1 | // (C) Copyright 2007-2009 Andrew Sutton |
2 | // |
3 | // Use, modification and distribution are subject to the |
4 | // Boost Software License, Version 1.0 (See accompanying file |
5 | // LICENSE_1_0.txt or http://www.boost.org/LICENSE_1_0.txt) |
6 | |
7 | #ifndef BOOST_GRAPH_CLIQUE_HPP |
8 | #define BOOST_GRAPH_CLIQUE_HPP |
9 | |
10 | #include <vector> |
11 | #include <deque> |
12 | #include <boost/config.hpp> |
13 | |
14 | #include <boost/concept/assert.hpp> |
15 | |
16 | #include <boost/graph/graph_concepts.hpp> |
17 | #include <boost/graph/lookup_edge.hpp> |
18 | |
19 | #include <boost/concept/detail/concept_def.hpp> |
20 | namespace boost { |
21 | namespace concepts { |
22 | BOOST_concept(CliqueVisitor,(Visitor)(Clique)(Graph)) |
23 | { |
24 | BOOST_CONCEPT_USAGE(CliqueVisitor) |
25 | { |
26 | vis.clique(k, g); |
27 | } |
28 | private: |
29 | Visitor vis; |
30 | Graph g; |
31 | Clique k; |
32 | }; |
33 | } /* namespace concepts */ |
34 | using concepts::CliqueVisitorConcept; |
35 | } /* namespace boost */ |
36 | #include <boost/concept/detail/concept_undef.hpp> |
37 | |
38 | namespace boost |
39 | { |
40 | // The algorithm implemented in this paper is based on the so-called |
41 | // Algorithm 457, published as: |
42 | // |
43 | // @article{362367, |
44 | // author = {Coen Bron and Joep Kerbosch}, |
45 | // title = {Algorithm 457: finding all cliques of an undirected graph}, |
46 | // journal = {Communications of the ACM}, |
47 | // volume = {16}, |
48 | // number = {9}, |
49 | // year = {1973}, |
50 | // issn = {0001-0782}, |
51 | // pages = {575--577}, |
52 | // doi = {http://doi.acm.org/10.1145/362342.362367}, |
53 | // publisher = {ACM Press}, |
54 | // address = {New York, NY, USA}, |
55 | // } |
56 | // |
57 | // Sort of. This implementation is adapted from the 1st version of the |
58 | // algorithm and does not implement the candidate selection optimization |
59 | // described as published - it could, it just doesn't yet. |
60 | // |
61 | // The algorithm is given as proportional to (3.14)^(n/3) power. This is |
62 | // not the same as O(...), but based on time measures and approximation. |
63 | // |
64 | // Unfortunately, this implementation may be less efficient on non- |
65 | // AdjacencyMatrix modeled graphs due to the non-constant implementation |
66 | // of the edge(u,v,g) functions. |
67 | // |
68 | // TODO: It might be worthwhile to provide functionality for passing |
69 | // a connectivity matrix to improve the efficiency of those lookups |
70 | // when needed. This could simply be passed as a BooleanMatrix |
71 | // s.t. edge(u,v,B) returns true or false. This could easily be |
72 | // abstracted for adjacency matricies. |
73 | // |
74 | // The following paper is interesting for a number of reasons. First, |
75 | // it lists a number of other such algorithms and second, it describes |
76 | // a new algorithm (that does not appear to require the edge(u,v,g) |
77 | // function and appears fairly efficient. It is probably worth investigating. |
78 | // |
79 | // @article{DBLP:journals/tcs/TomitaTT06, |
80 | // author = {Etsuji Tomita and Akira Tanaka and Haruhisa Takahashi}, |
81 | // title = {The worst-case time complexity for generating all maximal cliques and computational experiments}, |
82 | // journal = {Theor. Comput. Sci.}, |
83 | // volume = {363}, |
84 | // number = {1}, |
85 | // year = {2006}, |
86 | // pages = {28-42} |
87 | // ee = {http://dx.doi.org/10.1016/j.tcs.2006.06.015} |
88 | // } |
89 | |
90 | /** |
91 | * The default clique_visitor supplies an empty visitation function. |
92 | */ |
93 | struct clique_visitor |
94 | { |
95 | template <typename VertexSet, typename Graph> |
96 | void clique(const VertexSet&, Graph&) |
97 | { } |
98 | }; |
99 | |
100 | /** |
101 | * The max_clique_visitor records the size of the maximum clique (but not the |
102 | * clique itself). |
103 | */ |
104 | struct max_clique_visitor |
105 | { |
106 | max_clique_visitor(std::size_t& max) |
107 | : maximum(max) |
108 | { } |
109 | |
110 | template <typename Clique, typename Graph> |
111 | inline void clique(const Clique& p, const Graph& g) |
112 | { |
113 | BOOST_USING_STD_MAX(); |
114 | maximum = max BOOST_PREVENT_MACRO_SUBSTITUTION (maximum, p.size()); |
115 | } |
116 | std::size_t& maximum; |
117 | }; |
118 | |
119 | inline max_clique_visitor find_max_clique(std::size_t& max) |
120 | { return max_clique_visitor(max); } |
121 | |
122 | namespace detail |
123 | { |
124 | template <typename Graph> |
125 | inline bool |
126 | is_connected_to_clique(const Graph& g, |
127 | typename graph_traits<Graph>::vertex_descriptor u, |
128 | typename graph_traits<Graph>::vertex_descriptor v, |
129 | typename graph_traits<Graph>::undirected_category) |
130 | { |
131 | return lookup_edge(u, v, g).second; |
132 | } |
133 | |
134 | template <typename Graph> |
135 | inline bool |
136 | is_connected_to_clique(const Graph& g, |
137 | typename graph_traits<Graph>::vertex_descriptor u, |
138 | typename graph_traits<Graph>::vertex_descriptor v, |
139 | typename graph_traits<Graph>::directed_category) |
140 | { |
141 | // Note that this could alternate between using an || to determine |
142 | // full connectivity. I believe that this should produce strongly |
143 | // connected components. Note that using && instead of || will |
144 | // change the results to a fully connected subgraph (i.e., symmetric |
145 | // edges between all vertices s.t., if a->b, then b->a. |
146 | return lookup_edge(u, v, g).second && lookup_edge(v, u, g).second; |
147 | } |
148 | |
149 | template <typename Graph, typename Container> |
150 | inline void |
151 | filter_unconnected_vertices(const Graph& g, |
152 | typename graph_traits<Graph>::vertex_descriptor v, |
153 | const Container& in, |
154 | Container& out) |
155 | { |
156 | BOOST_CONCEPT_ASSERT(( GraphConcept<Graph> )); |
157 | |
158 | typename graph_traits<Graph>::directed_category cat; |
159 | typename Container::const_iterator i, end = in.end(); |
160 | for(i = in.begin(); i != end; ++i) { |
161 | if(is_connected_to_clique(g, v, *i, cat)) { |
162 | out.push_back(*i); |
163 | } |
164 | } |
165 | } |
166 | |
167 | template < |
168 | typename Graph, |
169 | typename Clique, // compsub type |
170 | typename Container, // candidates/not type |
171 | typename Visitor> |
172 | void extend_clique(const Graph& g, |
173 | Clique& clique, |
174 | Container& cands, |
175 | Container& nots, |
176 | Visitor vis, |
177 | std::size_t min) |
178 | { |
179 | BOOST_CONCEPT_ASSERT(( GraphConcept<Graph> )); |
180 | BOOST_CONCEPT_ASSERT(( CliqueVisitorConcept<Visitor,Clique,Graph> )); |
181 | typedef typename graph_traits<Graph>::vertex_descriptor Vertex; |
182 | |
183 | // Is there vertex in nots that is connected to all vertices |
184 | // in the candidate set? If so, no clique can ever be found. |
185 | // This could be broken out into a separate function. |
186 | { |
187 | typename Container::iterator ni, nend = nots.end(); |
188 | typename Container::iterator ci, cend = cands.end(); |
189 | for(ni = nots.begin(); ni != nend; ++ni) { |
190 | for(ci = cands.begin(); ci != cend; ++ci) { |
191 | // if we don't find an edge, then we're okay. |
192 | if(!lookup_edge(*ni, *ci, g).second) break; |
193 | } |
194 | // if we iterated all the way to the end, then *ni |
195 | // is connected to all *ci |
196 | if(ci == cend) break; |
197 | } |
198 | // if we broke early, we found *ni connected to all *ci |
199 | if(ni != nend) return; |
200 | } |
201 | |
202 | // TODO: the original algorithm 457 describes an alternative |
203 | // (albeit really complicated) mechanism for selecting candidates. |
204 | // The given optimizaiton seeks to bring about the above |
205 | // condition sooner (i.e., there is a vertex in the not set |
206 | // that is connected to all candidates). unfortunately, the |
207 | // method they give for doing this is fairly unclear. |
208 | |
209 | // basically, for every vertex in not, we should know how many |
210 | // vertices it is disconnected from in the candidate set. if |
211 | // we fix some vertex in the not set, then we want to keep |
212 | // choosing vertices that are not connected to that fixed vertex. |
213 | // apparently, by selecting fix point with the minimum number |
214 | // of disconnections (i.e., the maximum number of connections |
215 | // within the candidate set), then the previous condition wil |
216 | // be reached sooner. |
217 | |
218 | // there's some other stuff about using the number of disconnects |
219 | // as a counter, but i'm jot really sure i followed it. |
220 | |
221 | // TODO: If we min-sized cliques to visit, then theoretically, we |
222 | // should be able to stop recursing if the clique falls below that |
223 | // size - maybe? |
224 | |
225 | // otherwise, iterate over candidates and and test |
226 | // for maxmimal cliquiness. |
227 | typename Container::iterator i, j; |
228 | for(i = cands.begin(); i != cands.end(); ) { |
229 | Vertex candidate = *i; |
230 | |
231 | // add the candidate to the clique (keeping the iterator!) |
232 | // typename Clique::iterator ci = clique.insert(clique.end(), candidate); |
233 | clique.push_back(candidate); |
234 | |
235 | // remove it from the candidate set |
236 | i = cands.erase(i); |
237 | |
238 | // build new candidate and not sets by removing all vertices |
239 | // that are not connected to the current candidate vertex. |
240 | // these actually invert the operation, adding them to the new |
241 | // sets if the vertices are connected. its semantically the same. |
242 | Container new_cands, new_nots; |
243 | filter_unconnected_vertices(g, candidate, cands, new_cands); |
244 | filter_unconnected_vertices(g, candidate, nots, new_nots); |
245 | |
246 | if(new_cands.empty() && new_nots.empty()) { |
247 | // our current clique is maximal since there's nothing |
248 | // that's connected that we haven't already visited. If |
249 | // the clique is below our radar, then we won't visit it. |
250 | if(clique.size() >= min) { |
251 | vis.clique(clique, g); |
252 | } |
253 | } |
254 | else { |
255 | // recurse to explore the new candidates |
256 | extend_clique(g, clique, new_cands, new_nots, vis, min); |
257 | } |
258 | |
259 | // we're done with this vertex, so we need to move it |
260 | // to the nots, and remove the candidate from the clique. |
261 | nots.push_back(candidate); |
262 | clique.pop_back(); |
263 | } |
264 | } |
265 | } /* namespace detail */ |
266 | |
267 | template <typename Graph, typename Visitor> |
268 | inline void |
269 | bron_kerbosch_all_cliques(const Graph& g, Visitor vis, std::size_t min) |
270 | { |
271 | BOOST_CONCEPT_ASSERT(( IncidenceGraphConcept<Graph> )); |
272 | BOOST_CONCEPT_ASSERT(( VertexListGraphConcept<Graph> )); |
273 | BOOST_CONCEPT_ASSERT(( AdjacencyMatrixConcept<Graph> )); // Structural requirement only |
274 | typedef typename graph_traits<Graph>::vertex_descriptor Vertex; |
275 | typedef typename graph_traits<Graph>::vertex_iterator VertexIterator; |
276 | typedef std::vector<Vertex> VertexSet; |
277 | typedef std::deque<Vertex> Clique; |
278 | BOOST_CONCEPT_ASSERT(( CliqueVisitorConcept<Visitor,Clique,Graph> )); |
279 | |
280 | // NOTE: We're using a deque to implement the clique, because it provides |
281 | // constant inserts and removals at the end and also a constant size. |
282 | |
283 | VertexIterator i, end; |
284 | boost::tie(i, end) = vertices(g); |
285 | VertexSet cands(i, end); // start with all vertices as candidates |
286 | VertexSet nots; // start with no vertices visited |
287 | |
288 | Clique clique; // the first clique is an empty vertex set |
289 | detail::extend_clique(g, clique, cands, nots, vis, min); |
290 | } |
291 | |
292 | // NOTE: By default the minimum number of vertices per clique is set at 2 |
293 | // because singleton cliques aren't really very interesting. |
294 | template <typename Graph, typename Visitor> |
295 | inline void |
296 | bron_kerbosch_all_cliques(const Graph& g, Visitor vis) |
297 | { bron_kerbosch_all_cliques(g, vis, 2); } |
298 | |
299 | template <typename Graph> |
300 | inline std::size_t |
301 | bron_kerbosch_clique_number(const Graph& g) |
302 | { |
303 | std::size_t ret = 0; |
304 | bron_kerbosch_all_cliques(g, find_max_clique(max&: ret)); |
305 | return ret; |
306 | } |
307 | |
308 | } /* namespace boost */ |
309 | |
310 | #endif |
311 | |