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) 2003-2008 |
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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_PR_BIPARTITE_MATCHING |
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20 | #define LEMON_PR_BIPARTITE_MATCHING |
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21 | |
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22 | #include <lemon/graph_utils.h> |
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23 | #include <lemon/iterable_maps.h> |
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24 | #include <iostream> |
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25 | #include <queue> |
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26 | #include <lemon/elevator.h> |
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27 | |
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28 | ///\ingroup matching |
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29 | ///\file |
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30 | ///\brief Push-prelabel maximum matching algorithms in bipartite graphs. |
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31 | /// |
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32 | namespace lemon { |
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33 | |
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34 | ///Max cardinality matching algorithm based on push-relabel principle |
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35 | |
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36 | ///\ingroup matching |
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37 | ///Bipartite Max Cardinality Matching algorithm. This class uses the |
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38 | ///push-relabel principle which in several cases has better runtime |
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39 | ///performance than the augmenting path solutions. |
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40 | /// |
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41 | ///\author Alpar Juttner |
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42 | template<class Graph> |
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43 | class PrBipartiteMatching { |
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44 | typedef typename Graph::Node Node; |
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45 | typedef typename Graph::ANodeIt ANodeIt; |
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46 | typedef typename Graph::BNodeIt BNodeIt; |
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47 | typedef typename Graph::UEdge UEdge; |
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48 | typedef typename Graph::UEdgeIt UEdgeIt; |
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49 | typedef typename Graph::IncEdgeIt IncEdgeIt; |
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50 | |
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51 | const Graph &_g; |
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52 | int _node_num; |
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53 | int _matching_size; |
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54 | int _empty_level; |
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55 | |
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56 | typename Graph::template ANodeMap<typename Graph::UEdge> _matching; |
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57 | Elevator<Graph,typename Graph::BNode> _levels; |
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58 | typename Graph::template BNodeMap<int> _cov; |
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59 | |
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60 | public: |
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61 | |
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62 | /// Constructor |
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63 | |
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64 | /// Constructor |
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65 | /// |
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66 | PrBipartiteMatching(const Graph &g) : |
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67 | _g(g), |
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68 | _node_num(countBNodes(g)), |
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69 | _matching(g), |
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70 | _levels(g,_node_num), |
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71 | _cov(g,0) |
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72 | { |
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73 | } |
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74 | |
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75 | /// \name Execution control |
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76 | /// The simplest way to execute the algorithm is to use one of the |
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77 | /// member functions called \c run(). \n |
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78 | /// If you need more control on the execution, first |
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79 | /// you must call \ref init() and then one variant of the start() |
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80 | /// member. |
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81 | |
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82 | /// @{ |
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83 | |
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84 | ///Initialize the data structures |
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85 | |
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86 | ///This function constructs a prematching first, which is a |
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87 | ///regular matching on the A-side of the graph, but on the B-side |
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88 | ///each node could cover more matching edges. After that, the |
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89 | ///B-nodes which multiple matched, will be pushed into the lowest |
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90 | ///level of the Elevator. The remaning B-nodes will be pushed to |
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91 | ///the consequent levels respect to a Bfs on following graph: the |
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92 | ///nodes are the B-nodes of the original bipartite graph and two |
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93 | ///nodes are adjacent if a node can pass over a matching edge to |
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94 | ///an other node. The source of the Bfs are the lowest level |
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95 | ///nodes. Last, the reached B-nodes without covered matching edge |
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96 | ///becomes active. |
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97 | void init() { |
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98 | _matching_size=0; |
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99 | _empty_level=_node_num; |
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100 | for(ANodeIt n(_g);n!=INVALID;++n) |
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101 | if((_matching[n]=IncEdgeIt(_g,n))!=INVALID) |
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102 | ++_cov[_g.bNode(_matching[n])]; |
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103 | |
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104 | std::queue<Node> q; |
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105 | _levels.initStart(); |
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106 | for(BNodeIt n(_g);n!=INVALID;++n) |
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107 | if(_cov[n]>1) { |
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108 | _levels.initAddItem(n); |
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109 | q.push(n); |
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110 | } |
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111 | int hlev=0; |
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112 | while(!q.empty()) { |
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113 | Node n=q.front(); |
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114 | q.pop(); |
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115 | int nlev=_levels[n]+1; |
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116 | for(IncEdgeIt e(_g,n);e!=INVALID;++e) { |
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117 | Node m=_g.runningNode(e); |
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118 | if(e==_matching[m]) { |
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119 | for(IncEdgeIt f(_g,m);f!=INVALID;++f) { |
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120 | Node r=_g.runningNode(f); |
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121 | if(_levels[r]>nlev) { |
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122 | for(;nlev>hlev;hlev++) |
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123 | _levels.initNewLevel(); |
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124 | _levels.initAddItem(r); |
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125 | q.push(r); |
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126 | } |
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127 | } |
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128 | } |
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129 | } |
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130 | } |
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131 | _levels.initFinish(); |
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132 | for(BNodeIt n(_g);n!=INVALID;++n) |
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133 | if(_cov[n]<1&&_levels[n]<_node_num) |
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134 | _levels.activate(n); |
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135 | } |
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136 | |
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137 | ///Start the main phase of the algorithm |
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138 | |
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139 | ///This algorithm calculates the maximum matching with the |
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140 | ///push-relabel principle. This function should be called just |
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141 | ///after the init() function which already set the initial |
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142 | ///prematching, the level function on the B-nodes and the active, |
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143 | ///ie. unmatched B-nodes. |
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144 | /// |
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145 | ///The algorithm always takes highest active B-node, and it try to |
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146 | ///find a B-node which is eligible to pass over one of it's |
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147 | ///matching edge. This condition holds when the B-node is one |
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148 | ///level lower, and the opposite node of it's matching edge is |
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149 | ///adjacent to the highest active node. In this case the current |
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150 | ///node steals the matching edge and becomes inactive. If there is |
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151 | ///not eligible node then the highest active node should be lift |
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152 | ///to the next proper level. |
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153 | /// |
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154 | ///The nodes should not lift higher than the number of the |
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155 | ///B-nodes, if a node reach this level it remains unmatched. If |
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156 | ///during the execution one level becomes empty the nodes above it |
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157 | ///can be deactivated and lift to the highest level. |
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158 | void start() { |
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159 | Node act; |
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160 | Node bact=INVALID; |
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161 | Node last_activated=INVALID; |
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162 | while((act=_levels.highestActive())!=INVALID) { |
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163 | last_activated=INVALID; |
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164 | int actlevel=_levels[act]; |
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165 | |
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166 | UEdge bedge=INVALID; |
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167 | int nlevel=_node_num; |
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168 | { |
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169 | int nnlevel; |
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170 | for(IncEdgeIt tbedge(_g,act); |
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171 | tbedge!=INVALID && nlevel>=actlevel; |
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172 | ++tbedge) |
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173 | if((nnlevel=_levels[_g.bNode(_matching[_g.runningNode(tbedge)])])< |
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174 | nlevel) |
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175 | { |
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176 | nlevel=nnlevel; |
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177 | bedge=tbedge; |
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178 | } |
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179 | } |
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180 | if(nlevel<_node_num) { |
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181 | if(nlevel>=actlevel) |
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182 | _levels.liftHighestActive(nlevel+1); |
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183 | bact=_g.bNode(_matching[_g.aNode(bedge)]); |
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184 | if(--_cov[bact]<1) { |
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185 | _levels.activate(bact); |
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186 | last_activated=bact; |
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187 | } |
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188 | _matching[_g.aNode(bedge)]=bedge; |
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189 | _cov[act]=1; |
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190 | _levels.deactivate(act); |
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191 | } |
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192 | else { |
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193 | _levels.liftHighestActiveToTop(); |
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194 | } |
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195 | |
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196 | if(_levels.emptyLevel(actlevel)) |
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197 | _levels.liftToTop(actlevel); |
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198 | } |
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199 | |
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200 | for(ANodeIt n(_g);n!=INVALID;++n) { |
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201 | if (_matching[n]==INVALID)continue; |
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202 | if (_cov[_g.bNode(_matching[n])]>1) { |
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203 | _cov[_g.bNode(_matching[n])]--; |
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204 | _matching[n]=INVALID; |
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205 | } else { |
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206 | ++_matching_size; |
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207 | } |
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208 | } |
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209 | } |
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210 | |
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211 | ///Start the algorithm to find a perfect matching |
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212 | |
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213 | ///This function is close to identical to the simple start() |
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214 | ///member function but it calculates just perfect matching. |
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215 | ///However, the perfect property is only checked on the B-side of |
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216 | ///the graph |
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217 | /// |
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218 | ///The main difference between the two function is the handling of |
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219 | ///the empty levels. The simple start() function let the nodes |
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220 | ///above the empty levels unmatched while this variant if it find |
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221 | ///an empty level immediately terminates and gives back false |
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222 | ///return value. |
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223 | bool startPerfect() { |
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224 | Node act; |
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225 | Node bact=INVALID; |
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226 | Node last_activated=INVALID; |
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227 | while((act=_levels.highestActive())!=INVALID) { |
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228 | last_activated=INVALID; |
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229 | int actlevel=_levels[act]; |
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230 | |
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231 | UEdge bedge=INVALID; |
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232 | int nlevel=_node_num; |
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233 | { |
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234 | int nnlevel; |
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235 | for(IncEdgeIt tbedge(_g,act); |
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236 | tbedge!=INVALID && nlevel>=actlevel; |
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237 | ++tbedge) |
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238 | if((nnlevel=_levels[_g.bNode(_matching[_g.runningNode(tbedge)])])< |
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239 | nlevel) |
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240 | { |
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241 | nlevel=nnlevel; |
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242 | bedge=tbedge; |
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243 | } |
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244 | } |
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245 | if(nlevel<_node_num) { |
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246 | if(nlevel>=actlevel) |
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247 | _levels.liftHighestActive(nlevel+1); |
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248 | bact=_g.bNode(_matching[_g.aNode(bedge)]); |
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249 | if(--_cov[bact]<1) { |
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250 | _levels.activate(bact); |
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251 | last_activated=bact; |
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252 | } |
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253 | _matching[_g.aNode(bedge)]=bedge; |
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254 | _cov[act]=1; |
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255 | _levels.deactivate(act); |
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256 | } |
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257 | else { |
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258 | _levels.liftHighestActiveToTop(); |
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259 | } |
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260 | |
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261 | if(_levels.emptyLevel(actlevel)) |
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262 | _empty_level=actlevel; |
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263 | return false; |
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264 | } |
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265 | _matching_size = _node_num; |
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266 | return true; |
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267 | } |
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268 | |
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269 | ///Runs the algorithm |
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270 | |
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271 | ///Just a shortcut for the next code: |
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272 | ///\code |
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273 | /// init(); |
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274 | /// start(); |
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275 | ///\endcode |
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276 | void run() { |
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277 | init(); |
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278 | start(); |
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279 | } |
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280 | |
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281 | ///Runs the algorithm to find a perfect matching |
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282 | |
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283 | ///Just a shortcut for the next code: |
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284 | ///\code |
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285 | /// init(); |
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286 | /// startPerfect(); |
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287 | ///\endcode |
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288 | /// |
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289 | ///\note If the two nodesets of the graph have different size then |
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290 | ///this algorithm checks the perfect property on the B-side. |
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291 | bool runPerfect() { |
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292 | init(); |
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293 | return startPerfect(); |
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294 | } |
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295 | |
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296 | ///Runs the algorithm to find a perfect matching |
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297 | |
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298 | ///Just a shortcut for the next code: |
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299 | ///\code |
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300 | /// init(); |
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301 | /// startPerfect(); |
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302 | ///\endcode |
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303 | /// |
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304 | ///\note It checks that the size of the two nodesets are equal. |
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305 | bool checkedRunPerfect() { |
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306 | if (countANodes(_g) != _node_num) return false; |
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307 | init(); |
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308 | return startPerfect(); |
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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 | /// \name Query Functions |
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314 | /// The result of the %Matching algorithm can be obtained using these |
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315 | /// functions.\n |
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316 | /// Before the use of these functions, |
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317 | /// either run() or start() must be called. |
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318 | ///@{ |
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319 | |
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320 | ///Set true all matching uedge in the map. |
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321 | |
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322 | ///Set true all matching uedge in the map. It does not change the |
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323 | ///value mapped to the other uedges. |
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324 | ///\return The number of the matching edges. |
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325 | template <typename MatchingMap> |
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326 | int quickMatching(MatchingMap& mm) const { |
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327 | for (ANodeIt n(_g);n!=INVALID;++n) { |
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328 | if (_matching[n]!=INVALID) mm.set(_matching[n],true); |
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329 | } |
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330 | return _matching_size; |
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331 | } |
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332 | |
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333 | ///Set true all matching uedge in the map and the others to false. |
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334 | |
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335 | ///Set true all matching uedge in the map and the others to false. |
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336 | ///\return The number of the matching edges. |
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337 | template<class MatchingMap> |
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338 | int matching(MatchingMap& mm) const { |
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339 | for (UEdgeIt e(_g);e!=INVALID;++e) { |
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340 | mm.set(e,e==_matching[_g.aNode(e)]); |
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341 | } |
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342 | return _matching_size; |
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343 | } |
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344 | |
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345 | ///Gives back the matching in an ANodeMap. |
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346 | |
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347 | ///Gives back the matching in an ANodeMap. The parameter should |
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348 | ///be a write ANodeMap of UEdge values. |
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349 | ///\return The number of the matching edges. |
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350 | template<class MatchingMap> |
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351 | int aMatching(MatchingMap& mm) const { |
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352 | for (ANodeIt n(_g);n!=INVALID;++n) { |
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353 | mm.set(n,_matching[n]); |
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354 | } |
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355 | return _matching_size; |
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356 | } |
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357 | |
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358 | ///Gives back the matching in a BNodeMap. |
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359 | |
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360 | ///Gives back the matching in a BNodeMap. The parameter should |
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361 | ///be a write BNodeMap of UEdge values. |
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362 | ///\return The number of the matching edges. |
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363 | template<class MatchingMap> |
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364 | int bMatching(MatchingMap& mm) const { |
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365 | for (BNodeIt n(_g);n!=INVALID;++n) { |
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366 | mm.set(n,INVALID); |
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367 | } |
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368 | for (ANodeIt n(_g);n!=INVALID;++n) { |
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369 | if (_matching[n]!=INVALID) |
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370 | mm.set(_g.bNode(_matching[n]),_matching[n]); |
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371 | } |
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372 | return _matching_size; |
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373 | } |
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374 | |
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375 | |
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376 | ///Returns true if the given uedge is in the matching. |
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377 | |
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378 | ///It returns true if the given uedge is in the matching. |
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379 | /// |
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380 | bool matchingEdge(const UEdge& e) const { |
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381 | return _matching[_g.aNode(e)]==e; |
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382 | } |
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383 | |
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384 | ///Returns the matching edge from the node. |
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385 | |
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386 | ///Returns the matching edge from the node. If there is not such |
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387 | ///edge it gives back \c INVALID. |
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388 | ///\note If the parameter node is a B-node then the running time is |
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389 | ///propotional to the degree of the node. |
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390 | UEdge matchingEdge(const Node& n) const { |
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391 | if (_g.aNode(n)) { |
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392 | return _matching[n]; |
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393 | } else { |
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394 | for (IncEdgeIt e(_g,n);e!=INVALID;++e) |
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395 | if (e==_matching[_g.aNode(e)]) return e; |
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396 | return INVALID; |
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397 | } |
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398 | } |
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399 | |
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400 | ///Gives back the number of the matching edges. |
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401 | |
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402 | ///Gives back the number of the matching edges. |
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403 | int matchingSize() const { |
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404 | return _matching_size; |
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405 | } |
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406 | |
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407 | ///Gives back a barrier on the A-nodes |
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408 | |
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409 | ///The barrier is s subset of the nodes on the same side of the |
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410 | ///graph. If we tried to find a perfect matching and it failed |
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411 | ///then the barrier size will be greater than its neighbours. If |
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412 | ///the maximum matching searched then the barrier size minus its |
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413 | ///neighbours will be exactly the unmatched nodes on the A-side. |
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414 | ///\retval bar A WriteMap on the ANodes with bool value. |
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415 | template<class BarrierMap> |
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416 | void aBarrier(BarrierMap &bar) const |
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417 | { |
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418 | for(ANodeIt n(_g);n!=INVALID;++n) |
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419 | bar.set(n,_matching[n]==INVALID || |
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420 | _levels[_g.bNode(_matching[n])]<_empty_level); |
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421 | } |
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422 | |
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423 | ///Gives back a barrier on the B-nodes |
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424 | |
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425 | ///The barrier is s subset of the nodes on the same side of the |
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426 | ///graph. If we tried to find a perfect matching and it failed |
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427 | ///then the barrier size will be greater than its neighbours. If |
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428 | ///the maximum matching searched then the barrier size minus its |
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429 | ///neighbours will be exactly the unmatched nodes on the B-side. |
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430 | ///\retval bar A WriteMap on the BNodes with bool value. |
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431 | template<class BarrierMap> |
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432 | void bBarrier(BarrierMap &bar) const |
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433 | { |
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434 | for(BNodeIt n(_g);n!=INVALID;++n) bar.set(n,_levels[n]>=_empty_level); |
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435 | } |
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436 | |
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437 | ///Returns a minimum covering of the nodes. |
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438 | |
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439 | ///The minimum covering set problem is the dual solution of the |
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440 | ///maximum bipartite matching. It provides a solution for this |
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441 | ///problem what is proof of the optimality of the matching. |
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442 | ///\param covering NodeMap of bool values, the nodes of the cover |
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443 | ///set will set true while the others false. |
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444 | ///\return The size of the cover set. |
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445 | ///\note This function can be called just after the algorithm have |
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446 | ///already found a matching. |
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447 | template<class CoverMap> |
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448 | int coverSet(CoverMap& covering) const { |
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449 | int ret=0; |
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450 | for(BNodeIt n(_g);n!=INVALID;++n) { |
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451 | if (_levels[n]<_empty_level) { covering.set(n,true); ++ret; } |
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452 | else covering.set(n,false); |
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453 | } |
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454 | for(ANodeIt n(_g);n!=INVALID;++n) { |
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455 | if (_matching[n]!=INVALID && |
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456 | _levels[_g.bNode(_matching[n])]>=_empty_level) |
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457 | { covering.set(n,true); ++ret; } |
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458 | else covering.set(n,false); |
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459 | } |
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460 | return ret; |
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461 | } |
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462 | |
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463 | |
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464 | /// @} |
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465 | |
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466 | }; |
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467 | |
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468 | |
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469 | ///Maximum cardinality of the matchings in a bipartite graph |
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470 | |
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471 | ///\ingroup matching |
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472 | ///This function finds the maximum cardinality of the matchings |
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473 | ///in a bipartite graph \c g. |
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474 | ///\param g An undirected bipartite graph. |
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475 | ///\return The cardinality of the maximum matching. |
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476 | /// |
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477 | ///\note The the implementation is based |
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478 | ///on the push-relabel principle. |
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479 | template<class Graph> |
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480 | int prBipartiteMatching(const Graph &g) |
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481 | { |
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482 | PrBipartiteMatching<Graph> bpm(g); |
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483 | bpm.run(); |
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484 | return bpm.matchingSize(); |
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485 | } |
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486 | |
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487 | ///Maximum cardinality matching in a bipartite graph |
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488 | |
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489 | ///\ingroup matching |
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490 | ///This function finds a maximum cardinality matching |
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491 | ///in a bipartite graph \c g. |
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492 | ///\param g An undirected bipartite graph. |
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493 | ///\retval matching A write ANodeMap of value type \c UEdge. |
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494 | /// The found edges will be returned in this map, |
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495 | /// i.e. for an \c ANode \c n the edge <tt>matching[n]</tt> is the one |
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496 | /// that covers the node \c n. |
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497 | ///\return The cardinality of the maximum matching. |
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498 | /// |
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499 | ///\note The the implementation is based |
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500 | ///on the push-relabel principle. |
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501 | template<class Graph,class MT> |
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502 | int prBipartiteMatching(const Graph &g,MT &matching) |
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503 | { |
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504 | PrBipartiteMatching<Graph> bpm(g); |
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505 | bpm.run(); |
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506 | bpm.aMatching(matching); |
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507 | return bpm.matchingSize(); |
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508 | } |
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509 | |
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510 | ///Maximum cardinality matching in a bipartite graph |
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511 | |
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512 | ///\ingroup matching |
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513 | ///This function finds a maximum cardinality matching |
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514 | ///in a bipartite graph \c g. |
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515 | ///\param g An undirected bipartite graph. |
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516 | ///\retval matching A write ANodeMap of value type \c UEdge. |
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517 | /// The found edges will be returned in this map, |
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518 | /// i.e. for an \c ANode \c n the edge <tt>matching[n]</tt> is the one |
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519 | /// that covers the node \c n. |
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520 | ///\retval barrier A \c bool WriteMap on the BNodes. The map will be set |
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521 | /// exactly once for each BNode. The nodes with \c true value represent |
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522 | /// a barrier \e B, i.e. the cardinality of \e B minus the number of its |
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523 | /// neighbor is equal to the number of the <tt>BNode</tt>s minus the |
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524 | /// cardinality of the maximum matching. |
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525 | ///\return The cardinality of the maximum matching. |
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526 | /// |
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527 | ///\note The the implementation is based |
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528 | ///on the push-relabel principle. |
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529 | template<class Graph,class MT, class GT> |
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530 | int prBipartiteMatching(const Graph &g,MT &matching,GT &barrier) |
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531 | { |
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532 | PrBipartiteMatching<Graph> bpm(g); |
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533 | bpm.run(); |
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534 | bpm.aMatching(matching); |
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535 | bpm.bBarrier(barrier); |
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536 | return bpm.matchingSize(); |
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537 | } |
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538 | |
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539 | ///Perfect matching in a bipartite graph |
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540 | |
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541 | ///\ingroup matching |
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542 | ///This function checks whether the bipartite graph \c g |
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543 | ///has a perfect matching. |
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544 | ///\param g An undirected bipartite graph. |
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545 | ///\return \c true iff \c g has a perfect matching. |
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546 | /// |
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547 | ///\note The the implementation is based |
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548 | ///on the push-relabel principle. |
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549 | template<class Graph> |
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550 | bool prPerfectBipartiteMatching(const Graph &g) |
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551 | { |
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552 | PrBipartiteMatching<Graph> bpm(g); |
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553 | bpm.run(); |
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554 | return bpm.checkedRunPerfect(); |
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555 | } |
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556 | |
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557 | ///Perfect matching in a bipartite graph |
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558 | |
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559 | ///\ingroup matching |
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560 | ///This function finds a perfect matching in a bipartite graph \c g. |
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561 | ///\param g An undirected bipartite graph. |
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562 | ///\retval matching A write ANodeMap of value type \c UEdge. |
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563 | /// The found edges will be returned in this map, |
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564 | /// i.e. for an \c ANode \c n the edge <tt>matching[n]</tt> is the one |
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565 | /// that covers the node \c n. |
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566 | /// The values are unchanged if the graph |
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567 | /// has no perfect matching. |
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568 | ///\return \c true iff \c g has a perfect matching. |
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569 | /// |
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570 | ///\note The the implementation is based |
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571 | ///on the push-relabel principle. |
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572 | template<class Graph,class MT> |
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573 | bool prPerfectBipartiteMatching(const Graph &g,MT &matching) |
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574 | { |
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575 | PrBipartiteMatching<Graph> bpm(g); |
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576 | bool ret = bpm.checkedRunPerfect(); |
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577 | if (ret) bpm.aMatching(matching); |
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578 | return ret; |
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579 | } |
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580 | |
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581 | ///Perfect matching in a bipartite graph |
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582 | |
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583 | ///\ingroup matching |
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584 | ///This function finds a perfect matching in a bipartite graph \c g. |
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585 | ///\param g An undirected bipartite graph. |
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586 | ///\retval matching A write ANodeMap of value type \c UEdge. |
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587 | /// The found edges will be returned in this map, |
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588 | /// i.e. for an \c ANode \c n the edge <tt>matching[n]</tt> is the one |
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589 | /// that covers the node \c n. |
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590 | /// The values are unchanged if the graph |
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591 | /// has no perfect matching. |
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592 | ///\retval barrier A \c bool WriteMap on the BNodes. The map will only |
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593 | /// be set if \c g has no perfect matching. In this case it is set |
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594 | /// exactly once for each BNode. The nodes with \c true value represent |
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595 | /// a barrier, i.e. a subset \e B a of BNodes with the property that |
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596 | /// the cardinality of \e B is greater than the number of its neighbors. |
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597 | ///\return \c true iff \c g has a perfect matching. |
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598 | /// |
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599 | ///\note The the implementation is based |
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600 | ///on the push-relabel principle. |
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601 | template<class Graph,class MT, class GT> |
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602 | bool prPerfectBipartiteMatching(const Graph &g,MT &matching,GT &barrier) |
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603 | { |
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604 | PrBipartiteMatching<Graph> bpm(g); |
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605 | bool ret=bpm.checkedRunPerfect(); |
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606 | if(ret) |
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607 | bpm.aMatching(matching); |
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608 | else |
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609 | bpm.bBarrier(barrier); |
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610 | return ret; |
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611 | } |
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612 | } |
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613 | |
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614 | #endif |
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