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) 2003-2013 |
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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_GOMORY_HU_TREE_H |
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20 | #define LEMON_GOMORY_HU_TREE_H |
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21 | |
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22 | #include <limits> |
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23 | |
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24 | #include <lemon/core.h> |
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25 | #include <lemon/preflow.h> |
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26 | #include <lemon/concept_check.h> |
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27 | #include <lemon/concepts/maps.h> |
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28 | |
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29 | /// \ingroup min_cut |
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30 | /// \file |
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31 | /// \brief Gomory-Hu cut tree in graphs. |
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32 | |
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33 | namespace lemon { |
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34 | |
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35 | /// \ingroup min_cut |
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36 | /// |
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37 | /// \brief Gomory-Hu cut tree algorithm |
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38 | /// |
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39 | /// The Gomory-Hu tree is a tree on the node set of a given graph, but it |
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40 | /// may contain edges which are not in the original graph. It has the |
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41 | /// property that the minimum capacity edge of the path between two nodes |
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42 | /// in this tree has the same weight as the minimum cut in the graph |
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43 | /// between these nodes. Moreover the components obtained by removing |
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44 | /// this edge from the tree determine the corresponding minimum cut. |
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45 | /// Therefore once this tree is computed, the minimum cut between any pair |
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46 | /// of nodes can easily be obtained. |
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47 | /// |
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48 | /// The algorithm calculates \e n-1 distinct minimum cuts (currently with |
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49 | /// the \ref Preflow algorithm), thus it has \f$O(n^3\sqrt{m})\f$ overall |
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50 | /// time complexity. It calculates a rooted Gomory-Hu tree. |
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51 | /// The structure of the tree and the edge weights can be |
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52 | /// obtained using \c predNode(), \c predValue() and \c rootDist(). |
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53 | /// The functions \c minCutMap() and \c minCutValue() calculate |
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54 | /// the minimum cut and the minimum cut value between any two nodes |
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55 | /// in the graph. You can also list (iterate on) the nodes and the |
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56 | /// edges of the cuts using \c MinCutNodeIt and \c MinCutEdgeIt. |
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57 | /// |
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58 | /// \tparam GR The type of the undirected graph the algorithm runs on. |
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59 | /// \tparam CAP The type of the edge map containing the capacities. |
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60 | /// The default map type is \ref concepts::Graph::EdgeMap "GR::EdgeMap<int>". |
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61 | #ifdef DOXYGEN |
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62 | template <typename GR, |
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63 | typename CAP> |
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64 | #else |
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65 | template <typename GR, |
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66 | typename CAP = typename GR::template EdgeMap<int> > |
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67 | #endif |
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68 | class GomoryHu { |
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69 | public: |
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70 | |
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71 | /// The graph type of the algorithm |
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72 | typedef GR Graph; |
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73 | /// The capacity map type of the algorithm |
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74 | typedef CAP Capacity; |
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75 | /// The value type of capacities |
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76 | typedef typename Capacity::Value Value; |
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77 | |
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78 | private: |
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79 | |
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80 | TEMPLATE_GRAPH_TYPEDEFS(Graph); |
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81 | |
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82 | const Graph& _graph; |
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83 | const Capacity& _capacity; |
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84 | |
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85 | Node _root; |
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86 | typename Graph::template NodeMap<Node>* _pred; |
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87 | typename Graph::template NodeMap<Value>* _weight; |
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88 | typename Graph::template NodeMap<int>* _order; |
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89 | |
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90 | void createStructures() { |
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91 | if (!_pred) { |
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92 | _pred = new typename Graph::template NodeMap<Node>(_graph); |
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93 | } |
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94 | if (!_weight) { |
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95 | _weight = new typename Graph::template NodeMap<Value>(_graph); |
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96 | } |
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97 | if (!_order) { |
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98 | _order = new typename Graph::template NodeMap<int>(_graph); |
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99 | } |
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100 | } |
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101 | |
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102 | void destroyStructures() { |
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103 | if (_pred) { |
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104 | delete _pred; |
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105 | } |
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106 | if (_weight) { |
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107 | delete _weight; |
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108 | } |
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109 | if (_order) { |
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110 | delete _order; |
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111 | } |
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112 | } |
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113 | |
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114 | public: |
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115 | |
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116 | /// \brief Constructor |
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117 | /// |
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118 | /// Constructor. |
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119 | /// \param graph The undirected graph the algorithm runs on. |
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120 | /// \param capacity The edge capacity map. |
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121 | GomoryHu(const Graph& graph, const Capacity& capacity) |
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122 | : _graph(graph), _capacity(capacity), |
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123 | _pred(0), _weight(0), _order(0) |
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124 | { |
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125 | checkConcept<concepts::ReadMap<Edge, Value>, Capacity>(); |
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126 | } |
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127 | |
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128 | |
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129 | /// \brief Destructor |
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130 | /// |
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131 | /// Destructor. |
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132 | ~GomoryHu() { |
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133 | destroyStructures(); |
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134 | } |
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135 | |
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136 | private: |
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137 | |
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138 | // Initialize the internal data structures |
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139 | void init() { |
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140 | createStructures(); |
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141 | |
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142 | _root = NodeIt(_graph); |
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143 | for (NodeIt n(_graph); n != INVALID; ++n) { |
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144 | (*_pred)[n] = _root; |
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145 | (*_order)[n] = -1; |
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146 | } |
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147 | (*_pred)[_root] = INVALID; |
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148 | (*_weight)[_root] = std::numeric_limits<Value>::max(); |
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149 | } |
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150 | |
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151 | |
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152 | // Start the algorithm |
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153 | void start() { |
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154 | Preflow<Graph, Capacity> fa(_graph, _capacity, _root, INVALID); |
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155 | |
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156 | for (NodeIt n(_graph); n != INVALID; ++n) { |
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157 | if (n == _root) continue; |
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158 | |
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159 | Node pn = (*_pred)[n]; |
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160 | fa.source(n); |
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161 | fa.target(pn); |
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162 | |
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163 | fa.runMinCut(); |
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164 | |
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165 | (*_weight)[n] = fa.flowValue(); |
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166 | |
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167 | for (NodeIt nn(_graph); nn != INVALID; ++nn) { |
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168 | if (nn != n && fa.minCut(nn) && (*_pred)[nn] == pn) { |
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169 | (*_pred)[nn] = n; |
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170 | } |
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171 | } |
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172 | if ((*_pred)[pn] != INVALID && fa.minCut((*_pred)[pn])) { |
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173 | (*_pred)[n] = (*_pred)[pn]; |
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174 | (*_pred)[pn] = n; |
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175 | (*_weight)[n] = (*_weight)[pn]; |
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176 | (*_weight)[pn] = fa.flowValue(); |
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177 | } |
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178 | } |
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179 | |
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180 | (*_order)[_root] = 0; |
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181 | int index = 1; |
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182 | |
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183 | for (NodeIt n(_graph); n != INVALID; ++n) { |
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184 | std::vector<Node> st; |
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185 | Node nn = n; |
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186 | while ((*_order)[nn] == -1) { |
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187 | st.push_back(nn); |
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188 | nn = (*_pred)[nn]; |
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189 | } |
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190 | while (!st.empty()) { |
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191 | (*_order)[st.back()] = index++; |
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192 | st.pop_back(); |
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193 | } |
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194 | } |
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195 | } |
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196 | |
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197 | public: |
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198 | |
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199 | ///\name Execution Control |
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200 | |
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201 | ///@{ |
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202 | |
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203 | /// \brief Run the Gomory-Hu algorithm. |
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204 | /// |
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205 | /// This function runs the Gomory-Hu algorithm. |
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206 | void run() { |
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207 | init(); |
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208 | start(); |
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209 | } |
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210 | |
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211 | /// @} |
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212 | |
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213 | ///\name Query Functions |
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214 | ///The results of the algorithm can be obtained using these |
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215 | ///functions.\n |
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216 | ///\ref run() should be called before using them.\n |
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217 | ///See also \ref MinCutNodeIt and \ref MinCutEdgeIt. |
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218 | |
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219 | ///@{ |
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220 | |
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221 | /// \brief Return the predecessor node in the Gomory-Hu tree. |
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222 | /// |
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223 | /// This function returns the predecessor node of the given node |
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224 | /// in the Gomory-Hu tree. |
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225 | /// If \c node is the root of the tree, then it returns \c INVALID. |
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226 | /// |
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227 | /// \pre \ref run() must be called before using this function. |
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228 | Node predNode(const Node& node) const { |
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229 | return (*_pred)[node]; |
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230 | } |
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231 | |
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232 | /// \brief Return the weight of the predecessor edge in the |
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233 | /// Gomory-Hu tree. |
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234 | /// |
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235 | /// This function returns the weight of the predecessor edge of the |
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236 | /// given node in the Gomory-Hu tree. |
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237 | /// If \c node is the root of the tree, the result is undefined. |
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238 | /// |
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239 | /// \pre \ref run() must be called before using this function. |
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240 | Value predValue(const Node& node) const { |
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241 | return (*_weight)[node]; |
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242 | } |
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243 | |
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244 | /// \brief Return the distance from the root node in the Gomory-Hu tree. |
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245 | /// |
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246 | /// This function returns the distance of the given node from the root |
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247 | /// node in the Gomory-Hu tree. |
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248 | /// |
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249 | /// \pre \ref run() must be called before using this function. |
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250 | int rootDist(const Node& node) const { |
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251 | return (*_order)[node]; |
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252 | } |
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253 | |
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254 | /// \brief Return the minimum cut value between two nodes |
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255 | /// |
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256 | /// This function returns the minimum cut value between the nodes |
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257 | /// \c s and \c t. |
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258 | /// It finds the nearest common ancestor of the given nodes in the |
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259 | /// Gomory-Hu tree and calculates the minimum weight edge on the |
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260 | /// paths to the ancestor. |
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261 | /// |
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262 | /// \pre \ref run() must be called before using this function. |
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263 | Value minCutValue(const Node& s, const Node& t) const { |
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264 | Node sn = s, tn = t; |
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265 | Value value = std::numeric_limits<Value>::max(); |
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266 | |
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267 | while (sn != tn) { |
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268 | if ((*_order)[sn] < (*_order)[tn]) { |
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269 | if ((*_weight)[tn] <= value) value = (*_weight)[tn]; |
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270 | tn = (*_pred)[tn]; |
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271 | } else { |
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272 | if ((*_weight)[sn] <= value) value = (*_weight)[sn]; |
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273 | sn = (*_pred)[sn]; |
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274 | } |
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275 | } |
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276 | return value; |
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277 | } |
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278 | |
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279 | /// \brief Return the minimum cut between two nodes |
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280 | /// |
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281 | /// This function returns the minimum cut between the nodes \c s and \c t |
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282 | /// in the \c cutMap parameter by setting the nodes in the component of |
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283 | /// \c s to \c true and the other nodes to \c false. |
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284 | /// |
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285 | /// For higher level interfaces see MinCutNodeIt and MinCutEdgeIt. |
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286 | /// |
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287 | /// \param s The base node. |
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288 | /// \param t The node you want to separate from node \c s. |
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289 | /// \param cutMap The cut will be returned in this map. |
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290 | /// It must be a \c bool (or convertible) \ref concepts::ReadWriteMap |
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291 | /// "ReadWriteMap" on the graph nodes. |
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292 | /// |
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293 | /// \return The value of the minimum cut between \c s and \c t. |
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294 | /// |
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295 | /// \pre \ref run() must be called before using this function. |
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296 | template <typename CutMap> |
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297 | Value minCutMap(const Node& s, |
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298 | const Node& t, |
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299 | CutMap& cutMap |
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300 | ) const { |
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301 | Node sn = s, tn = t; |
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302 | bool s_root=false; |
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303 | Node rn = INVALID; |
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304 | Value value = std::numeric_limits<Value>::max(); |
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305 | |
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306 | while (sn != tn) { |
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307 | if ((*_order)[sn] < (*_order)[tn]) { |
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308 | if ((*_weight)[tn] <= value) { |
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309 | rn = tn; |
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310 | s_root = false; |
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311 | value = (*_weight)[tn]; |
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312 | } |
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313 | tn = (*_pred)[tn]; |
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314 | } else { |
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315 | if ((*_weight)[sn] <= value) { |
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316 | rn = sn; |
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317 | s_root = true; |
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318 | value = (*_weight)[sn]; |
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319 | } |
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320 | sn = (*_pred)[sn]; |
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321 | } |
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322 | } |
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323 | |
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324 | typename Graph::template NodeMap<bool> reached(_graph, false); |
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325 | reached[_root] = true; |
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326 | cutMap.set(_root, !s_root); |
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327 | reached[rn] = true; |
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328 | cutMap.set(rn, s_root); |
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329 | |
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330 | std::vector<Node> st; |
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331 | for (NodeIt n(_graph); n != INVALID; ++n) { |
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332 | st.clear(); |
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333 | Node nn = n; |
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334 | while (!reached[nn]) { |
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335 | st.push_back(nn); |
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336 | nn = (*_pred)[nn]; |
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337 | } |
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338 | while (!st.empty()) { |
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339 | cutMap.set(st.back(), cutMap[nn]); |
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340 | st.pop_back(); |
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341 | } |
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342 | } |
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343 | |
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344 | return value; |
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345 | } |
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346 | |
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347 | ///@} |
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348 | |
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349 | friend class MinCutNodeIt; |
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350 | |
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351 | /// Iterate on the nodes of a minimum cut |
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352 | |
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353 | /// This iterator class lists the nodes of a minimum cut found by |
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354 | /// GomoryHu. Before using it, you must allocate a GomoryHu class |
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355 | /// and call its \ref GomoryHu::run() "run()" method. |
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356 | /// |
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357 | /// This example counts the nodes in the minimum cut separating \c s from |
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358 | /// \c t. |
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359 | /// \code |
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360 | /// GomoryHu<Graph> gom(g, capacities); |
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361 | /// gom.run(); |
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362 | /// int cnt=0; |
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363 | /// for(GomoryHu<Graph>::MinCutNodeIt n(gom,s,t); n!=INVALID; ++n) ++cnt; |
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364 | /// \endcode |
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365 | class MinCutNodeIt |
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366 | { |
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367 | bool _side; |
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368 | typename Graph::NodeIt _node_it; |
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369 | typename Graph::template NodeMap<bool> _cut; |
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370 | public: |
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371 | /// Constructor |
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372 | |
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373 | /// Constructor. |
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374 | /// |
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375 | MinCutNodeIt(GomoryHu const &gomory, |
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376 | ///< The GomoryHu class. You must call its |
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377 | /// run() method |
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378 | /// before initializing this iterator. |
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379 | const Node& s, ///< The base node. |
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380 | const Node& t, |
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381 | ///< The node you want to separate from node \c s. |
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382 | bool side=true |
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383 | ///< If it is \c true (default) then the iterator lists |
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384 | /// the nodes of the component containing \c s, |
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385 | /// otherwise it lists the other component. |
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386 | /// \note As the minimum cut is not always unique, |
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387 | /// \code |
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388 | /// MinCutNodeIt(gomory, s, t, true); |
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389 | /// \endcode |
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390 | /// and |
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391 | /// \code |
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392 | /// MinCutNodeIt(gomory, t, s, false); |
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393 | /// \endcode |
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394 | /// does not necessarily give the same set of nodes. |
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395 | /// However, it is ensured that |
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396 | /// \code |
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397 | /// MinCutNodeIt(gomory, s, t, true); |
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398 | /// \endcode |
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399 | /// and |
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400 | /// \code |
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401 | /// MinCutNodeIt(gomory, s, t, false); |
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402 | /// \endcode |
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403 | /// together list each node exactly once. |
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404 | ) |
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405 | : _side(side), _cut(gomory._graph) |
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406 | { |
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407 | gomory.minCutMap(s,t,_cut); |
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408 | for(_node_it=typename Graph::NodeIt(gomory._graph); |
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409 | _node_it!=INVALID && _cut[_node_it]!=_side; |
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410 | ++_node_it) {} |
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411 | } |
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412 | /// Conversion to \c Node |
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413 | |
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414 | /// Conversion to \c Node. |
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415 | /// |
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416 | operator typename Graph::Node() const |
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417 | { |
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418 | return _node_it; |
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419 | } |
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420 | bool operator==(Invalid) { return _node_it==INVALID; } |
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421 | bool operator!=(Invalid) { return _node_it!=INVALID; } |
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422 | /// Next node |
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423 | |
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424 | /// Next node. |
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425 | /// |
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426 | MinCutNodeIt &operator++() |
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427 | { |
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428 | for(++_node_it;_node_it!=INVALID&&_cut[_node_it]!=_side;++_node_it) {} |
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429 | return *this; |
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430 | } |
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431 | /// Postfix incrementation |
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432 | |
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433 | /// Postfix incrementation. |
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434 | /// |
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435 | /// \warning This incrementation |
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436 | /// returns a \c Node, not a \c MinCutNodeIt, as one may |
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437 | /// expect. |
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438 | typename Graph::Node operator++(int) |
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439 | { |
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440 | typename Graph::Node n=*this; |
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441 | ++(*this); |
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442 | return n; |
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443 | } |
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444 | }; |
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445 | |
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446 | friend class MinCutEdgeIt; |
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447 | |
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448 | /// Iterate on the edges of a minimum cut |
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449 | |
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450 | /// This iterator class lists the edges of a minimum cut found by |
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451 | /// GomoryHu. Before using it, you must allocate a GomoryHu class |
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452 | /// and call its \ref GomoryHu::run() "run()" method. |
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453 | /// |
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454 | /// This example computes the value of the minimum cut separating \c s from |
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455 | /// \c t. |
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456 | /// \code |
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457 | /// GomoryHu<Graph> gom(g, capacities); |
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458 | /// gom.run(); |
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459 | /// int value=0; |
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460 | /// for(GomoryHu<Graph>::MinCutEdgeIt e(gom,s,t); e!=INVALID; ++e) |
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461 | /// value+=capacities[e]; |
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462 | /// \endcode |
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463 | /// The result will be the same as the value returned by |
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464 | /// \ref GomoryHu::minCutValue() "gom.minCutValue(s,t)". |
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465 | class MinCutEdgeIt |
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466 | { |
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467 | bool _side; |
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468 | const Graph &_graph; |
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469 | typename Graph::NodeIt _node_it; |
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470 | typename Graph::OutArcIt _arc_it; |
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471 | typename Graph::template NodeMap<bool> _cut; |
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472 | void step() |
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473 | { |
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474 | ++_arc_it; |
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475 | while(_node_it!=INVALID && _arc_it==INVALID) |
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476 | { |
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477 | for(++_node_it;_node_it!=INVALID&&!_cut[_node_it];++_node_it) {} |
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478 | if(_node_it!=INVALID) |
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479 | _arc_it=typename Graph::OutArcIt(_graph,_node_it); |
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480 | } |
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481 | } |
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482 | |
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483 | public: |
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484 | /// Constructor |
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485 | |
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486 | /// Constructor. |
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487 | /// |
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488 | MinCutEdgeIt(GomoryHu const &gomory, |
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489 | ///< The GomoryHu class. You must call its |
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490 | /// run() method |
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491 | /// before initializing this iterator. |
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492 | const Node& s, ///< The base node. |
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493 | const Node& t, |
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494 | ///< The node you want to separate from node \c s. |
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495 | bool side=true |
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496 | ///< If it is \c true (default) then the listed arcs |
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497 | /// will be oriented from the |
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498 | /// nodes of the component containing \c s, |
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499 | /// otherwise they will be oriented in the opposite |
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500 | /// direction. |
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501 | ) |
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502 | : _graph(gomory._graph), _cut(_graph) |
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503 | { |
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504 | gomory.minCutMap(s,t,_cut); |
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505 | if(!side) |
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506 | for(typename Graph::NodeIt n(_graph);n!=INVALID;++n) |
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507 | _cut[n]=!_cut[n]; |
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508 | |
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509 | for(_node_it=typename Graph::NodeIt(_graph); |
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510 | _node_it!=INVALID && !_cut[_node_it]; |
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511 | ++_node_it) {} |
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512 | _arc_it = _node_it!=INVALID ? |
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513 | typename Graph::OutArcIt(_graph,_node_it) : INVALID; |
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514 | while(_node_it!=INVALID && _arc_it == INVALID) |
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515 | { |
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516 | for(++_node_it; _node_it!=INVALID&&!_cut[_node_it]; ++_node_it) {} |
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517 | if(_node_it!=INVALID) |
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518 | _arc_it= typename Graph::OutArcIt(_graph,_node_it); |
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519 | } |
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520 | while(_arc_it!=INVALID && _cut[_graph.target(_arc_it)]) step(); |
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521 | } |
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522 | /// Conversion to \c Arc |
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523 | |
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524 | /// Conversion to \c Arc. |
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525 | /// |
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526 | operator typename Graph::Arc() const |
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527 | { |
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528 | return _arc_it; |
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529 | } |
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530 | /// Conversion to \c Edge |
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531 | |
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532 | /// Conversion to \c Edge. |
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533 | /// |
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534 | operator typename Graph::Edge() const |
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535 | { |
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536 | return _arc_it; |
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537 | } |
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538 | bool operator==(Invalid) { return _node_it==INVALID; } |
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539 | bool operator!=(Invalid) { return _node_it!=INVALID; } |
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540 | /// Next edge |
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541 | |
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542 | /// Next edge. |
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543 | /// |
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544 | MinCutEdgeIt &operator++() |
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545 | { |
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546 | step(); |
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547 | while(_arc_it!=INVALID && _cut[_graph.target(_arc_it)]) step(); |
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548 | return *this; |
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549 | } |
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550 | /// Postfix incrementation |
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551 | |
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552 | /// Postfix incrementation. |
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553 | /// |
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554 | /// \warning This incrementation |
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555 | /// returns an \c Arc, not a \c MinCutEdgeIt, as one may expect. |
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556 | typename Graph::Arc operator++(int) |
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557 | { |
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558 | typename Graph::Arc e=*this; |
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559 | ++(*this); |
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560 | return e; |
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561 | } |
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562 | }; |
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563 | |
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564 | }; |
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565 | |
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566 | } |
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567 | |
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568 | #endif |
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