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) 2003-2009 |
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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_SUURBALLE_H |
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20 | #define LEMON_SUURBALLE_H |
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21 | |
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22 | ///\ingroup shortest_path |
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23 | ///\file |
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24 | ///\brief An algorithm for finding arc-disjoint paths between two |
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25 | /// nodes having minimum total length. |
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26 | |
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27 | #include <vector> |
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28 | #include <limits> |
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29 | #include <lemon/bin_heap.h> |
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30 | #include <lemon/path.h> |
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31 | #include <lemon/list_graph.h> |
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32 | #include <lemon/maps.h> |
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33 | |
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34 | namespace lemon { |
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35 | |
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36 | /// \addtogroup shortest_path |
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37 | /// @{ |
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38 | |
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39 | /// \brief Algorithm for finding arc-disjoint paths between two nodes |
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40 | /// having minimum total length. |
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41 | /// |
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42 | /// \ref lemon::Suurballe "Suurballe" implements an algorithm for |
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43 | /// finding arc-disjoint paths having minimum total length (cost) |
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44 | /// from a given source node to a given target node in a digraph. |
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45 | /// |
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46 | /// Note that this problem is a special case of the \ref min_cost_flow |
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47 | /// "minimum cost flow problem". This implementation is actually an |
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48 | /// efficient specialized version of the \ref CapacityScaling |
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49 | /// "successive shortest path" algorithm directly for this problem. |
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50 | /// Therefore this class provides query functions for flow values and |
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51 | /// node potentials (the dual solution) just like the minimum cost flow |
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52 | /// algorithms. |
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53 | /// |
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54 | /// \tparam GR The digraph type the algorithm runs on. |
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55 | /// \tparam LEN The type of the length map. |
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56 | /// The default value is <tt>GR::ArcMap<int></tt>. |
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57 | /// |
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58 | /// \warning Length values should be \e non-negative. |
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59 | /// |
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60 | /// \note For finding \e node-disjoint paths, this algorithm can be used |
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61 | /// along with the \ref SplitNodes adaptor. |
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62 | #ifdef DOXYGEN |
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63 | template <typename GR, typename LEN> |
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64 | #else |
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65 | template < typename GR, |
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66 | typename LEN = typename GR::template ArcMap<int> > |
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67 | #endif |
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68 | class Suurballe |
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69 | { |
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70 | TEMPLATE_DIGRAPH_TYPEDEFS(GR); |
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71 | |
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72 | typedef ConstMap<Arc, int> ConstArcMap; |
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73 | typedef typename GR::template NodeMap<Arc> PredMap; |
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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 digraph the algorithm runs on. |
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78 | typedef GR Digraph; |
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79 | /// The type of the length map. |
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80 | typedef LEN LengthMap; |
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81 | /// The type of the lengths. |
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82 | typedef typename LengthMap::Value Length; |
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83 | #ifdef DOXYGEN |
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84 | /// The type of the flow map. |
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85 | typedef GR::ArcMap<int> FlowMap; |
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86 | /// The type of the potential map. |
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87 | typedef GR::NodeMap<Length> PotentialMap; |
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88 | #else |
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89 | /// The type of the flow map. |
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90 | typedef typename Digraph::template ArcMap<int> FlowMap; |
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91 | /// The type of the potential map. |
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92 | typedef typename Digraph::template NodeMap<Length> PotentialMap; |
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93 | #endif |
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94 | |
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95 | /// The type of the path structures. |
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96 | typedef SimplePath<GR> Path; |
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97 | |
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98 | private: |
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99 | |
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100 | // ResidualDijkstra is a special implementation of the |
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101 | // Dijkstra algorithm for finding shortest paths in the |
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102 | // residual network with respect to the reduced arc lengths |
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103 | // and modifying the node potentials according to the |
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104 | // distance of the nodes. |
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105 | class ResidualDijkstra |
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106 | { |
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107 | typedef typename Digraph::template NodeMap<int> HeapCrossRef; |
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108 | typedef BinHeap<Length, HeapCrossRef> Heap; |
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109 | |
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110 | private: |
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111 | |
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112 | const Digraph &_graph; |
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113 | const LengthMap &_length; |
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114 | const FlowMap &_flow; |
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115 | PotentialMap &_pi; |
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116 | PredMap &_pred; |
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117 | Node _s; |
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118 | Node _t; |
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119 | |
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120 | PotentialMap _dist; |
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121 | std::vector<Node> _proc_nodes; |
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122 | |
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123 | public: |
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124 | |
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125 | // Constructor |
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126 | ResidualDijkstra(Suurballe &srb) : |
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127 | _graph(srb._graph), _length(srb._length), |
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128 | _flow(*srb._flow), _pi(*srb._potential), _pred(srb._pred), |
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129 | _s(srb._s), _t(srb._t), _dist(_graph) {} |
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130 | |
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131 | // Run the algorithm and return true if a path is found |
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132 | // from the source node to the target node. |
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133 | bool run(int cnt) { |
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134 | return cnt == 0 ? startFirst() : start(); |
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135 | } |
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136 | |
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137 | private: |
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138 | |
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139 | // Execute the algorithm for the first time (the flow and potential |
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140 | // functions have to be identically zero). |
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141 | bool startFirst() { |
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142 | HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP); |
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143 | Heap heap(heap_cross_ref); |
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144 | heap.push(_s, 0); |
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145 | _pred[_s] = INVALID; |
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146 | _proc_nodes.clear(); |
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147 | |
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148 | // Process nodes |
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149 | while (!heap.empty() && heap.top() != _t) { |
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150 | Node u = heap.top(), v; |
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151 | Length d = heap.prio(), dn; |
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152 | _dist[u] = heap.prio(); |
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153 | _proc_nodes.push_back(u); |
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154 | heap.pop(); |
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155 | |
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156 | // Traverse outgoing arcs |
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157 | for (OutArcIt e(_graph, u); e != INVALID; ++e) { |
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158 | v = _graph.target(e); |
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159 | switch(heap.state(v)) { |
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160 | case Heap::PRE_HEAP: |
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161 | heap.push(v, d + _length[e]); |
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162 | _pred[v] = e; |
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163 | break; |
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164 | case Heap::IN_HEAP: |
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165 | dn = d + _length[e]; |
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166 | if (dn < heap[v]) { |
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167 | heap.decrease(v, dn); |
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168 | _pred[v] = e; |
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169 | } |
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170 | break; |
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171 | case Heap::POST_HEAP: |
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172 | break; |
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173 | } |
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174 | } |
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175 | } |
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176 | if (heap.empty()) return false; |
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177 | |
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178 | // Update potentials of processed nodes |
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179 | Length t_dist = heap.prio(); |
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180 | for (int i = 0; i < int(_proc_nodes.size()); ++i) |
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181 | _pi[_proc_nodes[i]] = _dist[_proc_nodes[i]] - t_dist; |
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182 | return true; |
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183 | } |
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184 | |
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185 | // Execute the algorithm. |
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186 | bool start() { |
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187 | HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP); |
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188 | Heap heap(heap_cross_ref); |
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189 | heap.push(_s, 0); |
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190 | _pred[_s] = INVALID; |
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191 | _proc_nodes.clear(); |
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192 | |
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193 | // Process nodes |
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194 | while (!heap.empty() && heap.top() != _t) { |
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195 | Node u = heap.top(), v; |
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196 | Length d = heap.prio() + _pi[u], dn; |
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197 | _dist[u] = heap.prio(); |
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198 | _proc_nodes.push_back(u); |
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199 | heap.pop(); |
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200 | |
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201 | // Traverse outgoing arcs |
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202 | for (OutArcIt e(_graph, u); e != INVALID; ++e) { |
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203 | if (_flow[e] == 0) { |
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204 | v = _graph.target(e); |
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205 | switch(heap.state(v)) { |
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206 | case Heap::PRE_HEAP: |
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207 | heap.push(v, d + _length[e] - _pi[v]); |
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208 | _pred[v] = e; |
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209 | break; |
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210 | case Heap::IN_HEAP: |
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211 | dn = d + _length[e] - _pi[v]; |
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212 | if (dn < heap[v]) { |
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213 | heap.decrease(v, dn); |
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214 | _pred[v] = e; |
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215 | } |
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216 | break; |
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217 | case Heap::POST_HEAP: |
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218 | break; |
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219 | } |
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220 | } |
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221 | } |
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222 | |
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223 | // Traverse incoming arcs |
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224 | for (InArcIt e(_graph, u); e != INVALID; ++e) { |
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225 | if (_flow[e] == 1) { |
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226 | v = _graph.source(e); |
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227 | switch(heap.state(v)) { |
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228 | case Heap::PRE_HEAP: |
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229 | heap.push(v, d - _length[e] - _pi[v]); |
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230 | _pred[v] = e; |
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231 | break; |
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232 | case Heap::IN_HEAP: |
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233 | dn = d - _length[e] - _pi[v]; |
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234 | if (dn < heap[v]) { |
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235 | heap.decrease(v, dn); |
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236 | _pred[v] = e; |
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237 | } |
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238 | break; |
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239 | case Heap::POST_HEAP: |
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240 | break; |
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241 | } |
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242 | } |
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243 | } |
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244 | } |
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245 | if (heap.empty()) return false; |
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246 | |
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247 | // Update potentials of processed nodes |
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248 | Length t_dist = heap.prio(); |
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249 | for (int i = 0; i < int(_proc_nodes.size()); ++i) |
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250 | _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - t_dist; |
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251 | return true; |
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252 | } |
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253 | |
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254 | }; //class ResidualDijkstra |
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255 | |
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256 | private: |
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257 | |
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258 | // The digraph the algorithm runs on |
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259 | const Digraph &_graph; |
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260 | // The length map |
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261 | const LengthMap &_length; |
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262 | |
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263 | // Arc map of the current flow |
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264 | FlowMap *_flow; |
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265 | bool _local_flow; |
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266 | // Node map of the current potentials |
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267 | PotentialMap *_potential; |
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268 | bool _local_potential; |
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269 | |
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270 | // The source node |
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271 | Node _s; |
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272 | // The target node |
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273 | Node _t; |
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274 | |
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275 | // Container to store the found paths |
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276 | std::vector<Path> _paths; |
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277 | int _path_num; |
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278 | |
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279 | // The pred arc map |
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280 | PredMap _pred; |
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281 | |
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282 | public: |
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283 | |
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284 | /// \brief Constructor. |
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285 | /// |
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286 | /// Constructor. |
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287 | /// |
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288 | /// \param graph The digraph the algorithm runs on. |
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289 | /// \param length The length (cost) values of the arcs. |
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290 | Suurballe( const Digraph &graph, |
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291 | const LengthMap &length ) : |
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292 | _graph(graph), _length(length), _flow(0), _local_flow(false), |
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293 | _potential(0), _local_potential(false), _pred(graph) |
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294 | {} |
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295 | |
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296 | /// Destructor. |
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297 | ~Suurballe() { |
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298 | if (_local_flow) delete _flow; |
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299 | if (_local_potential) delete _potential; |
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300 | } |
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301 | |
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302 | /// \brief Set the flow map. |
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303 | /// |
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304 | /// This function sets the flow map. |
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305 | /// If it is not used before calling \ref run() or \ref init(), |
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306 | /// an instance will be allocated automatically. The destructor |
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307 | /// deallocates this automatically allocated map, of course. |
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308 | /// |
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309 | /// The found flow contains only 0 and 1 values, since it is the |
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310 | /// union of the found arc-disjoint paths. |
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311 | /// |
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312 | /// \return <tt>(*this)</tt> |
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313 | Suurballe& flowMap(FlowMap &map) { |
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314 | if (_local_flow) { |
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315 | delete _flow; |
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316 | _local_flow = false; |
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317 | } |
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318 | _flow = ↦ |
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319 | return *this; |
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320 | } |
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321 | |
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322 | /// \brief Set the potential map. |
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323 | /// |
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324 | /// This function sets the potential map. |
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325 | /// If it is not used before calling \ref run() or \ref init(), |
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326 | /// an instance will be allocated automatically. The destructor |
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327 | /// deallocates this automatically allocated map, of course. |
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328 | /// |
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329 | /// The node potentials provide the dual solution of the underlying |
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330 | /// \ref min_cost_flow "minimum cost flow problem". |
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331 | /// |
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332 | /// \return <tt>(*this)</tt> |
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333 | Suurballe& potentialMap(PotentialMap &map) { |
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334 | if (_local_potential) { |
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335 | delete _potential; |
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336 | _local_potential = false; |
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337 | } |
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338 | _potential = ↦ |
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339 | return *this; |
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340 | } |
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341 | |
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342 | /// \name Execution Control |
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343 | /// The simplest way to execute the algorithm is to call the run() |
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344 | /// function. |
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345 | /// \n |
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346 | /// If you only need the flow that is the union of the found |
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347 | /// arc-disjoint paths, you may call init() and findFlow(). |
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348 | |
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349 | /// @{ |
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350 | |
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351 | /// \brief Run the algorithm. |
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352 | /// |
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353 | /// This function runs the algorithm. |
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354 | /// |
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355 | /// \param s The source node. |
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356 | /// \param t The target node. |
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357 | /// \param k The number of paths to be found. |
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358 | /// |
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359 | /// \return \c k if there are at least \c k arc-disjoint paths from |
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360 | /// \c s to \c t in the digraph. Otherwise it returns the number of |
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361 | /// arc-disjoint paths found. |
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362 | /// |
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363 | /// \note Apart from the return value, <tt>s.run(s, t, k)</tt> is |
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364 | /// just a shortcut of the following code. |
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365 | /// \code |
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366 | /// s.init(s); |
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367 | /// s.findFlow(t, k); |
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368 | /// s.findPaths(); |
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369 | /// \endcode |
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370 | int run(const Node& s, const Node& t, int k = 2) { |
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371 | init(s); |
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372 | findFlow(t, k); |
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373 | findPaths(); |
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374 | return _path_num; |
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375 | } |
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376 | |
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377 | /// \brief Initialize the algorithm. |
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378 | /// |
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379 | /// This function initializes the algorithm. |
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380 | /// |
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381 | /// \param s The source node. |
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382 | void init(const Node& s) { |
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383 | _s = s; |
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384 | |
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385 | // Initialize maps |
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386 | if (!_flow) { |
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387 | _flow = new FlowMap(_graph); |
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388 | _local_flow = true; |
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389 | } |
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390 | if (!_potential) { |
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391 | _potential = new PotentialMap(_graph); |
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392 | _local_potential = true; |
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393 | } |
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394 | for (ArcIt e(_graph); e != INVALID; ++e) (*_flow)[e] = 0; |
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395 | for (NodeIt n(_graph); n != INVALID; ++n) (*_potential)[n] = 0; |
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396 | } |
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397 | |
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398 | /// \brief Execute the algorithm to find an optimal flow. |
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399 | /// |
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400 | /// This function executes the successive shortest path algorithm to |
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401 | /// find a minimum cost flow, which is the union of \c k (or less) |
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402 | /// arc-disjoint paths. |
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403 | /// |
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404 | /// \param t The target node. |
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405 | /// \param k The number of paths to be found. |
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406 | /// |
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407 | /// \return \c k if there are at least \c k arc-disjoint paths from |
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408 | /// the source node to the given node \c t in the digraph. |
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409 | /// Otherwise it returns the number of arc-disjoint paths found. |
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410 | /// |
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411 | /// \pre \ref init() must be called before using this function. |
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412 | int findFlow(const Node& t, int k = 2) { |
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413 | _t = t; |
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414 | ResidualDijkstra dijkstra(*this); |
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415 | |
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416 | // Find shortest paths |
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417 | _path_num = 0; |
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418 | while (_path_num < k) { |
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419 | // Run Dijkstra |
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420 | if (!dijkstra.run(_path_num)) break; |
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421 | ++_path_num; |
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422 | |
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423 | // Set the flow along the found shortest path |
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424 | Node u = _t; |
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425 | Arc e; |
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426 | while ((e = _pred[u]) != INVALID) { |
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427 | if (u == _graph.target(e)) { |
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428 | (*_flow)[e] = 1; |
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429 | u = _graph.source(e); |
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430 | } else { |
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431 | (*_flow)[e] = 0; |
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432 | u = _graph.target(e); |
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433 | } |
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434 | } |
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435 | } |
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436 | return _path_num; |
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437 | } |
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438 | |
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439 | /// \brief Compute the paths from the flow. |
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440 | /// |
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441 | /// This function computes arc-disjoint paths from the found minimum |
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442 | /// cost flow, which is the union of them. |
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443 | /// |
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444 | /// \pre \ref init() and \ref findFlow() must be called before using |
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445 | /// this function. |
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446 | void findPaths() { |
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447 | FlowMap res_flow(_graph); |
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448 | for(ArcIt a(_graph); a != INVALID; ++a) res_flow[a] = (*_flow)[a]; |
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449 | |
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450 | _paths.clear(); |
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451 | _paths.resize(_path_num); |
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452 | for (int i = 0; i < _path_num; ++i) { |
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453 | Node n = _s; |
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454 | while (n != _t) { |
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455 | OutArcIt e(_graph, n); |
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456 | for ( ; res_flow[e] == 0; ++e) ; |
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457 | n = _graph.target(e); |
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458 | _paths[i].addBack(e); |
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459 | res_flow[e] = 0; |
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460 | } |
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461 | } |
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462 | } |
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463 | |
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464 | /// @} |
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465 | |
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466 | /// \name Query Functions |
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467 | /// The results of the algorithm can be obtained using these |
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468 | /// functions. |
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469 | /// \n The algorithm should be executed before using them. |
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470 | |
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471 | /// @{ |
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472 | |
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473 | /// \brief Return the total length of the found paths. |
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474 | /// |
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475 | /// This function returns the total length of the found paths, i.e. |
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476 | /// the total cost of the found flow. |
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477 | /// The complexity of the function is O(e). |
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478 | /// |
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479 | /// \pre \ref run() or \ref findFlow() must be called before using |
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480 | /// this function. |
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481 | Length totalLength() const { |
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482 | Length c = 0; |
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483 | for (ArcIt e(_graph); e != INVALID; ++e) |
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484 | c += (*_flow)[e] * _length[e]; |
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485 | return c; |
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486 | } |
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487 | |
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488 | /// \brief Return the flow value on the given arc. |
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489 | /// |
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490 | /// This function returns the flow value on the given arc. |
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491 | /// It is \c 1 if the arc is involved in one of the found arc-disjoint |
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492 | /// paths, otherwise it is \c 0. |
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493 | /// |
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494 | /// \pre \ref run() or \ref findFlow() must be called before using |
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495 | /// this function. |
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496 | int flow(const Arc& arc) const { |
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497 | return (*_flow)[arc]; |
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498 | } |
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499 | |
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500 | /// \brief Return a const reference to an arc map storing the |
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501 | /// found flow. |
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502 | /// |
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503 | /// This function returns a const reference to an arc map storing |
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504 | /// the flow that is the union of the found arc-disjoint paths. |
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505 | /// |
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506 | /// \pre \ref run() or \ref findFlow() must be called before using |
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507 | /// this function. |
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508 | const FlowMap& flowMap() const { |
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509 | return *_flow; |
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510 | } |
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511 | |
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512 | /// \brief Return the potential of the given node. |
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513 | /// |
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514 | /// This function returns the potential of the given node. |
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515 | /// The node potentials provide the dual solution of the |
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516 | /// underlying \ref min_cost_flow "minimum cost flow problem". |
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517 | /// |
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518 | /// \pre \ref run() or \ref findFlow() must be called before using |
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519 | /// this function. |
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520 | Length potential(const Node& node) const { |
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521 | return (*_potential)[node]; |
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522 | } |
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523 | |
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524 | /// \brief Return a const reference to a node map storing the |
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525 | /// found potentials (the dual solution). |
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526 | /// |
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527 | /// This function returns a const reference to a node map storing |
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528 | /// the found potentials that provide the dual solution of the |
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529 | /// underlying \ref min_cost_flow "minimum cost flow problem". |
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530 | /// |
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531 | /// \pre \ref run() or \ref findFlow() must be called before using |
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532 | /// this function. |
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533 | const PotentialMap& potentialMap() const { |
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534 | return *_potential; |
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535 | } |
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536 | |
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537 | /// \brief Return the number of the found paths. |
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538 | /// |
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539 | /// This function returns the number of the found paths. |
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540 | /// |
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541 | /// \pre \ref run() or \ref findFlow() must be called before using |
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542 | /// this function. |
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543 | int pathNum() const { |
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544 | return _path_num; |
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545 | } |
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546 | |
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547 | /// \brief Return a const reference to the specified path. |
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548 | /// |
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549 | /// This function returns a const reference to the specified path. |
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550 | /// |
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551 | /// \param i The function returns the <tt>i</tt>-th path. |
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552 | /// \c i must be between \c 0 and <tt>%pathNum()-1</tt>. |
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553 | /// |
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554 | /// \pre \ref run() or \ref findPaths() must be called before using |
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555 | /// this function. |
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556 | const Path& path(int i) const { |
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557 | return _paths[i]; |
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558 | } |
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559 | |
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560 | /// @} |
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561 | |
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562 | }; //class Suurballe |
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563 | |
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564 | ///@} |
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565 | |
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566 | } //namespace lemon |
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567 | |
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568 | #endif //LEMON_SUURBALLE_H |
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