1 | /* -*- C++ -*- |
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2 | * lemon/min_cost_flow.h - Part of LEMON, a generic C++ optimization library |
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3 | * |
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4 | * Copyright (C) 2005 Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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5 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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6 | * |
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7 | * Permission to use, modify and distribute this software is granted |
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8 | * provided that this copyright notice appears in all copies. For |
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9 | * precise terms see the accompanying LICENSE file. |
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10 | * |
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11 | * This software is provided "AS IS" with no warranty of any kind, |
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12 | * express or implied, and with no claim as to its suitability for any |
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13 | * purpose. |
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14 | * |
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15 | */ |
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16 | |
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17 | #ifndef LEMON_MIN_COST_FLOW_H |
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18 | #define LEMON_MIN_COST_FLOW_H |
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19 | |
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20 | ///\ingroup flowalgs |
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21 | ///\file |
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22 | ///\brief An algorithm for finding a flow of value \c k (for small values of \c k) having minimal total cost |
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23 | |
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24 | |
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25 | #include <lemon/dijkstra.h> |
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26 | #include <lemon/graph_adaptor.h> |
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27 | #include <lemon/maps.h> |
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28 | #include <vector> |
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29 | |
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30 | namespace lemon { |
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31 | |
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32 | /// \addtogroup flowalgs |
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33 | /// @{ |
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34 | |
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35 | ///\brief Implementation of an algorithm for finding a flow of value \c k |
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36 | ///(for small values of \c k) having minimal total cost between 2 nodes |
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37 | /// |
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38 | /// |
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39 | /// The class \ref lemon::MinCostFlow "MinCostFlow" implements an |
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40 | /// algorithm for finding a flow of value \c k having minimal total |
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41 | /// cost from a given source node to a given target node in a |
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42 | /// directed graph with a cost function on the edges. To |
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43 | /// this end, the edge-capacities and edge-costs have to be |
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44 | /// nonnegative. The edge-capacities should be integers, but the |
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45 | /// edge-costs can be integers, reals or of other comparable |
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46 | /// numeric type. This algorithm is intended to be used only for |
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47 | /// small values of \c k, since it is only polynomial in k, not in |
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48 | /// the length of k (which is log k): in order to find the minimum |
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49 | /// cost flow of value \c k it finds the minimum cost flow of value |
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50 | /// \c i for every \c i between 0 and \c k. |
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51 | /// |
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52 | ///\param Graph The directed graph type the algorithm runs on. |
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53 | ///\param LengthMap The type of the length map. |
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54 | ///\param CapacityMap The capacity map type. |
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55 | /// |
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56 | ///\author Attila Bernath |
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57 | template <typename Graph, typename LengthMap, typename CapacityMap> |
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58 | class MinCostFlow { |
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59 | |
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60 | typedef typename LengthMap::Value Length; |
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61 | |
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62 | //Warning: this should be integer type |
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63 | typedef typename CapacityMap::Value Capacity; |
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64 | |
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65 | typedef typename Graph::Node Node; |
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66 | typedef typename Graph::NodeIt NodeIt; |
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67 | typedef typename Graph::Edge Edge; |
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68 | typedef typename Graph::OutEdgeIt OutEdgeIt; |
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69 | typedef typename Graph::template EdgeMap<int> EdgeIntMap; |
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70 | |
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71 | typedef ResGraphAdaptor<const Graph,int,CapacityMap,EdgeIntMap> ResGW; |
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72 | typedef typename ResGW::Edge ResGraphEdge; |
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73 | |
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74 | protected: |
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75 | |
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76 | const Graph& g; |
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77 | const LengthMap& length; |
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78 | const CapacityMap& capacity; |
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79 | |
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80 | EdgeIntMap flow; |
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81 | typedef typename Graph::template NodeMap<Length> PotentialMap; |
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82 | PotentialMap potential; |
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83 | |
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84 | Node s; |
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85 | Node t; |
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86 | |
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87 | Length total_length; |
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88 | |
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89 | class ModLengthMap { |
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90 | typedef typename Graph::template NodeMap<Length> NodeMap; |
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91 | const ResGW& g; |
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92 | const LengthMap &length; |
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93 | const NodeMap &pot; |
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94 | public : |
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95 | typedef typename LengthMap::Key Key; |
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96 | typedef typename LengthMap::Value Value; |
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97 | |
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98 | ModLengthMap(const ResGW& _g, |
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99 | const LengthMap &_length, const NodeMap &_pot) : |
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100 | g(_g), /*rev(_rev),*/ length(_length), pot(_pot) { } |
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101 | |
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102 | Value operator[](typename ResGW::Edge e) const { |
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103 | if (g.forward(e)) |
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104 | return length[e]-(pot[g.target(e)]-pot[g.source(e)]); |
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105 | else |
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106 | return -length[e]-(pot[g.target(e)]-pot[g.source(e)]); |
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107 | } |
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108 | |
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109 | }; //ModLengthMap |
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110 | |
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111 | ResGW res_graph; |
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112 | ModLengthMap mod_length; |
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113 | Dijkstra<ResGW, ModLengthMap> dijkstra; |
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114 | |
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115 | public : |
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116 | |
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117 | /*! \brief The constructor of the class. |
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118 | |
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119 | \param _g The directed graph the algorithm runs on. |
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120 | \param _length The length (cost) of the edges. |
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121 | \param _cap The capacity of the edges. |
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122 | \param _s Source node. |
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123 | \param _t Target node. |
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124 | */ |
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125 | MinCostFlow(Graph& _g, LengthMap& _length, CapacityMap& _cap, |
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126 | Node _s, Node _t) : |
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127 | g(_g), length(_length), capacity(_cap), flow(_g), potential(_g), |
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128 | s(_s), t(_t), |
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129 | res_graph(g, capacity, flow), |
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130 | mod_length(res_graph, length, potential), |
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131 | dijkstra(res_graph, mod_length) { |
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132 | reset(); |
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133 | } |
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134 | |
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135 | /*! Tries to augment the flow between s and t by 1. |
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136 | The return value shows if the augmentation is successful. |
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137 | */ |
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138 | bool augment() { |
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139 | dijkstra.run(s); |
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140 | if (!dijkstra.reached(t)) { |
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141 | |
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142 | //Unsuccessful augmentation. |
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143 | return false; |
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144 | } else { |
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145 | |
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146 | //We have to change the potential |
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147 | for(typename ResGW::NodeIt n(res_graph); n!=INVALID; ++n) |
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148 | potential.set(n, potential[n]+dijkstra.distMap()[n]); |
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149 | |
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150 | //Augmenting on the shortest path |
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151 | Node n=t; |
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152 | ResGraphEdge e; |
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153 | while (n!=s){ |
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154 | e = dijkstra.predEdge(n); |
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155 | n = dijkstra.predNode(n); |
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156 | res_graph.augment(e,1); |
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157 | //Let's update the total length |
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158 | if (res_graph.forward(e)) |
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159 | total_length += length[e]; |
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160 | else |
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161 | total_length -= length[e]; |
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162 | } |
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163 | |
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164 | return true; |
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165 | } |
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166 | } |
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167 | |
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168 | /*! \brief Runs the algorithm. |
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169 | |
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170 | Runs the algorithm. |
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171 | Returns k if there is a flow of value at least k from s to t. |
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172 | Otherwise it returns the maximum value of a flow from s to t. |
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173 | |
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174 | \param k The value of the flow we are looking for. |
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175 | |
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176 | \todo May be it does make sense to be able to start with a nonzero |
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177 | feasible primal-dual solution pair as well. |
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178 | |
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179 | \todo If the actual flow value is bigger than k, then everything is |
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180 | cleared and the algorithm starts from zero flow. Is it a good approach? |
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181 | */ |
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182 | int run(int k) { |
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183 | if (flowValue()>k) reset(); |
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184 | while (flowValue()<k && augment()) { } |
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185 | return flowValue(); |
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186 | } |
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187 | |
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188 | /*! \brief The class is reset to zero flow and potential. |
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189 | The class is reset to zero flow and potential. |
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190 | */ |
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191 | void reset() { |
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192 | total_length=0; |
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193 | for (typename Graph::EdgeIt e(g); e!=INVALID; ++e) flow.set(e, 0); |
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194 | for (typename Graph::NodeIt n(g); n!=INVALID; ++n) potential.set(n, 0); |
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195 | } |
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196 | |
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197 | /*! Returns the value of the actual flow. |
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198 | */ |
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199 | int flowValue() const { |
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200 | int i=0; |
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201 | for (typename Graph::OutEdgeIt e(g, s); e!=INVALID; ++e) i+=flow[e]; |
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202 | for (typename Graph::InEdgeIt e(g, s); e!=INVALID; ++e) i-=flow[e]; |
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203 | return i; |
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204 | } |
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205 | |
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206 | /// Total cost of the found flow. |
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207 | |
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208 | /// This function gives back the total cost of the found flow. |
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209 | Length totalLength(){ |
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210 | return total_length; |
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211 | } |
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212 | |
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213 | ///Returns a const reference to the EdgeMap \c flow. |
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214 | |
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215 | ///Returns a const reference to the EdgeMap \c flow. |
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216 | const EdgeIntMap &getFlow() const { return flow;} |
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217 | |
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218 | /*! \brief Returns a const reference to the NodeMap \c potential (the dual solution). |
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219 | |
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220 | Returns a const reference to the NodeMap \c potential (the dual solution). |
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221 | */ |
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222 | const PotentialMap &getPotential() const { return potential;} |
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223 | |
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224 | /*! \brief Checking the complementary slackness optimality criteria. |
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225 | |
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226 | This function checks, whether the given flow and potential |
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227 | satisfy the complementary slackness conditions (i.e. these are optimal). |
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228 | This function only checks optimality, doesn't bother with feasibility. |
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229 | For testing purpose. |
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230 | */ |
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231 | bool checkComplementarySlackness(){ |
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232 | Length mod_pot; |
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233 | Length fl_e; |
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234 | for(typename Graph::EdgeIt e(g); e!=INVALID; ++e) { |
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235 | //C^{\Pi}_{i,j} |
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236 | mod_pot = length[e]-potential[g.target(e)]+potential[g.source(e)]; |
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237 | fl_e = flow[e]; |
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238 | if (0<fl_e && fl_e<capacity[e]) { |
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239 | /// \todo better comparison is needed for real types, moreover, |
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240 | /// this comparison here is superfluous. |
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241 | if (mod_pot != 0) |
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242 | return false; |
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243 | } |
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244 | else { |
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245 | if (mod_pot > 0 && fl_e != 0) |
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246 | return false; |
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247 | if (mod_pot < 0 && fl_e != capacity[e]) |
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248 | return false; |
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249 | } |
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250 | } |
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251 | return true; |
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252 | } |
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253 | |
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254 | }; //class MinCostFlow |
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255 | |
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256 | ///@} |
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257 | |
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258 | } //namespace lemon |
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259 | |
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260 | #endif //LEMON_MIN_COST_FLOW_H |
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