[1077] | 1 | /* -*- C++ -*- |
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| 2 | * src/lemon/max_matching.h - Part of LEMON, a generic C++ optimization library |
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| 3 | * |
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| 4 | * Copyright (C) 2004 Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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| 5 | * (Egervary Combinatorial Optimization Research Group, EGRES). |
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| 6 | * |
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| 7 | * Permission to use, modify and distribute this software is granted |
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| 8 | * provided that this copyright notice appears in all copies. For |
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| 9 | * precise terms see the accompanying LICENSE file. |
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| 10 | * |
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| 11 | * This software is provided "AS IS" with no warranty of any kind, |
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| 12 | * express or implied, and with no claim as to its suitability for any |
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| 13 | * purpose. |
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| 14 | * |
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| 15 | */ |
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| 16 | |
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| 17 | #ifndef LEMON_MAX_MATCHING_H |
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| 18 | #define LEMON_MAX_MATCHING_H |
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| 19 | |
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| 20 | #include <queue> |
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| 21 | #include <invalid.h> |
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| 22 | #include <unionfind.h> |
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| 23 | #include <lemon/graph_utils.h> |
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| 24 | |
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| 25 | ///\ingroup galgs |
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| 26 | ///\file |
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| 27 | ///\brief Maximum matching algorithm. |
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| 28 | |
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| 29 | namespace lemon { |
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| 30 | |
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| 31 | /// \addtogroup galgs |
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| 32 | /// @{ |
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| 33 | |
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| 34 | ///Edmonds' alternating forest maximum matching algorithm. |
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| 35 | |
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| 36 | ///This class provides Edmonds' alternating forest matching |
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| 37 | ///algorithm. The starting matching (if any) can be passed to the |
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| 38 | ///algorithm using read-in functions \ref readNMapNode, \ref |
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| 39 | ///readNMapEdge or \ref readEMapBool depending on the container. The |
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| 40 | ///resulting maximum matching can be attained by write-out functions |
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| 41 | ///\ref writeNMapNode, \ref writeNMapEdge or \ref writeEMapBool |
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| 42 | ///depending on the preferred container. |
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| 43 | /// |
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| 44 | ///The dual side of a matching is a map of the nodes to |
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| 45 | ///MaxMatching::pos_enum, having values D, A and C showing the |
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| 46 | ///Gallai-Edmonds decomposition of the graph. The nodes in D induce |
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| 47 | ///a graph with factor-critical components, the nodes in A form the |
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| 48 | ///barrier, and the nodes in C induce a graph having a perfect |
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| 49 | ///matching. This decomposition can be attained by calling \ref |
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[1090] | 50 | ///writePos after running the algorithm. |
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[1077] | 51 | /// |
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| 52 | ///\param Graph The undirected graph type the algorithm runs on. |
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| 53 | /// |
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| 54 | ///\author Jacint Szabo |
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| 55 | template <typename Graph> |
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| 56 | class MaxMatching { |
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| 57 | typedef typename Graph::Node Node; |
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| 58 | typedef typename Graph::Edge Edge; |
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| 59 | typedef typename Graph::UndirEdgeIt UndirEdgeIt; |
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| 60 | typedef typename Graph::NodeIt NodeIt; |
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| 61 | typedef typename Graph::IncEdgeIt IncEdgeIt; |
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| 62 | |
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| 63 | typedef UnionFindEnum<Node, Graph::template NodeMap> UFE; |
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| 64 | |
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| 65 | public: |
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| 66 | |
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| 67 | ///Indicates the Gallai-Edmonds decomposition of the graph. |
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| 68 | |
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| 69 | ///Indicates the Gallai-Edmonds decomposition of the graph, which |
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| 70 | ///shows an upper bound on the size of a maximum matching. The |
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| 71 | ///nodes with pos_enum \c D induce a graph with factor-critical |
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| 72 | ///components, the nodes in \c A form the canonical barrier, and the |
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| 73 | ///nodes in \c C induce a graph having a perfect matching. |
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| 74 | enum pos_enum { |
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| 75 | D=0, |
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| 76 | A=1, |
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| 77 | C=2 |
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| 78 | }; |
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| 79 | |
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| 80 | private: |
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| 81 | |
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| 82 | static const int HEUR_density=2; |
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| 83 | const Graph& g; |
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| 84 | typename Graph::template NodeMap<Node> mate; |
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| 85 | typename Graph::template NodeMap<pos_enum> position; |
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| 86 | |
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| 87 | public: |
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| 88 | |
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[1090] | 89 | MaxMatching(const Graph& _g) : g(_g), mate(_g,INVALID), position(_g) {} |
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[1077] | 90 | |
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| 91 | ///Runs Edmonds' algorithm. |
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| 92 | |
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| 93 | ///Runs Edmonds' algorithm for sparse graphs (number of edges < |
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| 94 | ///2*number of nodes), and a heuristical Edmonds' algorithm with a |
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[1090] | 95 | ///heuristic of postponing shrinks for dense graphs. |
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[1077] | 96 | inline void run(); |
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| 97 | |
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| 98 | ///Runs Edmonds' algorithm. |
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| 99 | |
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| 100 | ///If heur=0 it runs Edmonds' algorithm. If heur=1 it runs |
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| 101 | ///Edmonds' algorithm with a heuristic of postponing shrinks, |
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[1090] | 102 | ///giving a faster algorithm for dense graphs. |
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[1077] | 103 | void runEdmonds( int heur ); |
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| 104 | |
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| 105 | ///Finds a greedy matching starting from the actual matching. |
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| 106 | |
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| 107 | ///Starting form the actual matching stored, it finds a maximal |
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| 108 | ///greedy matching. |
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| 109 | void greedyMatching(); |
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| 110 | |
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| 111 | ///Returns the size of the actual matching stored. |
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| 112 | |
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| 113 | ///Returns the size of the actual matching stored. After \ref |
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| 114 | ///run() it returns the size of a maximum matching in the graph. |
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| 115 | int size() const; |
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| 116 | |
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| 117 | ///Resets the actual matching to the empty matching. |
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| 118 | |
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| 119 | ///Resets the actual matching to the empty matching. |
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| 120 | /// |
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| 121 | void resetMatching(); |
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| 122 | |
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| 123 | ///Reads a matching from a \c Node map of \c Nodes. |
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| 124 | |
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| 125 | ///Reads a matching from a \c Node map of \c Nodes. This map must be \e |
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| 126 | ///symmetric, i.e. if \c map[u]==v then \c map[v]==u must hold, and |
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| 127 | ///\c uv will be an edge of the matching. |
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| 128 | template<typename NMapN> |
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| 129 | void readNMapNode(NMapN& map) { |
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| 130 | for(NodeIt v(g); v!=INVALID; ++v) { |
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| 131 | mate.set(v,map[v]); |
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| 132 | } |
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| 133 | } |
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| 134 | |
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| 135 | ///Writes the stored matching to a \c Node map of \c Nodes. |
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| 136 | |
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| 137 | ///Writes the stored matching to a \c Node map of \c Nodes. The |
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| 138 | ///resulting map will be \e symmetric, i.e. if \c map[u]==v then \c |
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| 139 | ///map[v]==u will hold, and now \c uv is an edge of the matching. |
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| 140 | template<typename NMapN> |
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| 141 | void writeNMapNode (NMapN& map) const { |
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| 142 | for(NodeIt v(g); v!=INVALID; ++v) { |
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| 143 | map.set(v,mate[v]); |
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| 144 | } |
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| 145 | } |
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| 146 | |
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| 147 | ///Reads a matching from a \c Node map of \c Edges. |
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| 148 | |
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| 149 | ///Reads a matching from a \c Node map of incident \c Edges. This |
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| 150 | ///map must have the property that if \c G.target(map[u])==v then \c |
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| 151 | ///G.target(map[v])==u must hold, and now this edge is an edge of |
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| 152 | ///the matching. |
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| 153 | template<typename NMapE> |
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| 154 | void readNMapEdge(NMapE& map) { |
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| 155 | for(NodeIt v(g); v!=INVALID; ++v) { |
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| 156 | Edge e=map[v]; |
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| 157 | if ( g.valid(e) ) |
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| 158 | g.source(e) == v ? mate.set(v,g.target(e)) : mate.set(v,g.source(e)); |
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| 159 | } |
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| 160 | } |
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| 161 | |
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| 162 | ///Writes the matching stored to a \c Node map of \c Edges. |
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| 163 | |
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| 164 | ///Writes the stored matching to a \c Node map of incident \c |
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| 165 | ///Edges. This map will have the property that if \c |
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| 166 | ///g.target(map[u])==v then \c g.target(map[v])==u holds, and now this |
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| 167 | ///edge is an edge of the matching. |
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| 168 | template<typename NMapE> |
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| 169 | void writeNMapEdge (NMapE& map) const { |
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| 170 | typename Graph::template NodeMap<bool> todo(g,true); |
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| 171 | for(NodeIt v(g); v!=INVALID; ++v) { |
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| 172 | if ( todo[v] && mate[v]!=INVALID ) { |
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| 173 | Node u=mate[v]; |
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| 174 | for(IncEdgeIt e(g,v); e!=INVALID; ++e) { |
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| 175 | if ( g.target(e) == u ) { |
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| 176 | map.set(u,e); |
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| 177 | map.set(v,e); |
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| 178 | todo.set(u,false); |
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| 179 | todo.set(v,false); |
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| 180 | break; |
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| 181 | } |
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| 182 | } |
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| 183 | } |
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| 184 | } |
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| 185 | } |
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| 186 | |
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| 187 | |
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| 188 | ///Reads a matching from an \c Edge map of \c bools. |
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| 189 | |
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| 190 | ///Reads a matching from an \c Edge map of \c bools. This map must |
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| 191 | ///have the property that there are no two adjacent edges \c e, \c |
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| 192 | ///f with \c map[e]==map[f]==true. The edges \c e with \c |
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| 193 | ///map[e]==true form the matching. |
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| 194 | template<typename EMapB> |
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| 195 | void readEMapBool(EMapB& map) { |
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| 196 | for(UndirEdgeIt e(g); e!=INVALID; ++e) { |
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| 197 | if ( map[e] ) { |
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| 198 | Node u=g.source(e); |
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| 199 | Node v=g.target(e); |
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| 200 | mate.set(u,v); |
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| 201 | mate.set(v,u); |
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| 202 | } |
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| 203 | } |
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| 204 | } |
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| 205 | |
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| 206 | |
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| 207 | ///Writes the matching stored to an \c Edge map of \c bools. |
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| 208 | |
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| 209 | ///Writes the matching stored to an \c Edge map of \c bools. This |
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| 210 | ///map will have the property that there are no two adjacent edges |
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| 211 | ///\c e, \c f with \c map[e]==map[f]==true. The edges \c e with \c |
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| 212 | ///map[e]==true form the matching. |
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| 213 | template<typename EMapB> |
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| 214 | void writeEMapBool (EMapB& map) const { |
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| 215 | for(UndirEdgeIt e(g); e!=INVALID; ++e) map.set(e,false); |
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| 216 | |
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| 217 | typename Graph::template NodeMap<bool> todo(g,true); |
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| 218 | for(NodeIt v(g); v!=INVALID; ++v) { |
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| 219 | if ( todo[v] && mate[v]!=INVALID ) { |
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| 220 | Node u=mate[v]; |
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| 221 | for(IncEdgeIt e(g,v); e!=INVALID; ++e) { |
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| 222 | if ( g.target(e) == u ) { |
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| 223 | map.set(e,true); |
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| 224 | todo.set(u,false); |
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| 225 | todo.set(v,false); |
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| 226 | break; |
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| 227 | } |
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| 228 | } |
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| 229 | } |
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| 230 | } |
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| 231 | } |
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| 232 | |
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| 233 | |
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| 234 | ///Writes the canonical decomposition of the graph after running |
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| 235 | ///the algorithm. |
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| 236 | |
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[1090] | 237 | ///After calling any run methods of the class, it writes the |
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| 238 | ///Gallai-Edmonds canonical decomposition of the graph. \c map |
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| 239 | ///must be a node map of \ref pos_enum 's. |
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[1077] | 240 | template<typename NMapEnum> |
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| 241 | void writePos (NMapEnum& map) const { |
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| 242 | for(NodeIt v(g); v!=INVALID; ++v) map.set(v,position[v]); |
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| 243 | } |
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| 244 | |
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| 245 | private: |
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| 246 | |
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| 247 | void lateShrink(Node v, typename Graph::template NodeMap<Node>& ear, |
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| 248 | UFE& blossom, UFE& tree); |
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| 249 | |
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| 250 | void normShrink(Node v, typename Graph::NodeMap<Node>& ear, |
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| 251 | UFE& blossom, UFE& tree); |
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| 252 | |
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| 253 | bool noShrinkStep(Node x, typename Graph::NodeMap<Node>& ear, |
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| 254 | UFE& blossom, UFE& tree, std::queue<Node>& Q); |
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| 255 | |
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| 256 | void shrinkStep(Node& top, Node& middle, Node& bottom, typename Graph::NodeMap<Node>& ear, |
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| 257 | UFE& blossom, UFE& tree, std::queue<Node>& Q); |
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| 258 | |
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| 259 | void augment(Node x, typename Graph::NodeMap<Node>& ear, |
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| 260 | UFE& blossom, UFE& tree); |
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| 261 | |
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| 262 | }; |
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| 263 | |
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| 264 | |
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| 265 | // ********************************************************************** |
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| 266 | // IMPLEMENTATIONS |
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| 267 | // ********************************************************************** |
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| 268 | |
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| 269 | |
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| 270 | template <typename Graph> |
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| 271 | void MaxMatching<Graph>::run() { |
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| 272 | if ( countUndirEdges(g) < HEUR_density*countNodes(g) ) { |
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| 273 | greedyMatching(); |
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| 274 | runEdmonds(0); |
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| 275 | } else runEdmonds(1); |
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| 276 | } |
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| 277 | |
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| 278 | |
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| 279 | template <typename Graph> |
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| 280 | void MaxMatching<Graph>::runEdmonds( int heur=1 ) { |
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[1090] | 281 | |
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| 282 | for(NodeIt v(g); v!=INVALID; ++v) |
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| 283 | position.set(v,C); |
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| 284 | |
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[1077] | 285 | typename Graph::template NodeMap<Node> ear(g,INVALID); |
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| 286 | //undefined for the base nodes of the blossoms (i.e. for the |
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| 287 | //representative elements of UFE blossom) and for the nodes in C |
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| 288 | |
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| 289 | typename UFE::MapType blossom_base(g); |
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| 290 | UFE blossom(blossom_base); |
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| 291 | typename UFE::MapType tree_base(g); |
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| 292 | UFE tree(tree_base); |
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| 293 | |
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| 294 | for(NodeIt v(g); v!=INVALID; ++v) { |
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| 295 | if ( position[v]==C && mate[v]==INVALID ) { |
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| 296 | blossom.insert(v); |
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| 297 | tree.insert(v); |
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| 298 | position.set(v,D); |
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| 299 | if ( heur == 1 ) lateShrink( v, ear, blossom, tree ); |
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| 300 | else normShrink( v, ear, blossom, tree ); |
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| 301 | } |
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| 302 | } |
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| 303 | } |
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| 304 | |
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| 305 | |
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| 306 | template <typename Graph> |
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| 307 | void MaxMatching<Graph>::lateShrink(Node v, typename Graph::template NodeMap<Node>& ear, |
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| 308 | UFE& blossom, UFE& tree) { |
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| 309 | |
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| 310 | std::queue<Node> Q; //queue of the totally unscanned nodes |
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| 311 | Q.push(v); |
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| 312 | std::queue<Node> R; |
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| 313 | //queue of the nodes which must be scanned for a possible shrink |
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| 314 | |
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| 315 | while ( !Q.empty() ) { |
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| 316 | Node x=Q.front(); |
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| 317 | Q.pop(); |
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| 318 | if ( noShrinkStep( x, ear, blossom, tree, Q ) ) return; |
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| 319 | else R.push(x); |
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| 320 | } |
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| 321 | |
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| 322 | while ( !R.empty() ) { |
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| 323 | Node x=R.front(); |
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| 324 | R.pop(); |
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| 325 | |
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| 326 | for( IncEdgeIt e(g,x); e!=INVALID ; ++e ) { |
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| 327 | Node y=g.target(e); |
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| 328 | |
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| 329 | if ( position[y] == D && blossom.find(x) != blossom.find(y) ) { |
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| 330 | //x and y must be in the same tree |
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| 331 | |
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| 332 | typename Graph::template NodeMap<bool> path(g,false); |
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| 333 | |
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| 334 | Node b=blossom.find(x); |
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| 335 | path.set(b,true); |
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| 336 | b=mate[b]; |
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| 337 | while ( b!=INVALID ) { |
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| 338 | b=blossom.find(ear[b]); |
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| 339 | path.set(b,true); |
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| 340 | b=mate[b]; |
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| 341 | } //going till the root |
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| 342 | |
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| 343 | Node top=y; |
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| 344 | Node middle=blossom.find(top); |
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| 345 | Node bottom=x; |
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| 346 | while ( !path[middle] ) |
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| 347 | shrinkStep(top, middle, bottom, ear, blossom, tree, Q); |
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| 348 | |
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| 349 | Node base=middle; |
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| 350 | top=x; |
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| 351 | middle=blossom.find(top); |
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| 352 | bottom=y; |
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| 353 | Node blossom_base=blossom.find(base); |
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| 354 | while ( middle!=blossom_base ) |
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| 355 | shrinkStep(top, middle, bottom, ear, blossom, tree, Q); |
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| 356 | |
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| 357 | blossom.makeRep(base); |
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| 358 | } // if shrink is needed |
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| 359 | |
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| 360 | while ( !Q.empty() ) { |
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| 361 | Node x=Q.front(); |
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| 362 | Q.pop(); |
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| 363 | if ( noShrinkStep(x, ear, blossom, tree, Q) ) return; |
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| 364 | else R.push(x); |
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| 365 | } |
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| 366 | } //for e |
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| 367 | } // while ( !R.empty() ) |
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| 368 | } |
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| 369 | |
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| 370 | |
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| 371 | template <typename Graph> |
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| 372 | void MaxMatching<Graph>::normShrink(Node v, typename Graph::NodeMap<Node>& ear, |
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| 373 | UFE& blossom, UFE& tree) { |
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| 374 | |
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| 375 | std::queue<Node> Q; //queue of the unscanned nodes |
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| 376 | Q.push(v); |
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| 377 | while ( !Q.empty() ) { |
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| 378 | |
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| 379 | Node x=Q.front(); |
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| 380 | Q.pop(); |
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| 381 | |
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| 382 | for( IncEdgeIt e(g,x); e!=INVALID; ++e ) { |
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| 383 | Node y=g.target(e); |
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| 384 | |
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| 385 | switch ( position[y] ) { |
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| 386 | case D: //x and y must be in the same tree |
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| 387 | |
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| 388 | if ( blossom.find(x) != blossom.find(y) ) { //shrink |
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| 389 | typename Graph::template NodeMap<bool> path(g,false); |
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| 390 | |
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| 391 | Node b=blossom.find(x); |
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| 392 | path.set(b,true); |
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| 393 | b=mate[b]; |
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| 394 | while ( b!=INVALID ) { |
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| 395 | b=blossom.find(ear[b]); |
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| 396 | path.set(b,true); |
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| 397 | b=mate[b]; |
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| 398 | } //going till the root |
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| 399 | |
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| 400 | Node top=y; |
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| 401 | Node middle=blossom.find(top); |
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| 402 | Node bottom=x; |
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| 403 | while ( !path[middle] ) |
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| 404 | shrinkStep(top, middle, bottom, ear, blossom, tree, Q); |
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| 405 | |
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| 406 | Node base=middle; |
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| 407 | top=x; |
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| 408 | middle=blossom.find(top); |
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| 409 | bottom=y; |
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| 410 | Node blossom_base=blossom.find(base); |
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| 411 | while ( middle!=blossom_base ) |
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| 412 | shrinkStep(top, middle, bottom, ear, blossom, tree, Q); |
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| 413 | |
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| 414 | blossom.makeRep(base); |
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| 415 | } |
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| 416 | break; |
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| 417 | case C: |
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| 418 | if ( mate[y]!=INVALID ) { //grow |
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| 419 | |
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| 420 | ear.set(y,x); |
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| 421 | Node w=mate[y]; |
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| 422 | blossom.insert(w); |
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| 423 | position.set(y,A); |
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| 424 | position.set(w,D); |
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| 425 | tree.insert(y); |
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| 426 | tree.insert(w); |
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| 427 | tree.join(y,blossom.find(x)); |
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| 428 | tree.join(w,y); |
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| 429 | Q.push(w); |
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| 430 | } else { //augment |
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| 431 | augment(x, ear, blossom, tree); |
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| 432 | mate.set(x,y); |
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| 433 | mate.set(y,x); |
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| 434 | return; |
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| 435 | } //if |
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| 436 | break; |
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| 437 | default: break; |
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| 438 | } |
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| 439 | } |
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| 440 | } |
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| 441 | } |
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| 442 | |
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| 443 | template <typename Graph> |
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| 444 | void MaxMatching<Graph>::greedyMatching() { |
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| 445 | for(NodeIt v(g); v!=INVALID; ++v) |
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| 446 | if ( mate[v]==INVALID ) { |
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| 447 | for( IncEdgeIt e(g,v); e!=INVALID ; ++e ) { |
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| 448 | Node y=g.target(e); |
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| 449 | if ( mate[y]==INVALID && y!=v ) { |
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| 450 | mate.set(v,y); |
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| 451 | mate.set(y,v); |
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| 452 | break; |
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| 453 | } |
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| 454 | } |
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| 455 | } |
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| 456 | } |
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| 457 | |
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| 458 | template <typename Graph> |
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| 459 | int MaxMatching<Graph>::size() const { |
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| 460 | int s=0; |
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| 461 | for(NodeIt v(g); v!=INVALID; ++v) { |
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| 462 | if ( mate[v]!=INVALID ) { |
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| 463 | ++s; |
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| 464 | } |
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| 465 | } |
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| 466 | return (int)s/2; |
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| 467 | } |
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| 468 | |
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| 469 | template <typename Graph> |
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| 470 | void MaxMatching<Graph>::resetMatching() { |
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| 471 | for(NodeIt v(g); v!=INVALID; ++v) |
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| 472 | mate.set(v,INVALID); |
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| 473 | } |
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| 474 | |
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| 475 | template <typename Graph> |
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| 476 | bool MaxMatching<Graph>::noShrinkStep(Node x, typename Graph::NodeMap<Node>& ear, |
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| 477 | UFE& blossom, UFE& tree, std::queue<Node>& Q) { |
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| 478 | for( IncEdgeIt e(g,x); e!= INVALID; ++e ) { |
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| 479 | Node y=g.target(e); |
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| 480 | |
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| 481 | if ( position[y]==C ) { |
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| 482 | if ( mate[y]!=INVALID ) { //grow |
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| 483 | ear.set(y,x); |
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| 484 | Node w=mate[y]; |
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| 485 | blossom.insert(w); |
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| 486 | position.set(y,A); |
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| 487 | position.set(w,D); |
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| 488 | tree.insert(y); |
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| 489 | tree.insert(w); |
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| 490 | tree.join(y,blossom.find(x)); |
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| 491 | tree.join(w,y); |
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| 492 | Q.push(w); |
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| 493 | } else { //augment |
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| 494 | augment(x, ear, blossom, tree); |
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| 495 | mate.set(x,y); |
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| 496 | mate.set(y,x); |
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| 497 | return true; |
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| 498 | } |
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| 499 | } |
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| 500 | } |
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| 501 | return false; |
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| 502 | } |
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| 503 | |
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| 504 | template <typename Graph> |
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| 505 | void MaxMatching<Graph>::shrinkStep(Node& top, Node& middle, Node& bottom, typename Graph::NodeMap<Node>& ear, |
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| 506 | UFE& blossom, UFE& tree, std::queue<Node>& Q) { |
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| 507 | ear.set(top,bottom); |
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| 508 | Node t=top; |
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| 509 | while ( t!=middle ) { |
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| 510 | Node u=mate[t]; |
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| 511 | t=ear[u]; |
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| 512 | ear.set(t,u); |
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| 513 | } |
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| 514 | bottom=mate[middle]; |
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| 515 | position.set(bottom,D); |
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| 516 | Q.push(bottom); |
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| 517 | top=ear[bottom]; |
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| 518 | Node oldmiddle=middle; |
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| 519 | middle=blossom.find(top); |
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| 520 | tree.erase(bottom); |
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| 521 | tree.erase(oldmiddle); |
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| 522 | blossom.insert(bottom); |
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| 523 | blossom.join(bottom, oldmiddle); |
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| 524 | blossom.join(top, oldmiddle); |
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| 525 | } |
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| 526 | |
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| 527 | template <typename Graph> |
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| 528 | void MaxMatching<Graph>::augment(Node x, typename Graph::NodeMap<Node>& ear, |
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| 529 | UFE& blossom, UFE& tree) { |
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| 530 | Node v=mate[x]; |
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| 531 | while ( v!=INVALID ) { |
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| 532 | |
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| 533 | Node u=ear[v]; |
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| 534 | mate.set(v,u); |
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| 535 | Node tmp=v; |
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| 536 | v=mate[u]; |
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| 537 | mate.set(u,tmp); |
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| 538 | } |
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| 539 | typename UFE::ItemIt it; |
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| 540 | for (tree.first(it,blossom.find(x)); tree.valid(it); tree.next(it)) { |
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| 541 | if ( position[it] == D ) { |
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| 542 | typename UFE::ItemIt b_it; |
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| 543 | for (blossom.first(b_it,it); blossom.valid(b_it); blossom.next(b_it)) { |
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| 544 | position.set( b_it ,C); |
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| 545 | } |
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| 546 | blossom.eraseClass(it); |
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| 547 | } else position.set( it ,C); |
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| 548 | } |
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| 549 | tree.eraseClass(x); |
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| 550 | |
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| 551 | } |
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| 552 | |
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| 553 | /// @} |
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| 554 | |
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| 555 | } //END OF NAMESPACE LEMON |
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| 556 | |
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| 557 | #endif //EDMONDS_H |
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