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