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_CHRISTOFIDES_TSP_H |
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20 | #define LEMON_CHRISTOFIDES_TSP_H |
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21 | |
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22 | /// \ingroup tsp |
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23 | /// \file |
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24 | /// \brief Christofides algorithm for symmetric TSP |
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25 | |
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26 | #include <lemon/full_graph.h> |
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27 | #include <lemon/smart_graph.h> |
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28 | #include <lemon/kruskal.h> |
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29 | #include <lemon/matching.h> |
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30 | #include <lemon/euler.h> |
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31 | |
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32 | namespace lemon { |
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33 | |
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34 | /// \ingroup tsp |
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35 | /// |
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36 | /// \brief Christofides algorithm for symmetric TSP. |
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37 | /// |
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38 | /// ChristofidesTsp implements Christofides' heuristic for solving |
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39 | /// symmetric \ref tsp "TSP". |
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40 | /// |
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41 | /// This a well-known approximation method for the TSP problem with |
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42 | /// metric cost function. |
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43 | /// It has a guaranteed approximation factor of 3/2 (i.e. it finds a tour |
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44 | /// whose total cost is at most 3/2 of the optimum), but it usually |
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45 | /// provides better solutions in practice. |
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46 | /// This implementation runs in O(n<sup>3</sup>log(n)) time. |
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47 | /// |
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48 | /// The algorithm starts with a \ref spantree "minimum cost spanning tree" and |
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49 | /// finds a \ref MaxWeightedPerfectMatching "minimum cost perfect matching" |
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50 | /// in the subgraph induced by the nodes that have odd degree in the |
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51 | /// spanning tree. |
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52 | /// Finally, it constructs the tour from the \ref EulerIt "Euler traversal" |
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53 | /// of the union of the spanning tree and the matching. |
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54 | /// During this last step, the algorithm simply skips the visited nodes |
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55 | /// (i.e. creates shortcuts) assuming that the triangle inequality holds |
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56 | /// for the cost function. |
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57 | /// |
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58 | /// \tparam CM Type of the cost map. |
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59 | /// |
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60 | /// \warning CM::Value must be a signed number type. |
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61 | template <typename CM> |
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62 | class ChristofidesTsp |
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63 | { |
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64 | public: |
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65 | |
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66 | /// Type of the cost map |
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67 | typedef CM CostMap; |
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68 | /// Type of the edge costs |
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69 | typedef typename CM::Value Cost; |
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70 | |
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71 | private: |
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72 | |
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73 | GRAPH_TYPEDEFS(FullGraph); |
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74 | |
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75 | const FullGraph &_gr; |
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76 | const CostMap &_cost; |
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77 | std::vector<Node> _path; |
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78 | Cost _sum; |
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79 | |
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80 | public: |
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81 | |
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82 | /// \brief Constructor |
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83 | /// |
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84 | /// Constructor. |
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85 | /// \param gr The \ref FullGraph "full graph" the algorithm runs on. |
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86 | /// \param cost The cost map. |
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87 | ChristofidesTsp(const FullGraph &gr, const CostMap &cost) |
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88 | : _gr(gr), _cost(cost) {} |
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89 | |
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90 | /// \name Execution Control |
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91 | /// @{ |
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92 | |
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93 | /// \brief Runs the algorithm. |
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94 | /// |
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95 | /// This function runs the algorithm. |
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96 | /// |
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97 | /// \return The total cost of the found tour. |
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98 | Cost run() { |
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99 | _path.clear(); |
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100 | |
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101 | if (_gr.nodeNum() == 0) return _sum = 0; |
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102 | else if (_gr.nodeNum() == 1) { |
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103 | _path.push_back(_gr(0)); |
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104 | return _sum = 0; |
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105 | } |
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106 | else if (_gr.nodeNum() == 2) { |
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107 | _path.push_back(_gr(0)); |
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108 | _path.push_back(_gr(1)); |
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109 | return _sum = 2 * _cost[_gr.edge(_gr(0), _gr(1))]; |
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110 | } |
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111 | |
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112 | // Compute min. cost spanning tree |
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113 | std::vector<Edge> tree; |
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114 | kruskal(_gr, _cost, std::back_inserter(tree)); |
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115 | |
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116 | FullGraph::NodeMap<int> deg(_gr, 0); |
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117 | for (int i = 0; i != int(tree.size()); ++i) { |
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118 | Edge e = tree[i]; |
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119 | ++deg[_gr.u(e)]; |
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120 | ++deg[_gr.v(e)]; |
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121 | } |
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122 | |
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123 | // Copy the induced subgraph of odd nodes |
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124 | std::vector<Node> odd_nodes; |
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125 | for (NodeIt u(_gr); u != INVALID; ++u) { |
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126 | if (deg[u] % 2 == 1) odd_nodes.push_back(u); |
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127 | } |
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128 | |
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129 | SmartGraph sgr; |
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130 | SmartGraph::EdgeMap<Cost> scost(sgr); |
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131 | for (int i = 0; i != int(odd_nodes.size()); ++i) { |
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132 | sgr.addNode(); |
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133 | } |
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134 | for (int i = 0; i != int(odd_nodes.size()); ++i) { |
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135 | for (int j = 0; j != int(odd_nodes.size()); ++j) { |
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136 | if (j == i) continue; |
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137 | SmartGraph::Edge e = |
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138 | sgr.addEdge(sgr.nodeFromId(i), sgr.nodeFromId(j)); |
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139 | scost[e] = -_cost[_gr.edge(odd_nodes[i], odd_nodes[j])]; |
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140 | } |
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141 | } |
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142 | |
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143 | // Compute min. cost perfect matching |
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144 | MaxWeightedPerfectMatching<SmartGraph, SmartGraph::EdgeMap<Cost> > |
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145 | mwpm(sgr, scost); |
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146 | mwpm.run(); |
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147 | |
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148 | for (SmartGraph::EdgeIt e(sgr); e != INVALID; ++e) { |
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149 | if (mwpm.matching(e)) { |
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150 | tree.push_back( _gr.edge(odd_nodes[sgr.id(sgr.u(e))], |
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151 | odd_nodes[sgr.id(sgr.v(e))]) ); |
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152 | } |
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153 | } |
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154 | |
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155 | // Join the spanning tree and the matching |
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156 | sgr.clear(); |
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157 | for (int i = 0; i != _gr.nodeNum(); ++i) { |
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158 | sgr.addNode(); |
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159 | } |
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160 | for (int i = 0; i != int(tree.size()); ++i) { |
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161 | int ui = _gr.id(_gr.u(tree[i])), |
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162 | vi = _gr.id(_gr.v(tree[i])); |
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163 | sgr.addEdge(sgr.nodeFromId(ui), sgr.nodeFromId(vi)); |
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164 | } |
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165 | |
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166 | // Compute the tour from the Euler traversal |
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167 | SmartGraph::NodeMap<bool> visited(sgr, false); |
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168 | for (EulerIt<SmartGraph> e(sgr); e != INVALID; ++e) { |
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169 | SmartGraph::Node n = sgr.target(e); |
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170 | if (!visited[n]) { |
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171 | _path.push_back(_gr(sgr.id(n))); |
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172 | visited[n] = true; |
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173 | } |
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174 | } |
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175 | |
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176 | _sum = _cost[_gr.edge(_path.back(), _path.front())]; |
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177 | for (int i = 0; i < int(_path.size())-1; ++i) { |
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178 | _sum += _cost[_gr.edge(_path[i], _path[i+1])]; |
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179 | } |
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180 | |
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181 | return _sum; |
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182 | } |
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183 | |
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184 | /// @} |
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185 | |
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186 | /// \name Query Functions |
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187 | /// @{ |
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188 | |
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189 | /// \brief The total cost of the found tour. |
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190 | /// |
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191 | /// This function returns the total cost of the found tour. |
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192 | /// |
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193 | /// \pre run() must be called before using this function. |
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194 | Cost tourCost() const { |
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195 | return _sum; |
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196 | } |
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197 | |
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198 | /// \brief Returns a const reference to the node sequence of the |
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199 | /// found tour. |
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200 | /// |
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201 | /// This function returns a const reference to a vector |
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202 | /// that stores the node sequence of the found tour. |
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203 | /// |
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204 | /// \pre run() must be called before using this function. |
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205 | const std::vector<Node>& tourNodes() const { |
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206 | return _path; |
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207 | } |
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208 | |
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209 | /// \brief Gives back the node sequence of the found tour. |
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210 | /// |
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211 | /// This function copies the node sequence of the found tour into |
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212 | /// an STL container through the given output iterator. The |
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213 | /// <tt>value_type</tt> of the container must be <tt>FullGraph::Node</tt>. |
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214 | /// For example, |
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215 | /// \code |
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216 | /// std::vector<FullGraph::Node> nodes(countNodes(graph)); |
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217 | /// tsp.tourNodes(nodes.begin()); |
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218 | /// \endcode |
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219 | /// or |
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220 | /// \code |
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221 | /// std::list<FullGraph::Node> nodes; |
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222 | /// tsp.tourNodes(std::back_inserter(nodes)); |
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223 | /// \endcode |
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224 | /// |
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225 | /// \pre run() must be called before using this function. |
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226 | template <typename Iterator> |
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227 | void tourNodes(Iterator out) const { |
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228 | std::copy(_path.begin(), _path.end(), out); |
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229 | } |
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230 | |
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231 | /// \brief Gives back the found tour as a path. |
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232 | /// |
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233 | /// This function copies the found tour as a list of arcs/edges into |
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234 | /// the given \ref lemon::concepts::Path "path structure". |
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235 | /// |
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236 | /// \pre run() must be called before using this function. |
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237 | template <typename Path> |
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238 | void tour(Path &path) const { |
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239 | path.clear(); |
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240 | for (int i = 0; i < int(_path.size()) - 1; ++i) { |
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241 | path.addBack(_gr.arc(_path[i], _path[i+1])); |
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242 | } |
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243 | if (int(_path.size()) >= 2) { |
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244 | path.addBack(_gr.arc(_path.back(), _path.front())); |
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245 | } |
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246 | } |
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247 | |
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248 | /// @} |
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249 | |
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250 | }; |
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251 | |
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252 | }; // namespace lemon |
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253 | |
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254 | #endif |
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