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