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_NETWORK_SIMPLEX_H |
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20 | #define LEMON_NETWORK_SIMPLEX_H |
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21 | |
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22 | /// \ingroup min_cost_flow |
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23 | /// |
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24 | /// \file |
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25 | /// \brief Network simplex algorithm for finding a minimum cost flow. |
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26 | |
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27 | #include <vector> |
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28 | #include <limits> |
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29 | |
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30 | #include <lemon/graph_adaptor.h> |
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31 | #include <lemon/graph_utils.h> |
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32 | #include <lemon/smart_graph.h> |
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33 | #include <lemon/math.h> |
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34 | |
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35 | namespace lemon { |
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36 | |
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37 | /// \addtogroup min_cost_flow |
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38 | /// @{ |
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39 | |
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40 | /// \brief Implementation of the network simplex algorithm for |
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41 | /// finding a minimum cost flow. |
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42 | /// |
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43 | /// \ref NetworkSimplex implements the network simplex algorithm for |
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44 | /// finding a minimum cost flow. |
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45 | /// |
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46 | /// \tparam Graph The directed graph type the algorithm runs on. |
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47 | /// \tparam LowerMap The type of the lower bound map. |
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48 | /// \tparam CapacityMap The type of the capacity (upper bound) map. |
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49 | /// \tparam CostMap The type of the cost (length) map. |
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50 | /// \tparam SupplyMap The type of the supply map. |
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51 | /// |
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52 | /// \warning |
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53 | /// - Edge capacities and costs should be \e non-negative \e integers. |
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54 | /// - Supply values should be \e signed \e integers. |
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55 | /// - \c LowerMap::Value must be convertible to \c CapacityMap::Value. |
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56 | /// - \c CapacityMap::Value and \c SupplyMap::Value must be |
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57 | /// convertible to each other. |
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58 | /// - All value types must be convertible to \c CostMap::Value, which |
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59 | /// must be signed type. |
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60 | /// |
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61 | /// \note \ref NetworkSimplex provides six different pivot rule |
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62 | /// implementations that significantly affect the efficiency of the |
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63 | /// algorithm. |
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64 | /// By default a combined pivot rule is used, which is the fastest |
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65 | /// implementation according to our benchmark tests. |
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66 | /// Another pivot rule can be selected using \ref run() function |
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67 | /// with the proper parameter. |
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68 | /// |
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69 | /// \author Peter Kovacs |
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70 | |
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71 | template < typename Graph, |
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72 | typename LowerMap = typename Graph::template EdgeMap<int>, |
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73 | typename CapacityMap = typename Graph::template EdgeMap<int>, |
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74 | typename CostMap = typename Graph::template EdgeMap<int>, |
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75 | typename SupplyMap = typename Graph::template NodeMap<int> > |
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76 | class NetworkSimplex |
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77 | { |
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78 | typedef typename CapacityMap::Value Capacity; |
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79 | typedef typename CostMap::Value Cost; |
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80 | typedef typename SupplyMap::Value Supply; |
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81 | |
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82 | typedef SmartGraph SGraph; |
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83 | GRAPH_TYPEDEFS(typename SGraph); |
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84 | |
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85 | typedef typename SGraph::template EdgeMap<Capacity> SCapacityMap; |
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86 | typedef typename SGraph::template EdgeMap<Cost> SCostMap; |
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87 | typedef typename SGraph::template NodeMap<Supply> SSupplyMap; |
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88 | typedef typename SGraph::template NodeMap<Cost> SPotentialMap; |
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89 | |
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90 | typedef typename SGraph::template NodeMap<int> IntNodeMap; |
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91 | typedef typename SGraph::template NodeMap<bool> BoolNodeMap; |
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92 | typedef typename SGraph::template NodeMap<Node> NodeNodeMap; |
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93 | typedef typename SGraph::template NodeMap<Edge> EdgeNodeMap; |
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94 | typedef typename SGraph::template EdgeMap<int> IntEdgeMap; |
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95 | |
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96 | typedef typename Graph::template NodeMap<Node> NodeRefMap; |
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97 | typedef typename Graph::template EdgeMap<Edge> EdgeRefMap; |
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98 | |
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99 | public: |
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100 | |
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101 | /// The type of the flow map. |
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102 | typedef typename Graph::template EdgeMap<Capacity> FlowMap; |
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103 | /// The type of the potential map. |
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104 | typedef typename Graph::template NodeMap<Cost> PotentialMap; |
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105 | |
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106 | public: |
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107 | |
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108 | /// Enum type to select the pivot rule used by \ref run(). |
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109 | enum PivotRuleEnum { |
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110 | FIRST_ELIGIBLE_PIVOT, |
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111 | BEST_ELIGIBLE_PIVOT, |
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112 | BLOCK_SEARCH_PIVOT, |
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113 | LIMITED_SEARCH_PIVOT, |
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114 | CANDIDATE_LIST_PIVOT, |
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115 | COMBINED_PIVOT |
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116 | }; |
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117 | |
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118 | private: |
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119 | |
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120 | /// \brief Map adaptor class for handling reduced edge costs. |
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121 | /// |
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122 | /// Map adaptor class for handling reduced edge costs. |
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123 | class ReducedCostMap : public MapBase<Edge, Cost> |
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124 | { |
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125 | private: |
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126 | |
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127 | const SGraph &_gr; |
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128 | const SCostMap &_cost_map; |
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129 | const SPotentialMap &_pot_map; |
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130 | |
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131 | public: |
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132 | |
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133 | ///\e |
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134 | ReducedCostMap( const SGraph &gr, |
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135 | const SCostMap &cost_map, |
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136 | const SPotentialMap &pot_map ) : |
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137 | _gr(gr), _cost_map(cost_map), _pot_map(pot_map) {} |
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138 | |
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139 | ///\e |
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140 | Cost operator[](const Edge &e) const { |
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141 | return _cost_map[e] + _pot_map[_gr.source(e)] |
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142 | - _pot_map[_gr.target(e)]; |
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143 | } |
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144 | |
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145 | }; //class ReducedCostMap |
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146 | |
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147 | private: |
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148 | |
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149 | /// \brief Implementation of the "First Eligible" pivot rule for the |
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150 | /// \ref NetworkSimplex "network simplex" algorithm. |
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151 | /// |
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152 | /// This class implements the "First Eligible" pivot rule |
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153 | /// for the \ref NetworkSimplex "network simplex" algorithm. |
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154 | class FirstEligiblePivotRule |
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155 | { |
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156 | private: |
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157 | |
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158 | NetworkSimplex &_ns; |
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159 | EdgeIt _next_edge; |
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160 | |
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161 | public: |
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162 | |
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163 | /// Constructor. |
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164 | FirstEligiblePivotRule(NetworkSimplex &ns) : |
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165 | _ns(ns), _next_edge(ns._graph) {} |
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166 | |
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167 | /// Finds the next entering edge. |
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168 | bool findEnteringEdge() { |
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169 | for (EdgeIt e = _next_edge; e != INVALID; ++e) { |
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170 | if (_ns._state[e] * _ns._red_cost[e] < 0) { |
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171 | _ns._in_edge = e; |
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172 | _next_edge = ++e; |
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173 | return true; |
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174 | } |
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175 | } |
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176 | for (EdgeIt e(_ns._graph); e != _next_edge; ++e) { |
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177 | if (_ns._state[e] * _ns._red_cost[e] < 0) { |
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178 | _ns._in_edge = e; |
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179 | _next_edge = ++e; |
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180 | return true; |
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181 | } |
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182 | } |
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183 | return false; |
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184 | } |
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185 | }; //class FirstEligiblePivotRule |
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186 | |
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187 | /// \brief Implementation of the "Best Eligible" pivot rule for the |
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188 | /// \ref NetworkSimplex "network simplex" algorithm. |
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189 | /// |
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190 | /// This class implements the "Best Eligible" pivot rule |
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191 | /// for the \ref NetworkSimplex "network simplex" algorithm. |
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192 | class BestEligiblePivotRule |
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193 | { |
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194 | private: |
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195 | |
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196 | NetworkSimplex &_ns; |
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197 | |
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198 | public: |
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199 | |
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200 | /// Constructor. |
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201 | BestEligiblePivotRule(NetworkSimplex &ns) : _ns(ns) {} |
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202 | |
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203 | /// Finds the next entering edge. |
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204 | bool findEnteringEdge() { |
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205 | Cost min = 0; |
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206 | for (EdgeIt e(_ns._graph); e != INVALID; ++e) { |
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207 | if (_ns._state[e] * _ns._red_cost[e] < min) { |
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208 | min = _ns._state[e] * _ns._red_cost[e]; |
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209 | _ns._in_edge = e; |
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210 | } |
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211 | } |
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212 | return min < 0; |
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213 | } |
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214 | }; //class BestEligiblePivotRule |
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215 | |
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216 | /// \brief Implementation of the "Block Search" pivot rule for the |
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217 | /// \ref NetworkSimplex "network simplex" algorithm. |
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218 | /// |
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219 | /// This class implements the "Block Search" pivot rule |
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220 | /// for the \ref NetworkSimplex "network simplex" algorithm. |
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221 | class BlockSearchPivotRule |
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222 | { |
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223 | private: |
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224 | |
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225 | NetworkSimplex &_ns; |
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226 | EdgeIt _next_edge, _min_edge; |
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227 | int _block_size; |
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228 | |
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229 | static const int MIN_BLOCK_SIZE = 10; |
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230 | |
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231 | public: |
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232 | |
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233 | /// Constructor. |
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234 | BlockSearchPivotRule(NetworkSimplex &ns) : |
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235 | _ns(ns), _next_edge(ns._graph), _min_edge(ns._graph) |
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236 | { |
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237 | _block_size = 2 * int(sqrt(countEdges(_ns._graph))); |
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238 | if (_block_size < MIN_BLOCK_SIZE) _block_size = MIN_BLOCK_SIZE; |
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239 | } |
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240 | |
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241 | /// Finds the next entering edge. |
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242 | bool findEnteringEdge() { |
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243 | Cost curr, min = 0; |
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244 | int cnt = 0; |
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245 | for (EdgeIt e = _next_edge; e != INVALID; ++e) { |
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246 | if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) { |
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247 | min = curr; |
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248 | _min_edge = e; |
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249 | } |
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250 | if (++cnt == _block_size) { |
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251 | if (min < 0) break; |
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252 | cnt = 0; |
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253 | } |
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254 | } |
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255 | if (min == 0) { |
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256 | for (EdgeIt e(_ns._graph); e != _next_edge; ++e) { |
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257 | if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) { |
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258 | min = curr; |
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259 | _min_edge = e; |
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260 | } |
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261 | if (++cnt == _block_size) { |
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262 | if (min < 0) break; |
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263 | cnt = 0; |
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264 | } |
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265 | } |
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266 | } |
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267 | _ns._in_edge = _min_edge; |
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268 | _next_edge = ++_min_edge; |
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269 | return min < 0; |
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270 | } |
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271 | }; //class BlockSearchPivotRule |
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272 | |
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273 | /// \brief Implementation of the "Limited Search" pivot rule for the |
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274 | /// \ref NetworkSimplex "network simplex" algorithm. |
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275 | /// |
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276 | /// This class implements the "Limited Search" pivot rule |
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277 | /// for the \ref NetworkSimplex "network simplex" algorithm. |
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278 | class LimitedSearchPivotRule |
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279 | { |
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280 | private: |
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281 | |
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282 | NetworkSimplex &_ns; |
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283 | EdgeIt _next_edge, _min_edge; |
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284 | int _sample_size; |
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285 | |
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286 | static const int MIN_SAMPLE_SIZE = 10; |
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287 | static const double SAMPLE_SIZE_FACTOR = 0.0015; |
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288 | |
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289 | public: |
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290 | |
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291 | /// Constructor. |
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292 | LimitedSearchPivotRule(NetworkSimplex &ns) : |
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293 | _ns(ns), _next_edge(ns._graph), _min_edge(ns._graph) |
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294 | { |
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295 | _sample_size = int(SAMPLE_SIZE_FACTOR * countEdges(_ns._graph)); |
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296 | if (_sample_size < MIN_SAMPLE_SIZE) |
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297 | _sample_size = MIN_SAMPLE_SIZE; |
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298 | } |
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299 | |
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300 | /// Finds the next entering edge. |
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301 | bool findEnteringEdge() { |
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302 | Cost curr, min = 0; |
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303 | int cnt = 0; |
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304 | for (EdgeIt e = _next_edge; e != INVALID; ++e) { |
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305 | if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) { |
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306 | min = curr; |
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307 | _min_edge = e; |
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308 | } |
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309 | if (curr < 0 && ++cnt == _sample_size) break; |
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310 | } |
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311 | if (min == 0) { |
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312 | for (EdgeIt e(_ns._graph); e != _next_edge; ++e) { |
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313 | if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) { |
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314 | min = curr; |
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315 | _min_edge = e; |
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316 | } |
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317 | if (curr < 0 && ++cnt == _sample_size) break; |
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318 | } |
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319 | } |
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320 | _ns._in_edge = _min_edge; |
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321 | _next_edge = ++_min_edge; |
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322 | return min < 0; |
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323 | } |
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324 | }; //class LimitedSearchPivotRule |
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325 | |
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326 | /// \brief Implementation of the "Candidate List" pivot rule for the |
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327 | /// \ref NetworkSimplex "network simplex" algorithm. |
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328 | /// |
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329 | /// This class implements the "Candidate List" pivot rule |
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330 | /// for the \ref NetworkSimplex "network simplex" algorithm. |
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331 | class CandidateListPivotRule |
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332 | { |
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333 | private: |
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334 | |
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335 | NetworkSimplex &_ns; |
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336 | |
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337 | // The list of candidate edges. |
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338 | std::vector<Edge> _candidates; |
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339 | // The maximum length of the edge list. |
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340 | int _list_length; |
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341 | // The maximum number of minor iterations between two major |
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342 | // itarations. |
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343 | int _minor_limit; |
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344 | |
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345 | int _minor_count; |
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346 | EdgeIt _next_edge; |
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347 | |
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348 | static const double LIST_LENGTH_FACTOR = 0.002; |
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349 | static const double MINOR_LIMIT_FACTOR = 0.1; |
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350 | static const int MIN_LIST_LENGTH = 10; |
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351 | static const int MIN_MINOR_LIMIT = 2; |
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352 | |
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353 | public: |
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354 | |
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355 | /// Constructor. |
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356 | CandidateListPivotRule(NetworkSimplex &ns) : |
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357 | _ns(ns), _next_edge(ns._graph) |
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358 | { |
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359 | int edge_num = countEdges(_ns._graph); |
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360 | _minor_count = 0; |
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361 | _list_length = int(edge_num * LIST_LENGTH_FACTOR); |
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362 | if (_list_length < MIN_LIST_LENGTH) |
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363 | _list_length = MIN_LIST_LENGTH; |
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364 | _minor_limit = int(_list_length * MINOR_LIMIT_FACTOR); |
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365 | if (_minor_limit < MIN_MINOR_LIMIT) |
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366 | _minor_limit = MIN_MINOR_LIMIT; |
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367 | } |
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368 | |
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369 | /// Finds the next entering edge. |
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370 | bool findEnteringEdge() { |
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371 | Cost min, curr; |
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372 | if (_minor_count < _minor_limit && _candidates.size() > 0) { |
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373 | // Minor iteration |
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374 | ++_minor_count; |
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375 | Edge e; |
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376 | min = 0; |
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377 | for (int i = 0; i < int(_candidates.size()); ++i) { |
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378 | e = _candidates[i]; |
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379 | if ((curr = _ns._state[e] * _ns._red_cost[e]) < min) { |
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380 | min = curr; |
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381 | _ns._in_edge = e; |
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382 | } |
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383 | } |
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384 | if (min < 0) return true; |
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385 | } |
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386 | |
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387 | // Major iteration |
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388 | _candidates.clear(); |
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389 | EdgeIt e = _next_edge; |
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390 | min = 0; |
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391 | for ( ; e != INVALID; ++e) { |
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392 | if ((curr = _ns._state[e] * _ns._red_cost[e]) < 0) { |
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393 | _candidates.push_back(e); |
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394 | if (curr < min) { |
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395 | min = curr; |
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396 | _ns._in_edge = e; |
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397 | } |
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398 | if (int(_candidates.size()) == _list_length) break; |
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399 | } |
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400 | } |
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401 | if (int(_candidates.size()) < _list_length) { |
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402 | for (e = EdgeIt(_ns._graph); e != _next_edge; ++e) { |
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403 | if ((curr = _ns._state[e] * _ns._red_cost[e]) < 0) { |
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404 | _candidates.push_back(e); |
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405 | if (curr < min) { |
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406 | min = curr; |
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407 | _ns._in_edge = e; |
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408 | } |
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409 | if (int(_candidates.size()) == _list_length) break; |
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410 | } |
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411 | } |
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412 | } |
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413 | if (_candidates.size() == 0) return false; |
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414 | _minor_count = 1; |
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415 | _next_edge = ++e; |
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416 | return true; |
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417 | } |
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418 | }; //class CandidateListPivotRule |
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419 | |
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420 | private: |
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421 | |
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422 | // State constants for edges |
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423 | enum EdgeStateEnum { |
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424 | STATE_UPPER = -1, |
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425 | STATE_TREE = 0, |
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426 | STATE_LOWER = 1 |
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427 | }; |
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428 | |
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429 | // Constant for the combined pivot rule. |
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430 | static const int COMBINED_PIVOT_MAX_DEG = 5; |
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431 | |
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432 | private: |
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433 | |
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434 | // The directed graph the algorithm runs on |
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435 | SGraph _graph; |
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436 | // The original graph |
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437 | const Graph &_graph_ref; |
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438 | // The original lower bound map |
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439 | const LowerMap *_lower; |
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440 | // The capacity map |
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441 | SCapacityMap _capacity; |
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442 | // The cost map |
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443 | SCostMap _cost; |
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444 | // The supply map |
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445 | SSupplyMap _supply; |
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446 | bool _valid_supply; |
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447 | |
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448 | // Edge map of the current flow |
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449 | SCapacityMap _flow; |
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450 | // Node map of the current potentials |
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451 | SPotentialMap _potential; |
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452 | |
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453 | // The depth node map of the spanning tree structure |
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454 | IntNodeMap _depth; |
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455 | // The parent node map of the spanning tree structure |
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456 | NodeNodeMap _parent; |
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457 | // The pred_edge node map of the spanning tree structure |
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458 | EdgeNodeMap _pred_edge; |
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459 | // The thread node map of the spanning tree structure |
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460 | NodeNodeMap _thread; |
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461 | // The forward node map of the spanning tree structure |
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462 | BoolNodeMap _forward; |
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463 | // The state edge map |
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464 | IntEdgeMap _state; |
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465 | // The root node of the starting spanning tree |
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466 | Node _root; |
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467 | |
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468 | // The reduced cost map |
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469 | ReducedCostMap _red_cost; |
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470 | |
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471 | // Members for handling the original graph |
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472 | FlowMap _flow_result; |
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473 | PotentialMap _potential_result; |
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474 | NodeRefMap _node_ref; |
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475 | EdgeRefMap _edge_ref; |
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476 | |
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477 | // The entering edge of the current pivot iteration. |
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478 | Edge _in_edge; |
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479 | |
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480 | // Temporary nodes used in the current pivot iteration. |
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481 | Node join, u_in, v_in, u_out, v_out; |
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482 | Node right, first, second, last; |
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483 | Node stem, par_stem, new_stem; |
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484 | // The maximum augment amount along the found cycle in the current |
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485 | // pivot iteration. |
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486 | Capacity delta; |
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487 | |
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488 | public : |
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489 | |
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490 | /// \brief General constructor of the class (with lower bounds). |
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491 | /// |
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492 | /// General constructor of the class (with lower bounds). |
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493 | /// |
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494 | /// \param graph The directed graph the algorithm runs on. |
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495 | /// \param lower The lower bounds of the edges. |
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496 | /// \param capacity The capacities (upper bounds) of the edges. |
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497 | /// \param cost The cost (length) values of the edges. |
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498 | /// \param supply The supply values of the nodes (signed). |
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499 | NetworkSimplex( const Graph &graph, |
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500 | const LowerMap &lower, |
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501 | const CapacityMap &capacity, |
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502 | const CostMap &cost, |
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503 | const SupplyMap &supply ) : |
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504 | _graph(), _graph_ref(graph), _lower(&lower), _capacity(_graph), |
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505 | _cost(_graph), _supply(_graph), _flow(_graph), |
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506 | _potential(_graph), _depth(_graph), _parent(_graph), |
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507 | _pred_edge(_graph), _thread(_graph), _forward(_graph), |
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508 | _state(_graph), _red_cost(_graph, _cost, _potential), |
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509 | _flow_result(graph), _potential_result(graph), |
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510 | _node_ref(graph), _edge_ref(graph) |
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511 | { |
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512 | // Checking the sum of supply values |
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513 | Supply sum = 0; |
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514 | for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) |
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515 | sum += supply[n]; |
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516 | if (!(_valid_supply = sum == 0)) return; |
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517 | |
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518 | // Copying _graph_ref to _graph |
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519 | _graph.reserveNode(countNodes(_graph_ref) + 1); |
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520 | _graph.reserveEdge(countEdges(_graph_ref) + countNodes(_graph_ref)); |
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521 | copyGraph(_graph, _graph_ref) |
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522 | .edgeMap(_cost, cost) |
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523 | .nodeRef(_node_ref) |
---|
524 | .edgeRef(_edge_ref) |
---|
525 | .run(); |
---|
526 | |
---|
527 | // Removing non-zero lower bounds |
---|
528 | for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) { |
---|
529 | _capacity[_edge_ref[e]] = capacity[e] - lower[e]; |
---|
530 | } |
---|
531 | for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) { |
---|
532 | Supply s = supply[n]; |
---|
533 | for (typename Graph::InEdgeIt e(_graph_ref, n); e != INVALID; ++e) |
---|
534 | s += lower[e]; |
---|
535 | for (typename Graph::OutEdgeIt e(_graph_ref, n); e != INVALID; ++e) |
---|
536 | s -= lower[e]; |
---|
537 | _supply[_node_ref[n]] = s; |
---|
538 | } |
---|
539 | } |
---|
540 | |
---|
541 | /// \brief General constructor of the class (without lower bounds). |
---|
542 | /// |
---|
543 | /// General constructor of the class (without lower bounds). |
---|
544 | /// |
---|
545 | /// \param graph The directed graph the algorithm runs on. |
---|
546 | /// \param capacity The capacities (upper bounds) of the edges. |
---|
547 | /// \param cost The cost (length) values of the edges. |
---|
548 | /// \param supply The supply values of the nodes (signed). |
---|
549 | NetworkSimplex( const Graph &graph, |
---|
550 | const CapacityMap &capacity, |
---|
551 | const CostMap &cost, |
---|
552 | const SupplyMap &supply ) : |
---|
553 | _graph(), _graph_ref(graph), _lower(NULL), _capacity(_graph), |
---|
554 | _cost(_graph), _supply(_graph), _flow(_graph), |
---|
555 | _potential(_graph), _depth(_graph), _parent(_graph), |
---|
556 | _pred_edge(_graph), _thread(_graph), _forward(_graph), |
---|
557 | _state(_graph), _red_cost(_graph, _cost, _potential), |
---|
558 | _flow_result(graph), _potential_result(graph), |
---|
559 | _node_ref(graph), _edge_ref(graph) |
---|
560 | { |
---|
561 | // Checking the sum of supply values |
---|
562 | Supply sum = 0; |
---|
563 | for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) |
---|
564 | sum += supply[n]; |
---|
565 | if (!(_valid_supply = sum == 0)) return; |
---|
566 | |
---|
567 | // Copying _graph_ref to graph |
---|
568 | copyGraph(_graph, _graph_ref) |
---|
569 | .edgeMap(_capacity, capacity) |
---|
570 | .edgeMap(_cost, cost) |
---|
571 | .nodeMap(_supply, supply) |
---|
572 | .nodeRef(_node_ref) |
---|
573 | .edgeRef(_edge_ref) |
---|
574 | .run(); |
---|
575 | } |
---|
576 | |
---|
577 | /// \brief Simple constructor of the class (with lower bounds). |
---|
578 | /// |
---|
579 | /// Simple constructor of the class (with lower bounds). |
---|
580 | /// |
---|
581 | /// \param graph The directed graph the algorithm runs on. |
---|
582 | /// \param lower The lower bounds of the edges. |
---|
583 | /// \param capacity The capacities (upper bounds) of the edges. |
---|
584 | /// \param cost The cost (length) values of the edges. |
---|
585 | /// \param s The source node. |
---|
586 | /// \param t The target node. |
---|
587 | /// \param flow_value The required amount of flow from node \c s |
---|
588 | /// to node \c t (i.e. the supply of \c s and the demand of \c t). |
---|
589 | NetworkSimplex( const Graph &graph, |
---|
590 | const LowerMap &lower, |
---|
591 | const CapacityMap &capacity, |
---|
592 | const CostMap &cost, |
---|
593 | typename Graph::Node s, |
---|
594 | typename Graph::Node t, |
---|
595 | typename SupplyMap::Value flow_value ) : |
---|
596 | _graph(), _graph_ref(graph), _lower(&lower), _capacity(_graph), |
---|
597 | _cost(_graph), _supply(_graph), _flow(_graph), |
---|
598 | _potential(_graph), _depth(_graph), _parent(_graph), |
---|
599 | _pred_edge(_graph), _thread(_graph), _forward(_graph), |
---|
600 | _state(_graph), _red_cost(_graph, _cost, _potential), |
---|
601 | _flow_result(graph), _potential_result(graph), |
---|
602 | _node_ref(graph), _edge_ref(graph) |
---|
603 | { |
---|
604 | // Copying _graph_ref to graph |
---|
605 | copyGraph(_graph, _graph_ref) |
---|
606 | .edgeMap(_cost, cost) |
---|
607 | .nodeRef(_node_ref) |
---|
608 | .edgeRef(_edge_ref) |
---|
609 | .run(); |
---|
610 | |
---|
611 | // Removing non-zero lower bounds |
---|
612 | for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) { |
---|
613 | _capacity[_edge_ref[e]] = capacity[e] - lower[e]; |
---|
614 | } |
---|
615 | for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) { |
---|
616 | Supply sum = 0; |
---|
617 | if (n == s) sum = flow_value; |
---|
618 | if (n == t) sum = -flow_value; |
---|
619 | for (typename Graph::InEdgeIt e(_graph_ref, n); e != INVALID; ++e) |
---|
620 | sum += lower[e]; |
---|
621 | for (typename Graph::OutEdgeIt e(_graph_ref, n); e != INVALID; ++e) |
---|
622 | sum -= lower[e]; |
---|
623 | _supply[_node_ref[n]] = sum; |
---|
624 | } |
---|
625 | _valid_supply = true; |
---|
626 | } |
---|
627 | |
---|
628 | /// \brief Simple constructor of the class (without lower bounds). |
---|
629 | /// |
---|
630 | /// Simple constructor of the class (without lower bounds). |
---|
631 | /// |
---|
632 | /// \param graph The directed graph the algorithm runs on. |
---|
633 | /// \param capacity The capacities (upper bounds) of the edges. |
---|
634 | /// \param cost The cost (length) values of the edges. |
---|
635 | /// \param s The source node. |
---|
636 | /// \param t The target node. |
---|
637 | /// \param flow_value The required amount of flow from node \c s |
---|
638 | /// to node \c t (i.e. the supply of \c s and the demand of \c t). |
---|
639 | NetworkSimplex( const Graph &graph, |
---|
640 | const CapacityMap &capacity, |
---|
641 | const CostMap &cost, |
---|
642 | typename Graph::Node s, |
---|
643 | typename Graph::Node t, |
---|
644 | typename SupplyMap::Value flow_value ) : |
---|
645 | _graph(), _graph_ref(graph), _lower(NULL), _capacity(_graph), |
---|
646 | _cost(_graph), _supply(_graph, 0), _flow(_graph), |
---|
647 | _potential(_graph), _depth(_graph), _parent(_graph), |
---|
648 | _pred_edge(_graph), _thread(_graph), _forward(_graph), |
---|
649 | _state(_graph), _red_cost(_graph, _cost, _potential), |
---|
650 | _flow_result(graph), _potential_result(graph), |
---|
651 | _node_ref(graph), _edge_ref(graph) |
---|
652 | { |
---|
653 | // Copying _graph_ref to graph |
---|
654 | copyGraph(_graph, _graph_ref) |
---|
655 | .edgeMap(_capacity, capacity) |
---|
656 | .edgeMap(_cost, cost) |
---|
657 | .nodeRef(_node_ref) |
---|
658 | .edgeRef(_edge_ref) |
---|
659 | .run(); |
---|
660 | _supply[_node_ref[s]] = flow_value; |
---|
661 | _supply[_node_ref[t]] = -flow_value; |
---|
662 | _valid_supply = true; |
---|
663 | } |
---|
664 | |
---|
665 | /// \brief Runs the algorithm. |
---|
666 | /// |
---|
667 | /// Runs the algorithm. |
---|
668 | /// |
---|
669 | /// \param pivot_rule The pivot rule that is used during the |
---|
670 | /// algorithm. |
---|
671 | /// |
---|
672 | /// The available pivot rules: |
---|
673 | /// |
---|
674 | /// - FIRST_ELIGIBLE_PIVOT The next eligible edge is selected in |
---|
675 | /// a wraparound fashion in every iteration |
---|
676 | /// (\ref FirstEligiblePivotRule). |
---|
677 | /// |
---|
678 | /// - BEST_ELIGIBLE_PIVOT The best eligible edge is selected in |
---|
679 | /// every iteration (\ref BestEligiblePivotRule). |
---|
680 | /// |
---|
681 | /// - BLOCK_SEARCH_PIVOT A specified number of edges are examined in |
---|
682 | /// every iteration in a wraparound fashion and the best eligible |
---|
683 | /// edge is selected from this block (\ref BlockSearchPivotRule). |
---|
684 | /// |
---|
685 | /// - LIMITED_SEARCH_PIVOT A specified number of eligible edges are |
---|
686 | /// examined in every iteration in a wraparound fashion and the best |
---|
687 | /// one is selected from them (\ref LimitedSearchPivotRule). |
---|
688 | /// |
---|
689 | /// - CANDIDATE_LIST_PIVOT In major iterations a candidate list is |
---|
690 | /// built from eligible edges and it is used for edge selection in |
---|
691 | /// the following minor iterations (\ref CandidateListPivotRule). |
---|
692 | /// |
---|
693 | /// - COMBINED_PIVOT This is a combined version of the two fastest |
---|
694 | /// pivot rules. |
---|
695 | /// For rather sparse graphs \ref LimitedSearchPivotRule |
---|
696 | /// "Limited Search" implementation is used, otherwise |
---|
697 | /// \ref BlockSearchPivotRule "Block Search" pivot rule is used. |
---|
698 | /// According to our benchmark tests this combined method is the |
---|
699 | /// most efficient. |
---|
700 | /// |
---|
701 | /// \return \c true if a feasible flow can be found. |
---|
702 | bool run(PivotRuleEnum pivot_rule = COMBINED_PIVOT) { |
---|
703 | return init() && start(pivot_rule); |
---|
704 | } |
---|
705 | |
---|
706 | /// \brief Returns a const reference to the edge map storing the |
---|
707 | /// found flow. |
---|
708 | /// |
---|
709 | /// Returns a const reference to the edge map storing the found flow. |
---|
710 | /// |
---|
711 | /// \pre \ref run() must be called before using this function. |
---|
712 | const FlowMap& flowMap() const { |
---|
713 | return _flow_result; |
---|
714 | } |
---|
715 | |
---|
716 | /// \brief Returns a const reference to the node map storing the |
---|
717 | /// found potentials (the dual solution). |
---|
718 | /// |
---|
719 | /// Returns a const reference to the node map storing the found |
---|
720 | /// potentials (the dual solution). |
---|
721 | /// |
---|
722 | /// \pre \ref run() must be called before using this function. |
---|
723 | const PotentialMap& potentialMap() const { |
---|
724 | return _potential_result; |
---|
725 | } |
---|
726 | |
---|
727 | /// \brief Returns the total cost of the found flow. |
---|
728 | /// |
---|
729 | /// Returns the total cost of the found flow. The complexity of the |
---|
730 | /// function is \f$ O(e) \f$. |
---|
731 | /// |
---|
732 | /// \pre \ref run() must be called before using this function. |
---|
733 | Cost totalCost() const { |
---|
734 | Cost c = 0; |
---|
735 | for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) |
---|
736 | c += _flow_result[e] * _cost[_edge_ref[e]]; |
---|
737 | return c; |
---|
738 | } |
---|
739 | |
---|
740 | private: |
---|
741 | |
---|
742 | /// \brief Extends the underlaying graph and initializes all the |
---|
743 | /// node and edge maps. |
---|
744 | bool init() { |
---|
745 | if (!_valid_supply) return false; |
---|
746 | |
---|
747 | // Initializing state and flow maps |
---|
748 | for (EdgeIt e(_graph); e != INVALID; ++e) { |
---|
749 | _flow[e] = 0; |
---|
750 | _state[e] = STATE_LOWER; |
---|
751 | } |
---|
752 | |
---|
753 | // Adding an artificial root node to the graph |
---|
754 | _root = _graph.addNode(); |
---|
755 | _parent[_root] = INVALID; |
---|
756 | _pred_edge[_root] = INVALID; |
---|
757 | _depth[_root] = 0; |
---|
758 | _supply[_root] = 0; |
---|
759 | _potential[_root] = 0; |
---|
760 | |
---|
761 | // Adding artificial edges to the graph and initializing the node |
---|
762 | // maps of the spanning tree data structure |
---|
763 | Node last = _root; |
---|
764 | Edge e; |
---|
765 | Cost max_cost = std::numeric_limits<Cost>::max() / 4; |
---|
766 | for (NodeIt u(_graph); u != INVALID; ++u) { |
---|
767 | if (u == _root) continue; |
---|
768 | _thread[last] = u; |
---|
769 | last = u; |
---|
770 | _parent[u] = _root; |
---|
771 | _depth[u] = 1; |
---|
772 | if (_supply[u] >= 0) { |
---|
773 | e = _graph.addEdge(u, _root); |
---|
774 | _flow[e] = _supply[u]; |
---|
775 | _forward[u] = true; |
---|
776 | _potential[u] = -max_cost; |
---|
777 | } else { |
---|
778 | e = _graph.addEdge(_root, u); |
---|
779 | _flow[e] = -_supply[u]; |
---|
780 | _forward[u] = false; |
---|
781 | _potential[u] = max_cost; |
---|
782 | } |
---|
783 | _cost[e] = max_cost; |
---|
784 | _capacity[e] = std::numeric_limits<Capacity>::max(); |
---|
785 | _state[e] = STATE_TREE; |
---|
786 | _pred_edge[u] = e; |
---|
787 | } |
---|
788 | _thread[last] = _root; |
---|
789 | |
---|
790 | return true; |
---|
791 | } |
---|
792 | |
---|
793 | /// Finds the join node. |
---|
794 | Node findJoinNode() { |
---|
795 | Node u = _graph.source(_in_edge); |
---|
796 | Node v = _graph.target(_in_edge); |
---|
797 | while (u != v) { |
---|
798 | if (_depth[u] == _depth[v]) { |
---|
799 | u = _parent[u]; |
---|
800 | v = _parent[v]; |
---|
801 | } |
---|
802 | else if (_depth[u] > _depth[v]) u = _parent[u]; |
---|
803 | else v = _parent[v]; |
---|
804 | } |
---|
805 | return u; |
---|
806 | } |
---|
807 | |
---|
808 | /// \brief Finds the leaving edge of the cycle. Returns \c true if |
---|
809 | /// the leaving edge is not the same as the entering edge. |
---|
810 | bool findLeavingEdge() { |
---|
811 | // Initializing first and second nodes according to the direction |
---|
812 | // of the cycle |
---|
813 | if (_state[_in_edge] == STATE_LOWER) { |
---|
814 | first = _graph.source(_in_edge); |
---|
815 | second = _graph.target(_in_edge); |
---|
816 | } else { |
---|
817 | first = _graph.target(_in_edge); |
---|
818 | second = _graph.source(_in_edge); |
---|
819 | } |
---|
820 | delta = _capacity[_in_edge]; |
---|
821 | bool result = false; |
---|
822 | Capacity d; |
---|
823 | Edge e; |
---|
824 | |
---|
825 | // Searching the cycle along the path form the first node to the |
---|
826 | // root node |
---|
827 | for (Node u = first; u != join; u = _parent[u]) { |
---|
828 | e = _pred_edge[u]; |
---|
829 | d = _forward[u] ? _flow[e] : _capacity[e] - _flow[e]; |
---|
830 | if (d < delta) { |
---|
831 | delta = d; |
---|
832 | u_out = u; |
---|
833 | u_in = first; |
---|
834 | v_in = second; |
---|
835 | result = true; |
---|
836 | } |
---|
837 | } |
---|
838 | // Searching the cycle along the path form the second node to the |
---|
839 | // root node |
---|
840 | for (Node u = second; u != join; u = _parent[u]) { |
---|
841 | e = _pred_edge[u]; |
---|
842 | d = _forward[u] ? _capacity[e] - _flow[e] : _flow[e]; |
---|
843 | if (d <= delta) { |
---|
844 | delta = d; |
---|
845 | u_out = u; |
---|
846 | u_in = second; |
---|
847 | v_in = first; |
---|
848 | result = true; |
---|
849 | } |
---|
850 | } |
---|
851 | return result; |
---|
852 | } |
---|
853 | |
---|
854 | /// Changes \c flow and \c state edge maps. |
---|
855 | void changeFlows(bool change) { |
---|
856 | // Augmenting along the cycle |
---|
857 | if (delta > 0) { |
---|
858 | Capacity val = _state[_in_edge] * delta; |
---|
859 | _flow[_in_edge] += val; |
---|
860 | for (Node u = _graph.source(_in_edge); u != join; u = _parent[u]) { |
---|
861 | _flow[_pred_edge[u]] += _forward[u] ? -val : val; |
---|
862 | } |
---|
863 | for (Node u = _graph.target(_in_edge); u != join; u = _parent[u]) { |
---|
864 | _flow[_pred_edge[u]] += _forward[u] ? val : -val; |
---|
865 | } |
---|
866 | } |
---|
867 | // Updating the state of the entering and leaving edges |
---|
868 | if (change) { |
---|
869 | _state[_in_edge] = STATE_TREE; |
---|
870 | _state[_pred_edge[u_out]] = |
---|
871 | (_flow[_pred_edge[u_out]] == 0) ? STATE_LOWER : STATE_UPPER; |
---|
872 | } else { |
---|
873 | _state[_in_edge] = -_state[_in_edge]; |
---|
874 | } |
---|
875 | } |
---|
876 | |
---|
877 | /// Updates \c thread and \c parent node maps. |
---|
878 | void updateThreadParent() { |
---|
879 | Node u; |
---|
880 | v_out = _parent[u_out]; |
---|
881 | |
---|
882 | // Handling the case when join and v_out coincide |
---|
883 | bool par_first = false; |
---|
884 | if (join == v_out) { |
---|
885 | for (u = join; u != u_in && u != v_in; u = _thread[u]) ; |
---|
886 | if (u == v_in) { |
---|
887 | par_first = true; |
---|
888 | while (_thread[u] != u_out) u = _thread[u]; |
---|
889 | first = u; |
---|
890 | } |
---|
891 | } |
---|
892 | |
---|
893 | // Finding the last successor of u_in (u) and the node after it |
---|
894 | // (right) according to the thread index |
---|
895 | for (u = u_in; _depth[_thread[u]] > _depth[u_in]; u = _thread[u]) ; |
---|
896 | right = _thread[u]; |
---|
897 | if (_thread[v_in] == u_out) { |
---|
898 | for (last = u; _depth[last] > _depth[u_out]; last = _thread[last]) ; |
---|
899 | if (last == u_out) last = _thread[last]; |
---|
900 | } |
---|
901 | else last = _thread[v_in]; |
---|
902 | |
---|
903 | // Updating stem nodes |
---|
904 | _thread[v_in] = stem = u_in; |
---|
905 | par_stem = v_in; |
---|
906 | while (stem != u_out) { |
---|
907 | _thread[u] = new_stem = _parent[stem]; |
---|
908 | |
---|
909 | // Finding the node just before the stem node (u) according to |
---|
910 | // the original thread index |
---|
911 | for (u = new_stem; _thread[u] != stem; u = _thread[u]) ; |
---|
912 | _thread[u] = right; |
---|
913 | |
---|
914 | // Changing the parent node of stem and shifting stem and |
---|
915 | // par_stem nodes |
---|
916 | _parent[stem] = par_stem; |
---|
917 | par_stem = stem; |
---|
918 | stem = new_stem; |
---|
919 | |
---|
920 | // Finding the last successor of stem (u) and the node after it |
---|
921 | // (right) according to the thread index |
---|
922 | for (u = stem; _depth[_thread[u]] > _depth[stem]; u = _thread[u]) ; |
---|
923 | right = _thread[u]; |
---|
924 | } |
---|
925 | _parent[u_out] = par_stem; |
---|
926 | _thread[u] = last; |
---|
927 | |
---|
928 | if (join == v_out && par_first) { |
---|
929 | if (first != v_in) _thread[first] = right; |
---|
930 | } else { |
---|
931 | for (u = v_out; _thread[u] != u_out; u = _thread[u]) ; |
---|
932 | _thread[u] = right; |
---|
933 | } |
---|
934 | } |
---|
935 | |
---|
936 | /// Updates \c pred_edge and \c forward node maps. |
---|
937 | void updatePredEdge() { |
---|
938 | Node u = u_out, v; |
---|
939 | while (u != u_in) { |
---|
940 | v = _parent[u]; |
---|
941 | _pred_edge[u] = _pred_edge[v]; |
---|
942 | _forward[u] = !_forward[v]; |
---|
943 | u = v; |
---|
944 | } |
---|
945 | _pred_edge[u_in] = _in_edge; |
---|
946 | _forward[u_in] = (u_in == _graph.source(_in_edge)); |
---|
947 | } |
---|
948 | |
---|
949 | /// Updates \c depth and \c potential node maps. |
---|
950 | void updateDepthPotential() { |
---|
951 | _depth[u_in] = _depth[v_in] + 1; |
---|
952 | _potential[u_in] = _forward[u_in] ? |
---|
953 | _potential[v_in] - _cost[_pred_edge[u_in]] : |
---|
954 | _potential[v_in] + _cost[_pred_edge[u_in]]; |
---|
955 | |
---|
956 | Node u = _thread[u_in], v; |
---|
957 | while (true) { |
---|
958 | v = _parent[u]; |
---|
959 | if (v == INVALID) break; |
---|
960 | _depth[u] = _depth[v] + 1; |
---|
961 | _potential[u] = _forward[u] ? |
---|
962 | _potential[v] - _cost[_pred_edge[u]] : |
---|
963 | _potential[v] + _cost[_pred_edge[u]]; |
---|
964 | if (_depth[u] <= _depth[v_in]) break; |
---|
965 | u = _thread[u]; |
---|
966 | } |
---|
967 | } |
---|
968 | |
---|
969 | /// Executes the algorithm. |
---|
970 | bool start(PivotRuleEnum pivot_rule) { |
---|
971 | switch (pivot_rule) { |
---|
972 | case FIRST_ELIGIBLE_PIVOT: |
---|
973 | return start<FirstEligiblePivotRule>(); |
---|
974 | case BEST_ELIGIBLE_PIVOT: |
---|
975 | return start<BestEligiblePivotRule>(); |
---|
976 | case BLOCK_SEARCH_PIVOT: |
---|
977 | return start<BlockSearchPivotRule>(); |
---|
978 | case LIMITED_SEARCH_PIVOT: |
---|
979 | return start<LimitedSearchPivotRule>(); |
---|
980 | case CANDIDATE_LIST_PIVOT: |
---|
981 | return start<CandidateListPivotRule>(); |
---|
982 | case COMBINED_PIVOT: |
---|
983 | if ( countEdges(_graph) / countNodes(_graph) <= |
---|
984 | COMBINED_PIVOT_MAX_DEG ) |
---|
985 | return start<LimitedSearchPivotRule>(); |
---|
986 | else |
---|
987 | return start<BlockSearchPivotRule>(); |
---|
988 | } |
---|
989 | return false; |
---|
990 | } |
---|
991 | |
---|
992 | template<class PivotRuleImplementation> |
---|
993 | bool start() { |
---|
994 | PivotRuleImplementation pivot(*this); |
---|
995 | |
---|
996 | // Executing the network simplex algorithm |
---|
997 | while (pivot.findEnteringEdge()) { |
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998 | join = findJoinNode(); |
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999 | bool change = findLeavingEdge(); |
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1000 | changeFlows(change); |
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1001 | if (change) { |
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1002 | updateThreadParent(); |
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1003 | updatePredEdge(); |
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1004 | updateDepthPotential(); |
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1005 | } |
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1006 | } |
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1007 | |
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1008 | // Checking if the flow amount equals zero on all the artificial |
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1009 | // edges |
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1010 | for (InEdgeIt e(_graph, _root); e != INVALID; ++e) |
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1011 | if (_flow[e] > 0) return false; |
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1012 | for (OutEdgeIt e(_graph, _root); e != INVALID; ++e) |
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1013 | if (_flow[e] > 0) return false; |
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1014 | |
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1015 | // Copying flow values to _flow_result |
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1016 | if (_lower) { |
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1017 | for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) |
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1018 | _flow_result[e] = (*_lower)[e] + _flow[_edge_ref[e]]; |
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1019 | } else { |
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1020 | for (typename Graph::EdgeIt e(_graph_ref); e != INVALID; ++e) |
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1021 | _flow_result[e] = _flow[_edge_ref[e]]; |
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1022 | } |
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1023 | // Copying potential values to _potential_result |
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1024 | for (typename Graph::NodeIt n(_graph_ref); n != INVALID; ++n) |
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1025 | _potential_result[n] = _potential[_node_ref[n]]; |
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1026 | |
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1027 | return true; |
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1028 | } |
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1029 | |
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1030 | }; //class NetworkSimplex |
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1031 | |
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1032 | ///@} |
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1033 | |
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1034 | } //namespace lemon |
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1035 | |
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1036 | #endif //LEMON_NETWORK_SIMPLEX_H |
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