1 | // -*- C++ -*- |
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2 | #ifndef LEMON_MAX_MATCHING_H |
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3 | #define LEMON_MAX_MATCHING_H |
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4 | |
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5 | ///\ingroup galgs |
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6 | ///\file |
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7 | ///\brief Maximum matching algorithm. |
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8 | |
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9 | #include <queue> |
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10 | |
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11 | #include <invalid.h> |
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12 | #include <unionfind.h> |
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13 | |
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14 | namespace lemon { |
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15 | |
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16 | /// \addtogroup galgs |
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17 | /// @{ |
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18 | |
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19 | ///Maximum matching algorithms class. |
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20 | |
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21 | ///This class provides Edmonds' alternating forest matching |
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22 | ///algorithm. The starting matching (if any) can be passed to the |
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23 | ///algorithm using read-in functions \ref readNMapNode, \ref |
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24 | ///readNMapEdge or \ref readEMapBool depending on the container. The |
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25 | ///resulting maximum matching can be attained by write-out functions |
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26 | ///\ref writeNMapNode, \ref writeNMapEdge or \ref writeEMapBool |
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27 | ///depending on the preferred container. |
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28 | /// |
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29 | ///The dual side of a mathcing is a map of the nodes to |
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30 | ///MaxMatching::pos_enum, having values D, A and C showing the |
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31 | ///Gallai-Edmonds decomposition of the graph. The nodes in D induce |
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32 | ///a graph with factor-critical components, the nodes in A form the |
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33 | ///barrier, and the nodes in C induce a graph having a perfect |
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34 | ///matching. This decomposition can be attained by calling \ref |
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35 | ///writePos after running the algorithm. Before subsequent runs, |
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36 | ///the function \ref resetPos() must be called. |
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37 | /// |
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38 | ///\param Graph The undirected graph type the algorithm runs on. |
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39 | /// |
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40 | ///\author Jacint Szabo |
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41 | template <typename Graph> |
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42 | class MaxMatching { |
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43 | typedef typename Graph::Node Node; |
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44 | typedef typename Graph::Edge Edge; |
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45 | typedef typename Graph::EdgeIt EdgeIt; |
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46 | typedef typename Graph::NodeIt NodeIt; |
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47 | typedef typename Graph::OutEdgeIt OutEdgeIt; |
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48 | |
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49 | typedef UnionFindEnum<Node, Graph::template NodeMap> UFE; |
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50 | |
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51 | public: |
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52 | |
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53 | ///Indicates the Gallai-Edmonds decomposition of the graph. |
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54 | |
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55 | ///Indicates the Gallai-Edmonds decomposition of the graph, which |
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56 | ///shows an upper bound on the size of a maximum matching. The |
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57 | ///nodes with pos_enum \c D induce a graph with factor-critical |
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58 | ///components, the nodes in \c A form the canonical barrier, and the |
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59 | ///nodes in \c C induce a graph having a perfect matching. |
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60 | enum pos_enum { |
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61 | D=0, |
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62 | A=1, |
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63 | C=2 |
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64 | }; |
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65 | |
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66 | private: |
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67 | |
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68 | const Graph& G; |
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69 | typename Graph::template NodeMap<Node> mate; |
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70 | typename Graph::template NodeMap<pos_enum> position; |
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71 | |
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72 | public: |
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73 | |
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74 | MaxMatching(const Graph& _G) : G(_G), mate(_G,INVALID), position(_G,C) {} |
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75 | |
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76 | ///Runs Edmonds' algorithm. |
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77 | |
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78 | ///Runs Edmonds' algorithm for sparse graphs (edgeNum >= |
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79 | ///2*nodeNum), and a heuristical Edmonds' algorithm with a |
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80 | ///heuristic of postponing shrinks for dense graphs. \pre Before |
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81 | ///the subsequent calls \ref resetPos must be called. |
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82 | inline void run(); |
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83 | |
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84 | ///Runs Edmonds' algorithm. |
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85 | |
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86 | ///If heur=0 it runs Edmonds' algorithm. If heur=1 it runs |
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87 | ///Edmonds' algorithm with a heuristic of postponing shrinks, |
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88 | ///giving a faster algorithm for dense graphs. \pre Before the |
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89 | ///subsequent calls \ref resetPos must be called. |
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90 | void runEdmonds( int heur ); |
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91 | |
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92 | ///Finds a greedy matching starting from the actual matching. |
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93 | |
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94 | ///Starting form the actual matching stored, it finds a maximal |
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95 | ///greedy matching. |
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96 | void greedyMatching(); |
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97 | |
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98 | ///Returns the size of the actual matching stored. |
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99 | |
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100 | ///Returns the size of the actual matching stored. After \ref |
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101 | ///run() it returns the size of a maximum matching in the graph. |
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102 | int size () const; |
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103 | |
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104 | ///Resets the map storing the Gallai-Edmonds decomposition. |
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105 | |
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106 | ///Resets the map storing the Gallai-Edmonds decomposition of the |
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107 | ///graph, making it possible to run the algorithm. Must be called |
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108 | ///before all runs of the Edmonds algorithm, except for the first |
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109 | ///run. |
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110 | void resetPos(); |
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111 | |
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112 | ///Resets the actual matching to the empty matching. |
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113 | |
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114 | ///Resets the actual matching to the empty matching. |
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115 | /// |
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116 | void resetMatching(); |
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117 | |
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118 | ///Reads a matching from a \c Node map of \c Nodes. |
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119 | |
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120 | ///Reads a matching from a \c Node map of \c Nodes. This map must be \e |
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121 | ///symmetric, i.e. if \c map[u]=v then \c map[v]=u must hold, and |
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122 | ///now \c uv is an edge of the matching. |
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123 | template<typename NMapN> |
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124 | void readNMapNode(NMapN& map) { |
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125 | NodeIt v; |
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126 | for( G.first(v); G.valid(v); G.next(v)) { |
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127 | mate.set(v,map[v]); |
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128 | } |
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129 | } |
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130 | |
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131 | ///Writes the stored matching to a \c Node map of \c Nodes. |
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132 | |
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133 | ///Writes the stored matching to a \c Node map of \c Nodes. The |
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134 | ///resulting map will be \e symmetric, i.e. if \c map[u]=v then \c |
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135 | ///map[v]=u will hold, and now \c uv is an edge of the matching. |
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136 | template<typename NMapN> |
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137 | void writeNMapNode (NMapN& map) const { |
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138 | NodeIt v; |
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139 | for( G.first(v); G.valid(v); G.next(v)) { |
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140 | map.set(v,mate[v]); |
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141 | } |
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142 | } |
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143 | |
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144 | ///Reads a matching from a \c Node map of \c Edges. |
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145 | |
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146 | ///Reads a matching from a \c Node map of incident \c Edges. This |
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147 | ///map must have the property that if \c G.bNode(map[u])=v then \c |
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148 | ///G.bNode(map[v])=u must hold, and now this edge is an edge of |
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149 | ///the matching. |
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150 | template<typename NMapE> |
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151 | void readNMapEdge(NMapE& map) { |
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152 | NodeIt v; |
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153 | for( G.first(v); G.valid(v); G.next(v)) { |
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154 | Edge e=map[v]; |
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155 | if ( G.valid(e) ) |
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156 | G.tail(e) == v ? mate.set(v,G.head(e)) : mate.set(v,G.tail(e)); |
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157 | } |
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158 | } |
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159 | |
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160 | ///Writes the matching stored to a \c Node map of \c Edges. |
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161 | |
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162 | ///Writes the stored matching to a \c Node map of incident \c |
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163 | ///Edges. This map will have the property that if \c |
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164 | ///G.bNode(map[u])=v then \c G.bNode(map[v])=u holds, and now this |
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165 | ///edge is an edge of the matching. |
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166 | template<typename NMapE> |
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167 | void writeNMapEdge (NMapE& map) const { |
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168 | typename Graph::template NodeMap<bool> todo(G,false); |
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169 | NodeIt v; |
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170 | for( G.first(v); G.valid(v); G.next(v)) { |
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171 | if ( mate[v]!=INVALID ) todo.set(v,true); |
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172 | } |
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173 | NodeIt e; |
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174 | for( G.first(e); G.valid(e); G.next(e)) { |
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175 | if ( todo[G.head(e)] && todo[G.tail(e)] ) { |
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176 | Node u=G.tail(e); |
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177 | Node v=G.head(e); |
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178 | if ( mate[u]=v && mate[v]=u ) { |
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179 | map.set(u,e); |
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180 | map.set(v,e); |
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181 | todo.set(u,false); |
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182 | todo.set(v,false); |
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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 | EdgeIt e; |
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197 | for( G.first(e); G.valid(e); G.next(e)) { |
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198 | if ( G.valid(e) ) { |
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199 | Node u=G.tail(e); |
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200 | Node v=G.head(e); |
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201 | mate.set(u,v); |
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202 | mate.set(v,u); |
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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 | |
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208 | ///Writes the matching stored to an \c Edge map of \c bools. |
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209 | |
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210 | ///Writes the matching stored to an \c Edge map of \c bools. This |
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211 | ///map will have the property that there are no two adjacent edges |
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212 | ///\c e, \c f with \c map[e]=map[f]=true. The edges \c e with \c |
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213 | ///map[e]=true form the matching. |
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214 | template<typename EMapB> |
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215 | void writeEMapBool (EMapB& map) const { |
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216 | typename Graph::template NodeMap<bool> todo(G,false); |
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217 | NodeIt v; |
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218 | for( G.first(v); G.valid(v); G.next(v)) { |
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219 | if ( mate[v]!=INVALID ) todo.set(v,true); |
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220 | } |
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221 | |
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222 | NodeIt e; |
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223 | for( G.first(e); G.valid(e); G.next(e)) { |
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224 | map.set(e,false); |
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225 | if ( todo[G.head(e)] && todo[G.tail(e)] ) { |
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226 | Node u=G.tail(e); |
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227 | Node v=G.head(e); |
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228 | if ( mate[u]=v && mate[v]=u ) { |
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229 | map.set(e,true); |
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230 | todo.set(u,false); |
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231 | todo.set(v,false); |
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232 | } |
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233 | } |
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234 | } |
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235 | } |
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236 | |
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237 | ///Writes the canonical decomposition of the graph after running |
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238 | ///the algorithm. |
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239 | |
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240 | ///After calling any run methods of the class, and before calling |
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241 | ///\ref resetPos(), it writes the Gallai-Edmonds canonical |
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242 | ///decomposition of the graph. \c map must be a node map |
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243 | ///of \ref pos_enum 's. |
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244 | template<typename NMapEnum> |
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245 | void writePos (NMapEnum& map) const { |
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246 | NodeIt v; |
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247 | for( G.first(v); G.valid(v); G.next(v)) map.set(v,position[v]); |
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248 | } |
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249 | |
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250 | private: |
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251 | |
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252 | void lateShrink(Node v, typename Graph::template NodeMap<Node>& ear, |
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253 | UFE& blossom, UFE& tree); |
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254 | |
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255 | void normShrink(Node v, typename Graph::NodeMap<Node>& ear, |
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256 | UFE& blossom, UFE& tree); |
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257 | |
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258 | bool noShrinkStep(Node x, typename Graph::NodeMap<Node>& ear, |
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259 | UFE& blossom, UFE& tree, std::queue<Node>& Q); |
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260 | |
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261 | void shrinkStep(Node& top, Node& middle, Node& bottom, typename Graph::NodeMap<Node>& ear, |
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262 | UFE& blossom, UFE& tree, std::queue<Node>& Q); |
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263 | |
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264 | void augment(Node x, typename Graph::NodeMap<Node>& ear, |
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265 | UFE& blossom, UFE& tree); |
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266 | |
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267 | }; |
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268 | |
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269 | |
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270 | // ********************************************************************** |
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271 | // IMPLEMENTATIONS |
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272 | // ********************************************************************** |
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273 | |
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274 | |
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275 | template <typename Graph> |
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276 | void MaxMatching<Graph>::run() { |
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277 | if ( G.edgeNum() > 2*G.nodeNum() ) { |
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278 | greedyMatching(); |
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279 | runEdmonds(1); |
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280 | } else runEdmonds(0); |
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281 | } |
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282 | |
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283 | template <typename Graph> |
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284 | void MaxMatching<Graph>::runEdmonds( int heur=1 ) { |
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285 | |
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286 | typename Graph::template NodeMap<Node> ear(G,INVALID); |
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287 | //undefined for the base nodes of the blossoms (i.e. for the |
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288 | //representative elements of UFE blossom) and for the nodes in C |
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289 | |
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290 | typename UFE::MapType blossom_base(G); |
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291 | UFE blossom(blossom_base); |
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292 | typename UFE::MapType tree_base(G); |
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293 | UFE tree(tree_base); |
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294 | |
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295 | NodeIt v; |
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296 | for( G.first(v); G.valid(v); G.next(v) ) { |
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297 | if ( position[v]==C && mate[v]==INVALID ) { |
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298 | blossom.insert(v); |
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299 | tree.insert(v); |
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300 | position.set(v,D); |
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301 | if ( heur == 1 ) lateShrink( v, ear, blossom, tree ); |
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302 | else normShrink( v, ear, blossom, tree ); |
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303 | } |
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304 | } |
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305 | } |
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306 | |
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307 | template <typename Graph> |
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308 | void MaxMatching<Graph>::lateShrink(Node v, typename Graph::template NodeMap<Node>& ear, |
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309 | UFE& blossom, UFE& tree) { |
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310 | |
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311 | std::queue<Node> Q; //queue of the totally unscanned nodes |
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312 | Q.push(v); |
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313 | std::queue<Node> R; |
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314 | //queue of the nodes which must be scanned for a possible shrink |
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315 | |
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316 | while ( !Q.empty() ) { |
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317 | Node x=Q.front(); |
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318 | Q.pop(); |
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319 | if ( noShrinkStep( x, ear, blossom, tree, Q ) ) return; |
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320 | else R.push(x); |
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321 | } |
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322 | |
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323 | while ( !R.empty() ) { |
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324 | Node x=R.front(); |
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325 | R.pop(); |
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326 | |
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327 | OutEdgeIt e; |
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328 | for( G.first(e,x); G.valid(e); G.next(e) ) { |
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329 | Node y=G.bNode(e); |
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330 | |
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331 | if ( position[y] == D && blossom.find(x) != blossom.find(y) ) { |
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332 | //x and y must be in the same tree |
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333 | |
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334 | typename Graph::template NodeMap<bool> path(G,false); |
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335 | |
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336 | Node b=blossom.find(x); |
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337 | path.set(b,true); |
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338 | b=mate[b]; |
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339 | while ( b!=INVALID ) { |
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340 | b=blossom.find(ear[b]); |
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341 | path.set(b,true); |
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342 | b=mate[b]; |
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343 | } //going till the root |
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344 | |
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345 | Node top=y; |
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346 | Node middle=blossom.find(top); |
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347 | Node bottom=x; |
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348 | while ( !path[middle] ) |
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349 | shrinkStep(top, middle, bottom, ear, blossom, tree, Q); |
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350 | |
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351 | Node base=middle; |
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352 | top=x; |
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353 | middle=blossom.find(top); |
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354 | bottom=y; |
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355 | Node blossom_base=blossom.find(base); |
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356 | while ( middle!=blossom_base ) |
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357 | shrinkStep(top, middle, bottom, ear, blossom, tree, Q); |
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358 | |
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359 | blossom.makeRep(base); |
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360 | } // if shrink is needed |
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361 | |
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362 | while ( !Q.empty() ) { |
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363 | Node x=Q.front(); |
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364 | Q.pop(); |
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365 | if ( noShrinkStep(x, ear, blossom, tree, Q) ) return; |
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366 | else R.push(x); |
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367 | } |
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368 | } //for e |
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369 | } // while ( !R.empty() ) |
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370 | } |
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371 | |
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372 | template <typename Graph> |
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373 | void MaxMatching<Graph>::normShrink(Node v, typename Graph::NodeMap<Node>& ear, |
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374 | UFE& blossom, UFE& tree) { |
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375 | |
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376 | std::queue<Node> Q; //queue of the unscanned nodes |
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377 | Q.push(v); |
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378 | while ( !Q.empty() ) { |
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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 | OutEdgeIt e; |
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383 | for( G.first(e,x); G.valid(e); G.next(e) ) { |
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384 | Node y=G.bNode(e); |
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385 | |
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386 | switch ( position[y] ) { |
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387 | case D: //x and y must be in the same tree |
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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 | ear.set(y,x); |
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420 | Node w=mate[y]; |
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421 | blossom.insert(w); |
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422 | position.set(y,A); |
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423 | position.set(w,D); |
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424 | tree.insert(y); |
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425 | tree.insert(w); |
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426 | tree.join(y,blossom.find(x)); |
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427 | tree.join(w,y); |
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428 | Q.push(w); |
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429 | } else { //augment |
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430 | augment(x, ear, blossom, tree); |
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431 | mate.set(x,y); |
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432 | mate.set(y,x); |
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433 | return; |
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434 | } //if |
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435 | break; |
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436 | default: break; |
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437 | } |
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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 | template <typename Graph> |
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443 | void MaxMatching<Graph>::greedyMatching() { |
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444 | NodeIt v; |
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445 | for( G.first(v); G.valid(v); G.next(v) ) |
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446 | if ( mate[v]==INVALID ) { |
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447 | OutEdgeIt e; |
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448 | for( G.first(e,v); G.valid(e); G.next(e) ) { |
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449 | Node y=G.bNode(e); |
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450 | if ( mate[y]==INVALID && y!=v ) { |
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451 | mate.set(v,y); |
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452 | mate.set(y,v); |
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453 | break; |
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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 | |
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459 | template <typename Graph> |
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460 | int MaxMatching<Graph>::size() const { |
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461 | int s=0; |
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462 | NodeIt v; |
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463 | for(G.first(v); G.valid(v); G.next(v) ) { |
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464 | if ( G.valid(mate[v]) ) { |
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465 | ++s; |
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466 | } |
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467 | } |
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468 | return (int)s/2; |
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469 | } |
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470 | |
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471 | template <typename Graph> |
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472 | void MaxMatching<Graph>::resetPos() { |
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473 | NodeIt v; |
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474 | for( G.first(v); G.valid(v); G.next(v)) |
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475 | position.set(v,C); |
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476 | } |
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477 | |
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478 | template <typename Graph> |
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479 | void MaxMatching<Graph>::resetMatching() { |
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480 | NodeIt v; |
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481 | for( G.first(v); G.valid(v); G.next(v)) |
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482 | mate.set(v,INVALID); |
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483 | } |
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484 | |
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485 | template <typename Graph> |
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486 | bool MaxMatching<Graph>::noShrinkStep(Node x, typename Graph::NodeMap<Node>& ear, |
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487 | UFE& blossom, UFE& tree, std::queue<Node>& Q) { |
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488 | OutEdgeIt e; |
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489 | for( G.first(e,x); G.valid(e); G.next(e) ) { |
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490 | Node y=G.bNode(e); |
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491 | |
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492 | if ( position[y]==C ) { |
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493 | if ( mate[y]!=INVALID ) { //grow |
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494 | ear.set(y,x); |
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495 | Node w=mate[y]; |
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496 | blossom.insert(w); |
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497 | position.set(y,A); |
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498 | position.set(w,D); |
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499 | tree.insert(y); |
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500 | tree.insert(w); |
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501 | tree.join(y,blossom.find(x)); |
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502 | tree.join(w,y); |
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503 | Q.push(w); |
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504 | } else { //augment |
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505 | augment(x, ear, blossom, tree); |
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506 | mate.set(x,y); |
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507 | mate.set(y,x); |
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508 | return true; |
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509 | } |
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510 | } |
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511 | } |
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512 | return false; |
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513 | } |
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514 | |
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515 | template <typename Graph> |
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516 | void MaxMatching<Graph>::shrinkStep(Node& top, Node& middle, Node& bottom, typename Graph::NodeMap<Node>& ear, |
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517 | UFE& blossom, UFE& tree, std::queue<Node>& Q) { |
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518 | ear.set(top,bottom); |
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519 | Node t=top; |
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520 | while ( t!=middle ) { |
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521 | Node u=mate[t]; |
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522 | t=ear[u]; |
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523 | ear.set(t,u); |
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524 | } |
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525 | bottom=mate[middle]; |
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526 | position.set(bottom,D); |
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527 | Q.push(bottom); |
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528 | top=ear[bottom]; |
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529 | Node oldmiddle=middle; |
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530 | middle=blossom.find(top); |
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531 | tree.erase(bottom); |
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532 | tree.erase(oldmiddle); |
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533 | blossom.insert(bottom); |
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534 | blossom.join(bottom, oldmiddle); |
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535 | blossom.join(top, oldmiddle); |
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536 | } |
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537 | |
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538 | template <typename Graph> |
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539 | void MaxMatching<Graph>::augment(Node x, typename Graph::NodeMap<Node>& ear, |
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540 | UFE& blossom, UFE& tree) { |
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541 | Node v=mate[x]; |
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542 | while ( G.valid(v) ) { |
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543 | |
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544 | Node u=ear[v]; |
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545 | mate.set(v,u); |
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546 | Node tmp=v; |
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547 | v=mate[u]; |
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548 | mate.set(u,tmp); |
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549 | } |
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550 | typename UFE::ItemIt it; |
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551 | for (tree.first(it,blossom.find(x)); tree.valid(it); tree.next(it)) { |
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552 | if ( position[it] == D ) { |
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553 | typename UFE::ItemIt b_it; |
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554 | for (blossom.first(b_it,it); blossom.valid(b_it); blossom.next(b_it)) { |
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555 | position.set( b_it ,C); |
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556 | } |
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557 | blossom.eraseClass(it); |
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558 | } else position.set( it ,C); |
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559 | } |
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560 | tree.eraseClass(x); |
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561 | } |
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562 | |
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563 | |
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564 | |
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565 | /// @} |
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566 | |
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567 | } //END OF NAMESPACE LEMON |
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568 | |
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569 | #endif //EDMONDS_H |
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