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_CYCLE_CANCELING_H |
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20 | #define LEMON_CYCLE_CANCELING_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 Cycle-canceling 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 <lemon/adaptors.h> |
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29 | #include <lemon/path.h> |
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30 | |
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31 | #include <lemon/circulation.h> |
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32 | #include <lemon/bellman_ford.h> |
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33 | #include <lemon/howard.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 a cycle-canceling algorithm for |
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41 | /// finding a minimum cost flow. |
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42 | /// |
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43 | /// \ref CycleCanceling implements a cycle-canceling algorithm for |
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44 | /// finding a minimum cost flow. |
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45 | /// |
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46 | /// \tparam Digraph The digraph 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 | /// - Arc 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 | /// - The value types of the maps should be convertible to each other. |
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56 | /// - \c CostMap::Value must be signed type. |
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57 | /// |
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58 | /// \note By default the \ref BellmanFord "Bellman-Ford" algorithm is |
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59 | /// used for negative cycle detection with limited iteration number. |
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60 | /// However \ref CycleCanceling also provides the "Minimum Mean |
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61 | /// Cycle-Canceling" algorithm, which is \e strongly \e polynomial, |
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62 | /// but rather slower in practice. |
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63 | /// To use this version of the algorithm, call \ref run() with \c true |
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64 | /// parameter. |
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65 | /// |
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66 | /// \author Peter Kovacs |
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67 | template < typename Digraph, |
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68 | typename LowerMap = typename Digraph::template ArcMap<int>, |
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69 | typename CapacityMap = typename Digraph::template ArcMap<int>, |
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70 | typename CostMap = typename Digraph::template ArcMap<int>, |
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71 | typename SupplyMap = typename Digraph::template NodeMap<int> > |
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72 | class CycleCanceling |
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73 | { |
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74 | TEMPLATE_DIGRAPH_TYPEDEFS(Digraph); |
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75 | |
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76 | typedef typename CapacityMap::Value Capacity; |
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77 | typedef typename CostMap::Value Cost; |
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78 | typedef typename SupplyMap::Value Supply; |
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79 | typedef typename Digraph::template ArcMap<Capacity> CapacityArcMap; |
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80 | typedef typename Digraph::template NodeMap<Supply> SupplyNodeMap; |
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81 | |
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82 | typedef ResidualDigraph< const Digraph, |
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83 | CapacityArcMap, CapacityArcMap > ResDigraph; |
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84 | typedef typename ResDigraph::Node ResNode; |
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85 | typedef typename ResDigraph::NodeIt ResNodeIt; |
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86 | typedef typename ResDigraph::Arc ResArc; |
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87 | typedef typename ResDigraph::ArcIt ResArcIt; |
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88 | |
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89 | public: |
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90 | |
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91 | /// The type of the flow map. |
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92 | typedef typename Digraph::template ArcMap<Capacity> FlowMap; |
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93 | /// The type of the potential map. |
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94 | typedef typename Digraph::template NodeMap<Cost> PotentialMap; |
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95 | |
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96 | private: |
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97 | |
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98 | /// \brief Map adaptor class for handling residual arc costs. |
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99 | /// |
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100 | /// Map adaptor class for handling residual arc costs. |
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101 | class ResidualCostMap : public MapBase<ResArc, Cost> |
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102 | { |
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103 | private: |
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104 | |
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105 | const CostMap &_cost_map; |
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106 | |
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107 | public: |
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108 | |
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109 | ///\e |
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110 | ResidualCostMap(const CostMap &cost_map) : _cost_map(cost_map) {} |
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111 | |
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112 | ///\e |
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113 | Cost operator[](const ResArc &e) const { |
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114 | return ResDigraph::forward(e) ? _cost_map[e] : -_cost_map[e]; |
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115 | } |
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116 | |
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117 | }; //class ResidualCostMap |
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118 | |
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119 | private: |
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120 | |
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121 | // The maximum number of iterations for the first execution of the |
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122 | // Bellman-Ford algorithm. It should be at least 2. |
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123 | static const int BF_FIRST_LIMIT = 2; |
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124 | // The iteration limit for the Bellman-Ford algorithm is multiplied |
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125 | // by BF_LIMIT_FACTOR/100 in every round. |
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126 | static const int BF_LIMIT_FACTOR = 150; |
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127 | |
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128 | private: |
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129 | |
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130 | // The digraph the algorithm runs on |
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131 | const Digraph &_graph; |
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132 | // The original lower bound map |
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133 | const LowerMap *_lower; |
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134 | // The modified capacity map |
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135 | CapacityArcMap _capacity; |
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136 | // The original cost map |
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137 | const CostMap &_cost; |
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138 | // The modified supply map |
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139 | SupplyNodeMap _supply; |
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140 | bool _valid_supply; |
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141 | |
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142 | // Arc map of the current flow |
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143 | FlowMap *_flow; |
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144 | bool _local_flow; |
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145 | // Node map of the current potentials |
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146 | PotentialMap *_potential; |
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147 | bool _local_potential; |
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148 | |
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149 | // The residual digraph |
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150 | ResDigraph *_res_graph; |
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151 | // The residual cost map |
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152 | ResidualCostMap _res_cost; |
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153 | |
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154 | public: |
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155 | |
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156 | /// \brief General constructor (with lower bounds). |
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157 | /// |
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158 | /// General constructor (with lower bounds). |
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159 | /// |
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160 | /// \param digraph The digraph the algorithm runs on. |
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161 | /// \param lower The lower bounds of the arcs. |
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162 | /// \param capacity The capacities (upper bounds) of the arcs. |
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163 | /// \param cost The cost (length) values of the arcs. |
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164 | /// \param supply The supply values of the nodes (signed). |
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165 | CycleCanceling( const Digraph &digraph, |
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166 | const LowerMap &lower, |
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167 | const CapacityMap &capacity, |
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168 | const CostMap &cost, |
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169 | const SupplyMap &supply ) : |
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170 | _graph(digraph), _lower(&lower), _capacity(digraph), _cost(cost), |
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171 | _supply(digraph), _flow(NULL), _local_flow(false), |
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172 | _potential(NULL), _local_potential(false), |
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173 | _res_graph(NULL), _res_cost(_cost) |
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174 | { |
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175 | // Check the sum of supply values |
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176 | Supply sum = 0; |
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177 | for (NodeIt n(_graph); n != INVALID; ++n) { |
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178 | _supply[n] = supply[n]; |
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179 | sum += _supply[n]; |
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180 | } |
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181 | _valid_supply = sum == 0; |
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182 | |
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183 | // Remove non-zero lower bounds |
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184 | for (ArcIt e(_graph); e != INVALID; ++e) { |
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185 | _capacity[e] = capacity[e]; |
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186 | if (lower[e] != 0) { |
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187 | _capacity[e] -= lower[e]; |
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188 | _supply[_graph.source(e)] -= lower[e]; |
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189 | _supply[_graph.target(e)] += lower[e]; |
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190 | } |
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191 | } |
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192 | } |
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193 | /* |
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194 | /// \brief General constructor (without lower bounds). |
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195 | /// |
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196 | /// General constructor (without lower bounds). |
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197 | /// |
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198 | /// \param digraph The digraph the algorithm runs on. |
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199 | /// \param capacity The capacities (upper bounds) of the arcs. |
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200 | /// \param cost The cost (length) values of the arcs. |
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201 | /// \param supply The supply values of the nodes (signed). |
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202 | CycleCanceling( const Digraph &digraph, |
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203 | const CapacityMap &capacity, |
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204 | const CostMap &cost, |
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205 | const SupplyMap &supply ) : |
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206 | _graph(digraph), _lower(NULL), _capacity(capacity), _cost(cost), |
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207 | _supply(supply), _flow(NULL), _local_flow(false), |
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208 | _potential(NULL), _local_potential(false), _res_graph(NULL), |
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209 | _res_cost(_cost) |
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210 | { |
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211 | // Check the sum of supply values |
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212 | Supply sum = 0; |
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213 | for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n]; |
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214 | _valid_supply = sum == 0; |
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215 | } |
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216 | |
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217 | /// \brief Simple constructor (with lower bounds). |
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218 | /// |
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219 | /// Simple constructor (with lower bounds). |
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220 | /// |
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221 | /// \param digraph The digraph the algorithm runs on. |
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222 | /// \param lower The lower bounds of the arcs. |
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223 | /// \param capacity The capacities (upper bounds) of the arcs. |
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224 | /// \param cost The cost (length) values of the arcs. |
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225 | /// \param s The source node. |
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226 | /// \param t The target node. |
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227 | /// \param flow_value The required amount of flow from node \c s |
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228 | /// to node \c t (i.e. the supply of \c s and the demand of \c t). |
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229 | CycleCanceling( const Digraph &digraph, |
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230 | const LowerMap &lower, |
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231 | const CapacityMap &capacity, |
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232 | const CostMap &cost, |
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233 | Node s, Node t, |
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234 | Supply flow_value ) : |
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235 | _graph(digraph), _lower(&lower), _capacity(capacity), _cost(cost), |
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236 | _supply(digraph, 0), _flow(NULL), _local_flow(false), |
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237 | _potential(NULL), _local_potential(false), _res_graph(NULL), |
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238 | _res_cost(_cost) |
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239 | { |
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240 | // Remove non-zero lower bounds |
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241 | _supply[s] = flow_value; |
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242 | _supply[t] = -flow_value; |
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243 | for (ArcIt e(_graph); e != INVALID; ++e) { |
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244 | if (lower[e] != 0) { |
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245 | _capacity[e] -= lower[e]; |
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246 | _supply[_graph.source(e)] -= lower[e]; |
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247 | _supply[_graph.target(e)] += lower[e]; |
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248 | } |
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249 | } |
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250 | _valid_supply = true; |
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251 | } |
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252 | |
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253 | /// \brief Simple constructor (without lower bounds). |
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254 | /// |
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255 | /// Simple constructor (without lower bounds). |
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256 | /// |
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257 | /// \param digraph The digraph the algorithm runs on. |
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258 | /// \param capacity The capacities (upper bounds) of the arcs. |
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259 | /// \param cost The cost (length) values of the arcs. |
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260 | /// \param s The source node. |
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261 | /// \param t The target node. |
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262 | /// \param flow_value The required amount of flow from node \c s |
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263 | /// to node \c t (i.e. the supply of \c s and the demand of \c t). |
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264 | CycleCanceling( const Digraph &digraph, |
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265 | const CapacityMap &capacity, |
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266 | const CostMap &cost, |
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267 | Node s, Node t, |
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268 | Supply flow_value ) : |
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269 | _graph(digraph), _lower(NULL), _capacity(capacity), _cost(cost), |
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270 | _supply(digraph, 0), _flow(NULL), _local_flow(false), |
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271 | _potential(NULL), _local_potential(false), _res_graph(NULL), |
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272 | _res_cost(_cost) |
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273 | { |
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274 | _supply[s] = flow_value; |
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275 | _supply[t] = -flow_value; |
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276 | _valid_supply = true; |
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277 | } |
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278 | */ |
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279 | /// Destructor. |
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280 | ~CycleCanceling() { |
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281 | if (_local_flow) delete _flow; |
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282 | if (_local_potential) delete _potential; |
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283 | delete _res_graph; |
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284 | } |
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285 | |
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286 | /// \brief Set the flow map. |
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287 | /// |
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288 | /// Set the flow map. |
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289 | /// |
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290 | /// \return \c (*this) |
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291 | CycleCanceling& flowMap(FlowMap &map) { |
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292 | if (_local_flow) { |
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293 | delete _flow; |
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294 | _local_flow = false; |
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295 | } |
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296 | _flow = ↦ |
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297 | return *this; |
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298 | } |
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299 | |
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300 | /// \brief Set the potential map. |
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301 | /// |
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302 | /// Set the potential map. |
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303 | /// |
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304 | /// \return \c (*this) |
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305 | CycleCanceling& potentialMap(PotentialMap &map) { |
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306 | if (_local_potential) { |
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307 | delete _potential; |
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308 | _local_potential = false; |
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309 | } |
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310 | _potential = ↦ |
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311 | return *this; |
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312 | } |
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313 | |
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314 | /// \name Execution control |
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315 | |
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316 | /// @{ |
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317 | |
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318 | /// \brief Run the algorithm. |
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319 | /// |
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320 | /// Run the algorithm. |
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321 | /// |
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322 | /// \param min_mean_cc Set this parameter to \c true to run the |
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323 | /// "Minimum Mean Cycle-Canceling" algorithm, which is strongly |
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324 | /// polynomial, but rather slower in practice. |
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325 | /// |
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326 | /// \return \c true if a feasible flow can be found. |
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327 | bool run(bool min_mean_cc = false) { |
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328 | return init() && start(min_mean_cc); |
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329 | } |
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330 | |
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331 | /// @} |
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332 | |
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333 | /// \name Query Functions |
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334 | /// The result of the algorithm can be obtained using these |
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335 | /// functions.\n |
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336 | /// \ref lemon::CycleCanceling::run() "run()" must be called before |
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337 | /// using them. |
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338 | |
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339 | /// @{ |
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340 | |
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341 | /// \brief Return a const reference to the arc map storing the |
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342 | /// found flow. |
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343 | /// |
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344 | /// Return a const reference to the arc map storing the found flow. |
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345 | /// |
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346 | /// \pre \ref run() must be called before using this function. |
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347 | const FlowMap& flowMap() const { |
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348 | return *_flow; |
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349 | } |
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350 | |
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351 | /// \brief Return a const reference to the node map storing the |
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352 | /// found potentials (the dual solution). |
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353 | /// |
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354 | /// Return a const reference to the node map storing the found |
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355 | /// potentials (the dual solution). |
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356 | /// |
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357 | /// \pre \ref run() must be called before using this function. |
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358 | const PotentialMap& potentialMap() const { |
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359 | return *_potential; |
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360 | } |
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361 | |
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362 | /// \brief Return the flow on the given arc. |
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363 | /// |
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364 | /// Return the flow on the given arc. |
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365 | /// |
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366 | /// \pre \ref run() must be called before using this function. |
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367 | Capacity flow(const Arc& arc) const { |
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368 | return (*_flow)[arc]; |
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369 | } |
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370 | |
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371 | /// \brief Return the potential of the given node. |
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372 | /// |
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373 | /// Return the potential of the given node. |
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374 | /// |
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375 | /// \pre \ref run() must be called before using this function. |
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376 | Cost potential(const Node& node) const { |
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377 | return (*_potential)[node]; |
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378 | } |
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379 | |
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380 | /// \brief Return the total cost of the found flow. |
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381 | /// |
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382 | /// Return the total cost of the found flow. The complexity of the |
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383 | /// function is \f$ O(e) \f$. |
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384 | /// |
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385 | /// \pre \ref run() must be called before using this function. |
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386 | Cost totalCost() const { |
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387 | Cost c = 0; |
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388 | for (ArcIt e(_graph); e != INVALID; ++e) |
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389 | c += (*_flow)[e] * _cost[e]; |
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390 | return c; |
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391 | } |
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392 | |
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393 | /// @} |
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394 | |
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395 | private: |
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396 | |
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397 | /// Initialize the algorithm. |
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398 | bool init() { |
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399 | if (!_valid_supply) return false; |
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400 | |
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401 | // Initializing flow and potential maps |
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402 | if (!_flow) { |
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403 | _flow = new FlowMap(_graph); |
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404 | _local_flow = true; |
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405 | } |
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406 | if (!_potential) { |
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407 | _potential = new PotentialMap(_graph); |
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408 | _local_potential = true; |
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409 | } |
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410 | |
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411 | _res_graph = new ResDigraph(_graph, _capacity, *_flow); |
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412 | |
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413 | // Finding a feasible flow using Circulation |
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414 | Circulation< Digraph, ConstMap<Arc, Capacity>, CapacityArcMap, |
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415 | SupplyMap > |
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416 | circulation( _graph, constMap<Arc>(Capacity(0)), _capacity, |
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417 | _supply ); |
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418 | return circulation.flowMap(*_flow).run(); |
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419 | } |
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420 | |
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421 | bool start(bool min_mean_cc) { |
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422 | if (min_mean_cc) |
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423 | startMinMean(); |
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424 | else |
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425 | start(); |
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426 | |
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427 | // Handling non-zero lower bounds |
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428 | if (_lower) { |
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429 | for (ArcIt e(_graph); e != INVALID; ++e) |
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430 | (*_flow)[e] += (*_lower)[e]; |
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431 | } |
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432 | return true; |
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433 | } |
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434 | |
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435 | /// \brief Execute the algorithm using \ref BellmanFord. |
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436 | /// |
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437 | /// Execute the algorithm using the \ref BellmanFord |
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438 | /// "Bellman-Ford" algorithm for negative cycle detection with |
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439 | /// successively larger limit for the number of iterations. |
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440 | void start() { |
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441 | typename BellmanFord<ResDigraph, ResidualCostMap>::PredMap pred(*_res_graph); |
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442 | typename ResDigraph::template NodeMap<int> visited(*_res_graph); |
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443 | std::vector<ResArc> cycle; |
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444 | int node_num = countNodes(_graph); |
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445 | |
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446 | int length_bound = BF_FIRST_LIMIT; |
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447 | bool optimal = false; |
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448 | while (!optimal) { |
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449 | BellmanFord<ResDigraph, ResidualCostMap> bf(*_res_graph, _res_cost); |
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450 | bf.predMap(pred); |
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451 | bf.init(0); |
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452 | int iter_num = 0; |
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453 | bool cycle_found = false; |
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454 | while (!cycle_found) { |
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455 | int curr_iter_num = iter_num + length_bound <= node_num ? |
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456 | length_bound : node_num - iter_num; |
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457 | iter_num += curr_iter_num; |
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458 | int real_iter_num = curr_iter_num; |
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459 | for (int i = 0; i < curr_iter_num; ++i) { |
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460 | if (bf.processNextWeakRound()) { |
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461 | real_iter_num = i; |
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462 | break; |
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463 | } |
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464 | } |
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465 | if (real_iter_num < curr_iter_num) { |
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466 | // Optimal flow is found |
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467 | optimal = true; |
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468 | // Setting node potentials |
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469 | for (NodeIt n(_graph); n != INVALID; ++n) |
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470 | (*_potential)[n] = bf.dist(n); |
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471 | break; |
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472 | } else { |
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473 | // Searching for node disjoint negative cycles |
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474 | for (ResNodeIt n(*_res_graph); n != INVALID; ++n) |
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475 | visited[n] = 0; |
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476 | int id = 0; |
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477 | for (ResNodeIt n(*_res_graph); n != INVALID; ++n) { |
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478 | if (visited[n] > 0) continue; |
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479 | visited[n] = ++id; |
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480 | ResNode u = pred[n] == INVALID ? |
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481 | INVALID : _res_graph->source(pred[n]); |
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482 | while (u != INVALID && visited[u] == 0) { |
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483 | visited[u] = id; |
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484 | u = pred[u] == INVALID ? |
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485 | INVALID : _res_graph->source(pred[u]); |
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486 | } |
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487 | if (u != INVALID && visited[u] == id) { |
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488 | // Finding the negative cycle |
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489 | cycle_found = true; |
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490 | cycle.clear(); |
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491 | ResArc e = pred[u]; |
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492 | cycle.push_back(e); |
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493 | Capacity d = _res_graph->residualCapacity(e); |
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494 | while (_res_graph->source(e) != u) { |
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495 | cycle.push_back(e = pred[_res_graph->source(e)]); |
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496 | if (_res_graph->residualCapacity(e) < d) |
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497 | d = _res_graph->residualCapacity(e); |
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498 | } |
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499 | |
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500 | // Augmenting along the cycle |
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501 | for (int i = 0; i < int(cycle.size()); ++i) |
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502 | _res_graph->augment(cycle[i], d); |
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503 | } |
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504 | } |
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505 | } |
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506 | |
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507 | if (!cycle_found) |
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508 | length_bound = length_bound * BF_LIMIT_FACTOR / 100; |
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509 | } |
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510 | } |
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511 | } |
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512 | |
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513 | /// \brief Execute the algorithm using \ref Howard. |
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514 | /// |
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515 | /// Execute the algorithm using \ref Howard for negative |
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516 | /// cycle detection. |
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517 | void startMinMean() { |
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518 | typedef Path<ResDigraph> ResPath; |
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519 | Howard<ResDigraph, ResidualCostMap> mmc(*_res_graph, _res_cost); |
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520 | ResPath cycle; |
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521 | |
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522 | mmc.cycle(cycle); |
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523 | if (mmc.findMinMean()) { |
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524 | while (mmc.cycleLength() < 0) { |
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525 | // Finding the cycle |
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526 | mmc.findCycle(); |
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527 | |
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528 | // Finding the largest flow amount that can be augmented |
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529 | // along the cycle |
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530 | Capacity delta = 0; |
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531 | for (typename ResPath::ArcIt e(cycle); e != INVALID; ++e) { |
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532 | if (delta == 0 || _res_graph->residualCapacity(e) < delta) |
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533 | delta = _res_graph->residualCapacity(e); |
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534 | } |
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535 | |
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536 | // Augmenting along the cycle |
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537 | for (typename ResPath::ArcIt e(cycle); e != INVALID; ++e) |
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538 | _res_graph->augment(e, delta); |
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539 | |
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540 | // Finding the minimum cycle mean for the modified residual |
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541 | // digraph |
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542 | if (!mmc.findMinMean()) break; |
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543 | } |
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544 | } |
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545 | |
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546 | // Computing node potentials |
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547 | BellmanFord<ResDigraph, ResidualCostMap> bf(*_res_graph, _res_cost); |
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548 | bf.init(0); bf.start(); |
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549 | for (NodeIt n(_graph); n != INVALID; ++n) |
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550 | (*_potential)[n] = bf.dist(n); |
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551 | } |
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552 | |
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553 | }; //class CycleCanceling |
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554 | |
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555 | ///@} |
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556 | |
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557 | } //namespace lemon |
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558 | |
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559 | #endif //LEMON_CYCLE_CANCELING_H |
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