1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
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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) 2003-2010 |
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6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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8 | * |
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9 | * Permission to use, modify and distribute this software is granted |
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10 | * provided that this copyright notice appears in all copies. For |
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11 | * precise terms see the accompanying LICENSE file. |
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12 | * |
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13 | * This software is provided "AS IS" with no warranty of any kind, |
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14 | * express or implied, and with no claim as to its suitability for any |
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15 | * purpose. |
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16 | * |
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17 | */ |
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18 | |
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19 | #ifndef LEMON_NETWORK_SIMPLEX_H |
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20 | #define LEMON_NETWORK_SIMPLEX_H |
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21 | |
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22 | /// \ingroup min_cost_flow_algs |
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23 | /// |
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24 | /// \file |
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25 | /// \brief Network Simplex algorithm for finding a minimum cost flow. |
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26 | |
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27 | #include <vector> |
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28 | #include <limits> |
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29 | #include <algorithm> |
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30 | |
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31 | #include <lemon/core.h> |
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32 | #include <lemon/math.h> |
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33 | |
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34 | namespace lemon { |
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35 | |
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36 | /// \addtogroup min_cost_flow_algs |
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37 | /// @{ |
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38 | |
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39 | /// \brief Implementation of the primal Network Simplex algorithm |
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40 | /// for finding a \ref min_cost_flow "minimum cost flow". |
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41 | /// |
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42 | /// \ref NetworkSimplex implements the primal Network Simplex algorithm |
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43 | /// for finding a \ref min_cost_flow "minimum cost flow" |
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44 | /// \ref amo93networkflows, \ref dantzig63linearprog, |
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45 | /// \ref kellyoneill91netsimplex. |
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46 | /// This algorithm is a highly efficient specialized version of the |
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47 | /// linear programming simplex method directly for the minimum cost |
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48 | /// flow problem. |
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49 | /// |
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50 | /// In general, %NetworkSimplex is the fastest implementation available |
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51 | /// in LEMON for this problem. |
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52 | /// Moreover, it supports both directions of the supply/demand inequality |
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53 | /// constraints. For more information, see \ref SupplyType. |
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54 | /// |
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55 | /// Most of the parameters of the problem (except for the digraph) |
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56 | /// can be given using separate functions, and the algorithm can be |
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57 | /// executed using the \ref run() function. If some parameters are not |
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58 | /// specified, then default values will be used. |
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59 | /// |
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60 | /// \tparam GR The digraph type the algorithm runs on. |
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61 | /// \tparam V The number type used for flow amounts, capacity bounds |
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62 | /// and supply values in the algorithm. By default, it is \c int. |
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63 | /// \tparam C The number type used for costs and potentials in the |
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64 | /// algorithm. By default, it is the same as \c V. |
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65 | /// |
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66 | /// \warning Both number types must be signed and all input data must |
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67 | /// be integer. |
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68 | /// |
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69 | /// \note %NetworkSimplex provides five different pivot rule |
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70 | /// implementations, from which the most efficient one is used |
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71 | /// by default. For more information, see \ref PivotRule. |
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72 | template <typename GR, typename V = int, typename C = V> |
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73 | class NetworkSimplex |
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74 | { |
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75 | public: |
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76 | |
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77 | /// The type of the flow amounts, capacity bounds and supply values |
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78 | typedef V Value; |
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79 | /// The type of the arc costs |
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80 | typedef C Cost; |
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81 | |
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82 | public: |
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83 | |
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84 | /// \brief Problem type constants for the \c run() function. |
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85 | /// |
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86 | /// Enum type containing the problem type constants that can be |
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87 | /// returned by the \ref run() function of the algorithm. |
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88 | enum ProblemType { |
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89 | /// The problem has no feasible solution (flow). |
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90 | INFEASIBLE, |
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91 | /// The problem has optimal solution (i.e. it is feasible and |
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92 | /// bounded), and the algorithm has found optimal flow and node |
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93 | /// potentials (primal and dual solutions). |
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94 | OPTIMAL, |
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95 | /// The objective function of the problem is unbounded, i.e. |
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96 | /// there is a directed cycle having negative total cost and |
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97 | /// infinite upper bound. |
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98 | UNBOUNDED |
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99 | }; |
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100 | |
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101 | /// \brief Constants for selecting the type of the supply constraints. |
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102 | /// |
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103 | /// Enum type containing constants for selecting the supply type, |
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104 | /// i.e. the direction of the inequalities in the supply/demand |
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105 | /// constraints of the \ref min_cost_flow "minimum cost flow problem". |
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106 | /// |
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107 | /// The default supply type is \c GEQ, the \c LEQ type can be |
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108 | /// selected using \ref supplyType(). |
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109 | /// The equality form is a special case of both supply types. |
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110 | enum SupplyType { |
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111 | /// This option means that there are <em>"greater or equal"</em> |
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112 | /// supply/demand constraints in the definition of the problem. |
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113 | GEQ, |
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114 | /// This option means that there are <em>"less or equal"</em> |
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115 | /// supply/demand constraints in the definition of the problem. |
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116 | LEQ |
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117 | }; |
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118 | |
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119 | /// \brief Constants for selecting the pivot rule. |
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120 | /// |
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121 | /// Enum type containing constants for selecting the pivot rule for |
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122 | /// the \ref run() function. |
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123 | /// |
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124 | /// \ref NetworkSimplex provides five different pivot rule |
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125 | /// implementations that significantly affect the running time |
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126 | /// of the algorithm. |
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127 | /// By default, \ref BLOCK_SEARCH "Block Search" is used, which |
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128 | /// proved to be the most efficient and the most robust on various |
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129 | /// test inputs. |
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130 | /// However, another pivot rule can be selected using the \ref run() |
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131 | /// function with the proper parameter. |
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132 | enum PivotRule { |
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133 | |
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134 | /// The \e First \e Eligible pivot rule. |
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135 | /// The next eligible arc is selected in a wraparound fashion |
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136 | /// in every iteration. |
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137 | FIRST_ELIGIBLE, |
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138 | |
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139 | /// The \e Best \e Eligible pivot rule. |
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140 | /// The best eligible arc is selected in every iteration. |
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141 | BEST_ELIGIBLE, |
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142 | |
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143 | /// The \e Block \e Search pivot rule. |
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144 | /// A specified number of arcs are examined in every iteration |
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145 | /// in a wraparound fashion and the best eligible arc is selected |
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146 | /// from this block. |
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147 | BLOCK_SEARCH, |
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148 | |
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149 | /// The \e Candidate \e List pivot rule. |
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150 | /// In a major iteration a candidate list is built from eligible arcs |
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151 | /// in a wraparound fashion and in the following minor iterations |
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152 | /// the best eligible arc is selected from this list. |
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153 | CANDIDATE_LIST, |
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154 | |
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155 | /// The \e Altering \e Candidate \e List pivot rule. |
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156 | /// It is a modified version of the Candidate List method. |
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157 | /// It keeps only the several best eligible arcs from the former |
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158 | /// candidate list and extends this list in every iteration. |
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159 | ALTERING_LIST |
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160 | }; |
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161 | |
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162 | private: |
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163 | |
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164 | TEMPLATE_DIGRAPH_TYPEDEFS(GR); |
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165 | |
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166 | typedef std::vector<int> IntVector; |
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167 | typedef std::vector<Value> ValueVector; |
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168 | typedef std::vector<Cost> CostVector; |
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169 | typedef std::vector<signed char> CharVector; |
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170 | // Note: vector<signed char> is used instead of vector<ArcState> and |
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171 | // vector<ArcDirection> for efficiency reasons |
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172 | |
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173 | // State constants for arcs |
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174 | enum ArcState { |
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175 | STATE_UPPER = -1, |
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176 | STATE_TREE = 0, |
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177 | STATE_LOWER = 1 |
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178 | }; |
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179 | |
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180 | // Direction constants for tree arcs |
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181 | enum ArcDirection { |
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182 | DIR_DOWN = -1, |
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183 | DIR_UP = 1 |
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184 | }; |
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185 | |
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186 | private: |
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187 | |
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188 | // Data related to the underlying digraph |
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189 | const GR &_graph; |
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190 | int _node_num; |
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191 | int _arc_num; |
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192 | int _all_arc_num; |
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193 | int _search_arc_num; |
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194 | |
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195 | // Parameters of the problem |
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196 | bool _have_lower; |
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197 | SupplyType _stype; |
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198 | Value _sum_supply; |
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199 | |
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200 | // Data structures for storing the digraph |
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201 | IntNodeMap _node_id; |
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202 | IntArcMap _arc_id; |
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203 | IntVector _source; |
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204 | IntVector _target; |
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205 | bool _arc_mixing; |
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206 | |
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207 | // Node and arc data |
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208 | ValueVector _lower; |
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209 | ValueVector _upper; |
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210 | ValueVector _cap; |
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211 | CostVector _cost; |
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212 | ValueVector _supply; |
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213 | ValueVector _flow; |
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214 | CostVector _pi; |
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215 | |
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216 | // Data for storing the spanning tree structure |
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217 | IntVector _parent; |
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218 | IntVector _pred; |
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219 | IntVector _thread; |
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220 | IntVector _rev_thread; |
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221 | IntVector _succ_num; |
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222 | IntVector _last_succ; |
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223 | CharVector _pred_dir; |
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224 | CharVector _state; |
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225 | IntVector _dirty_revs; |
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226 | int _root; |
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227 | |
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228 | // Temporary data used in the current pivot iteration |
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229 | int in_arc, join, u_in, v_in, u_out, v_out; |
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230 | Value delta; |
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231 | |
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232 | const Value MAX; |
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233 | |
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234 | public: |
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235 | |
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236 | /// \brief Constant for infinite upper bounds (capacities). |
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237 | /// |
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238 | /// Constant for infinite upper bounds (capacities). |
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239 | /// It is \c std::numeric_limits<Value>::infinity() if available, |
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240 | /// \c std::numeric_limits<Value>::max() otherwise. |
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241 | const Value INF; |
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242 | |
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243 | private: |
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244 | |
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245 | // Implementation of the First Eligible pivot rule |
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246 | class FirstEligiblePivotRule |
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247 | { |
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248 | private: |
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249 | |
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250 | // References to the NetworkSimplex class |
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251 | const IntVector &_source; |
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252 | const IntVector &_target; |
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253 | const CostVector &_cost; |
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254 | const CharVector &_state; |
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255 | const CostVector &_pi; |
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256 | int &_in_arc; |
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257 | int _search_arc_num; |
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258 | |
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259 | // Pivot rule data |
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260 | int _next_arc; |
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261 | |
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262 | public: |
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263 | |
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264 | // Constructor |
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265 | FirstEligiblePivotRule(NetworkSimplex &ns) : |
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266 | _source(ns._source), _target(ns._target), |
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267 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
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268 | _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num), |
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269 | _next_arc(0) |
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270 | {} |
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271 | |
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272 | // Find next entering arc |
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273 | bool findEnteringArc() { |
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274 | Cost c; |
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275 | for (int e = _next_arc; e != _search_arc_num; ++e) { |
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276 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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277 | if (c < 0) { |
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278 | _in_arc = e; |
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279 | _next_arc = e + 1; |
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280 | return true; |
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281 | } |
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282 | } |
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283 | for (int e = 0; e != _next_arc; ++e) { |
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284 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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285 | if (c < 0) { |
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286 | _in_arc = e; |
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287 | _next_arc = e + 1; |
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288 | return true; |
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289 | } |
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290 | } |
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291 | return false; |
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292 | } |
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293 | |
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294 | }; //class FirstEligiblePivotRule |
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295 | |
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296 | |
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297 | // Implementation of the Best Eligible pivot rule |
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298 | class BestEligiblePivotRule |
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299 | { |
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300 | private: |
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301 | |
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302 | // References to the NetworkSimplex class |
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303 | const IntVector &_source; |
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304 | const IntVector &_target; |
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305 | const CostVector &_cost; |
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306 | const CharVector &_state; |
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307 | const CostVector &_pi; |
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308 | int &_in_arc; |
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309 | int _search_arc_num; |
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310 | |
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311 | public: |
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312 | |
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313 | // Constructor |
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314 | BestEligiblePivotRule(NetworkSimplex &ns) : |
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315 | _source(ns._source), _target(ns._target), |
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316 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
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317 | _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num) |
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318 | {} |
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319 | |
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320 | // Find next entering arc |
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321 | bool findEnteringArc() { |
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322 | Cost c, min = 0; |
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323 | for (int e = 0; e != _search_arc_num; ++e) { |
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324 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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325 | if (c < min) { |
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326 | min = c; |
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327 | _in_arc = e; |
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328 | } |
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329 | } |
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330 | return min < 0; |
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331 | } |
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332 | |
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333 | }; //class BestEligiblePivotRule |
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334 | |
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335 | |
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336 | // Implementation of the Block Search pivot rule |
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337 | class BlockSearchPivotRule |
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338 | { |
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339 | private: |
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340 | |
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341 | // References to the NetworkSimplex class |
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342 | const IntVector &_source; |
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343 | const IntVector &_target; |
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344 | const CostVector &_cost; |
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345 | const CharVector &_state; |
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346 | const CostVector &_pi; |
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347 | int &_in_arc; |
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348 | int _search_arc_num; |
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349 | |
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350 | // Pivot rule data |
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351 | int _block_size; |
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352 | int _next_arc; |
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353 | |
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354 | public: |
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355 | |
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356 | // Constructor |
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357 | BlockSearchPivotRule(NetworkSimplex &ns) : |
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358 | _source(ns._source), _target(ns._target), |
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359 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
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360 | _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num), |
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361 | _next_arc(0) |
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362 | { |
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363 | // The main parameters of the pivot rule |
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364 | const double BLOCK_SIZE_FACTOR = 1.0; |
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365 | const int MIN_BLOCK_SIZE = 10; |
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366 | |
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367 | _block_size = std::max( int(BLOCK_SIZE_FACTOR * |
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368 | std::sqrt(double(_search_arc_num))), |
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369 | MIN_BLOCK_SIZE ); |
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370 | } |
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371 | |
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372 | // Find next entering arc |
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373 | bool findEnteringArc() { |
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374 | Cost c, min = 0; |
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375 | int cnt = _block_size; |
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376 | int e; |
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377 | for (e = _next_arc; e != _search_arc_num; ++e) { |
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378 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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379 | if (c < min) { |
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380 | min = c; |
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381 | _in_arc = e; |
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382 | } |
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383 | if (--cnt == 0) { |
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384 | if (min < 0) goto search_end; |
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385 | cnt = _block_size; |
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386 | } |
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387 | } |
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388 | for (e = 0; e != _next_arc; ++e) { |
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389 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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390 | if (c < min) { |
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391 | min = c; |
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392 | _in_arc = e; |
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393 | } |
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394 | if (--cnt == 0) { |
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395 | if (min < 0) goto search_end; |
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396 | cnt = _block_size; |
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397 | } |
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398 | } |
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399 | if (min >= 0) return false; |
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400 | |
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401 | search_end: |
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402 | _next_arc = e; |
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403 | return true; |
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404 | } |
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405 | |
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406 | }; //class BlockSearchPivotRule |
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407 | |
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408 | |
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409 | // Implementation of the Candidate List pivot rule |
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410 | class CandidateListPivotRule |
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411 | { |
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412 | private: |
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413 | |
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414 | // References to the NetworkSimplex class |
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415 | const IntVector &_source; |
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416 | const IntVector &_target; |
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417 | const CostVector &_cost; |
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418 | const CharVector &_state; |
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419 | const CostVector &_pi; |
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420 | int &_in_arc; |
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421 | int _search_arc_num; |
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422 | |
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423 | // Pivot rule data |
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424 | IntVector _candidates; |
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425 | int _list_length, _minor_limit; |
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426 | int _curr_length, _minor_count; |
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427 | int _next_arc; |
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428 | |
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429 | public: |
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430 | |
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431 | /// Constructor |
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432 | CandidateListPivotRule(NetworkSimplex &ns) : |
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433 | _source(ns._source), _target(ns._target), |
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434 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
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435 | _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num), |
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436 | _next_arc(0) |
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437 | { |
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438 | // The main parameters of the pivot rule |
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439 | const double LIST_LENGTH_FACTOR = 0.25; |
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440 | const int MIN_LIST_LENGTH = 10; |
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441 | const double MINOR_LIMIT_FACTOR = 0.1; |
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442 | const int MIN_MINOR_LIMIT = 3; |
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443 | |
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444 | _list_length = std::max( int(LIST_LENGTH_FACTOR * |
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445 | std::sqrt(double(_search_arc_num))), |
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446 | MIN_LIST_LENGTH ); |
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447 | _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length), |
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448 | MIN_MINOR_LIMIT ); |
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449 | _curr_length = _minor_count = 0; |
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450 | _candidates.resize(_list_length); |
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451 | } |
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452 | |
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453 | /// Find next entering arc |
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454 | bool findEnteringArc() { |
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455 | Cost min, c; |
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456 | int e; |
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457 | if (_curr_length > 0 && _minor_count < _minor_limit) { |
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458 | // Minor iteration: select the best eligible arc from the |
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459 | // current candidate list |
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460 | ++_minor_count; |
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461 | min = 0; |
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462 | for (int i = 0; i < _curr_length; ++i) { |
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463 | e = _candidates[i]; |
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464 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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465 | if (c < min) { |
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466 | min = c; |
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467 | _in_arc = e; |
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468 | } |
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469 | else if (c >= 0) { |
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470 | _candidates[i--] = _candidates[--_curr_length]; |
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471 | } |
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472 | } |
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473 | if (min < 0) return true; |
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474 | } |
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475 | |
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476 | // Major iteration: build a new candidate list |
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477 | min = 0; |
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478 | _curr_length = 0; |
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479 | for (e = _next_arc; e != _search_arc_num; ++e) { |
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480 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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481 | if (c < 0) { |
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482 | _candidates[_curr_length++] = e; |
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483 | if (c < min) { |
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484 | min = c; |
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485 | _in_arc = e; |
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486 | } |
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487 | if (_curr_length == _list_length) goto search_end; |
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488 | } |
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489 | } |
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490 | for (e = 0; e != _next_arc; ++e) { |
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491 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
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492 | if (c < 0) { |
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493 | _candidates[_curr_length++] = e; |
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494 | if (c < min) { |
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495 | min = c; |
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496 | _in_arc = e; |
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497 | } |
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498 | if (_curr_length == _list_length) goto search_end; |
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499 | } |
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500 | } |
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501 | if (_curr_length == 0) return false; |
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502 | |
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503 | search_end: |
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504 | _minor_count = 1; |
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505 | _next_arc = e; |
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506 | return true; |
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507 | } |
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508 | |
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509 | }; //class CandidateListPivotRule |
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510 | |
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511 | |
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512 | // Implementation of the Altering Candidate List pivot rule |
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513 | class AlteringListPivotRule |
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514 | { |
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515 | private: |
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516 | |
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517 | // References to the NetworkSimplex class |
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518 | const IntVector &_source; |
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519 | const IntVector &_target; |
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520 | const CostVector &_cost; |
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521 | const CharVector &_state; |
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522 | const CostVector &_pi; |
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523 | int &_in_arc; |
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524 | int _search_arc_num; |
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525 | |
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526 | // Pivot rule data |
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527 | int _block_size, _head_length, _curr_length; |
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528 | int _next_arc; |
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529 | IntVector _candidates; |
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530 | CostVector _cand_cost; |
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531 | |
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532 | // Functor class to compare arcs during sort of the candidate list |
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533 | class SortFunc |
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534 | { |
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535 | private: |
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536 | const CostVector &_map; |
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537 | public: |
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538 | SortFunc(const CostVector &map) : _map(map) {} |
---|
539 | bool operator()(int left, int right) { |
---|
540 | return _map[left] > _map[right]; |
---|
541 | } |
---|
542 | }; |
---|
543 | |
---|
544 | SortFunc _sort_func; |
---|
545 | |
---|
546 | public: |
---|
547 | |
---|
548 | // Constructor |
---|
549 | AlteringListPivotRule(NetworkSimplex &ns) : |
---|
550 | _source(ns._source), _target(ns._target), |
---|
551 | _cost(ns._cost), _state(ns._state), _pi(ns._pi), |
---|
552 | _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num), |
---|
553 | _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost) |
---|
554 | { |
---|
555 | // The main parameters of the pivot rule |
---|
556 | const double BLOCK_SIZE_FACTOR = 1.0; |
---|
557 | const int MIN_BLOCK_SIZE = 10; |
---|
558 | const double HEAD_LENGTH_FACTOR = 0.1; |
---|
559 | const int MIN_HEAD_LENGTH = 3; |
---|
560 | |
---|
561 | _block_size = std::max( int(BLOCK_SIZE_FACTOR * |
---|
562 | std::sqrt(double(_search_arc_num))), |
---|
563 | MIN_BLOCK_SIZE ); |
---|
564 | _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size), |
---|
565 | MIN_HEAD_LENGTH ); |
---|
566 | _candidates.resize(_head_length + _block_size); |
---|
567 | _curr_length = 0; |
---|
568 | } |
---|
569 | |
---|
570 | // Find next entering arc |
---|
571 | bool findEnteringArc() { |
---|
572 | // Check the current candidate list |
---|
573 | int e; |
---|
574 | Cost c; |
---|
575 | for (int i = 0; i != _curr_length; ++i) { |
---|
576 | e = _candidates[i]; |
---|
577 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
578 | if (c < 0) { |
---|
579 | _cand_cost[e] = c; |
---|
580 | } else { |
---|
581 | _candidates[i--] = _candidates[--_curr_length]; |
---|
582 | } |
---|
583 | } |
---|
584 | |
---|
585 | // Extend the list |
---|
586 | int cnt = _block_size; |
---|
587 | int limit = _head_length; |
---|
588 | |
---|
589 | for (e = _next_arc; e != _search_arc_num; ++e) { |
---|
590 | c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
591 | if (c < 0) { |
---|
592 | _cand_cost[e] = c; |
---|
593 | _candidates[_curr_length++] = e; |
---|
594 | } |
---|
595 | if (--cnt == 0) { |
---|
596 | if (_curr_length > limit) goto search_end; |
---|
597 | limit = 0; |
---|
598 | cnt = _block_size; |
---|
599 | } |
---|
600 | } |
---|
601 | for (e = 0; e != _next_arc; ++e) { |
---|
602 | _cand_cost[e] = _state[e] * |
---|
603 | (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); |
---|
604 | if (_cand_cost[e] < 0) { |
---|
605 | _candidates[_curr_length++] = e; |
---|
606 | } |
---|
607 | if (--cnt == 0) { |
---|
608 | if (_curr_length > limit) goto search_end; |
---|
609 | limit = 0; |
---|
610 | cnt = _block_size; |
---|
611 | } |
---|
612 | } |
---|
613 | if (_curr_length == 0) return false; |
---|
614 | |
---|
615 | search_end: |
---|
616 | |
---|
617 | // Make heap of the candidate list (approximating a partial sort) |
---|
618 | make_heap( _candidates.begin(), _candidates.begin() + _curr_length, |
---|
619 | _sort_func ); |
---|
620 | |
---|
621 | // Pop the first element of the heap |
---|
622 | _in_arc = _candidates[0]; |
---|
623 | _next_arc = e; |
---|
624 | pop_heap( _candidates.begin(), _candidates.begin() + _curr_length, |
---|
625 | _sort_func ); |
---|
626 | _curr_length = std::min(_head_length, _curr_length - 1); |
---|
627 | return true; |
---|
628 | } |
---|
629 | |
---|
630 | }; //class AlteringListPivotRule |
---|
631 | |
---|
632 | public: |
---|
633 | |
---|
634 | /// \brief Constructor. |
---|
635 | /// |
---|
636 | /// The constructor of the class. |
---|
637 | /// |
---|
638 | /// \param graph The digraph the algorithm runs on. |
---|
639 | /// \param arc_mixing Indicate if the arcs will be stored in a |
---|
640 | /// mixed order in the internal data structure. |
---|
641 | /// In general, it leads to similar performance as using the original |
---|
642 | /// arc order, but it makes the algorithm more robust and in special |
---|
643 | /// cases, even significantly faster. Therefore, it is enabled by default. |
---|
644 | NetworkSimplex(const GR& graph, bool arc_mixing = true) : |
---|
645 | _graph(graph), _node_id(graph), _arc_id(graph), |
---|
646 | _arc_mixing(arc_mixing), |
---|
647 | MAX(std::numeric_limits<Value>::max()), |
---|
648 | INF(std::numeric_limits<Value>::has_infinity ? |
---|
649 | std::numeric_limits<Value>::infinity() : MAX) |
---|
650 | { |
---|
651 | // Check the number types |
---|
652 | LEMON_ASSERT(std::numeric_limits<Value>::is_signed, |
---|
653 | "The flow type of NetworkSimplex must be signed"); |
---|
654 | LEMON_ASSERT(std::numeric_limits<Cost>::is_signed, |
---|
655 | "The cost type of NetworkSimplex must be signed"); |
---|
656 | |
---|
657 | // Reset data structures |
---|
658 | reset(); |
---|
659 | } |
---|
660 | |
---|
661 | /// \name Parameters |
---|
662 | /// The parameters of the algorithm can be specified using these |
---|
663 | /// functions. |
---|
664 | |
---|
665 | /// @{ |
---|
666 | |
---|
667 | /// \brief Set the lower bounds on the arcs. |
---|
668 | /// |
---|
669 | /// This function sets the lower bounds on the arcs. |
---|
670 | /// If it is not used before calling \ref run(), the lower bounds |
---|
671 | /// will be set to zero on all arcs. |
---|
672 | /// |
---|
673 | /// \param map An arc map storing the lower bounds. |
---|
674 | /// Its \c Value type must be convertible to the \c Value type |
---|
675 | /// of the algorithm. |
---|
676 | /// |
---|
677 | /// \return <tt>(*this)</tt> |
---|
678 | template <typename LowerMap> |
---|
679 | NetworkSimplex& lowerMap(const LowerMap& map) { |
---|
680 | _have_lower = true; |
---|
681 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
682 | _lower[_arc_id[a]] = map[a]; |
---|
683 | } |
---|
684 | return *this; |
---|
685 | } |
---|
686 | |
---|
687 | /// \brief Set the upper bounds (capacities) on the arcs. |
---|
688 | /// |
---|
689 | /// This function sets the upper bounds (capacities) on the arcs. |
---|
690 | /// If it is not used before calling \ref run(), the upper bounds |
---|
691 | /// will be set to \ref INF on all arcs (i.e. the flow value will be |
---|
692 | /// unbounded from above). |
---|
693 | /// |
---|
694 | /// \param map An arc map storing the upper bounds. |
---|
695 | /// Its \c Value type must be convertible to the \c Value type |
---|
696 | /// of the algorithm. |
---|
697 | /// |
---|
698 | /// \return <tt>(*this)</tt> |
---|
699 | template<typename UpperMap> |
---|
700 | NetworkSimplex& upperMap(const UpperMap& map) { |
---|
701 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
702 | _upper[_arc_id[a]] = map[a]; |
---|
703 | } |
---|
704 | return *this; |
---|
705 | } |
---|
706 | |
---|
707 | /// \brief Set the costs of the arcs. |
---|
708 | /// |
---|
709 | /// This function sets the costs of the arcs. |
---|
710 | /// If it is not used before calling \ref run(), the costs |
---|
711 | /// will be set to \c 1 on all arcs. |
---|
712 | /// |
---|
713 | /// \param map An arc map storing the costs. |
---|
714 | /// Its \c Value type must be convertible to the \c Cost type |
---|
715 | /// of the algorithm. |
---|
716 | /// |
---|
717 | /// \return <tt>(*this)</tt> |
---|
718 | template<typename CostMap> |
---|
719 | NetworkSimplex& costMap(const CostMap& map) { |
---|
720 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
721 | _cost[_arc_id[a]] = map[a]; |
---|
722 | } |
---|
723 | return *this; |
---|
724 | } |
---|
725 | |
---|
726 | /// \brief Set the supply values of the nodes. |
---|
727 | /// |
---|
728 | /// This function sets the supply values of the nodes. |
---|
729 | /// If neither this function nor \ref stSupply() is used before |
---|
730 | /// calling \ref run(), the supply of each node will be set to zero. |
---|
731 | /// |
---|
732 | /// \param map A node map storing the supply values. |
---|
733 | /// Its \c Value type must be convertible to the \c Value type |
---|
734 | /// of the algorithm. |
---|
735 | /// |
---|
736 | /// \return <tt>(*this)</tt> |
---|
737 | template<typename SupplyMap> |
---|
738 | NetworkSimplex& supplyMap(const SupplyMap& map) { |
---|
739 | for (NodeIt n(_graph); n != INVALID; ++n) { |
---|
740 | _supply[_node_id[n]] = map[n]; |
---|
741 | } |
---|
742 | return *this; |
---|
743 | } |
---|
744 | |
---|
745 | /// \brief Set single source and target nodes and a supply value. |
---|
746 | /// |
---|
747 | /// This function sets a single source node and a single target node |
---|
748 | /// and the required flow value. |
---|
749 | /// If neither this function nor \ref supplyMap() is used before |
---|
750 | /// calling \ref run(), the supply of each node will be set to zero. |
---|
751 | /// |
---|
752 | /// Using this function has the same effect as using \ref supplyMap() |
---|
753 | /// with such a map in which \c k is assigned to \c s, \c -k is |
---|
754 | /// assigned to \c t and all other nodes have zero supply value. |
---|
755 | /// |
---|
756 | /// \param s The source node. |
---|
757 | /// \param t The target node. |
---|
758 | /// \param k The required amount of flow from node \c s to node \c t |
---|
759 | /// (i.e. the supply of \c s and the demand of \c t). |
---|
760 | /// |
---|
761 | /// \return <tt>(*this)</tt> |
---|
762 | NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) { |
---|
763 | for (int i = 0; i != _node_num; ++i) { |
---|
764 | _supply[i] = 0; |
---|
765 | } |
---|
766 | _supply[_node_id[s]] = k; |
---|
767 | _supply[_node_id[t]] = -k; |
---|
768 | return *this; |
---|
769 | } |
---|
770 | |
---|
771 | /// \brief Set the type of the supply constraints. |
---|
772 | /// |
---|
773 | /// This function sets the type of the supply/demand constraints. |
---|
774 | /// If it is not used before calling \ref run(), the \ref GEQ supply |
---|
775 | /// type will be used. |
---|
776 | /// |
---|
777 | /// For more information, see \ref SupplyType. |
---|
778 | /// |
---|
779 | /// \return <tt>(*this)</tt> |
---|
780 | NetworkSimplex& supplyType(SupplyType supply_type) { |
---|
781 | _stype = supply_type; |
---|
782 | return *this; |
---|
783 | } |
---|
784 | |
---|
785 | /// @} |
---|
786 | |
---|
787 | /// \name Execution Control |
---|
788 | /// The algorithm can be executed using \ref run(). |
---|
789 | |
---|
790 | /// @{ |
---|
791 | |
---|
792 | /// \brief Run the algorithm. |
---|
793 | /// |
---|
794 | /// This function runs the algorithm. |
---|
795 | /// The paramters can be specified using functions \ref lowerMap(), |
---|
796 | /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), |
---|
797 | /// \ref supplyType(). |
---|
798 | /// For example, |
---|
799 | /// \code |
---|
800 | /// NetworkSimplex<ListDigraph> ns(graph); |
---|
801 | /// ns.lowerMap(lower).upperMap(upper).costMap(cost) |
---|
802 | /// .supplyMap(sup).run(); |
---|
803 | /// \endcode |
---|
804 | /// |
---|
805 | /// This function can be called more than once. All the given parameters |
---|
806 | /// are kept for the next call, unless \ref resetParams() or \ref reset() |
---|
807 | /// is used, thus only the modified parameters have to be set again. |
---|
808 | /// If the underlying digraph was also modified after the construction |
---|
809 | /// of the class (or the last \ref reset() call), then the \ref reset() |
---|
810 | /// function must be called. |
---|
811 | /// |
---|
812 | /// \param pivot_rule The pivot rule that will be used during the |
---|
813 | /// algorithm. For more information, see \ref PivotRule. |
---|
814 | /// |
---|
815 | /// \return \c INFEASIBLE if no feasible flow exists, |
---|
816 | /// \n \c OPTIMAL if the problem has optimal solution |
---|
817 | /// (i.e. it is feasible and bounded), and the algorithm has found |
---|
818 | /// optimal flow and node potentials (primal and dual solutions), |
---|
819 | /// \n \c UNBOUNDED if the objective function of the problem is |
---|
820 | /// unbounded, i.e. there is a directed cycle having negative total |
---|
821 | /// cost and infinite upper bound. |
---|
822 | /// |
---|
823 | /// \see ProblemType, PivotRule |
---|
824 | /// \see resetParams(), reset() |
---|
825 | ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) { |
---|
826 | if (!init()) return INFEASIBLE; |
---|
827 | return start(pivot_rule); |
---|
828 | } |
---|
829 | |
---|
830 | /// \brief Reset all the parameters that have been given before. |
---|
831 | /// |
---|
832 | /// This function resets all the paramaters that have been given |
---|
833 | /// before using functions \ref lowerMap(), \ref upperMap(), |
---|
834 | /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType(). |
---|
835 | /// |
---|
836 | /// It is useful for multiple \ref run() calls. Basically, all the given |
---|
837 | /// parameters are kept for the next \ref run() call, unless |
---|
838 | /// \ref resetParams() or \ref reset() is used. |
---|
839 | /// If the underlying digraph was also modified after the construction |
---|
840 | /// of the class or the last \ref reset() call, then the \ref reset() |
---|
841 | /// function must be used, otherwise \ref resetParams() is sufficient. |
---|
842 | /// |
---|
843 | /// For example, |
---|
844 | /// \code |
---|
845 | /// NetworkSimplex<ListDigraph> ns(graph); |
---|
846 | /// |
---|
847 | /// // First run |
---|
848 | /// ns.lowerMap(lower).upperMap(upper).costMap(cost) |
---|
849 | /// .supplyMap(sup).run(); |
---|
850 | /// |
---|
851 | /// // Run again with modified cost map (resetParams() is not called, |
---|
852 | /// // so only the cost map have to be set again) |
---|
853 | /// cost[e] += 100; |
---|
854 | /// ns.costMap(cost).run(); |
---|
855 | /// |
---|
856 | /// // Run again from scratch using resetParams() |
---|
857 | /// // (the lower bounds will be set to zero on all arcs) |
---|
858 | /// ns.resetParams(); |
---|
859 | /// ns.upperMap(capacity).costMap(cost) |
---|
860 | /// .supplyMap(sup).run(); |
---|
861 | /// \endcode |
---|
862 | /// |
---|
863 | /// \return <tt>(*this)</tt> |
---|
864 | /// |
---|
865 | /// \see reset(), run() |
---|
866 | NetworkSimplex& resetParams() { |
---|
867 | for (int i = 0; i != _node_num; ++i) { |
---|
868 | _supply[i] = 0; |
---|
869 | } |
---|
870 | for (int i = 0; i != _arc_num; ++i) { |
---|
871 | _lower[i] = 0; |
---|
872 | _upper[i] = INF; |
---|
873 | _cost[i] = 1; |
---|
874 | } |
---|
875 | _have_lower = false; |
---|
876 | _stype = GEQ; |
---|
877 | return *this; |
---|
878 | } |
---|
879 | |
---|
880 | /// \brief Reset the internal data structures and all the parameters |
---|
881 | /// that have been given before. |
---|
882 | /// |
---|
883 | /// This function resets the internal data structures and all the |
---|
884 | /// paramaters that have been given before using functions \ref lowerMap(), |
---|
885 | /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), |
---|
886 | /// \ref supplyType(). |
---|
887 | /// |
---|
888 | /// It is useful for multiple \ref run() calls. Basically, all the given |
---|
889 | /// parameters are kept for the next \ref run() call, unless |
---|
890 | /// \ref resetParams() or \ref reset() is used. |
---|
891 | /// If the underlying digraph was also modified after the construction |
---|
892 | /// of the class or the last \ref reset() call, then the \ref reset() |
---|
893 | /// function must be used, otherwise \ref resetParams() is sufficient. |
---|
894 | /// |
---|
895 | /// See \ref resetParams() for examples. |
---|
896 | /// |
---|
897 | /// \return <tt>(*this)</tt> |
---|
898 | /// |
---|
899 | /// \see resetParams(), run() |
---|
900 | NetworkSimplex& reset() { |
---|
901 | // Resize vectors |
---|
902 | _node_num = countNodes(_graph); |
---|
903 | _arc_num = countArcs(_graph); |
---|
904 | int all_node_num = _node_num + 1; |
---|
905 | int max_arc_num = _arc_num + 2 * _node_num; |
---|
906 | |
---|
907 | _source.resize(max_arc_num); |
---|
908 | _target.resize(max_arc_num); |
---|
909 | |
---|
910 | _lower.resize(_arc_num); |
---|
911 | _upper.resize(_arc_num); |
---|
912 | _cap.resize(max_arc_num); |
---|
913 | _cost.resize(max_arc_num); |
---|
914 | _supply.resize(all_node_num); |
---|
915 | _flow.resize(max_arc_num); |
---|
916 | _pi.resize(all_node_num); |
---|
917 | |
---|
918 | _parent.resize(all_node_num); |
---|
919 | _pred.resize(all_node_num); |
---|
920 | _pred_dir.resize(all_node_num); |
---|
921 | _thread.resize(all_node_num); |
---|
922 | _rev_thread.resize(all_node_num); |
---|
923 | _succ_num.resize(all_node_num); |
---|
924 | _last_succ.resize(all_node_num); |
---|
925 | _state.resize(max_arc_num); |
---|
926 | |
---|
927 | // Copy the graph |
---|
928 | int i = 0; |
---|
929 | for (NodeIt n(_graph); n != INVALID; ++n, ++i) { |
---|
930 | _node_id[n] = i; |
---|
931 | } |
---|
932 | if (_arc_mixing && _node_num > 1) { |
---|
933 | // Store the arcs in a mixed order |
---|
934 | const int skip = std::max(_arc_num / _node_num, 3); |
---|
935 | int i = 0, j = 0; |
---|
936 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
937 | _arc_id[a] = i; |
---|
938 | _source[i] = _node_id[_graph.source(a)]; |
---|
939 | _target[i] = _node_id[_graph.target(a)]; |
---|
940 | if ((i += skip) >= _arc_num) i = ++j; |
---|
941 | } |
---|
942 | } else { |
---|
943 | // Store the arcs in the original order |
---|
944 | int i = 0; |
---|
945 | for (ArcIt a(_graph); a != INVALID; ++a, ++i) { |
---|
946 | _arc_id[a] = i; |
---|
947 | _source[i] = _node_id[_graph.source(a)]; |
---|
948 | _target[i] = _node_id[_graph.target(a)]; |
---|
949 | } |
---|
950 | } |
---|
951 | |
---|
952 | // Reset parameters |
---|
953 | resetParams(); |
---|
954 | return *this; |
---|
955 | } |
---|
956 | |
---|
957 | /// @} |
---|
958 | |
---|
959 | /// \name Query Functions |
---|
960 | /// The results of the algorithm can be obtained using these |
---|
961 | /// functions.\n |
---|
962 | /// The \ref run() function must be called before using them. |
---|
963 | |
---|
964 | /// @{ |
---|
965 | |
---|
966 | /// \brief Return the total cost of the found flow. |
---|
967 | /// |
---|
968 | /// This function returns the total cost of the found flow. |
---|
969 | /// Its complexity is O(e). |
---|
970 | /// |
---|
971 | /// \note The return type of the function can be specified as a |
---|
972 | /// template parameter. For example, |
---|
973 | /// \code |
---|
974 | /// ns.totalCost<double>(); |
---|
975 | /// \endcode |
---|
976 | /// It is useful if the total cost cannot be stored in the \c Cost |
---|
977 | /// type of the algorithm, which is the default return type of the |
---|
978 | /// function. |
---|
979 | /// |
---|
980 | /// \pre \ref run() must be called before using this function. |
---|
981 | template <typename Number> |
---|
982 | Number totalCost() const { |
---|
983 | Number c = 0; |
---|
984 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
985 | int i = _arc_id[a]; |
---|
986 | c += Number(_flow[i]) * Number(_cost[i]); |
---|
987 | } |
---|
988 | return c; |
---|
989 | } |
---|
990 | |
---|
991 | #ifndef DOXYGEN |
---|
992 | Cost totalCost() const { |
---|
993 | return totalCost<Cost>(); |
---|
994 | } |
---|
995 | #endif |
---|
996 | |
---|
997 | /// \brief Return the flow on the given arc. |
---|
998 | /// |
---|
999 | /// This function returns the flow on the given arc. |
---|
1000 | /// |
---|
1001 | /// \pre \ref run() must be called before using this function. |
---|
1002 | Value flow(const Arc& a) const { |
---|
1003 | return _flow[_arc_id[a]]; |
---|
1004 | } |
---|
1005 | |
---|
1006 | /// \brief Return the flow map (the primal solution). |
---|
1007 | /// |
---|
1008 | /// This function copies the flow value on each arc into the given |
---|
1009 | /// map. The \c Value type of the algorithm must be convertible to |
---|
1010 | /// the \c Value type of the map. |
---|
1011 | /// |
---|
1012 | /// \pre \ref run() must be called before using this function. |
---|
1013 | template <typename FlowMap> |
---|
1014 | void flowMap(FlowMap &map) const { |
---|
1015 | for (ArcIt a(_graph); a != INVALID; ++a) { |
---|
1016 | map.set(a, _flow[_arc_id[a]]); |
---|
1017 | } |
---|
1018 | } |
---|
1019 | |
---|
1020 | /// \brief Return the potential (dual value) of the given node. |
---|
1021 | /// |
---|
1022 | /// This function returns the potential (dual value) of the |
---|
1023 | /// given node. |
---|
1024 | /// |
---|
1025 | /// \pre \ref run() must be called before using this function. |
---|
1026 | Cost potential(const Node& n) const { |
---|
1027 | return _pi[_node_id[n]]; |
---|
1028 | } |
---|
1029 | |
---|
1030 | /// \brief Return the potential map (the dual solution). |
---|
1031 | /// |
---|
1032 | /// This function copies the potential (dual value) of each node |
---|
1033 | /// into the given map. |
---|
1034 | /// The \c Cost type of the algorithm must be convertible to the |
---|
1035 | /// \c Value type of the map. |
---|
1036 | /// |
---|
1037 | /// \pre \ref run() must be called before using this function. |
---|
1038 | template <typename PotentialMap> |
---|
1039 | void potentialMap(PotentialMap &map) const { |
---|
1040 | for (NodeIt n(_graph); n != INVALID; ++n) { |
---|
1041 | map.set(n, _pi[_node_id[n]]); |
---|
1042 | } |
---|
1043 | } |
---|
1044 | |
---|
1045 | /// @} |
---|
1046 | |
---|
1047 | private: |
---|
1048 | |
---|
1049 | // Initialize internal data structures |
---|
1050 | bool init() { |
---|
1051 | if (_node_num == 0) return false; |
---|
1052 | |
---|
1053 | // Check the sum of supply values |
---|
1054 | _sum_supply = 0; |
---|
1055 | for (int i = 0; i != _node_num; ++i) { |
---|
1056 | _sum_supply += _supply[i]; |
---|
1057 | } |
---|
1058 | if ( !((_stype == GEQ && _sum_supply <= 0) || |
---|
1059 | (_stype == LEQ && _sum_supply >= 0)) ) return false; |
---|
1060 | |
---|
1061 | // Remove non-zero lower bounds |
---|
1062 | if (_have_lower) { |
---|
1063 | for (int i = 0; i != _arc_num; ++i) { |
---|
1064 | Value c = _lower[i]; |
---|
1065 | if (c >= 0) { |
---|
1066 | _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF; |
---|
1067 | } else { |
---|
1068 | _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF; |
---|
1069 | } |
---|
1070 | _supply[_source[i]] -= c; |
---|
1071 | _supply[_target[i]] += c; |
---|
1072 | } |
---|
1073 | } else { |
---|
1074 | for (int i = 0; i != _arc_num; ++i) { |
---|
1075 | _cap[i] = _upper[i]; |
---|
1076 | } |
---|
1077 | } |
---|
1078 | |
---|
1079 | // Initialize artifical cost |
---|
1080 | Cost ART_COST; |
---|
1081 | if (std::numeric_limits<Cost>::is_exact) { |
---|
1082 | ART_COST = std::numeric_limits<Cost>::max() / 2 + 1; |
---|
1083 | } else { |
---|
1084 | ART_COST = 0; |
---|
1085 | for (int i = 0; i != _arc_num; ++i) { |
---|
1086 | if (_cost[i] > ART_COST) ART_COST = _cost[i]; |
---|
1087 | } |
---|
1088 | ART_COST = (ART_COST + 1) * _node_num; |
---|
1089 | } |
---|
1090 | |
---|
1091 | // Initialize arc maps |
---|
1092 | for (int i = 0; i != _arc_num; ++i) { |
---|
1093 | _flow[i] = 0; |
---|
1094 | _state[i] = STATE_LOWER; |
---|
1095 | } |
---|
1096 | |
---|
1097 | // Set data for the artificial root node |
---|
1098 | _root = _node_num; |
---|
1099 | _parent[_root] = -1; |
---|
1100 | _pred[_root] = -1; |
---|
1101 | _thread[_root] = 0; |
---|
1102 | _rev_thread[0] = _root; |
---|
1103 | _succ_num[_root] = _node_num + 1; |
---|
1104 | _last_succ[_root] = _root - 1; |
---|
1105 | _supply[_root] = -_sum_supply; |
---|
1106 | _pi[_root] = 0; |
---|
1107 | |
---|
1108 | // Add artificial arcs and initialize the spanning tree data structure |
---|
1109 | if (_sum_supply == 0) { |
---|
1110 | // EQ supply constraints |
---|
1111 | _search_arc_num = _arc_num; |
---|
1112 | _all_arc_num = _arc_num + _node_num; |
---|
1113 | for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { |
---|
1114 | _parent[u] = _root; |
---|
1115 | _pred[u] = e; |
---|
1116 | _thread[u] = u + 1; |
---|
1117 | _rev_thread[u + 1] = u; |
---|
1118 | _succ_num[u] = 1; |
---|
1119 | _last_succ[u] = u; |
---|
1120 | _cap[e] = INF; |
---|
1121 | _state[e] = STATE_TREE; |
---|
1122 | if (_supply[u] >= 0) { |
---|
1123 | _pred_dir[u] = DIR_UP; |
---|
1124 | _pi[u] = 0; |
---|
1125 | _source[e] = u; |
---|
1126 | _target[e] = _root; |
---|
1127 | _flow[e] = _supply[u]; |
---|
1128 | _cost[e] = 0; |
---|
1129 | } else { |
---|
1130 | _pred_dir[u] = DIR_DOWN; |
---|
1131 | _pi[u] = ART_COST; |
---|
1132 | _source[e] = _root; |
---|
1133 | _target[e] = u; |
---|
1134 | _flow[e] = -_supply[u]; |
---|
1135 | _cost[e] = ART_COST; |
---|
1136 | } |
---|
1137 | } |
---|
1138 | } |
---|
1139 | else if (_sum_supply > 0) { |
---|
1140 | // LEQ supply constraints |
---|
1141 | _search_arc_num = _arc_num + _node_num; |
---|
1142 | int f = _arc_num + _node_num; |
---|
1143 | for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { |
---|
1144 | _parent[u] = _root; |
---|
1145 | _thread[u] = u + 1; |
---|
1146 | _rev_thread[u + 1] = u; |
---|
1147 | _succ_num[u] = 1; |
---|
1148 | _last_succ[u] = u; |
---|
1149 | if (_supply[u] >= 0) { |
---|
1150 | _pred_dir[u] = DIR_UP; |
---|
1151 | _pi[u] = 0; |
---|
1152 | _pred[u] = e; |
---|
1153 | _source[e] = u; |
---|
1154 | _target[e] = _root; |
---|
1155 | _cap[e] = INF; |
---|
1156 | _flow[e] = _supply[u]; |
---|
1157 | _cost[e] = 0; |
---|
1158 | _state[e] = STATE_TREE; |
---|
1159 | } else { |
---|
1160 | _pred_dir[u] = DIR_DOWN; |
---|
1161 | _pi[u] = ART_COST; |
---|
1162 | _pred[u] = f; |
---|
1163 | _source[f] = _root; |
---|
1164 | _target[f] = u; |
---|
1165 | _cap[f] = INF; |
---|
1166 | _flow[f] = -_supply[u]; |
---|
1167 | _cost[f] = ART_COST; |
---|
1168 | _state[f] = STATE_TREE; |
---|
1169 | _source[e] = u; |
---|
1170 | _target[e] = _root; |
---|
1171 | _cap[e] = INF; |
---|
1172 | _flow[e] = 0; |
---|
1173 | _cost[e] = 0; |
---|
1174 | _state[e] = STATE_LOWER; |
---|
1175 | ++f; |
---|
1176 | } |
---|
1177 | } |
---|
1178 | _all_arc_num = f; |
---|
1179 | } |
---|
1180 | else { |
---|
1181 | // GEQ supply constraints |
---|
1182 | _search_arc_num = _arc_num + _node_num; |
---|
1183 | int f = _arc_num + _node_num; |
---|
1184 | for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { |
---|
1185 | _parent[u] = _root; |
---|
1186 | _thread[u] = u + 1; |
---|
1187 | _rev_thread[u + 1] = u; |
---|
1188 | _succ_num[u] = 1; |
---|
1189 | _last_succ[u] = u; |
---|
1190 | if (_supply[u] <= 0) { |
---|
1191 | _pred_dir[u] = DIR_DOWN; |
---|
1192 | _pi[u] = 0; |
---|
1193 | _pred[u] = e; |
---|
1194 | _source[e] = _root; |
---|
1195 | _target[e] = u; |
---|
1196 | _cap[e] = INF; |
---|
1197 | _flow[e] = -_supply[u]; |
---|
1198 | _cost[e] = 0; |
---|
1199 | _state[e] = STATE_TREE; |
---|
1200 | } else { |
---|
1201 | _pred_dir[u] = DIR_UP; |
---|
1202 | _pi[u] = -ART_COST; |
---|
1203 | _pred[u] = f; |
---|
1204 | _source[f] = u; |
---|
1205 | _target[f] = _root; |
---|
1206 | _cap[f] = INF; |
---|
1207 | _flow[f] = _supply[u]; |
---|
1208 | _state[f] = STATE_TREE; |
---|
1209 | _cost[f] = ART_COST; |
---|
1210 | _source[e] = _root; |
---|
1211 | _target[e] = u; |
---|
1212 | _cap[e] = INF; |
---|
1213 | _flow[e] = 0; |
---|
1214 | _cost[e] = 0; |
---|
1215 | _state[e] = STATE_LOWER; |
---|
1216 | ++f; |
---|
1217 | } |
---|
1218 | } |
---|
1219 | _all_arc_num = f; |
---|
1220 | } |
---|
1221 | |
---|
1222 | return true; |
---|
1223 | } |
---|
1224 | |
---|
1225 | // Find the join node |
---|
1226 | void findJoinNode() { |
---|
1227 | int u = _source[in_arc]; |
---|
1228 | int v = _target[in_arc]; |
---|
1229 | while (u != v) { |
---|
1230 | if (_succ_num[u] < _succ_num[v]) { |
---|
1231 | u = _parent[u]; |
---|
1232 | } else { |
---|
1233 | v = _parent[v]; |
---|
1234 | } |
---|
1235 | } |
---|
1236 | join = u; |
---|
1237 | } |
---|
1238 | |
---|
1239 | // Find the leaving arc of the cycle and returns true if the |
---|
1240 | // leaving arc is not the same as the entering arc |
---|
1241 | bool findLeavingArc() { |
---|
1242 | // Initialize first and second nodes according to the direction |
---|
1243 | // of the cycle |
---|
1244 | int first, second; |
---|
1245 | if (_state[in_arc] == STATE_LOWER) { |
---|
1246 | first = _source[in_arc]; |
---|
1247 | second = _target[in_arc]; |
---|
1248 | } else { |
---|
1249 | first = _target[in_arc]; |
---|
1250 | second = _source[in_arc]; |
---|
1251 | } |
---|
1252 | delta = _cap[in_arc]; |
---|
1253 | int result = 0; |
---|
1254 | Value c, d; |
---|
1255 | int e; |
---|
1256 | |
---|
1257 | // Search the cycle form the first node to the join node |
---|
1258 | for (int u = first; u != join; u = _parent[u]) { |
---|
1259 | e = _pred[u]; |
---|
1260 | d = _flow[e]; |
---|
1261 | if (_pred_dir[u] == DIR_DOWN) { |
---|
1262 | c = _cap[e]; |
---|
1263 | d = c >= MAX ? INF : c - d; |
---|
1264 | } |
---|
1265 | if (d < delta) { |
---|
1266 | delta = d; |
---|
1267 | u_out = u; |
---|
1268 | result = 1; |
---|
1269 | } |
---|
1270 | } |
---|
1271 | |
---|
1272 | // Search the cycle form the second node to the join node |
---|
1273 | for (int u = second; u != join; u = _parent[u]) { |
---|
1274 | e = _pred[u]; |
---|
1275 | d = _flow[e]; |
---|
1276 | if (_pred_dir[u] == DIR_UP) { |
---|
1277 | c = _cap[e]; |
---|
1278 | d = c >= MAX ? INF : c - d; |
---|
1279 | } |
---|
1280 | if (d <= delta) { |
---|
1281 | delta = d; |
---|
1282 | u_out = u; |
---|
1283 | result = 2; |
---|
1284 | } |
---|
1285 | } |
---|
1286 | |
---|
1287 | if (result == 1) { |
---|
1288 | u_in = first; |
---|
1289 | v_in = second; |
---|
1290 | } else { |
---|
1291 | u_in = second; |
---|
1292 | v_in = first; |
---|
1293 | } |
---|
1294 | return result != 0; |
---|
1295 | } |
---|
1296 | |
---|
1297 | // Change _flow and _state vectors |
---|
1298 | void changeFlow(bool change) { |
---|
1299 | // Augment along the cycle |
---|
1300 | if (delta > 0) { |
---|
1301 | Value val = _state[in_arc] * delta; |
---|
1302 | _flow[in_arc] += val; |
---|
1303 | for (int u = _source[in_arc]; u != join; u = _parent[u]) { |
---|
1304 | _flow[_pred[u]] -= _pred_dir[u] * val; |
---|
1305 | } |
---|
1306 | for (int u = _target[in_arc]; u != join; u = _parent[u]) { |
---|
1307 | _flow[_pred[u]] += _pred_dir[u] * val; |
---|
1308 | } |
---|
1309 | } |
---|
1310 | // Update the state of the entering and leaving arcs |
---|
1311 | if (change) { |
---|
1312 | _state[in_arc] = STATE_TREE; |
---|
1313 | _state[_pred[u_out]] = |
---|
1314 | (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER; |
---|
1315 | } else { |
---|
1316 | _state[in_arc] = -_state[in_arc]; |
---|
1317 | } |
---|
1318 | } |
---|
1319 | |
---|
1320 | // Update the tree structure |
---|
1321 | void updateTreeStructure() { |
---|
1322 | int old_rev_thread = _rev_thread[u_out]; |
---|
1323 | int old_succ_num = _succ_num[u_out]; |
---|
1324 | int old_last_succ = _last_succ[u_out]; |
---|
1325 | v_out = _parent[u_out]; |
---|
1326 | |
---|
1327 | // Check if u_in and u_out coincide |
---|
1328 | if (u_in == u_out) { |
---|
1329 | // Update _parent, _pred, _pred_dir |
---|
1330 | _parent[u_in] = v_in; |
---|
1331 | _pred[u_in] = in_arc; |
---|
1332 | _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN; |
---|
1333 | |
---|
1334 | // Update _thread and _rev_thread |
---|
1335 | if (_thread[v_in] != u_out) { |
---|
1336 | int after = _thread[old_last_succ]; |
---|
1337 | _thread[old_rev_thread] = after; |
---|
1338 | _rev_thread[after] = old_rev_thread; |
---|
1339 | after = _thread[v_in]; |
---|
1340 | _thread[v_in] = u_out; |
---|
1341 | _rev_thread[u_out] = v_in; |
---|
1342 | _thread[old_last_succ] = after; |
---|
1343 | _rev_thread[after] = old_last_succ; |
---|
1344 | } |
---|
1345 | } else { |
---|
1346 | // Handle the case when old_rev_thread equals to v_in |
---|
1347 | // (it also means that join and v_out coincide) |
---|
1348 | int thread_continue = old_rev_thread == v_in ? |
---|
1349 | _thread[old_last_succ] : _thread[v_in]; |
---|
1350 | |
---|
1351 | // Update _thread and _parent along the stem nodes (i.e. the nodes |
---|
1352 | // between u_in and u_out, whose parent have to be changed) |
---|
1353 | int stem = u_in; // the current stem node |
---|
1354 | int par_stem = v_in; // the new parent of stem |
---|
1355 | int next_stem; // the next stem node |
---|
1356 | int last = _last_succ[u_in]; // the last successor of stem |
---|
1357 | int before, after = _thread[last]; |
---|
1358 | _thread[v_in] = u_in; |
---|
1359 | _dirty_revs.clear(); |
---|
1360 | _dirty_revs.push_back(v_in); |
---|
1361 | while (stem != u_out) { |
---|
1362 | // Insert the next stem node into the thread list |
---|
1363 | next_stem = _parent[stem]; |
---|
1364 | _thread[last] = next_stem; |
---|
1365 | _dirty_revs.push_back(last); |
---|
1366 | |
---|
1367 | // Remove the subtree of stem from the thread list |
---|
1368 | before = _rev_thread[stem]; |
---|
1369 | _thread[before] = after; |
---|
1370 | _rev_thread[after] = before; |
---|
1371 | |
---|
1372 | // Change the parent node and shift stem nodes |
---|
1373 | _parent[stem] = par_stem; |
---|
1374 | par_stem = stem; |
---|
1375 | stem = next_stem; |
---|
1376 | |
---|
1377 | // Update last and after |
---|
1378 | last = _last_succ[stem] == _last_succ[par_stem] ? |
---|
1379 | _rev_thread[par_stem] : _last_succ[stem]; |
---|
1380 | after = _thread[last]; |
---|
1381 | } |
---|
1382 | _parent[u_out] = par_stem; |
---|
1383 | _thread[last] = thread_continue; |
---|
1384 | _rev_thread[thread_continue] = last; |
---|
1385 | _last_succ[u_out] = last; |
---|
1386 | |
---|
1387 | // Remove the subtree of u_out from the thread list except for |
---|
1388 | // the case when old_rev_thread equals to v_in |
---|
1389 | if (old_rev_thread != v_in) { |
---|
1390 | _thread[old_rev_thread] = after; |
---|
1391 | _rev_thread[after] = old_rev_thread; |
---|
1392 | } |
---|
1393 | |
---|
1394 | // Update _rev_thread using the new _thread values |
---|
1395 | for (int i = 0; i != int(_dirty_revs.size()); ++i) { |
---|
1396 | int u = _dirty_revs[i]; |
---|
1397 | _rev_thread[_thread[u]] = u; |
---|
1398 | } |
---|
1399 | |
---|
1400 | // Update _pred, _pred_dir, _last_succ and _succ_num for the |
---|
1401 | // stem nodes from u_out to u_in |
---|
1402 | int tmp_sc = 0, tmp_ls = _last_succ[u_out]; |
---|
1403 | for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) { |
---|
1404 | _pred[u] = _pred[p]; |
---|
1405 | _pred_dir[u] = -_pred_dir[p]; |
---|
1406 | tmp_sc += _succ_num[u] - _succ_num[p]; |
---|
1407 | _succ_num[u] = tmp_sc; |
---|
1408 | _last_succ[p] = tmp_ls; |
---|
1409 | } |
---|
1410 | _pred[u_in] = in_arc; |
---|
1411 | _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN; |
---|
1412 | _succ_num[u_in] = old_succ_num; |
---|
1413 | } |
---|
1414 | |
---|
1415 | // Update _last_succ from v_in towards the root |
---|
1416 | int up_limit_out = _last_succ[join] == v_in ? join : -1; |
---|
1417 | int last_succ_out = _last_succ[u_out]; |
---|
1418 | for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) { |
---|
1419 | _last_succ[u] = last_succ_out; |
---|
1420 | } |
---|
1421 | |
---|
1422 | // Update _last_succ from v_out towards the root |
---|
1423 | if (join != old_rev_thread && v_in != old_rev_thread) { |
---|
1424 | for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; |
---|
1425 | u = _parent[u]) { |
---|
1426 | _last_succ[u] = old_rev_thread; |
---|
1427 | } |
---|
1428 | } |
---|
1429 | else if (last_succ_out != old_last_succ) { |
---|
1430 | for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; |
---|
1431 | u = _parent[u]) { |
---|
1432 | _last_succ[u] = last_succ_out; |
---|
1433 | } |
---|
1434 | } |
---|
1435 | |
---|
1436 | // Update _succ_num from v_in to join |
---|
1437 | for (int u = v_in; u != join; u = _parent[u]) { |
---|
1438 | _succ_num[u] += old_succ_num; |
---|
1439 | } |
---|
1440 | // Update _succ_num from v_out to join |
---|
1441 | for (int u = v_out; u != join; u = _parent[u]) { |
---|
1442 | _succ_num[u] -= old_succ_num; |
---|
1443 | } |
---|
1444 | } |
---|
1445 | |
---|
1446 | // Update potentials in the subtree that has been moved |
---|
1447 | void updatePotential() { |
---|
1448 | Cost sigma = _pi[v_in] - _pi[u_in] - |
---|
1449 | _pred_dir[u_in] * _cost[in_arc]; |
---|
1450 | int end = _thread[_last_succ[u_in]]; |
---|
1451 | for (int u = u_in; u != end; u = _thread[u]) { |
---|
1452 | _pi[u] += sigma; |
---|
1453 | } |
---|
1454 | } |
---|
1455 | |
---|
1456 | // Heuristic initial pivots |
---|
1457 | bool initialPivots() { |
---|
1458 | Value curr, total = 0; |
---|
1459 | std::vector<Node> supply_nodes, demand_nodes; |
---|
1460 | for (NodeIt u(_graph); u != INVALID; ++u) { |
---|
1461 | curr = _supply[_node_id[u]]; |
---|
1462 | if (curr > 0) { |
---|
1463 | total += curr; |
---|
1464 | supply_nodes.push_back(u); |
---|
1465 | } |
---|
1466 | else if (curr < 0) { |
---|
1467 | demand_nodes.push_back(u); |
---|
1468 | } |
---|
1469 | } |
---|
1470 | if (_sum_supply > 0) total -= _sum_supply; |
---|
1471 | if (total <= 0) return true; |
---|
1472 | |
---|
1473 | IntVector arc_vector; |
---|
1474 | if (_sum_supply >= 0) { |
---|
1475 | if (supply_nodes.size() == 1 && demand_nodes.size() == 1) { |
---|
1476 | // Perform a reverse graph search from the sink to the source |
---|
1477 | typename GR::template NodeMap<bool> reached(_graph, false); |
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1478 | Node s = supply_nodes[0], t = demand_nodes[0]; |
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1479 | std::vector<Node> stack; |
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1480 | reached[t] = true; |
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1481 | stack.push_back(t); |
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1482 | while (!stack.empty()) { |
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1483 | Node u, v = stack.back(); |
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1484 | stack.pop_back(); |
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1485 | if (v == s) break; |
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1486 | for (InArcIt a(_graph, v); a != INVALID; ++a) { |
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1487 | if (reached[u = _graph.source(a)]) continue; |
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1488 | int j = _arc_id[a]; |
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1489 | if (_cap[j] >= total) { |
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1490 | arc_vector.push_back(j); |
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1491 | reached[u] = true; |
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1492 | stack.push_back(u); |
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1493 | } |
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1494 | } |
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1495 | } |
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1496 | } else { |
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1497 | // Find the min. cost incomming arc for each demand node |
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1498 | for (int i = 0; i != int(demand_nodes.size()); ++i) { |
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1499 | Node v = demand_nodes[i]; |
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1500 | Cost c, min_cost = std::numeric_limits<Cost>::max(); |
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1501 | Arc min_arc = INVALID; |
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1502 | for (InArcIt a(_graph, v); a != INVALID; ++a) { |
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1503 | c = _cost[_arc_id[a]]; |
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1504 | if (c < min_cost) { |
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1505 | min_cost = c; |
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1506 | min_arc = a; |
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1507 | } |
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1508 | } |
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1509 | if (min_arc != INVALID) { |
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1510 | arc_vector.push_back(_arc_id[min_arc]); |
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1511 | } |
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1512 | } |
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1513 | } |
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1514 | } else { |
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1515 | // Find the min. cost outgoing arc for each supply node |
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1516 | for (int i = 0; i != int(supply_nodes.size()); ++i) { |
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1517 | Node u = supply_nodes[i]; |
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1518 | Cost c, min_cost = std::numeric_limits<Cost>::max(); |
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1519 | Arc min_arc = INVALID; |
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1520 | for (OutArcIt a(_graph, u); a != INVALID; ++a) { |
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1521 | c = _cost[_arc_id[a]]; |
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1522 | if (c < min_cost) { |
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1523 | min_cost = c; |
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1524 | min_arc = a; |
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1525 | } |
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1526 | } |
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1527 | if (min_arc != INVALID) { |
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1528 | arc_vector.push_back(_arc_id[min_arc]); |
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1529 | } |
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1530 | } |
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1531 | } |
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1532 | |
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1533 | // Perform heuristic initial pivots |
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1534 | for (int i = 0; i != int(arc_vector.size()); ++i) { |
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1535 | in_arc = arc_vector[i]; |
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1536 | if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] - |
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1537 | _pi[_target[in_arc]]) >= 0) continue; |
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1538 | findJoinNode(); |
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1539 | bool change = findLeavingArc(); |
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1540 | if (delta >= MAX) return false; |
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1541 | changeFlow(change); |
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1542 | if (change) { |
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1543 | updateTreeStructure(); |
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1544 | updatePotential(); |
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1545 | } |
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1546 | } |
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1547 | return true; |
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1548 | } |
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1549 | |
---|
1550 | // Execute the algorithm |
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1551 | ProblemType start(PivotRule pivot_rule) { |
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1552 | // Select the pivot rule implementation |
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1553 | switch (pivot_rule) { |
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1554 | case FIRST_ELIGIBLE: |
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1555 | return start<FirstEligiblePivotRule>(); |
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1556 | case BEST_ELIGIBLE: |
---|
1557 | return start<BestEligiblePivotRule>(); |
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1558 | case BLOCK_SEARCH: |
---|
1559 | return start<BlockSearchPivotRule>(); |
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1560 | case CANDIDATE_LIST: |
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1561 | return start<CandidateListPivotRule>(); |
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1562 | case ALTERING_LIST: |
---|
1563 | return start<AlteringListPivotRule>(); |
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1564 | } |
---|
1565 | return INFEASIBLE; // avoid warning |
---|
1566 | } |
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1567 | |
---|
1568 | template <typename PivotRuleImpl> |
---|
1569 | ProblemType start() { |
---|
1570 | PivotRuleImpl pivot(*this); |
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1571 | |
---|
1572 | // Perform heuristic initial pivots |
---|
1573 | if (!initialPivots()) return UNBOUNDED; |
---|
1574 | |
---|
1575 | // Execute the Network Simplex algorithm |
---|
1576 | while (pivot.findEnteringArc()) { |
---|
1577 | findJoinNode(); |
---|
1578 | bool change = findLeavingArc(); |
---|
1579 | if (delta >= MAX) return UNBOUNDED; |
---|
1580 | changeFlow(change); |
---|
1581 | if (change) { |
---|
1582 | updateTreeStructure(); |
---|
1583 | updatePotential(); |
---|
1584 | } |
---|
1585 | } |
---|
1586 | |
---|
1587 | // Check feasibility |
---|
1588 | for (int e = _search_arc_num; e != _all_arc_num; ++e) { |
---|
1589 | if (_flow[e] != 0) return INFEASIBLE; |
---|
1590 | } |
---|
1591 | |
---|
1592 | // Transform the solution and the supply map to the original form |
---|
1593 | if (_have_lower) { |
---|
1594 | for (int i = 0; i != _arc_num; ++i) { |
---|
1595 | Value c = _lower[i]; |
---|
1596 | if (c != 0) { |
---|
1597 | _flow[i] += c; |
---|
1598 | _supply[_source[i]] += c; |
---|
1599 | _supply[_target[i]] -= c; |
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1600 | } |
---|
1601 | } |
---|
1602 | } |
---|
1603 | |
---|
1604 | // Shift potentials to meet the requirements of the GEQ/LEQ type |
---|
1605 | // optimality conditions |
---|
1606 | if (_sum_supply == 0) { |
---|
1607 | if (_stype == GEQ) { |
---|
1608 | Cost max_pot = -std::numeric_limits<Cost>::max(); |
---|
1609 | for (int i = 0; i != _node_num; ++i) { |
---|
1610 | if (_pi[i] > max_pot) max_pot = _pi[i]; |
---|
1611 | } |
---|
1612 | if (max_pot > 0) { |
---|
1613 | for (int i = 0; i != _node_num; ++i) |
---|
1614 | _pi[i] -= max_pot; |
---|
1615 | } |
---|
1616 | } else { |
---|
1617 | Cost min_pot = std::numeric_limits<Cost>::max(); |
---|
1618 | for (int i = 0; i != _node_num; ++i) { |
---|
1619 | if (_pi[i] < min_pot) min_pot = _pi[i]; |
---|
1620 | } |
---|
1621 | if (min_pot < 0) { |
---|
1622 | for (int i = 0; i != _node_num; ++i) |
---|
1623 | _pi[i] -= min_pot; |
---|
1624 | } |
---|
1625 | } |
---|
1626 | } |
---|
1627 | |
---|
1628 | return OPTIMAL; |
---|
1629 | } |
---|
1630 | |
---|
1631 | }; //class NetworkSimplex |
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1632 | |
---|
1633 | ///@} |
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1634 | |
---|
1635 | } //namespace lemon |
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1636 | |
---|
1637 | #endif //LEMON_NETWORK_SIMPLEX_H |
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