1 | // -*- c++ -*- |
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2 | #ifndef HUGO_MINCOSTFLOWS_H |
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3 | #define HUGO_MINCOSTFLOWS_H |
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4 | |
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5 | ///\ingroup flowalgs |
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6 | ///\file |
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7 | ///\brief An algorithm for finding a flow of value \c k (for small values of \c k) having minimal total cost |
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8 | |
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9 | |
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10 | #include <hugo/dijkstra.h> |
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11 | #include <hugo/graph_wrapper.h> |
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12 | #include <hugo/maps.h> |
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13 | #include <vector> |
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14 | |
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15 | namespace hugo { |
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16 | |
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17 | /// \addtogroup flowalgs |
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18 | /// @{ |
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19 | |
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20 | ///\brief Implementation of an algorithm for finding a flow of value \c k |
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21 | ///(for small values of \c k) having minimal total cost between 2 nodes |
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22 | /// |
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23 | /// |
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24 | /// The class \ref hugo::MinCostFlows "MinCostFlows" implements |
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25 | /// an algorithm for finding a flow of value \c k |
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26 | /// having minimal total cost |
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27 | /// from a given source node to a given target node in an |
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28 | /// edge-weighted directed graph. To this end, |
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29 | /// the edge-capacities and edge-weitghs have to be nonnegative. |
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30 | /// The edge-capacities should be integers, but the edge-weights can be |
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31 | /// integers, reals or of other comparable numeric type. |
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32 | /// This algorithm is intended to use only for small values of \c k, |
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33 | /// since it is only polynomial in k, |
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34 | /// not in the length of k (which is log k). |
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35 | /// In order to find the minimum cost flow of value \c k it |
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36 | /// finds the minimum cost flow of value \c i for every |
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37 | /// \c i between 0 and \c k. |
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38 | /// |
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39 | ///\param Graph The directed graph type the algorithm runs on. |
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40 | ///\param LengthMap The type of the length map. |
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41 | ///\param CapacityMap The capacity map type. |
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42 | /// |
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43 | ///\author Attila Bernath |
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44 | template <typename Graph, typename LengthMap, typename CapacityMap> |
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45 | class MinCostFlows { |
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46 | |
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47 | typedef typename LengthMap::ValueType Length; |
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48 | |
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49 | //Warning: this should be integer type |
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50 | typedef typename CapacityMap::ValueType Capacity; |
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51 | |
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52 | typedef typename Graph::Node Node; |
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53 | typedef typename Graph::NodeIt NodeIt; |
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54 | typedef typename Graph::Edge Edge; |
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55 | typedef typename Graph::OutEdgeIt OutEdgeIt; |
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56 | typedef typename Graph::template EdgeMap<int> EdgeIntMap; |
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57 | |
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58 | |
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59 | typedef ResGraphWrapper<const Graph,int,CapacityMap,EdgeIntMap> ResGraphType; |
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60 | typedef typename ResGraphType::Edge ResGraphEdge; |
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61 | |
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62 | class ModLengthMap { |
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63 | typedef typename Graph::template NodeMap<Length> NodeMap; |
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64 | const ResGraphType& G; |
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65 | const LengthMap &ol; |
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66 | const NodeMap &pot; |
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67 | public : |
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68 | typedef typename LengthMap::KeyType KeyType; |
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69 | typedef typename LengthMap::ValueType ValueType; |
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70 | |
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71 | ValueType operator[](typename ResGraphType::Edge e) const { |
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72 | if (G.forward(e)) |
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73 | return ol[e]-(pot[G.head(e)]-pot[G.tail(e)]); |
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74 | else |
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75 | return -ol[e]-(pot[G.head(e)]-pot[G.tail(e)]); |
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76 | } |
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77 | |
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78 | ModLengthMap(const ResGraphType& _G, |
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79 | const LengthMap &o, const NodeMap &p) : |
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80 | G(_G), /*rev(_rev),*/ ol(o), pot(p){}; |
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81 | };//ModLengthMap |
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82 | |
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83 | |
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84 | protected: |
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85 | |
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86 | //Input |
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87 | const Graph& G; |
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88 | const LengthMap& length; |
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89 | const CapacityMap& capacity; |
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90 | |
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91 | |
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92 | //auxiliary variables |
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93 | |
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94 | //To store the flow |
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95 | EdgeIntMap flow; |
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96 | //To store the potential (dual variables) |
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97 | typedef typename Graph::template NodeMap<Length> PotentialMap; |
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98 | PotentialMap potential; |
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99 | |
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100 | |
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101 | Length total_length; |
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102 | |
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103 | |
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104 | public : |
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105 | |
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106 | /// The constructor of the class. |
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107 | |
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108 | ///\param _G The directed graph the algorithm runs on. |
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109 | ///\param _length The length (weight or cost) of the edges. |
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110 | ///\param _cap The capacity of the edges. |
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111 | MinCostFlows(Graph& _G, LengthMap& _length, CapacityMap& _cap) : G(_G), |
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112 | length(_length), capacity(_cap), flow(_G), potential(_G){ } |
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113 | |
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114 | |
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115 | ///Runs the algorithm. |
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116 | |
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117 | ///Runs the algorithm. |
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118 | ///Returns k if there is a flow of value at least k edge-disjoint |
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119 | ///from s to t. |
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120 | ///Otherwise it returns the maximum value of a flow from s to t. |
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121 | /// |
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122 | ///\param s The source node. |
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123 | ///\param t The target node. |
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124 | ///\param k The value of the flow we are looking for. |
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125 | /// |
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126 | ///\todo May be it does make sense to be able to start with a nonzero |
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127 | /// feasible primal-dual solution pair as well. |
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128 | int run(Node s, Node t, int k) { |
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129 | |
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130 | //Resetting variables from previous runs |
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131 | total_length = 0; |
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132 | |
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133 | for (typename Graph::EdgeIt e(G); e!=INVALID; ++e) flow.set(e, 0); |
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134 | |
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135 | //Initialize the potential to zero |
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136 | for (typename Graph::NodeIt n(G); n!=INVALID; ++n) potential.set(n, 0); |
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137 | |
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138 | |
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139 | //We need a residual graph |
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140 | ResGraphType res_graph(G, capacity, flow); |
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141 | |
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142 | |
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143 | ModLengthMap mod_length(res_graph, length, potential); |
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144 | |
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145 | Dijkstra<ResGraphType, ModLengthMap> dijkstra(res_graph, mod_length); |
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146 | |
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147 | int i; |
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148 | for (i=0; i<k; ++i){ |
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149 | dijkstra.run(s); |
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150 | if (!dijkstra.reached(t)){ |
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151 | //There are no flow of value k from s to t |
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152 | break; |
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153 | }; |
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154 | |
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155 | //We have to change the potential |
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156 | for(typename ResGraphType::NodeIt n(res_graph); n!=INVALID; ++n) |
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157 | potential[n] += dijkstra.distMap()[n]; |
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158 | |
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159 | |
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160 | //Augmenting on the sortest path |
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161 | Node n=t; |
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162 | ResGraphEdge e; |
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163 | while (n!=s){ |
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164 | e = dijkstra.pred(n); |
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165 | n = dijkstra.predNode(n); |
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166 | res_graph.augment(e,1); |
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167 | //Let's update the total length |
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168 | if (res_graph.forward(e)) |
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169 | total_length += length[e]; |
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170 | else |
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171 | total_length -= length[e]; |
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172 | } |
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173 | |
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174 | |
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175 | } |
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176 | |
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177 | |
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178 | return i; |
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179 | } |
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180 | |
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181 | |
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182 | |
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183 | /// Gives back the total weight of the found flow. |
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184 | |
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185 | ///This function gives back the total weight of the found flow. |
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186 | ///Assumes that \c run() has been run and nothing changed since then. |
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187 | Length totalLength(){ |
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188 | return total_length; |
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189 | } |
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190 | |
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191 | ///Returns a const reference to the EdgeMap \c flow. |
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192 | |
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193 | ///Returns a const reference to the EdgeMap \c flow. |
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194 | ///\pre \ref run() must |
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195 | ///be called before using this function. |
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196 | const EdgeIntMap &getFlow() const { return flow;} |
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197 | |
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198 | ///Returns a const reference to the NodeMap \c potential (the dual solution). |
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199 | |
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200 | ///Returns a const reference to the NodeMap \c potential (the dual solution). |
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201 | /// \pre \ref run() must be called before using this function. |
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202 | const PotentialMap &getPotential() const { return potential;} |
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203 | |
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204 | /// Checking the complementary slackness optimality criteria |
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205 | |
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206 | ///This function checks, whether the given solution is optimal |
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207 | ///If executed after the call of \c run() then it should return with true. |
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208 | ///This function only checks optimality, doesn't bother with feasibility. |
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209 | ///It is meant for testing purposes. |
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210 | /// |
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211 | bool checkComplementarySlackness(){ |
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212 | Length mod_pot; |
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213 | Length fl_e; |
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214 | for(typename Graph::EdgeIt e(G); e!=INVALID; ++e) { |
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215 | //C^{\Pi}_{i,j} |
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216 | mod_pot = length[e]-potential[G.head(e)]+potential[G.tail(e)]; |
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217 | fl_e = flow[e]; |
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218 | if (0<fl_e && fl_e<capacity[e]) { |
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219 | /// \todo better comparison is needed for real types, moreover, |
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220 | /// this comparison here is superfluous. |
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221 | if (mod_pot != 0) |
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222 | return false; |
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223 | } |
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224 | else { |
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225 | if (mod_pot > 0 && fl_e != 0) |
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226 | return false; |
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227 | if (mod_pot < 0 && fl_e != capacity[e]) |
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228 | return false; |
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229 | } |
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230 | } |
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231 | return true; |
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232 | } |
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233 | |
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234 | |
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235 | }; //class MinCostFlows |
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236 | |
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237 | ///@} |
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238 | |
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239 | } //namespace hugo |
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240 | |
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241 | #endif //HUGO_MINCOSTFLOWS_H |
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