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_CAPACITY_SCALING_H |
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20 | #define LEMON_CAPACITY_SCALING_H |
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21 | |
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22 | /// \ingroup min_cost_flow_algs |
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23 | /// |
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24 | /// \file |
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25 | /// \brief Capacity Scaling 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 | #include <lemon/core.h> |
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30 | #include <lemon/maps.h> |
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31 | #include <lemon/bin_heap.h> |
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32 | |
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33 | namespace lemon { |
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34 | |
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35 | /// \brief Default traits class of CapacityScaling algorithm. |
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36 | /// |
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37 | /// Default traits class of CapacityScaling algorithm. |
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38 | /// \tparam GR Digraph type. |
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39 | /// \tparam V The number type used for flow amounts, capacity bounds |
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40 | /// and supply values. By default it is \c int. |
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41 | /// \tparam C The number type used for costs and potentials. |
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42 | /// By default it is the same as \c V. |
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43 | template <typename GR, typename V = int, typename C = V> |
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44 | struct CapacityScalingDefaultTraits |
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45 | { |
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46 | /// The type of the digraph |
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47 | typedef GR Digraph; |
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48 | /// The type of the flow amounts, capacity bounds and supply values |
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49 | typedef V Value; |
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50 | /// The type of the arc costs |
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51 | typedef C Cost; |
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52 | |
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53 | /// \brief The type of the heap used for internal Dijkstra computations. |
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54 | /// |
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55 | /// The type of the heap used for internal Dijkstra computations. |
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56 | /// It must conform to the \ref lemon::concepts::Heap "Heap" concept, |
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57 | /// its priority type must be \c Cost and its cross reference type |
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58 | /// must be \ref RangeMap "RangeMap<int>". |
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59 | typedef BinHeap<Cost, RangeMap<int> > Heap; |
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60 | }; |
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61 | |
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62 | /// \addtogroup min_cost_flow_algs |
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63 | /// @{ |
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64 | |
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65 | /// \brief Implementation of the Capacity Scaling algorithm for |
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66 | /// finding a \ref min_cost_flow "minimum cost flow". |
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67 | /// |
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68 | /// \ref CapacityScaling implements the capacity scaling version |
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69 | /// of the successive shortest path algorithm for finding a |
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70 | /// \ref min_cost_flow "minimum cost flow" \ref amo93networkflows, |
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71 | /// \ref edmondskarp72theoretical. It is an efficient dual |
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72 | /// solution method. |
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73 | /// |
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74 | /// Most of the parameters of the problem (except for the digraph) |
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75 | /// can be given using separate functions, and the algorithm can be |
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76 | /// executed using the \ref run() function. If some parameters are not |
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77 | /// specified, then default values will be used. |
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78 | /// |
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79 | /// \tparam GR The digraph type the algorithm runs on. |
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80 | /// \tparam V The number type used for flow amounts, capacity bounds |
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81 | /// and supply values in the algorithm. By default it is \c int. |
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82 | /// \tparam C The number type used for costs and potentials in the |
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83 | /// algorithm. By default it is the same as \c V. |
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84 | /// |
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85 | /// \warning Both number types must be signed and all input data must |
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86 | /// be integer. |
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87 | /// \warning This algorithm does not support negative costs for such |
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88 | /// arcs that have infinite upper bound. |
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89 | #ifdef DOXYGEN |
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90 | template <typename GR, typename V, typename C, typename TR> |
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91 | #else |
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92 | template < typename GR, typename V = int, typename C = V, |
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93 | typename TR = CapacityScalingDefaultTraits<GR, V, C> > |
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94 | #endif |
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95 | class CapacityScaling |
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96 | { |
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97 | public: |
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98 | |
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99 | /// The type of the digraph |
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100 | typedef typename TR::Digraph Digraph; |
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101 | /// The type of the flow amounts, capacity bounds and supply values |
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102 | typedef typename TR::Value Value; |
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103 | /// The type of the arc costs |
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104 | typedef typename TR::Cost Cost; |
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105 | |
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106 | /// The type of the heap used for internal Dijkstra computations |
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107 | typedef typename TR::Heap Heap; |
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108 | |
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109 | /// The \ref CapacityScalingDefaultTraits "traits class" of the algorithm |
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110 | typedef TR Traits; |
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111 | |
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112 | public: |
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113 | |
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114 | /// \brief Problem type constants for the \c run() function. |
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115 | /// |
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116 | /// Enum type containing the problem type constants that can be |
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117 | /// returned by the \ref run() function of the algorithm. |
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118 | enum ProblemType { |
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119 | /// The problem has no feasible solution (flow). |
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120 | INFEASIBLE, |
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121 | /// The problem has optimal solution (i.e. it is feasible and |
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122 | /// bounded), and the algorithm has found optimal flow and node |
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123 | /// potentials (primal and dual solutions). |
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124 | OPTIMAL, |
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125 | /// The digraph contains an arc of negative cost and infinite |
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126 | /// upper bound. It means that the objective function is unbounded |
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127 | /// on that arc, however, note that it could actually be bounded |
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128 | /// over the feasible flows, but this algroithm cannot handle |
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129 | /// these cases. |
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130 | UNBOUNDED |
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131 | }; |
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132 | |
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133 | private: |
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134 | |
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135 | TEMPLATE_DIGRAPH_TYPEDEFS(GR); |
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136 | |
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137 | typedef std::vector<int> IntVector; |
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138 | typedef std::vector<char> BoolVector; |
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139 | typedef std::vector<Value> ValueVector; |
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140 | typedef std::vector<Cost> CostVector; |
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141 | |
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142 | private: |
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143 | |
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144 | // Data related to the underlying digraph |
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145 | const GR &_graph; |
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146 | int _node_num; |
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147 | int _arc_num; |
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148 | int _res_arc_num; |
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149 | int _root; |
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150 | |
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151 | // Parameters of the problem |
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152 | bool _have_lower; |
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153 | Value _sum_supply; |
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154 | |
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155 | // Data structures for storing the digraph |
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156 | IntNodeMap _node_id; |
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157 | IntArcMap _arc_idf; |
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158 | IntArcMap _arc_idb; |
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159 | IntVector _first_out; |
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160 | BoolVector _forward; |
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161 | IntVector _source; |
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162 | IntVector _target; |
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163 | IntVector _reverse; |
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164 | |
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165 | // Node and arc data |
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166 | ValueVector _lower; |
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167 | ValueVector _upper; |
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168 | CostVector _cost; |
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169 | ValueVector _supply; |
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170 | |
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171 | ValueVector _res_cap; |
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172 | CostVector _pi; |
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173 | ValueVector _excess; |
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174 | IntVector _excess_nodes; |
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175 | IntVector _deficit_nodes; |
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176 | |
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177 | Value _delta; |
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178 | int _factor; |
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179 | IntVector _pred; |
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180 | |
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181 | public: |
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182 | |
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183 | /// \brief Constant for infinite upper bounds (capacities). |
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184 | /// |
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185 | /// Constant for infinite upper bounds (capacities). |
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186 | /// It is \c std::numeric_limits<Value>::infinity() if available, |
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187 | /// \c std::numeric_limits<Value>::max() otherwise. |
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188 | const Value INF; |
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189 | |
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190 | private: |
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191 | |
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192 | // Special implementation of the Dijkstra algorithm for finding |
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193 | // shortest paths in the residual network of the digraph with |
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194 | // respect to the reduced arc costs and modifying the node |
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195 | // potentials according to the found distance labels. |
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196 | class ResidualDijkstra |
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197 | { |
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198 | private: |
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199 | |
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200 | int _node_num; |
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201 | bool _geq; |
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202 | const IntVector &_first_out; |
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203 | const IntVector &_target; |
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204 | const CostVector &_cost; |
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205 | const ValueVector &_res_cap; |
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206 | const ValueVector &_excess; |
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207 | CostVector &_pi; |
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208 | IntVector &_pred; |
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209 | |
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210 | IntVector _proc_nodes; |
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211 | CostVector _dist; |
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212 | |
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213 | public: |
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214 | |
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215 | ResidualDijkstra(CapacityScaling& cs) : |
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216 | _node_num(cs._node_num), _geq(cs._sum_supply < 0), |
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217 | _first_out(cs._first_out), _target(cs._target), _cost(cs._cost), |
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218 | _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi), |
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219 | _pred(cs._pred), _dist(cs._node_num) |
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220 | {} |
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221 | |
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222 | int run(int s, Value delta = 1) { |
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223 | RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP); |
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224 | Heap heap(heap_cross_ref); |
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225 | heap.push(s, 0); |
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226 | _pred[s] = -1; |
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227 | _proc_nodes.clear(); |
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228 | |
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229 | // Process nodes |
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230 | while (!heap.empty() && _excess[heap.top()] > -delta) { |
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231 | int u = heap.top(), v; |
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232 | Cost d = heap.prio() + _pi[u], dn; |
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233 | _dist[u] = heap.prio(); |
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234 | _proc_nodes.push_back(u); |
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235 | heap.pop(); |
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236 | |
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237 | // Traverse outgoing residual arcs |
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238 | int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1; |
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239 | for (int a = _first_out[u]; a != last_out; ++a) { |
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240 | if (_res_cap[a] < delta) continue; |
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241 | v = _target[a]; |
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242 | switch (heap.state(v)) { |
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243 | case Heap::PRE_HEAP: |
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244 | heap.push(v, d + _cost[a] - _pi[v]); |
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245 | _pred[v] = a; |
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246 | break; |
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247 | case Heap::IN_HEAP: |
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248 | dn = d + _cost[a] - _pi[v]; |
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249 | if (dn < heap[v]) { |
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250 | heap.decrease(v, dn); |
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251 | _pred[v] = a; |
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252 | } |
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253 | break; |
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254 | case Heap::POST_HEAP: |
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255 | break; |
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256 | } |
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257 | } |
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258 | } |
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259 | if (heap.empty()) return -1; |
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260 | |
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261 | // Update potentials of processed nodes |
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262 | int t = heap.top(); |
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263 | Cost dt = heap.prio(); |
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264 | for (int i = 0; i < int(_proc_nodes.size()); ++i) { |
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265 | _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt; |
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266 | } |
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267 | |
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268 | return t; |
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269 | } |
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270 | |
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271 | }; //class ResidualDijkstra |
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272 | |
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273 | public: |
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274 | |
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275 | /// \name Named Template Parameters |
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276 | /// @{ |
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277 | |
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278 | template <typename T> |
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279 | struct SetHeapTraits : public Traits { |
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280 | typedef T Heap; |
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281 | }; |
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282 | |
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283 | /// \brief \ref named-templ-param "Named parameter" for setting |
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284 | /// \c Heap type. |
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285 | /// |
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286 | /// \ref named-templ-param "Named parameter" for setting \c Heap |
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287 | /// type, which is used for internal Dijkstra computations. |
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288 | /// It must conform to the \ref lemon::concepts::Heap "Heap" concept, |
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289 | /// its priority type must be \c Cost and its cross reference type |
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290 | /// must be \ref RangeMap "RangeMap<int>". |
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291 | template <typename T> |
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292 | struct SetHeap |
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293 | : public CapacityScaling<GR, V, C, SetHeapTraits<T> > { |
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294 | typedef CapacityScaling<GR, V, C, SetHeapTraits<T> > Create; |
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295 | }; |
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296 | |
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297 | /// @} |
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298 | |
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299 | public: |
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300 | |
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301 | /// \brief Constructor. |
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302 | /// |
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303 | /// The constructor of the class. |
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304 | /// |
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305 | /// \param graph The digraph the algorithm runs on. |
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306 | CapacityScaling(const GR& graph) : |
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307 | _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph), |
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308 | INF(std::numeric_limits<Value>::has_infinity ? |
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309 | std::numeric_limits<Value>::infinity() : |
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310 | std::numeric_limits<Value>::max()) |
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311 | { |
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312 | // Check the number types |
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313 | LEMON_ASSERT(std::numeric_limits<Value>::is_signed, |
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314 | "The flow type of CapacityScaling must be signed"); |
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315 | LEMON_ASSERT(std::numeric_limits<Cost>::is_signed, |
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316 | "The cost type of CapacityScaling must be signed"); |
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317 | |
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318 | // Resize vectors |
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319 | _node_num = countNodes(_graph); |
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320 | _arc_num = countArcs(_graph); |
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321 | _res_arc_num = 2 * (_arc_num + _node_num); |
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322 | _root = _node_num; |
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323 | ++_node_num; |
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324 | |
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325 | _first_out.resize(_node_num + 1); |
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326 | _forward.resize(_res_arc_num); |
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327 | _source.resize(_res_arc_num); |
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328 | _target.resize(_res_arc_num); |
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329 | _reverse.resize(_res_arc_num); |
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330 | |
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331 | _lower.resize(_res_arc_num); |
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332 | _upper.resize(_res_arc_num); |
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333 | _cost.resize(_res_arc_num); |
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334 | _supply.resize(_node_num); |
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335 | |
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336 | _res_cap.resize(_res_arc_num); |
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337 | _pi.resize(_node_num); |
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338 | _excess.resize(_node_num); |
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339 | _pred.resize(_node_num); |
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340 | |
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341 | // Copy the graph |
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342 | int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1; |
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343 | for (NodeIt n(_graph); n != INVALID; ++n, ++i) { |
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344 | _node_id[n] = i; |
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345 | } |
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346 | i = 0; |
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347 | for (NodeIt n(_graph); n != INVALID; ++n, ++i) { |
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348 | _first_out[i] = j; |
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349 | for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) { |
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350 | _arc_idf[a] = j; |
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351 | _forward[j] = true; |
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352 | _source[j] = i; |
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353 | _target[j] = _node_id[_graph.runningNode(a)]; |
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354 | } |
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355 | for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) { |
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356 | _arc_idb[a] = j; |
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357 | _forward[j] = false; |
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358 | _source[j] = i; |
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359 | _target[j] = _node_id[_graph.runningNode(a)]; |
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360 | } |
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361 | _forward[j] = false; |
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362 | _source[j] = i; |
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363 | _target[j] = _root; |
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364 | _reverse[j] = k; |
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365 | _forward[k] = true; |
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366 | _source[k] = _root; |
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367 | _target[k] = i; |
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368 | _reverse[k] = j; |
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369 | ++j; ++k; |
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370 | } |
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371 | _first_out[i] = j; |
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372 | _first_out[_node_num] = k; |
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373 | for (ArcIt a(_graph); a != INVALID; ++a) { |
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374 | int fi = _arc_idf[a]; |
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375 | int bi = _arc_idb[a]; |
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376 | _reverse[fi] = bi; |
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377 | _reverse[bi] = fi; |
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378 | } |
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379 | |
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380 | // Reset parameters |
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381 | reset(); |
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382 | } |
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383 | |
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384 | /// \name Parameters |
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385 | /// The parameters of the algorithm can be specified using these |
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386 | /// functions. |
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387 | |
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388 | /// @{ |
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389 | |
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390 | /// \brief Set the lower bounds on the arcs. |
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391 | /// |
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392 | /// This function sets the lower bounds on the arcs. |
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393 | /// If it is not used before calling \ref run(), the lower bounds |
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394 | /// will be set to zero on all arcs. |
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395 | /// |
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396 | /// \param map An arc map storing the lower bounds. |
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397 | /// Its \c Value type must be convertible to the \c Value type |
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398 | /// of the algorithm. |
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399 | /// |
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400 | /// \return <tt>(*this)</tt> |
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401 | template <typename LowerMap> |
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402 | CapacityScaling& lowerMap(const LowerMap& map) { |
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403 | _have_lower = true; |
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404 | for (ArcIt a(_graph); a != INVALID; ++a) { |
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405 | _lower[_arc_idf[a]] = map[a]; |
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406 | _lower[_arc_idb[a]] = map[a]; |
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407 | } |
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408 | return *this; |
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409 | } |
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410 | |
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411 | /// \brief Set the upper bounds (capacities) on the arcs. |
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412 | /// |
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413 | /// This function sets the upper bounds (capacities) on the arcs. |
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414 | /// If it is not used before calling \ref run(), the upper bounds |
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415 | /// will be set to \ref INF on all arcs (i.e. the flow value will be |
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416 | /// unbounded from above). |
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417 | /// |
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418 | /// \param map An arc map storing the upper bounds. |
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419 | /// Its \c Value type must be convertible to the \c Value type |
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420 | /// of the algorithm. |
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421 | /// |
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422 | /// \return <tt>(*this)</tt> |
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423 | template<typename UpperMap> |
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424 | CapacityScaling& upperMap(const UpperMap& map) { |
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425 | for (ArcIt a(_graph); a != INVALID; ++a) { |
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426 | _upper[_arc_idf[a]] = map[a]; |
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427 | } |
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428 | return *this; |
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429 | } |
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430 | |
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431 | /// \brief Set the costs of the arcs. |
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432 | /// |
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433 | /// This function sets the costs of the arcs. |
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434 | /// If it is not used before calling \ref run(), the costs |
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435 | /// will be set to \c 1 on all arcs. |
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436 | /// |
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437 | /// \param map An arc map storing the costs. |
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438 | /// Its \c Value type must be convertible to the \c Cost type |
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439 | /// of the algorithm. |
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440 | /// |
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441 | /// \return <tt>(*this)</tt> |
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442 | template<typename CostMap> |
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443 | CapacityScaling& costMap(const CostMap& map) { |
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444 | for (ArcIt a(_graph); a != INVALID; ++a) { |
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445 | _cost[_arc_idf[a]] = map[a]; |
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446 | _cost[_arc_idb[a]] = -map[a]; |
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447 | } |
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448 | return *this; |
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449 | } |
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450 | |
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451 | /// \brief Set the supply values of the nodes. |
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452 | /// |
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453 | /// This function sets the supply values of the nodes. |
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454 | /// If neither this function nor \ref stSupply() is used before |
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455 | /// calling \ref run(), the supply of each node will be set to zero. |
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456 | /// |
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457 | /// \param map A node map storing the supply values. |
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458 | /// Its \c Value type must be convertible to the \c Value type |
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459 | /// of the algorithm. |
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460 | /// |
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461 | /// \return <tt>(*this)</tt> |
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462 | template<typename SupplyMap> |
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463 | CapacityScaling& supplyMap(const SupplyMap& map) { |
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464 | for (NodeIt n(_graph); n != INVALID; ++n) { |
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465 | _supply[_node_id[n]] = map[n]; |
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466 | } |
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467 | return *this; |
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468 | } |
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469 | |
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470 | /// \brief Set single source and target nodes and a supply value. |
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471 | /// |
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472 | /// This function sets a single source node and a single target node |
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473 | /// and the required flow value. |
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474 | /// If neither this function nor \ref supplyMap() is used before |
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475 | /// calling \ref run(), the supply of each node will be set to zero. |
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476 | /// |
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477 | /// Using this function has the same effect as using \ref supplyMap() |
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478 | /// with such a map in which \c k is assigned to \c s, \c -k is |
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479 | /// assigned to \c t and all other nodes have zero supply value. |
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480 | /// |
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481 | /// \param s The source node. |
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482 | /// \param t The target node. |
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483 | /// \param k The required amount of flow from node \c s to node \c t |
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484 | /// (i.e. the supply of \c s and the demand of \c t). |
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485 | /// |
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486 | /// \return <tt>(*this)</tt> |
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487 | CapacityScaling& stSupply(const Node& s, const Node& t, Value k) { |
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488 | for (int i = 0; i != _node_num; ++i) { |
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489 | _supply[i] = 0; |
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490 | } |
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491 | _supply[_node_id[s]] = k; |
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492 | _supply[_node_id[t]] = -k; |
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493 | return *this; |
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494 | } |
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495 | |
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496 | /// @} |
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497 | |
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498 | /// \name Execution control |
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499 | /// The algorithm can be executed using \ref run(). |
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500 | |
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501 | /// @{ |
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502 | |
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503 | /// \brief Run the algorithm. |
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504 | /// |
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505 | /// This function runs the algorithm. |
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506 | /// The paramters can be specified using functions \ref lowerMap(), |
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507 | /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(). |
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508 | /// For example, |
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509 | /// \code |
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510 | /// CapacityScaling<ListDigraph> cs(graph); |
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511 | /// cs.lowerMap(lower).upperMap(upper).costMap(cost) |
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512 | /// .supplyMap(sup).run(); |
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513 | /// \endcode |
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514 | /// |
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515 | /// This function can be called more than once. All the parameters |
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516 | /// that have been given are kept for the next call, unless |
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517 | /// \ref reset() is called, thus only the modified parameters |
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518 | /// have to be set again. See \ref reset() for examples. |
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519 | /// However, the underlying digraph must not be modified after this |
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520 | /// class have been constructed, since it copies and extends the graph. |
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521 | /// |
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522 | /// \param factor The capacity scaling factor. It must be larger than |
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523 | /// one to use scaling. If it is less or equal to one, then scaling |
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524 | /// will be disabled. |
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525 | /// |
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526 | /// \return \c INFEASIBLE if no feasible flow exists, |
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527 | /// \n \c OPTIMAL if the problem has optimal solution |
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528 | /// (i.e. it is feasible and bounded), and the algorithm has found |
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529 | /// optimal flow and node potentials (primal and dual solutions), |
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530 | /// \n \c UNBOUNDED if the digraph contains an arc of negative cost |
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531 | /// and infinite upper bound. It means that the objective function |
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532 | /// is unbounded on that arc, however, note that it could actually be |
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533 | /// bounded over the feasible flows, but this algroithm cannot handle |
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534 | /// these cases. |
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535 | /// |
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536 | /// \see ProblemType |
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537 | ProblemType run(int factor = 4) { |
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538 | _factor = factor; |
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539 | ProblemType pt = init(); |
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540 | if (pt != OPTIMAL) return pt; |
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541 | return start(); |
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542 | } |
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543 | |
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544 | /// \brief Reset all the parameters that have been given before. |
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545 | /// |
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546 | /// This function resets all the paramaters that have been given |
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547 | /// before using functions \ref lowerMap(), \ref upperMap(), |
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548 | /// \ref costMap(), \ref supplyMap(), \ref stSupply(). |
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549 | /// |
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550 | /// It is useful for multiple run() calls. If this function is not |
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551 | /// used, all the parameters given before are kept for the next |
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552 | /// \ref run() call. |
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553 | /// However, the underlying digraph must not be modified after this |
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554 | /// class have been constructed, since it copies and extends the graph. |
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555 | /// |
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556 | /// For example, |
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557 | /// \code |
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558 | /// CapacityScaling<ListDigraph> cs(graph); |
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559 | /// |
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560 | /// // First run |
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561 | /// cs.lowerMap(lower).upperMap(upper).costMap(cost) |
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562 | /// .supplyMap(sup).run(); |
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563 | /// |
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564 | /// // Run again with modified cost map (reset() is not called, |
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565 | /// // so only the cost map have to be set again) |
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566 | /// cost[e] += 100; |
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567 | /// cs.costMap(cost).run(); |
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568 | /// |
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569 | /// // Run again from scratch using reset() |
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570 | /// // (the lower bounds will be set to zero on all arcs) |
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571 | /// cs.reset(); |
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572 | /// cs.upperMap(capacity).costMap(cost) |
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573 | /// .supplyMap(sup).run(); |
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574 | /// \endcode |
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575 | /// |
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576 | /// \return <tt>(*this)</tt> |
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577 | CapacityScaling& reset() { |
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578 | for (int i = 0; i != _node_num; ++i) { |
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579 | _supply[i] = 0; |
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580 | } |
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581 | for (int j = 0; j != _res_arc_num; ++j) { |
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582 | _lower[j] = 0; |
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583 | _upper[j] = INF; |
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584 | _cost[j] = _forward[j] ? 1 : -1; |
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585 | } |
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586 | _have_lower = false; |
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587 | return *this; |
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588 | } |
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589 | |
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590 | /// @} |
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591 | |
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592 | /// \name Query Functions |
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593 | /// The results of the algorithm can be obtained using these |
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594 | /// functions.\n |
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595 | /// The \ref run() function must be called before using them. |
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596 | |
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597 | /// @{ |
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598 | |
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599 | /// \brief Return the total cost of the found flow. |
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600 | /// |
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601 | /// This function returns the total cost of the found flow. |
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602 | /// Its complexity is O(e). |
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603 | /// |
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604 | /// \note The return type of the function can be specified as a |
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605 | /// template parameter. For example, |
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606 | /// \code |
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607 | /// cs.totalCost<double>(); |
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608 | /// \endcode |
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609 | /// It is useful if the total cost cannot be stored in the \c Cost |
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610 | /// type of the algorithm, which is the default return type of the |
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611 | /// function. |
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612 | /// |
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613 | /// \pre \ref run() must be called before using this function. |
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614 | template <typename Number> |
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615 | Number totalCost() const { |
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616 | Number c = 0; |
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617 | for (ArcIt a(_graph); a != INVALID; ++a) { |
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618 | int i = _arc_idb[a]; |
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619 | c += static_cast<Number>(_res_cap[i]) * |
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620 | (-static_cast<Number>(_cost[i])); |
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621 | } |
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622 | return c; |
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623 | } |
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624 | |
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625 | #ifndef DOXYGEN |
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626 | Cost totalCost() const { |
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627 | return totalCost<Cost>(); |
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628 | } |
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629 | #endif |
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630 | |
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631 | /// \brief Return the flow on the given arc. |
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632 | /// |
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633 | /// This function returns the flow on the given arc. |
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634 | /// |
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635 | /// \pre \ref run() must be called before using this function. |
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636 | Value flow(const Arc& a) const { |
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637 | return _res_cap[_arc_idb[a]]; |
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638 | } |
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639 | |
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640 | /// \brief Return the flow map (the primal solution). |
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641 | /// |
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642 | /// This function copies the flow value on each arc into the given |
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643 | /// map. The \c Value type of the algorithm must be convertible to |
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644 | /// the \c Value type of the map. |
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645 | /// |
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646 | /// \pre \ref run() must be called before using this function. |
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647 | template <typename FlowMap> |
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648 | void flowMap(FlowMap &map) const { |
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649 | for (ArcIt a(_graph); a != INVALID; ++a) { |
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650 | map.set(a, _res_cap[_arc_idb[a]]); |
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651 | } |
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652 | } |
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653 | |
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654 | /// \brief Return the potential (dual value) of the given node. |
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655 | /// |
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656 | /// This function returns the potential (dual value) of the |
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657 | /// given node. |
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658 | /// |
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659 | /// \pre \ref run() must be called before using this function. |
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660 | Cost potential(const Node& n) const { |
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661 | return _pi[_node_id[n]]; |
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662 | } |
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663 | |
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664 | /// \brief Return the potential map (the dual solution). |
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665 | /// |
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666 | /// This function copies the potential (dual value) of each node |
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667 | /// into the given map. |
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668 | /// The \c Cost type of the algorithm must be convertible to the |
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669 | /// \c Value type of the map. |
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670 | /// |
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671 | /// \pre \ref run() must be called before using this function. |
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672 | template <typename PotentialMap> |
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673 | void potentialMap(PotentialMap &map) const { |
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674 | for (NodeIt n(_graph); n != INVALID; ++n) { |
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675 | map.set(n, _pi[_node_id[n]]); |
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676 | } |
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677 | } |
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678 | |
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679 | /// @} |
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680 | |
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681 | private: |
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682 | |
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683 | // Initialize the algorithm |
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684 | ProblemType init() { |
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685 | if (_node_num <= 1) return INFEASIBLE; |
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686 | |
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687 | // Check the sum of supply values |
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688 | _sum_supply = 0; |
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689 | for (int i = 0; i != _root; ++i) { |
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690 | _sum_supply += _supply[i]; |
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691 | } |
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692 | if (_sum_supply > 0) return INFEASIBLE; |
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693 | |
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694 | // Initialize vectors |
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695 | for (int i = 0; i != _root; ++i) { |
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696 | _pi[i] = 0; |
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697 | _excess[i] = _supply[i]; |
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698 | } |
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699 | |
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700 | // Remove non-zero lower bounds |
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701 | const Value MAX = std::numeric_limits<Value>::max(); |
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702 | int last_out; |
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703 | if (_have_lower) { |
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704 | for (int i = 0; i != _root; ++i) { |
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705 | last_out = _first_out[i+1]; |
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706 | for (int j = _first_out[i]; j != last_out; ++j) { |
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707 | if (_forward[j]) { |
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708 | Value c = _lower[j]; |
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709 | if (c >= 0) { |
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710 | _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF; |
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711 | } else { |
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712 | _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF; |
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713 | } |
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714 | _excess[i] -= c; |
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715 | _excess[_target[j]] += c; |
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716 | } else { |
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717 | _res_cap[j] = 0; |
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718 | } |
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719 | } |
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720 | } |
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721 | } else { |
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722 | for (int j = 0; j != _res_arc_num; ++j) { |
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723 | _res_cap[j] = _forward[j] ? _upper[j] : 0; |
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724 | } |
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725 | } |
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726 | |
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727 | // Handle negative costs |
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728 | for (int i = 0; i != _root; ++i) { |
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729 | last_out = _first_out[i+1] - 1; |
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730 | for (int j = _first_out[i]; j != last_out; ++j) { |
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731 | Value rc = _res_cap[j]; |
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732 | if (_cost[j] < 0 && rc > 0) { |
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733 | if (rc >= MAX) return UNBOUNDED; |
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734 | _excess[i] -= rc; |
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735 | _excess[_target[j]] += rc; |
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736 | _res_cap[j] = 0; |
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737 | _res_cap[_reverse[j]] += rc; |
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738 | } |
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739 | } |
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740 | } |
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741 | |
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742 | // Handle GEQ supply type |
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743 | if (_sum_supply < 0) { |
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744 | _pi[_root] = 0; |
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745 | _excess[_root] = -_sum_supply; |
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746 | for (int a = _first_out[_root]; a != _res_arc_num; ++a) { |
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747 | int ra = _reverse[a]; |
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748 | _res_cap[a] = -_sum_supply + 1; |
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749 | _res_cap[ra] = 0; |
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750 | _cost[a] = 0; |
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751 | _cost[ra] = 0; |
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752 | } |
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753 | } else { |
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754 | _pi[_root] = 0; |
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755 | _excess[_root] = 0; |
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756 | for (int a = _first_out[_root]; a != _res_arc_num; ++a) { |
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757 | int ra = _reverse[a]; |
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758 | _res_cap[a] = 1; |
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759 | _res_cap[ra] = 0; |
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760 | _cost[a] = 0; |
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761 | _cost[ra] = 0; |
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762 | } |
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763 | } |
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764 | |
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765 | // Initialize delta value |
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766 | if (_factor > 1) { |
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767 | // With scaling |
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768 | Value max_sup = 0, max_dem = 0; |
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769 | for (int i = 0; i != _node_num; ++i) { |
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770 | Value ex = _excess[i]; |
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771 | if ( ex > max_sup) max_sup = ex; |
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772 | if (-ex > max_dem) max_dem = -ex; |
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773 | } |
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774 | Value max_cap = 0; |
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775 | for (int j = 0; j != _res_arc_num; ++j) { |
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776 | if (_res_cap[j] > max_cap) max_cap = _res_cap[j]; |
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777 | } |
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778 | max_sup = std::min(std::min(max_sup, max_dem), max_cap); |
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779 | for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ; |
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780 | } else { |
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781 | // Without scaling |
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782 | _delta = 1; |
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783 | } |
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784 | |
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785 | return OPTIMAL; |
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786 | } |
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787 | |
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788 | ProblemType start() { |
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789 | // Execute the algorithm |
---|
790 | ProblemType pt; |
---|
791 | if (_delta > 1) |
---|
792 | pt = startWithScaling(); |
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793 | else |
---|
794 | pt = startWithoutScaling(); |
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795 | |
---|
796 | // Handle non-zero lower bounds |
---|
797 | if (_have_lower) { |
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798 | int limit = _first_out[_root]; |
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799 | for (int j = 0; j != limit; ++j) { |
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800 | if (!_forward[j]) _res_cap[j] += _lower[j]; |
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801 | } |
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802 | } |
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803 | |
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804 | // Shift potentials if necessary |
---|
805 | Cost pr = _pi[_root]; |
---|
806 | if (_sum_supply < 0 || pr > 0) { |
---|
807 | for (int i = 0; i != _node_num; ++i) { |
---|
808 | _pi[i] -= pr; |
---|
809 | } |
---|
810 | } |
---|
811 | |
---|
812 | return pt; |
---|
813 | } |
---|
814 | |
---|
815 | // Execute the capacity scaling algorithm |
---|
816 | ProblemType startWithScaling() { |
---|
817 | // Perform capacity scaling phases |
---|
818 | int s, t; |
---|
819 | ResidualDijkstra _dijkstra(*this); |
---|
820 | while (true) { |
---|
821 | // Saturate all arcs not satisfying the optimality condition |
---|
822 | int last_out; |
---|
823 | for (int u = 0; u != _node_num; ++u) { |
---|
824 | last_out = _sum_supply < 0 ? |
---|
825 | _first_out[u+1] : _first_out[u+1] - 1; |
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826 | for (int a = _first_out[u]; a != last_out; ++a) { |
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827 | int v = _target[a]; |
---|
828 | Cost c = _cost[a] + _pi[u] - _pi[v]; |
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829 | Value rc = _res_cap[a]; |
---|
830 | if (c < 0 && rc >= _delta) { |
---|
831 | _excess[u] -= rc; |
---|
832 | _excess[v] += rc; |
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833 | _res_cap[a] = 0; |
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834 | _res_cap[_reverse[a]] += rc; |
---|
835 | } |
---|
836 | } |
---|
837 | } |
---|
838 | |
---|
839 | // Find excess nodes and deficit nodes |
---|
840 | _excess_nodes.clear(); |
---|
841 | _deficit_nodes.clear(); |
---|
842 | for (int u = 0; u != _node_num; ++u) { |
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843 | Value ex = _excess[u]; |
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844 | if (ex >= _delta) _excess_nodes.push_back(u); |
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845 | if (ex <= -_delta) _deficit_nodes.push_back(u); |
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846 | } |
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847 | int next_node = 0, next_def_node = 0; |
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848 | |
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849 | // Find augmenting shortest paths |
---|
850 | while (next_node < int(_excess_nodes.size())) { |
---|
851 | // Check deficit nodes |
---|
852 | if (_delta > 1) { |
---|
853 | bool delta_deficit = false; |
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854 | for ( ; next_def_node < int(_deficit_nodes.size()); |
---|
855 | ++next_def_node ) { |
---|
856 | if (_excess[_deficit_nodes[next_def_node]] <= -_delta) { |
---|
857 | delta_deficit = true; |
---|
858 | break; |
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859 | } |
---|
860 | } |
---|
861 | if (!delta_deficit) break; |
---|
862 | } |
---|
863 | |
---|
864 | // Run Dijkstra in the residual network |
---|
865 | s = _excess_nodes[next_node]; |
---|
866 | if ((t = _dijkstra.run(s, _delta)) == -1) { |
---|
867 | if (_delta > 1) { |
---|
868 | ++next_node; |
---|
869 | continue; |
---|
870 | } |
---|
871 | return INFEASIBLE; |
---|
872 | } |
---|
873 | |
---|
874 | // Augment along a shortest path from s to t |
---|
875 | Value d = std::min(_excess[s], -_excess[t]); |
---|
876 | int u = t; |
---|
877 | int a; |
---|
878 | if (d > _delta) { |
---|
879 | while ((a = _pred[u]) != -1) { |
---|
880 | if (_res_cap[a] < d) d = _res_cap[a]; |
---|
881 | u = _source[a]; |
---|
882 | } |
---|
883 | } |
---|
884 | u = t; |
---|
885 | while ((a = _pred[u]) != -1) { |
---|
886 | _res_cap[a] -= d; |
---|
887 | _res_cap[_reverse[a]] += d; |
---|
888 | u = _source[a]; |
---|
889 | } |
---|
890 | _excess[s] -= d; |
---|
891 | _excess[t] += d; |
---|
892 | |
---|
893 | if (_excess[s] < _delta) ++next_node; |
---|
894 | } |
---|
895 | |
---|
896 | if (_delta == 1) break; |
---|
897 | _delta = _delta <= _factor ? 1 : _delta / _factor; |
---|
898 | } |
---|
899 | |
---|
900 | return OPTIMAL; |
---|
901 | } |
---|
902 | |
---|
903 | // Execute the successive shortest path algorithm |
---|
904 | ProblemType startWithoutScaling() { |
---|
905 | // Find excess nodes |
---|
906 | _excess_nodes.clear(); |
---|
907 | for (int i = 0; i != _node_num; ++i) { |
---|
908 | if (_excess[i] > 0) _excess_nodes.push_back(i); |
---|
909 | } |
---|
910 | if (_excess_nodes.size() == 0) return OPTIMAL; |
---|
911 | int next_node = 0; |
---|
912 | |
---|
913 | // Find shortest paths |
---|
914 | int s, t; |
---|
915 | ResidualDijkstra _dijkstra(*this); |
---|
916 | while ( _excess[_excess_nodes[next_node]] > 0 || |
---|
917 | ++next_node < int(_excess_nodes.size()) ) |
---|
918 | { |
---|
919 | // Run Dijkstra in the residual network |
---|
920 | s = _excess_nodes[next_node]; |
---|
921 | if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE; |
---|
922 | |
---|
923 | // Augment along a shortest path from s to t |
---|
924 | Value d = std::min(_excess[s], -_excess[t]); |
---|
925 | int u = t; |
---|
926 | int a; |
---|
927 | if (d > 1) { |
---|
928 | while ((a = _pred[u]) != -1) { |
---|
929 | if (_res_cap[a] < d) d = _res_cap[a]; |
---|
930 | u = _source[a]; |
---|
931 | } |
---|
932 | } |
---|
933 | u = t; |
---|
934 | while ((a = _pred[u]) != -1) { |
---|
935 | _res_cap[a] -= d; |
---|
936 | _res_cap[_reverse[a]] += d; |
---|
937 | u = _source[a]; |
---|
938 | } |
---|
939 | _excess[s] -= d; |
---|
940 | _excess[t] += d; |
---|
941 | } |
---|
942 | |
---|
943 | return OPTIMAL; |
---|
944 | } |
---|
945 | |
---|
946 | }; //class CapacityScaling |
---|
947 | |
---|
948 | ///@} |
---|
949 | |
---|
950 | } //namespace lemon |
---|
951 | |
---|
952 | #endif //LEMON_CAPACITY_SCALING_H |
---|