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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} | |
92f5a8d4 | 87 | // ee = {https://doi.org/10.1016/j.tcs.2006.06.015} |
7c673cae FG |
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(ret)); | |
305 | return ret; | |
306 | } | |
307 | ||
308 | } /* namespace boost */ | |
309 | ||
310 | #endif |