1 | /* -*- C++ -*- |
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2 | * |
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3 | * This file is a part of LEMON, a generic C++ optimization library |
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4 | * |
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5 | * Copyright (C) 2003-2007 |
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6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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8 | * |
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9 | * Permission to use, modify and distribute this software is granted |
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10 | * provided that this copyright notice appears in all copies. For |
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11 | * precise terms see the accompanying LICENSE file. |
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12 | * |
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13 | * This software is provided "AS IS" with no warranty of any kind, |
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14 | * express or implied, and with no claim as to its suitability for any |
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15 | * purpose. |
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16 | * |
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17 | */ |
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18 | |
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19 | #ifndef LEMON_MAX_MATCHING_H |
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20 | #define LEMON_MAX_MATCHING_H |
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21 | |
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22 | #include <queue> |
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23 | #include <lemon/bits/invalid.h> |
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24 | #include <lemon/unionfind.h> |
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25 | #include <lemon/graph_utils.h> |
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26 | |
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27 | ///\ingroup matching |
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28 | ///\file |
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29 | ///\brief Maximum matching algorithm in undirected graph. |
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30 | |
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31 | namespace lemon { |
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32 | |
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33 | /// \ingroup matching |
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34 | |
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35 | ///Edmonds' alternating forest maximum matching algorithm. |
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36 | |
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37 | ///This class provides Edmonds' alternating forest matching |
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38 | ///algorithm. The starting matching (if any) can be passed to the |
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39 | ///algorithm using read-in functions \ref readNMapNode, \ref |
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40 | ///readNMapEdge or \ref readEMapBool depending on the container. The |
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41 | ///resulting maximum matching can be attained by write-out functions |
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42 | ///\ref writeNMapNode, \ref writeNMapEdge or \ref writeEMapBool |
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43 | ///depending on the preferred container. |
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44 | /// |
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45 | ///The dual side of a matching is a map of the nodes to |
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46 | ///MaxMatching::pos_enum, having values D, A and C showing the |
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47 | ///Gallai-Edmonds decomposition of the graph. The nodes in D induce |
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48 | ///a graph with factor-critical components, the nodes in A form the |
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49 | ///barrier, and the nodes in C induce a graph having a perfect |
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50 | ///matching. This decomposition can be attained by calling \ref |
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51 | ///writePos after running the algorithm. |
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52 | /// |
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53 | ///\param Graph The undirected graph type the algorithm runs on. |
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54 | /// |
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55 | ///\author Jacint Szabo |
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56 | template <typename Graph> |
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57 | class MaxMatching { |
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58 | |
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59 | protected: |
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60 | |
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61 | typedef typename Graph::Node Node; |
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62 | typedef typename Graph::Edge Edge; |
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63 | typedef typename Graph::UEdge UEdge; |
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64 | typedef typename Graph::UEdgeIt UEdgeIt; |
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65 | typedef typename Graph::NodeIt NodeIt; |
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66 | typedef typename Graph::IncEdgeIt IncEdgeIt; |
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67 | |
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68 | typedef typename Graph::template NodeMap<int> UFECrossRef; |
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69 | typedef UnionFindEnum<UFECrossRef> UFE; |
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70 | |
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71 | public: |
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72 | |
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73 | ///Indicates the Gallai-Edmonds decomposition of the graph. |
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74 | |
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75 | ///Indicates the Gallai-Edmonds decomposition of the graph, which |
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76 | ///shows an upper bound on the size of a maximum matching. The |
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77 | ///nodes with pos_enum \c D induce a graph with factor-critical |
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78 | ///components, the nodes in \c A form the canonical barrier, and the |
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79 | ///nodes in \c C induce a graph having a perfect matching. |
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80 | enum pos_enum { |
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81 | D=0, |
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82 | A=1, |
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83 | C=2 |
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84 | }; |
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85 | |
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86 | protected: |
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87 | |
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88 | static const int HEUR_density=2; |
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89 | const Graph& g; |
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90 | typename Graph::template NodeMap<Node> _mate; |
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91 | typename Graph::template NodeMap<pos_enum> position; |
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92 | |
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93 | public: |
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94 | |
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95 | MaxMatching(const Graph& _g) : g(_g), _mate(_g,INVALID), position(_g) {} |
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96 | |
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97 | ///Runs Edmonds' algorithm. |
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98 | |
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99 | ///Runs Edmonds' algorithm for sparse graphs (number of edges < |
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100 | ///2*number of nodes), and a heuristical Edmonds' algorithm with a |
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101 | ///heuristic of postponing shrinks for dense graphs. |
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102 | void run() { |
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103 | if ( countUEdges(g) < HEUR_density*countNodes(g) ) { |
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104 | greedyMatching(); |
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105 | runEdmonds(0); |
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106 | } else runEdmonds(1); |
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107 | } |
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108 | |
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109 | |
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110 | ///Runs Edmonds' algorithm. |
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111 | |
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112 | ///If heur=0 it runs Edmonds' algorithm. If heur=1 it runs |
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113 | ///Edmonds' algorithm with a heuristic of postponing shrinks, |
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114 | ///giving a faster algorithm for dense graphs. |
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115 | void runEdmonds( int heur = 1 ) { |
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116 | |
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117 | //each vertex is put to C |
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118 | for(NodeIt v(g); v!=INVALID; ++v) |
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119 | position.set(v,C); |
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120 | |
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121 | typename Graph::template NodeMap<Node> ear(g,INVALID); |
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122 | //undefined for the base nodes of the blossoms (i.e. for the |
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123 | //representative elements of UFE blossom) and for the nodes in C |
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124 | |
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125 | UFECrossRef blossom_base(g); |
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126 | UFE blossom(blossom_base); |
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127 | |
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128 | UFECrossRef tree_base(g); |
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129 | UFE tree(tree_base); |
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130 | |
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131 | //If these UFE's would be members of the class then also |
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132 | //blossom_base and tree_base should be a member. |
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133 | |
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134 | //We build only one tree and the other vertices uncovered by the |
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135 | //matching belong to C. (They can be considered as singleton |
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136 | //trees.) If this tree can be augmented or no more |
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137 | //grow/augmentation/shrink is possible then we return to this |
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138 | //"for" cycle. |
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139 | for(NodeIt v(g); v!=INVALID; ++v) { |
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140 | if ( position[v]==C && _mate[v]==INVALID ) { |
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141 | blossom.insert(v); |
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142 | tree.insert(v); |
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143 | position.set(v,D); |
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144 | if ( heur == 1 ) lateShrink( v, ear, blossom, tree ); |
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145 | else normShrink( v, ear, blossom, tree ); |
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146 | } |
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147 | } |
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148 | } |
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149 | |
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150 | |
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151 | ///Finds a greedy matching starting from the actual matching. |
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152 | |
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153 | ///Starting form the actual matching stored, it finds a maximal |
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154 | ///greedy matching. |
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155 | void greedyMatching() { |
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156 | for(NodeIt v(g); v!=INVALID; ++v) |
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157 | if ( _mate[v]==INVALID ) { |
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158 | for( IncEdgeIt e(g,v); e!=INVALID ; ++e ) { |
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159 | Node y=g.runningNode(e); |
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160 | if ( _mate[y]==INVALID && y!=v ) { |
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161 | _mate.set(v,y); |
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162 | _mate.set(y,v); |
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163 | break; |
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164 | } |
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165 | } |
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166 | } |
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167 | } |
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168 | |
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169 | ///Returns the size of the actual matching stored. |
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170 | |
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171 | ///Returns the size of the actual matching stored. After \ref |
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172 | ///run() it returns the size of a maximum matching in the graph. |
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173 | int size() const { |
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174 | int s=0; |
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175 | for(NodeIt v(g); v!=INVALID; ++v) { |
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176 | if ( _mate[v]!=INVALID ) { |
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177 | ++s; |
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178 | } |
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179 | } |
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180 | return s/2; |
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181 | } |
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182 | |
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183 | |
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184 | ///Resets the actual matching to the empty matching. |
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185 | |
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186 | ///Resets the actual matching to the empty matching. |
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187 | /// |
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188 | void resetMatching() { |
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189 | for(NodeIt v(g); v!=INVALID; ++v) |
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190 | _mate.set(v,INVALID); |
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191 | } |
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192 | |
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193 | ///Returns the mate of a node in the actual matching. |
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194 | |
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195 | ///Returns the mate of a \c node in the actual matching. |
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196 | ///Returns INVALID if the \c node is not covered by the actual matching. |
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197 | Node mate(Node& node) const { |
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198 | return _mate[node]; |
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199 | } |
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200 | |
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201 | ///Reads a matching from a \c Node valued \c Node map. |
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202 | |
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203 | ///Reads a matching from a \c Node valued \c Node map. This map |
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204 | ///must be \e symmetric, i.e. if \c map[u]==v then \c map[v]==u |
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205 | ///must hold, and \c uv will be an edge of the matching. |
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206 | template<typename NMapN> |
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207 | void readNMapNode(NMapN& map) { |
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208 | for(NodeIt v(g); v!=INVALID; ++v) { |
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209 | _mate.set(v,map[v]); |
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210 | } |
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211 | } |
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212 | |
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213 | ///Writes the stored matching to a \c Node valued \c Node map. |
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214 | |
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215 | ///Writes the stored matching to a \c Node valued \c Node map. The |
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216 | ///resulting map will be \e symmetric, i.e. if \c map[u]==v then \c |
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217 | ///map[v]==u will hold, and now \c uv is an edge of the matching. |
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218 | template<typename NMapN> |
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219 | void writeNMapNode (NMapN& map) const { |
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220 | for(NodeIt v(g); v!=INVALID; ++v) { |
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221 | map.set(v,_mate[v]); |
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222 | } |
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223 | } |
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224 | |
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225 | ///Reads a matching from an \c UEdge valued \c Node map. |
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226 | |
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227 | ///Reads a matching from an \c UEdge valued \c Node map. \c |
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228 | ///map[v] must be an \c UEdge incident to \c v. This map must |
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229 | ///have the property that if \c g.oppositeNode(u,map[u])==v then |
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230 | ///\c \c g.oppositeNode(v,map[v])==u holds, and now some edge |
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231 | ///joining \c u to \c v will be an edge of the matching. |
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232 | template<typename NMapE> |
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233 | void readNMapEdge(NMapE& map) { |
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234 | for(NodeIt v(g); v!=INVALID; ++v) { |
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235 | UEdge e=map[v]; |
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236 | if ( e!=INVALID ) |
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237 | _mate.set(v,g.oppositeNode(v,e)); |
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238 | } |
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239 | } |
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240 | |
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241 | ///Writes the matching stored to an \c UEdge valued \c Node map. |
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242 | |
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243 | ///Writes the stored matching to an \c UEdge valued \c Node |
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244 | ///map. \c map[v] will be an \c UEdge incident to \c v. This |
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245 | ///map will have the property that if \c g.oppositeNode(u,map[u]) |
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246 | ///== v then \c map[u]==map[v] holds, and now this edge is an edge |
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247 | ///of the matching. |
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248 | template<typename NMapE> |
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249 | void writeNMapEdge (NMapE& map) const { |
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250 | typename Graph::template NodeMap<bool> todo(g,true); |
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251 | for(NodeIt v(g); v!=INVALID; ++v) { |
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252 | if ( todo[v] && _mate[v]!=INVALID ) { |
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253 | Node u=_mate[v]; |
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254 | for(IncEdgeIt e(g,v); e!=INVALID; ++e) { |
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255 | if ( g.runningNode(e) == u ) { |
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256 | map.set(u,e); |
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257 | map.set(v,e); |
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258 | todo.set(u,false); |
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259 | todo.set(v,false); |
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260 | break; |
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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 | |
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267 | |
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268 | ///Reads a matching from a \c bool valued \c Edge map. |
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269 | |
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270 | ///Reads a matching from a \c bool valued \c Edge map. This map |
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271 | ///must have the property that there are no two incident edges \c |
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272 | ///e, \c f with \c map[e]==map[f]==true. The edges \c e with \c |
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273 | ///map[e]==true form the matching. |
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274 | template<typename EMapB> |
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275 | void readEMapBool(EMapB& map) { |
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276 | for(UEdgeIt e(g); e!=INVALID; ++e) { |
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277 | if ( map[e] ) { |
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278 | Node u=g.source(e); |
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279 | Node v=g.target(e); |
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280 | _mate.set(u,v); |
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281 | _mate.set(v,u); |
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282 | } |
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283 | } |
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284 | } |
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285 | |
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286 | |
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287 | ///Writes the matching stored to a \c bool valued \c Edge map. |
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288 | |
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289 | ///Writes the matching stored to a \c bool valued \c Edge |
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290 | ///map. This map will have the property that there are no two |
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291 | ///incident edges \c e, \c f with \c map[e]==map[f]==true. The |
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292 | ///edges \c e with \c map[e]==true form the matching. |
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293 | template<typename EMapB> |
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294 | void writeEMapBool (EMapB& map) const { |
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295 | for(UEdgeIt e(g); e!=INVALID; ++e) map.set(e,false); |
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296 | |
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297 | typename Graph::template NodeMap<bool> todo(g,true); |
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298 | for(NodeIt v(g); v!=INVALID; ++v) { |
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299 | if ( todo[v] && _mate[v]!=INVALID ) { |
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300 | Node u=_mate[v]; |
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301 | for(IncEdgeIt e(g,v); e!=INVALID; ++e) { |
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302 | if ( g.runningNode(e) == u ) { |
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303 | map.set(e,true); |
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304 | todo.set(u,false); |
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305 | todo.set(v,false); |
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306 | break; |
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307 | } |
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308 | } |
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309 | } |
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310 | } |
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311 | } |
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312 | |
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313 | |
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314 | ///Writes the canonical decomposition of the graph after running |
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315 | ///the algorithm. |
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316 | |
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317 | ///After calling any run methods of the class, it writes the |
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318 | ///Gallai-Edmonds canonical decomposition of the graph. \c map |
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319 | ///must be a node map of \ref pos_enum 's. |
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320 | template<typename NMapEnum> |
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321 | void writePos (NMapEnum& map) const { |
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322 | for(NodeIt v(g); v!=INVALID; ++v) map.set(v,position[v]); |
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323 | } |
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324 | |
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325 | private: |
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326 | |
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327 | |
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328 | void lateShrink(Node v, typename Graph::template NodeMap<Node>& ear, |
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329 | UFE& blossom, UFE& tree); |
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330 | |
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331 | void normShrink(Node v, typename Graph::template NodeMap<Node>& ear, |
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332 | UFE& blossom, UFE& tree); |
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333 | |
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334 | void shrink(Node x,Node y, typename Graph::template NodeMap<Node>& ear, |
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335 | UFE& blossom, UFE& tree,std::queue<Node>& Q); |
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336 | |
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337 | void shrinkStep(Node& top, Node& middle, Node& bottom, |
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338 | typename Graph::template NodeMap<Node>& ear, |
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339 | UFE& blossom, UFE& tree, std::queue<Node>& Q); |
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340 | |
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341 | bool growOrAugment(Node& y, Node& x, typename Graph::template |
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342 | NodeMap<Node>& ear, UFE& blossom, UFE& tree, |
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343 | std::queue<Node>& Q); |
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344 | |
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345 | void augment(Node x, typename Graph::template NodeMap<Node>& ear, |
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346 | UFE& blossom, UFE& tree); |
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347 | |
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348 | }; |
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349 | |
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350 | |
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351 | // ********************************************************************** |
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352 | // IMPLEMENTATIONS |
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353 | // ********************************************************************** |
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354 | |
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355 | |
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356 | template <typename Graph> |
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357 | void MaxMatching<Graph>::lateShrink(Node v, typename Graph::template |
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358 | NodeMap<Node>& ear, UFE& blossom, |
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359 | UFE& tree) { |
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360 | //We have one tree which we grow, and also shrink but only if it cannot be |
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361 | //postponed. If we augment then we return to the "for" cycle of |
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362 | //runEdmonds(). |
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363 | |
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364 | std::queue<Node> Q; //queue of the totally unscanned nodes |
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365 | Q.push(v); |
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366 | std::queue<Node> R; |
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367 | //queue of the nodes which must be scanned for a possible shrink |
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368 | |
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369 | while ( !Q.empty() ) { |
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370 | Node x=Q.front(); |
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371 | Q.pop(); |
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372 | for( IncEdgeIt e(g,x); e!= INVALID; ++e ) { |
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373 | Node y=g.runningNode(e); |
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374 | //growOrAugment grows if y is covered by the matching and |
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375 | //augments if not. In this latter case it returns 1. |
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376 | if ( position[y]==C && growOrAugment(y, x, ear, blossom, tree, Q) ) |
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377 | return; |
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378 | } |
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379 | R.push(x); |
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380 | } |
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381 | |
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382 | while ( !R.empty() ) { |
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383 | Node x=R.front(); |
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384 | R.pop(); |
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385 | |
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386 | for( IncEdgeIt e(g,x); e!=INVALID ; ++e ) { |
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387 | Node y=g.runningNode(e); |
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388 | |
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389 | if ( position[y] == D && blossom.find(x) != blossom.find(y) ) |
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390 | //Recall that we have only one tree. |
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391 | shrink( x, y, ear, blossom, tree, Q); |
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392 | |
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393 | while ( !Q.empty() ) { |
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394 | Node z=Q.front(); |
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395 | Q.pop(); |
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396 | for( IncEdgeIt f(g,z); f!= INVALID; ++f ) { |
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397 | Node w=g.runningNode(f); |
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398 | //growOrAugment grows if y is covered by the matching and |
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399 | //augments if not. In this latter case it returns 1. |
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400 | if ( position[w]==C && growOrAugment(w, z, ear, blossom, tree, Q) ) |
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401 | return; |
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402 | } |
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403 | R.push(z); |
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404 | } |
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405 | } //for e |
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406 | } // while ( !R.empty() ) |
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407 | } |
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408 | |
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409 | |
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410 | template <typename Graph> |
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411 | void MaxMatching<Graph>::normShrink(Node v, |
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412 | typename Graph::template |
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413 | NodeMap<Node>& ear, |
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414 | UFE& blossom, UFE& tree) { |
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415 | //We have one tree, which we grow and shrink. If we augment then we |
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416 | //return to the "for" cycle of runEdmonds(). |
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417 | |
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418 | std::queue<Node> Q; //queue of the unscanned nodes |
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419 | Q.push(v); |
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420 | while ( !Q.empty() ) { |
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421 | |
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422 | Node x=Q.front(); |
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423 | Q.pop(); |
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424 | |
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425 | for( IncEdgeIt e(g,x); e!=INVALID; ++e ) { |
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426 | Node y=g.runningNode(e); |
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427 | |
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428 | switch ( position[y] ) { |
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429 | case D: //x and y must be in the same tree |
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430 | if ( blossom.find(x) != blossom.find(y) ) |
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431 | //x and y are in the same tree |
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432 | shrink( x, y, ear, blossom, tree, Q); |
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433 | break; |
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434 | case C: |
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435 | //growOrAugment grows if y is covered by the matching and |
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436 | //augments if not. In this latter case it returns 1. |
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437 | if ( growOrAugment(y, x, ear, blossom, tree, Q) ) return; |
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438 | break; |
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439 | default: break; |
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440 | } |
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441 | } |
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442 | } |
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443 | } |
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444 | |
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445 | |
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446 | template <typename Graph> |
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447 | void MaxMatching<Graph>::shrink(Node x,Node y, typename |
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448 | Graph::template NodeMap<Node>& ear, |
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449 | UFE& blossom, UFE& tree, std::queue<Node>& Q) { |
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450 | //x and y are the two adjacent vertices in two blossoms. |
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451 | |
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452 | typename Graph::template NodeMap<bool> path(g,false); |
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453 | |
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454 | Node b=blossom.find(x); |
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455 | path.set(b,true); |
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456 | b=_mate[b]; |
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457 | while ( b!=INVALID ) { |
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458 | b=blossom.find(ear[b]); |
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459 | path.set(b,true); |
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460 | b=_mate[b]; |
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461 | } //we go until the root through bases of blossoms and odd vertices |
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462 | |
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463 | Node top=y; |
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464 | Node middle=blossom.find(top); |
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465 | Node bottom=x; |
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466 | while ( !path[middle] ) |
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467 | shrinkStep(top, middle, bottom, ear, blossom, tree, Q); |
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468 | //Until we arrive to a node on the path, we update blossom, tree |
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469 | //and the positions of the odd nodes. |
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470 | |
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471 | Node base=middle; |
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472 | top=x; |
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473 | middle=blossom.find(top); |
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474 | bottom=y; |
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475 | Node blossom_base=blossom.find(base); |
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476 | while ( middle!=blossom_base ) |
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477 | shrinkStep(top, middle, bottom, ear, blossom, tree, Q); |
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478 | //Until we arrive to a node on the path, we update blossom, tree |
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479 | //and the positions of the odd nodes. |
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480 | |
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481 | blossom.makeRep(base); |
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482 | } |
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483 | |
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484 | |
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485 | |
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486 | template <typename Graph> |
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487 | void MaxMatching<Graph>::shrinkStep(Node& top, Node& middle, Node& bottom, |
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488 | typename Graph::template |
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489 | NodeMap<Node>& ear, |
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490 | UFE& blossom, UFE& tree, |
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491 | std::queue<Node>& Q) { |
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492 | //We traverse a blossom and update everything. |
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493 | |
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494 | ear.set(top,bottom); |
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495 | Node t=top; |
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496 | while ( t!=middle ) { |
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497 | Node u=_mate[t]; |
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498 | t=ear[u]; |
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499 | ear.set(t,u); |
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500 | } |
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501 | bottom=_mate[middle]; |
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502 | position.set(bottom,D); |
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503 | Q.push(bottom); |
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504 | top=ear[bottom]; |
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505 | Node oldmiddle=middle; |
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506 | middle=blossom.find(top); |
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507 | tree.erase(bottom); |
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508 | tree.erase(oldmiddle); |
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509 | blossom.insert(bottom); |
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510 | blossom.join(bottom, oldmiddle); |
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511 | blossom.join(top, oldmiddle); |
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512 | } |
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513 | |
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514 | |
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515 | template <typename Graph> |
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516 | bool MaxMatching<Graph>::growOrAugment(Node& y, Node& x, typename Graph::template |
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517 | NodeMap<Node>& ear, UFE& blossom, UFE& tree, |
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518 | std::queue<Node>& Q) { |
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519 | //x is in a blossom in the tree, y is outside. If y is covered by |
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520 | //the matching we grow, otherwise we augment. In this case we |
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521 | //return 1. |
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522 | |
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523 | if ( _mate[y]!=INVALID ) { //grow |
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524 | ear.set(y,x); |
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525 | Node w=_mate[y]; |
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526 | blossom.insert(w); |
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527 | position.set(y,A); |
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528 | position.set(w,D); |
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529 | tree.insert(y); |
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530 | tree.insert(w); |
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531 | tree.join(y,blossom.find(x)); |
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532 | tree.join(w,y); |
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533 | Q.push(w); |
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534 | } else { //augment |
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535 | augment(x, ear, blossom, tree); |
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536 | _mate.set(x,y); |
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537 | _mate.set(y,x); |
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538 | return true; |
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539 | } |
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540 | return false; |
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541 | } |
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542 | |
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543 | |
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544 | template <typename Graph> |
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545 | void MaxMatching<Graph>::augment(Node x, |
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546 | typename Graph::template NodeMap<Node>& ear, |
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547 | UFE& blossom, UFE& tree) { |
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548 | Node v=_mate[x]; |
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549 | while ( v!=INVALID ) { |
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550 | |
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551 | Node u=ear[v]; |
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552 | _mate.set(v,u); |
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553 | Node tmp=v; |
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554 | v=_mate[u]; |
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555 | _mate.set(u,tmp); |
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556 | } |
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557 | Node y=blossom.find(x); |
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558 | for (typename UFE::ItemIt tit(tree, y); tit != INVALID; ++tit) { |
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559 | if ( position[tit] == D ) { |
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560 | for (typename UFE::ItemIt bit(blossom, tit); bit != INVALID; ++bit) { |
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561 | position.set( bit ,C); |
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562 | } |
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563 | blossom.eraseClass(tit); |
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564 | } else position.set( tit ,C); |
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565 | } |
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566 | tree.eraseClass(y); |
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567 | |
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568 | } |
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569 | |
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570 | |
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571 | } //END OF NAMESPACE LEMON |
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572 | |
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573 | #endif //LEMON_MAX_MATCHING_H |
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