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900 lines
30 KiB
C++
900 lines
30 KiB
C++
//=======================================================================
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// Copyright (c) 2005 Aaron Windsor
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//
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// Distributed under the Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt or copy at
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// http://www.boost.org/LICENSE_1_0.txt)
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//
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//=======================================================================
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#ifndef BOOST_GRAPH_MAXIMUM_CARDINALITY_MATCHING_HPP
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#define BOOST_GRAPH_MAXIMUM_CARDINALITY_MATCHING_HPP
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#include <vector>
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#include <list>
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#include <deque>
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#include <algorithm> // for std::sort and std::stable_sort
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#include <utility> // for std::pair
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#include <boost/property_map/property_map.hpp>
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#include <boost/graph/graph_traits.hpp>
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#include <boost/graph/visitors.hpp>
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#include <boost/graph/depth_first_search.hpp>
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#include <boost/graph/filtered_graph.hpp>
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#include <boost/pending/disjoint_sets.hpp>
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#include <boost/assert.hpp>
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namespace boost
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{
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namespace graph { namespace detail {
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enum { V_EVEN, V_ODD, V_UNREACHED };
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} } // end namespace graph::detail
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template <typename Graph, typename MateMap, typename VertexIndexMap>
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typename graph_traits<Graph>::vertices_size_type
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matching_size(const Graph& g, MateMap mate, VertexIndexMap vm)
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{
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typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator_t;
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typedef typename graph_traits<Graph>::vertex_descriptor
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vertex_descriptor_t;
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typedef typename graph_traits<Graph>::vertices_size_type v_size_t;
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v_size_t size_of_matching = 0;
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vertex_iterator_t vi, vi_end;
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for(boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
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{
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vertex_descriptor_t v = *vi;
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if (get(mate,v) != graph_traits<Graph>::null_vertex()
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&& get(vm,v) < get(vm,get(mate,v)))
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++size_of_matching;
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}
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return size_of_matching;
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}
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template <typename Graph, typename MateMap>
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inline typename graph_traits<Graph>::vertices_size_type
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matching_size(const Graph& g, MateMap mate)
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{
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return matching_size(g, mate, get(vertex_index,g));
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}
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template <typename Graph, typename MateMap, typename VertexIndexMap>
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bool is_a_matching(const Graph& g, MateMap mate, VertexIndexMap)
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{
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typedef typename graph_traits<Graph>::vertex_descriptor
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vertex_descriptor_t;
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typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator_t;
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vertex_iterator_t vi, vi_end;
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for( boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
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{
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vertex_descriptor_t v = *vi;
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if (get(mate,v) != graph_traits<Graph>::null_vertex()
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&& v != get(mate,get(mate,v)))
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return false;
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}
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return true;
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}
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template <typename Graph, typename MateMap>
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inline bool is_a_matching(const Graph& g, MateMap mate)
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{
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return is_a_matching(g, mate, get(vertex_index,g));
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}
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//***************************************************************************
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//***************************************************************************
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// Maximum Cardinality Matching Functors
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//***************************************************************************
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//***************************************************************************
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template <typename Graph, typename MateMap,
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typename VertexIndexMap = dummy_property_map>
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struct no_augmenting_path_finder
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{
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no_augmenting_path_finder(const Graph&, MateMap, VertexIndexMap)
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{ }
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inline bool augment_matching() { return false; }
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template <typename PropertyMap>
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void get_current_matching(PropertyMap) {}
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};
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template <typename Graph, typename MateMap, typename VertexIndexMap>
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class edmonds_augmenting_path_finder
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{
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// This implementation of Edmonds' matching algorithm closely
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// follows Tarjan's description of the algorithm in "Data
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// Structures and Network Algorithms."
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public:
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//generates the type of an iterator property map from vertices to type X
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template <typename X>
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struct map_vertex_to_
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{
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typedef boost::iterator_property_map<typename std::vector<X>::iterator,
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VertexIndexMap> type;
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};
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typedef typename graph_traits<Graph>::vertex_descriptor
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vertex_descriptor_t;
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typedef typename std::pair< vertex_descriptor_t, vertex_descriptor_t >
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vertex_pair_t;
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typedef typename graph_traits<Graph>::edge_descriptor edge_descriptor_t;
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typedef typename graph_traits<Graph>::vertices_size_type v_size_t;
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typedef typename graph_traits<Graph>::edges_size_type e_size_t;
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typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator_t;
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typedef typename graph_traits<Graph>::out_edge_iterator
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out_edge_iterator_t;
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typedef typename std::deque<vertex_descriptor_t> vertex_list_t;
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typedef typename std::vector<edge_descriptor_t> edge_list_t;
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typedef typename map_vertex_to_<vertex_descriptor_t>::type
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vertex_to_vertex_map_t;
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typedef typename map_vertex_to_<int>::type vertex_to_int_map_t;
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typedef typename map_vertex_to_<vertex_pair_t>::type
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vertex_to_vertex_pair_map_t;
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typedef typename map_vertex_to_<v_size_t>::type vertex_to_vsize_map_t;
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typedef typename map_vertex_to_<e_size_t>::type vertex_to_esize_map_t;
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edmonds_augmenting_path_finder(const Graph& arg_g, MateMap arg_mate,
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VertexIndexMap arg_vm) :
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g(arg_g),
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vm(arg_vm),
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n_vertices(num_vertices(arg_g)),
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mate_vector(n_vertices),
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ancestor_of_v_vector(n_vertices),
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ancestor_of_w_vector(n_vertices),
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vertex_state_vector(n_vertices),
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origin_vector(n_vertices),
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pred_vector(n_vertices),
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bridge_vector(n_vertices),
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ds_parent_vector(n_vertices),
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ds_rank_vector(n_vertices),
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mate(mate_vector.begin(), vm),
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ancestor_of_v(ancestor_of_v_vector.begin(), vm),
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ancestor_of_w(ancestor_of_w_vector.begin(), vm),
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vertex_state(vertex_state_vector.begin(), vm),
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origin(origin_vector.begin(), vm),
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pred(pred_vector.begin(), vm),
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bridge(bridge_vector.begin(), vm),
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ds_parent_map(ds_parent_vector.begin(), vm),
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ds_rank_map(ds_rank_vector.begin(), vm),
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ds(ds_rank_map, ds_parent_map)
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{
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vertex_iterator_t vi, vi_end;
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for(boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
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mate[*vi] = get(arg_mate, *vi);
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}
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bool augment_matching()
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{
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//As an optimization, some of these values can be saved from one
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//iteration to the next instead of being re-initialized each
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//iteration, allowing for "lazy blossom expansion." This is not
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//currently implemented.
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e_size_t timestamp = 0;
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even_edges.clear();
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vertex_iterator_t vi, vi_end;
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for(boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
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{
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vertex_descriptor_t u = *vi;
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origin[u] = u;
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pred[u] = u;
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ancestor_of_v[u] = 0;
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ancestor_of_w[u] = 0;
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ds.make_set(u);
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if (mate[u] == graph_traits<Graph>::null_vertex())
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{
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vertex_state[u] = graph::detail::V_EVEN;
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out_edge_iterator_t ei, ei_end;
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for(boost::tie(ei,ei_end) = out_edges(u,g); ei != ei_end; ++ei)
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{
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if (target(*ei,g) != u)
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{
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even_edges.push_back( *ei );
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}
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}
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}
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else
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vertex_state[u] = graph::detail::V_UNREACHED;
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}
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//end initializations
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vertex_descriptor_t v,w,w_free_ancestor,v_free_ancestor;
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w_free_ancestor = graph_traits<Graph>::null_vertex();
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v_free_ancestor = graph_traits<Graph>::null_vertex();
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bool found_alternating_path = false;
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while(!even_edges.empty() && !found_alternating_path)
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{
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// since we push even edges onto the back of the list as
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// they're discovered, taking them off the back will search
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// for augmenting paths depth-first.
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edge_descriptor_t current_edge = even_edges.back();
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even_edges.pop_back();
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v = source(current_edge,g);
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w = target(current_edge,g);
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vertex_descriptor_t v_prime = origin[ds.find_set(v)];
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vertex_descriptor_t w_prime = origin[ds.find_set(w)];
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// because of the way we put all of the edges on the queue,
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// v_prime should be labeled V_EVEN; the following is a
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// little paranoid but it could happen...
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if (vertex_state[v_prime] != graph::detail::V_EVEN)
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{
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std::swap(v_prime,w_prime);
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std::swap(v,w);
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}
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if (vertex_state[w_prime] == graph::detail::V_UNREACHED)
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{
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vertex_state[w_prime] = graph::detail::V_ODD;
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vertex_descriptor_t w_prime_mate = mate[w_prime];
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vertex_state[w_prime_mate] = graph::detail::V_EVEN;
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out_edge_iterator_t ei, ei_end;
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for( boost::tie(ei,ei_end) = out_edges(w_prime_mate, g); ei != ei_end; ++ei)
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{
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if (target(*ei,g) != w_prime_mate)
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{
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even_edges.push_back(*ei);
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}
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}
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pred[w_prime] = v;
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}
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//w_prime == v_prime can happen below if we get an edge that has been
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//shrunk into a blossom
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else if (vertex_state[w_prime] == graph::detail::V_EVEN && w_prime != v_prime)
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{
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vertex_descriptor_t w_up = w_prime;
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vertex_descriptor_t v_up = v_prime;
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vertex_descriptor_t nearest_common_ancestor
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= graph_traits<Graph>::null_vertex();
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w_free_ancestor = graph_traits<Graph>::null_vertex();
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v_free_ancestor = graph_traits<Graph>::null_vertex();
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// We now need to distinguish between the case that
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// w_prime and v_prime share an ancestor under the
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// "parent" relation, in which case we've found a
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// blossom and should shrink it, or the case that
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// w_prime and v_prime both have distinct ancestors that
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// are free, in which case we've found an alternating
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// path between those two ancestors.
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++timestamp;
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while (nearest_common_ancestor == graph_traits<Graph>::null_vertex() &&
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(v_free_ancestor == graph_traits<Graph>::null_vertex() ||
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w_free_ancestor == graph_traits<Graph>::null_vertex()
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)
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)
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{
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ancestor_of_w[w_up] = timestamp;
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ancestor_of_v[v_up] = timestamp;
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if (w_free_ancestor == graph_traits<Graph>::null_vertex())
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w_up = parent(w_up);
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if (v_free_ancestor == graph_traits<Graph>::null_vertex())
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v_up = parent(v_up);
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if (mate[v_up] == graph_traits<Graph>::null_vertex())
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v_free_ancestor = v_up;
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if (mate[w_up] == graph_traits<Graph>::null_vertex())
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w_free_ancestor = w_up;
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if (ancestor_of_w[v_up] == timestamp)
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nearest_common_ancestor = v_up;
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else if (ancestor_of_v[w_up] == timestamp)
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nearest_common_ancestor = w_up;
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else if (v_free_ancestor == w_free_ancestor &&
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v_free_ancestor != graph_traits<Graph>::null_vertex())
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nearest_common_ancestor = v_up;
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}
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if (nearest_common_ancestor == graph_traits<Graph>::null_vertex())
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found_alternating_path = true; //to break out of the loop
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else
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{
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//shrink the blossom
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link_and_set_bridges(w_prime, nearest_common_ancestor, std::make_pair(w,v));
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link_and_set_bridges(v_prime, nearest_common_ancestor, std::make_pair(v,w));
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}
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}
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}
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if (!found_alternating_path)
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return false;
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// retrieve the augmenting path and put it in aug_path
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reversed_retrieve_augmenting_path(v, v_free_ancestor);
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retrieve_augmenting_path(w, w_free_ancestor);
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// augment the matching along aug_path
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vertex_descriptor_t a,b;
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while (!aug_path.empty())
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{
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a = aug_path.front();
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aug_path.pop_front();
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b = aug_path.front();
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aug_path.pop_front();
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mate[a] = b;
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mate[b] = a;
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}
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return true;
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}
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template <typename PropertyMap>
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void get_current_matching(PropertyMap pm)
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{
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vertex_iterator_t vi,vi_end;
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for(boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
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put(pm, *vi, mate[*vi]);
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}
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template <typename PropertyMap>
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void get_vertex_state_map(PropertyMap pm)
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{
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vertex_iterator_t vi,vi_end;
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for(boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
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put(pm, *vi, vertex_state[origin[ds.find_set(*vi)]]);
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}
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private:
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vertex_descriptor_t parent(vertex_descriptor_t x)
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{
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if (vertex_state[x] == graph::detail::V_EVEN
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&& mate[x] != graph_traits<Graph>::null_vertex())
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return mate[x];
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else if (vertex_state[x] == graph::detail::V_ODD)
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return origin[ds.find_set(pred[x])];
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else
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return x;
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}
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void link_and_set_bridges(vertex_descriptor_t x,
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vertex_descriptor_t stop_vertex,
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vertex_pair_t the_bridge)
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{
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for(vertex_descriptor_t v = x; v != stop_vertex; v = parent(v))
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{
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ds.union_set(v, stop_vertex);
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origin[ds.find_set(stop_vertex)] = stop_vertex;
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if (vertex_state[v] == graph::detail::V_ODD)
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{
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bridge[v] = the_bridge;
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out_edge_iterator_t oei, oei_end;
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for(boost::tie(oei, oei_end) = out_edges(v,g); oei != oei_end; ++oei)
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{
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if (target(*oei,g) != v)
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{
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even_edges.push_back(*oei);
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}
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}
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}
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}
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}
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// Since none of the STL containers support both constant-time
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// concatenation and reversal, the process of expanding an
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// augmenting path once we know one exists is a little more
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// complicated than it has to be. If we know the path is from v to
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// w, then the augmenting path is recursively defined as:
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//
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// path(v,w) = [v], if v = w
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// = concat([v, mate[v]], path(pred[mate[v]], w),
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// if v != w and vertex_state[v] == graph::detail::V_EVEN
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// = concat([v], reverse(path(x,mate[v])), path(y,w)),
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// if v != w, vertex_state[v] == graph::detail::V_ODD, and bridge[v] = (x,y)
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//
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// These next two mutually recursive functions implement this definition.
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void retrieve_augmenting_path(vertex_descriptor_t v, vertex_descriptor_t w)
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{
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if (v == w)
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aug_path.push_back(v);
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else if (vertex_state[v] == graph::detail::V_EVEN)
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{
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aug_path.push_back(v);
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aug_path.push_back(mate[v]);
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retrieve_augmenting_path(pred[mate[v]], w);
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}
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else //vertex_state[v] == graph::detail::V_ODD
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{
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aug_path.push_back(v);
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reversed_retrieve_augmenting_path(bridge[v].first, mate[v]);
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retrieve_augmenting_path(bridge[v].second, w);
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}
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}
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void reversed_retrieve_augmenting_path(vertex_descriptor_t v,
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vertex_descriptor_t w)
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{
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if (v == w)
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aug_path.push_back(v);
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else if (vertex_state[v] == graph::detail::V_EVEN)
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{
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reversed_retrieve_augmenting_path(pred[mate[v]], w);
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aug_path.push_back(mate[v]);
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aug_path.push_back(v);
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}
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else //vertex_state[v] == graph::detail::V_ODD
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{
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reversed_retrieve_augmenting_path(bridge[v].second, w);
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retrieve_augmenting_path(bridge[v].first, mate[v]);
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aug_path.push_back(v);
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}
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}
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//private data members
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const Graph& g;
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VertexIndexMap vm;
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v_size_t n_vertices;
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//storage for the property maps below
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std::vector<vertex_descriptor_t> mate_vector;
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std::vector<e_size_t> ancestor_of_v_vector;
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std::vector<e_size_t> ancestor_of_w_vector;
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std::vector<int> vertex_state_vector;
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std::vector<vertex_descriptor_t> origin_vector;
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std::vector<vertex_descriptor_t> pred_vector;
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std::vector<vertex_pair_t> bridge_vector;
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std::vector<vertex_descriptor_t> ds_parent_vector;
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std::vector<v_size_t> ds_rank_vector;
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//iterator property maps
|
|
vertex_to_vertex_map_t mate;
|
|
vertex_to_esize_map_t ancestor_of_v;
|
|
vertex_to_esize_map_t ancestor_of_w;
|
|
vertex_to_int_map_t vertex_state;
|
|
vertex_to_vertex_map_t origin;
|
|
vertex_to_vertex_map_t pred;
|
|
vertex_to_vertex_pair_map_t bridge;
|
|
vertex_to_vertex_map_t ds_parent_map;
|
|
vertex_to_vsize_map_t ds_rank_map;
|
|
|
|
vertex_list_t aug_path;
|
|
edge_list_t even_edges;
|
|
disjoint_sets< vertex_to_vsize_map_t, vertex_to_vertex_map_t > ds;
|
|
|
|
};
|
|
|
|
|
|
|
|
|
|
//***************************************************************************
|
|
//***************************************************************************
|
|
// Initial Matching Functors
|
|
//***************************************************************************
|
|
//***************************************************************************
|
|
|
|
template <typename Graph, typename MateMap>
|
|
struct greedy_matching
|
|
{
|
|
typedef typename graph_traits< Graph >::vertex_descriptor vertex_descriptor_t;
|
|
typedef typename graph_traits< Graph >::vertex_iterator vertex_iterator_t;
|
|
typedef typename graph_traits< Graph >::edge_descriptor edge_descriptor_t;
|
|
typedef typename graph_traits< Graph >::edge_iterator edge_iterator_t;
|
|
|
|
static void find_matching(const Graph& g, MateMap mate)
|
|
{
|
|
vertex_iterator_t vi, vi_end;
|
|
for(boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
|
|
put(mate, *vi, graph_traits<Graph>::null_vertex());
|
|
|
|
edge_iterator_t ei, ei_end;
|
|
for( boost::tie(ei, ei_end) = edges(g); ei != ei_end; ++ei)
|
|
{
|
|
edge_descriptor_t e = *ei;
|
|
vertex_descriptor_t u = source(e,g);
|
|
vertex_descriptor_t v = target(e,g);
|
|
|
|
if (u != v && get(mate,u) == get(mate,v))
|
|
//only way equality can hold is if
|
|
// mate[u] == mate[v] == null_vertex
|
|
{
|
|
put(mate,u,v);
|
|
put(mate,v,u);
|
|
}
|
|
}
|
|
}
|
|
};
|
|
|
|
|
|
|
|
|
|
template <typename Graph, typename MateMap>
|
|
struct extra_greedy_matching
|
|
{
|
|
// The "extra greedy matching" is formed by repeating the
|
|
// following procedure as many times as possible: Choose the
|
|
// unmatched vertex v of minimum non-zero degree. Choose the
|
|
// neighbor w of v which is unmatched and has minimum degree over
|
|
// all of v's neighbors. Add (u,v) to the matching. Ties for
|
|
// either choice are broken arbitrarily. This procedure takes time
|
|
// O(m log n), where m is the number of edges in the graph and n
|
|
// is the number of vertices.
|
|
|
|
typedef typename graph_traits< Graph >::vertex_descriptor
|
|
vertex_descriptor_t;
|
|
typedef typename graph_traits< Graph >::vertex_iterator vertex_iterator_t;
|
|
typedef typename graph_traits< Graph >::edge_descriptor edge_descriptor_t;
|
|
typedef typename graph_traits< Graph >::edge_iterator edge_iterator_t;
|
|
typedef std::pair<vertex_descriptor_t, vertex_descriptor_t> vertex_pair_t;
|
|
|
|
struct select_first
|
|
{
|
|
inline static vertex_descriptor_t select_vertex(const vertex_pair_t p)
|
|
{return p.first;}
|
|
};
|
|
|
|
struct select_second
|
|
{
|
|
inline static vertex_descriptor_t select_vertex(const vertex_pair_t p)
|
|
{return p.second;}
|
|
};
|
|
|
|
template <class PairSelector>
|
|
class less_than_by_degree
|
|
{
|
|
public:
|
|
less_than_by_degree(const Graph& g): m_g(g) {}
|
|
bool operator() (const vertex_pair_t x, const vertex_pair_t y)
|
|
{
|
|
return
|
|
out_degree(PairSelector::select_vertex(x), m_g)
|
|
< out_degree(PairSelector::select_vertex(y), m_g);
|
|
}
|
|
private:
|
|
const Graph& m_g;
|
|
};
|
|
|
|
|
|
static void find_matching(const Graph& g, MateMap mate)
|
|
{
|
|
typedef std::vector<std::pair<vertex_descriptor_t, vertex_descriptor_t> >
|
|
directed_edges_vector_t;
|
|
|
|
directed_edges_vector_t edge_list;
|
|
vertex_iterator_t vi, vi_end;
|
|
for(boost::tie(vi, vi_end) = vertices(g); vi != vi_end; ++vi)
|
|
put(mate, *vi, graph_traits<Graph>::null_vertex());
|
|
|
|
edge_iterator_t ei, ei_end;
|
|
for(boost::tie(ei, ei_end) = edges(g); ei != ei_end; ++ei)
|
|
{
|
|
edge_descriptor_t e = *ei;
|
|
vertex_descriptor_t u = source(e,g);
|
|
vertex_descriptor_t v = target(e,g);
|
|
if (u == v) continue;
|
|
edge_list.push_back(std::make_pair(u,v));
|
|
edge_list.push_back(std::make_pair(v,u));
|
|
}
|
|
|
|
//sort the edges by the degree of the target, then (using a
|
|
//stable sort) by degree of the source
|
|
std::sort(edge_list.begin(), edge_list.end(),
|
|
less_than_by_degree<select_second>(g));
|
|
std::stable_sort(edge_list.begin(), edge_list.end(),
|
|
less_than_by_degree<select_first>(g));
|
|
|
|
//construct the extra greedy matching
|
|
for(typename directed_edges_vector_t::const_iterator itr = edge_list.begin(); itr != edge_list.end(); ++itr)
|
|
{
|
|
if (get(mate,itr->first) == get(mate,itr->second))
|
|
//only way equality can hold is if mate[itr->first] == mate[itr->second] == null_vertex
|
|
{
|
|
put(mate, itr->first, itr->second);
|
|
put(mate, itr->second, itr->first);
|
|
}
|
|
}
|
|
}
|
|
};
|
|
|
|
|
|
|
|
|
|
template <typename Graph, typename MateMap>
|
|
struct empty_matching
|
|
{
|
|
typedef typename graph_traits< Graph >::vertex_iterator vertex_iterator_t;
|
|
|
|
static void find_matching(const Graph& g, MateMap mate)
|
|
{
|
|
vertex_iterator_t vi, vi_end;
|
|
for(boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
|
|
put(mate, *vi, graph_traits<Graph>::null_vertex());
|
|
}
|
|
};
|
|
|
|
|
|
|
|
|
|
//***************************************************************************
|
|
//***************************************************************************
|
|
// Matching Verifiers
|
|
//***************************************************************************
|
|
//***************************************************************************
|
|
|
|
namespace detail
|
|
{
|
|
|
|
template <typename SizeType>
|
|
class odd_components_counter : public dfs_visitor<>
|
|
// This depth-first search visitor will count the number of connected
|
|
// components with an odd number of vertices. It's used by
|
|
// maximum_matching_verifier.
|
|
{
|
|
public:
|
|
odd_components_counter(SizeType& c_count):
|
|
m_count(c_count)
|
|
{
|
|
m_count = 0;
|
|
}
|
|
|
|
template <class Vertex, class Graph>
|
|
void start_vertex(Vertex, Graph&)
|
|
{
|
|
m_parity = false;
|
|
}
|
|
|
|
template <class Vertex, class Graph>
|
|
void discover_vertex(Vertex, Graph&)
|
|
{
|
|
m_parity = !m_parity;
|
|
m_parity ? ++m_count : --m_count;
|
|
}
|
|
|
|
protected:
|
|
SizeType& m_count;
|
|
|
|
private:
|
|
bool m_parity;
|
|
|
|
};
|
|
|
|
}//namespace detail
|
|
|
|
|
|
|
|
|
|
template <typename Graph, typename MateMap,
|
|
typename VertexIndexMap = dummy_property_map>
|
|
struct no_matching_verifier
|
|
{
|
|
inline static bool
|
|
verify_matching(const Graph&, MateMap, VertexIndexMap)
|
|
{ return true;}
|
|
};
|
|
|
|
|
|
|
|
|
|
template <typename Graph, typename MateMap, typename VertexIndexMap>
|
|
struct maximum_cardinality_matching_verifier
|
|
{
|
|
|
|
template <typename X>
|
|
struct map_vertex_to_
|
|
{
|
|
typedef boost::iterator_property_map<typename std::vector<X>::iterator,
|
|
VertexIndexMap> type;
|
|
};
|
|
|
|
typedef typename graph_traits<Graph>::vertex_descriptor
|
|
vertex_descriptor_t;
|
|
typedef typename graph_traits<Graph>::vertices_size_type v_size_t;
|
|
typedef typename graph_traits<Graph>::vertex_iterator vertex_iterator_t;
|
|
typedef typename map_vertex_to_<int>::type vertex_to_int_map_t;
|
|
typedef typename map_vertex_to_<vertex_descriptor_t>::type
|
|
vertex_to_vertex_map_t;
|
|
|
|
|
|
template <typename VertexStateMap>
|
|
struct non_odd_vertex {
|
|
//this predicate is used to create a filtered graph that
|
|
//excludes vertices labeled "graph::detail::V_ODD"
|
|
non_odd_vertex() : vertex_state(0) { }
|
|
|
|
non_odd_vertex(VertexStateMap* arg_vertex_state)
|
|
: vertex_state(arg_vertex_state) { }
|
|
|
|
template <typename Vertex>
|
|
bool operator()(const Vertex& v) const
|
|
{
|
|
BOOST_ASSERT(vertex_state);
|
|
return get(*vertex_state, v) != graph::detail::V_ODD;
|
|
}
|
|
|
|
VertexStateMap* vertex_state;
|
|
};
|
|
|
|
|
|
static bool verify_matching(const Graph& g, MateMap mate, VertexIndexMap vm)
|
|
{
|
|
//For any graph G, let o(G) be the number of connected
|
|
//components in G of odd size. For a subset S of G's vertex set
|
|
//V(G), let (G - S) represent the subgraph of G induced by
|
|
//removing all vertices in S from G. Let M(G) be the size of the
|
|
//maximum cardinality matching in G. Then the Tutte-Berge
|
|
//formula guarantees that
|
|
//
|
|
// 2 * M(G) = min ( |V(G)| + |U| + o(G - U) )
|
|
//
|
|
//where the minimum is taken over all subsets U of
|
|
//V(G). Edmonds' algorithm finds a set U that achieves the
|
|
//minimum in the above formula, namely the vertices labeled
|
|
//"ODD." This function runs one iteration of Edmonds' algorithm
|
|
//to find U, then verifies that the size of the matching given
|
|
//by mate satisfies the Tutte-Berge formula.
|
|
|
|
//first, make sure it's a valid matching
|
|
if (!is_a_matching(g,mate,vm))
|
|
return false;
|
|
|
|
//We'll try to augment the matching once. This serves two
|
|
//purposes: first, if we find some augmenting path, the matching
|
|
//is obviously non-maximum. Second, running edmonds' algorithm
|
|
//on a graph with no augmenting path will create the
|
|
//Edmonds-Gallai decomposition that we need as a certificate of
|
|
//maximality - we can get it by looking at the vertex_state map
|
|
//that results.
|
|
edmonds_augmenting_path_finder<Graph,MateMap,VertexIndexMap>
|
|
augmentor(g,mate,vm);
|
|
if (augmentor.augment_matching())
|
|
return false;
|
|
|
|
std::vector<int> vertex_state_vector(num_vertices(g));
|
|
vertex_to_int_map_t vertex_state(vertex_state_vector.begin(), vm);
|
|
augmentor.get_vertex_state_map(vertex_state);
|
|
|
|
//count the number of graph::detail::V_ODD vertices
|
|
v_size_t num_odd_vertices = 0;
|
|
vertex_iterator_t vi, vi_end;
|
|
for(boost::tie(vi,vi_end) = vertices(g); vi != vi_end; ++vi)
|
|
if (vertex_state[*vi] == graph::detail::V_ODD)
|
|
++num_odd_vertices;
|
|
|
|
//count the number of connected components with odd cardinality
|
|
//in the graph without graph::detail::V_ODD vertices
|
|
non_odd_vertex<vertex_to_int_map_t> filter(&vertex_state);
|
|
filtered_graph<Graph, keep_all, non_odd_vertex<vertex_to_int_map_t> > fg(g, keep_all(), filter);
|
|
|
|
v_size_t num_odd_components;
|
|
detail::odd_components_counter<v_size_t> occ(num_odd_components);
|
|
depth_first_search(fg, visitor(occ).vertex_index_map(vm));
|
|
|
|
if (2 * matching_size(g,mate,vm) == num_vertices(g) + num_odd_vertices - num_odd_components)
|
|
return true;
|
|
else
|
|
return false;
|
|
}
|
|
};
|
|
|
|
|
|
|
|
|
|
template <typename Graph,
|
|
typename MateMap,
|
|
typename VertexIndexMap,
|
|
template <typename, typename, typename> class AugmentingPathFinder,
|
|
template <typename, typename> class InitialMatchingFinder,
|
|
template <typename, typename, typename> class MatchingVerifier>
|
|
bool matching(const Graph& g, MateMap mate, VertexIndexMap vm)
|
|
{
|
|
|
|
InitialMatchingFinder<Graph,MateMap>::find_matching(g,mate);
|
|
|
|
AugmentingPathFinder<Graph,MateMap,VertexIndexMap> augmentor(g,mate,vm);
|
|
bool not_maximum_yet = true;
|
|
while(not_maximum_yet)
|
|
{
|
|
not_maximum_yet = augmentor.augment_matching();
|
|
}
|
|
augmentor.get_current_matching(mate);
|
|
|
|
return MatchingVerifier<Graph,MateMap,VertexIndexMap>::verify_matching(g,mate,vm);
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
template <typename Graph, typename MateMap, typename VertexIndexMap>
|
|
inline bool checked_edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate, VertexIndexMap vm)
|
|
{
|
|
return matching
|
|
< Graph, MateMap, VertexIndexMap,
|
|
edmonds_augmenting_path_finder, extra_greedy_matching, maximum_cardinality_matching_verifier>
|
|
(g, mate, vm);
|
|
}
|
|
|
|
|
|
|
|
|
|
template <typename Graph, typename MateMap>
|
|
inline bool checked_edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate)
|
|
{
|
|
return checked_edmonds_maximum_cardinality_matching(g, mate, get(vertex_index,g));
|
|
}
|
|
|
|
|
|
|
|
|
|
template <typename Graph, typename MateMap, typename VertexIndexMap>
|
|
inline void edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate, VertexIndexMap vm)
|
|
{
|
|
matching < Graph, MateMap, VertexIndexMap,
|
|
edmonds_augmenting_path_finder, extra_greedy_matching, no_matching_verifier>
|
|
(g, mate, vm);
|
|
}
|
|
|
|
|
|
|
|
|
|
template <typename Graph, typename MateMap>
|
|
inline void edmonds_maximum_cardinality_matching(const Graph& g, MateMap mate)
|
|
{
|
|
edmonds_maximum_cardinality_matching(g, mate, get(vertex_index,g));
|
|
}
|
|
|
|
}//namespace boost
|
|
|
|
#endif //BOOST_GRAPH_MAXIMUM_CARDINALITY_MATCHING_HPP
|