Edit: /usr/include/boost/graph/bron_kerbosch_all_cliques.hpp (11736B)
// (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
#include
#include
#include
#include
#include
#include
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
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 = {https://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)
{
BOOST_CONCEPT_ASSERT((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)
{
BOOST_CONCEPT_ASSERT((GraphConcept< Graph >));
BOOST_CONCEPT_ASSERT((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;
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)
{
BOOST_CONCEPT_ASSERT((IncidenceGraphConcept< Graph >));
BOOST_CONCEPT_ASSERT((VertexListGraphConcept< Graph >));
BOOST_CONCEPT_ASSERT(
(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;
BOOST_CONCEPT_ASSERT((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