{
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for (ci = cands.begin(); ci != cend; ++ci)
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{
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// if we don't find an edge, then we're okay.
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if (!lookup_edge(*ni, *ci, g).second)
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break;
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}
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// if we iterated all the way to the end, then *ni
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// is connected to all *ci
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if (ci == cend)
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break;
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}
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// if we broke early, we found *ni connected to all *ci
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if (ni != nend)
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return;
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}
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// TODO: the original algorithm 457 describes an alternative
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// (albeit really complicated) mechanism for selecting candidates.
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// The given optimizaiton seeks to bring about the above
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// condition sooner (i.e., there is a vertex in the not set
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// that is connected to all candidates). unfortunately, the
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// method they give for doing this is fairly unclear.
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// basically, for every vertex in not, we should know how many
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// vertices it is disconnected from in the candidate set. if
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// we fix some vertex in the not set, then we want to keep
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// choosing vertices that are not connected to that fixed vertex.
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// apparently, by selecting fix point with the minimum number
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// of disconnections (i.e., the maximum number of connections
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// within the candidate set), then the previous condition wil
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// be reached sooner.
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// there's some other stuff about using the number of disconnects
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// as a counter, but i'm jot really sure i followed it.
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// TODO: If we min-sized cliques to visit, then theoretically, we
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// should be able to stop recursing if the clique falls below that
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// size - maybe?
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// otherwise, iterate over candidates and and test
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// for maxmimal cliquiness.
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typename Container::iterator i, j;
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for (i = cands.begin(); i != cands.end();)
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{
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Vertex candidate = *i;
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// add the candidate to the clique (keeping the iterator!)
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// typename Clique::iterator ci = clique.insert(clique.end(),
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// candidate);
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clique.push_back(candidate);
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// remove it from the candidate set
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i = cands.erase(i);
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// build new candidate and not sets by removing all vertices
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// that are not connected to the current candidate vertex.
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// these actually invert the operation, adding them to the new
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// sets if the vertices are connected. its semantically the same.
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Container new_cands, new_nots;
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filter_unconnected_vertices(g, candidate, cands, new_cands);
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filter_unconnected_vertices(g, candidate, nots, new_nots);
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if (new_cands.empty() && new_nots.empty())
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{
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// our current clique is maximal since there's nothing
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// that's connected that we haven't already visited. If
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// the clique is below our radar, then we won't visit it.
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if (clique.size() >= min)
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{
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vis.clique(clique, g);
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}
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}
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else
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{
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// recurse to explore the new candidates
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extend_clique(g, clique, new_cands, new_nots, vis, min);
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}
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// we're done with this vertex, so we need to move it
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// to the nots, and remove the candidate from the clique.
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nots.push_back(candidate);
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clique.pop_back();
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}
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}
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} /* namespace detail */
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template < typename Graph, typename Visitor >
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inline void bron_kerbosch_all_cliques(
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const Graph& g, Visitor vis, std::size_t min)
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{
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BOOST_CONCEPT_ASSERT((IncidenceGraphConcept< Graph >));
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BOOST_CONCEPT_ASSERT((VertexListGraphConcept< Graph >));
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BOOST_CONCEPT_ASSERT(
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(AdjacencyMatrixConcept< Graph >)); // Structural requirement only
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typedef typename graph_traits< Graph >::vertex_descriptor Vertex;
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typedef typename graph_traits< Graph >::vertex_iterator VertexIterator;
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typedef std::vector< Vertex > VertexSet;
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typedef std::deque< Vertex > Clique;
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BOOST_CONCEPT_ASSERT((CliqueVisitorConcept< Visitor, Clique, Graph >));
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// NOTE: We're using a deque to implement the clique, because it provides
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// constant inserts and removals at the end and also a constant size.
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VertexIterator i, end;
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boost::tie(i, end) = vertices(g);
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VertexSet cands(i, end); // start with all vertices as candidates
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VertexSet nots; // start with no vertices visited
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Clique clique; // the first clique is an empty vertex set
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detail::extend_clique(g, clique, cands, nots, vis, min);
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}
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// NOTE: By default the minimum number of vertices per clique is set at 2
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// because singleton cliques aren't really very interesting.
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template < typename Graph, typename Visitor >
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inline void bron_kerbosch_all_cliques(const Graph& g, Visitor vis)
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{
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bron_kerbosch_all_cliques(g, vis, 2);
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}
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template < typename Graph >
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inline std::size_t bron_kerbosch_clique_number(const Graph& g)
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{
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std::size_t ret = 0;
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bron_kerbosch_all_cliques(g, find_max_clique(ret));
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return ret;
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}
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} /* namespace boost */
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#endif
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