1 | /* -*- C++ -*- |
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2 | * |
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3 | * This file is a part of LEMON, a generic C++ optimization library |
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4 | * |
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5 | * Copyright (C) 2003-2008 |
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6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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8 | * |
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9 | * Permission to use, modify and distribute this software is granted |
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10 | * provided that this copyright notice appears in all copies. For |
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11 | * precise terms see the accompanying LICENSE file. |
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12 | * |
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13 | * This software is provided "AS IS" with no warranty of any kind, |
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14 | * express or implied, and with no claim as to its suitability for any |
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15 | * purpose. |
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16 | * |
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17 | */ |
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18 | |
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19 | #ifndef LEMON_KRUSKAL_H |
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20 | #define LEMON_KRUSKAL_H |
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21 | |
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22 | #include <algorithm> |
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23 | #include <vector> |
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24 | #include <lemon/unionfind.h> |
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25 | #include <lemon/graph_utils.h> |
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26 | #include <lemon/maps.h> |
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27 | |
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28 | #include <lemon/radix_sort.h> |
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29 | |
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30 | #include <lemon/bits/utility.h> |
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31 | #include <lemon/bits/traits.h> |
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32 | |
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33 | ///\ingroup spantree |
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34 | ///\file |
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35 | ///\brief Kruskal's algorithm to compute a minimum cost tree |
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36 | /// |
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37 | ///Kruskal's algorithm to compute a minimum cost tree. |
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38 | /// |
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39 | |
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40 | namespace lemon { |
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41 | |
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42 | namespace _kruskal_bits { |
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43 | |
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44 | template <typename Map, typename Comp> |
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45 | struct MappedComp { |
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46 | |
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47 | typedef typename Map::Key Key; |
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48 | |
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49 | const Map& map; |
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50 | Comp comp; |
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51 | |
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52 | MappedComp(const Map& _map) |
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53 | : map(_map), comp() {} |
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54 | |
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55 | bool operator()(const Key& left, const Key& right) { |
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56 | return comp(map[left], map[right]); |
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57 | } |
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58 | |
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59 | }; |
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60 | |
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61 | } |
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62 | |
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63 | /// \brief Default traits class of Kruskal class. |
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64 | /// |
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65 | /// Default traits class of Kruskal class. |
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66 | /// \param _UGraph Undirected graph type. |
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67 | /// \param _CostMap Type of cost map. |
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68 | template <typename _UGraph, typename _CostMap> |
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69 | struct KruskalDefaultTraits{ |
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70 | |
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71 | /// \brief The graph type the algorithm runs on. |
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72 | typedef _UGraph UGraph; |
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73 | |
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74 | /// \brief The type of the map that stores the edge costs. |
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75 | /// |
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76 | /// The type of the map that stores the edge costs. |
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77 | /// It must meet the \ref concepts::ReadMap "ReadMap" concept. |
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78 | typedef _CostMap CostMap; |
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79 | |
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80 | /// \brief The type of the cost of the edges. |
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81 | typedef typename _CostMap::Value Value; |
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82 | |
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83 | /// \brief The type of the map that stores whether an edge is in the |
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84 | /// spanning tree or not. |
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85 | /// |
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86 | /// The type of the map that stores whether an edge is in the |
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87 | /// spanning tree or not. |
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88 | typedef typename _UGraph::template UEdgeMap<bool> TreeMap; |
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89 | |
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90 | /// \brief Instantiates a TreeMap. |
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91 | /// |
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92 | /// This function instantiates a \ref TreeMap. |
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93 | /// |
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94 | /// The first parameter is the graph, to which |
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95 | /// we would like to define the \ref TreeMap |
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96 | static TreeMap *createTreeMap(const _UGraph& graph){ |
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97 | return new TreeMap(graph); |
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98 | } |
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99 | |
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100 | template <typename Iterator> |
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101 | static void sort(Iterator begin, Iterator end, const CostMap& cost) { |
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102 | _kruskal_bits::MappedComp<CostMap, std::less<Value> > comp(cost); |
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103 | std::sort(begin, end, comp); |
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104 | } |
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105 | |
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106 | }; |
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107 | |
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108 | ///\ingroup spantree |
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109 | /// |
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110 | /// \brief Kruskal's algorithm to find a minimum cost tree of a graph. |
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111 | /// |
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112 | /// This class implements Kruskal's algorithm to find a minimum cost |
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113 | /// spanning tree. The |
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114 | /// |
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115 | /// \param _UGraph Undirected graph type. |
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116 | /// \param _CostMap Type of cost map. |
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117 | template <typename _UGraph, typename _CostMap, |
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118 | typename _Traits = KruskalDefaultTraits<_UGraph, _CostMap> > |
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119 | class Kruskal { |
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120 | public: |
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121 | |
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122 | typedef _Traits Traits; |
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123 | |
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124 | typedef typename _Traits::UGraph UGraph; |
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125 | typedef typename _Traits::CostMap CostMap; |
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126 | |
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127 | typedef typename _Traits::TreeMap TreeMap; |
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128 | |
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129 | typedef typename _Traits::Value Value; |
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130 | |
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131 | template <typename Comp> |
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132 | struct DefSortCompareTraits : public Traits { |
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133 | template <typename Iterator> |
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134 | static void sort(Iterator begin, Iterator end, const CostMap& cost) { |
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135 | _kruskal_bits::MappedComp<CostMap, Comp> comp(cost, Comp()); |
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136 | std::sort(begin, end, comp); |
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137 | } |
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138 | }; |
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139 | |
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140 | /// \brief \ref named-templ-param "Named parameter" for setting the |
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141 | /// comparator object of the standard sort |
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142 | /// |
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143 | /// \ref named-templ-param "Named parameter" for setting the |
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144 | /// comparator object of the standard sort |
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145 | template <typename Comp> |
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146 | struct DefSortCompare |
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147 | : public Kruskal<UGraph, CostMap, DefSortCompareTraits<Comp> > { |
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148 | typedef Kruskal<UGraph, CostMap, DefSortCompareTraits<Comp> > Create; |
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149 | }; |
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150 | |
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151 | struct DefRadixSortTraits : public Traits { |
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152 | template <typename Iterator> |
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153 | static void sort(Iterator begin, Iterator end, const CostMap& cost) { |
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154 | radixSort(begin, end, cost); |
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155 | } |
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156 | }; |
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157 | |
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158 | /// \brief \ref named-templ-param "Named parameter" for setting the |
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159 | /// sort function to radix sort |
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160 | /// |
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161 | /// \brief \ref named-templ-param "Named parameter" for setting the |
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162 | /// sort function to radix sort. The value type of the cost map should |
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163 | /// be integral, of course. |
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164 | struct DefRadixSort |
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165 | : public Kruskal<UGraph, CostMap, DefRadixSortTraits> { |
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166 | typedef Kruskal<UGraph, CostMap, DefRadixSortTraits> Create; |
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167 | }; |
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168 | |
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169 | template <class TM> |
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170 | struct DefTreeMapTraits : public Traits { |
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171 | typedef TM TreeMap; |
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172 | static TreeMap *createTreeMap(const UGraph &) { |
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173 | throw UninitializedParameter(); |
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174 | } |
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175 | }; |
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176 | |
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177 | /// \brief \ref named-templ-param "Named parameter" for setting |
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178 | /// TreeMap |
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179 | /// |
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180 | /// \ref named-templ-param "Named parameter" for setting TreeMap |
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181 | /// |
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182 | template <class TM> |
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183 | struct DefTreeMap |
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184 | : public Kruskal< UGraph, CostMap, DefTreeMapTraits<TM> > { |
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185 | typedef Kruskal< UGraph, CostMap, DefTreeMapTraits<TM> > Create; |
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186 | }; |
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187 | |
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188 | |
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189 | private: |
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190 | |
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191 | typedef typename UGraph::Node Node; |
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192 | typedef typename UGraph::NodeIt NodeIt; |
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193 | |
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194 | typedef typename UGraph::UEdge UEdge; |
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195 | typedef typename UGraph::UEdgeIt UEdgeIt; |
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196 | |
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197 | const UGraph& graph; |
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198 | const CostMap& cost; |
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199 | |
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200 | std::vector<UEdge> edges; |
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201 | |
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202 | typedef typename UGraph::template NodeMap<int> UfIndex; |
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203 | typedef UnionFind<UfIndex> Uf; |
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204 | UfIndex *ufi; |
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205 | Uf *uf; |
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206 | |
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207 | int index; |
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208 | |
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209 | void initStructures() { |
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210 | if (!_tree) { |
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211 | _tree = Traits::createTreeMap(graph); |
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212 | local_tree = true; |
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213 | } |
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214 | if (!uf) { |
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215 | ufi = new typename UGraph::template NodeMap<int>(graph); |
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216 | uf = new UnionFind<typename UGraph::template NodeMap<int> >(*ufi); |
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217 | } |
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218 | } |
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219 | |
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220 | void initUnionFind() { |
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221 | uf->clear(); |
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222 | for (NodeIt it(graph); it != INVALID; ++it) { |
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223 | uf->insert(it); |
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224 | } |
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225 | } |
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226 | |
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227 | bool local_tree; |
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228 | TreeMap* _tree; |
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229 | |
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230 | public: |
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231 | |
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232 | /// \brief Constructor |
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233 | /// |
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234 | /// Constructor of the algorithm. |
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235 | Kruskal(const UGraph& _graph, const CostMap& _cost) |
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236 | : graph(_graph), cost(_cost), |
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237 | ufi(0), uf(0), local_tree(false), _tree(0) {} |
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238 | |
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239 | /// \brief Destructor |
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240 | /// |
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241 | /// Destructor |
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242 | ~Kruskal() { |
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243 | if (local_tree) { |
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244 | delete _tree; |
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245 | } |
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246 | if (uf) { |
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247 | delete uf; |
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248 | delete ufi; |
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249 | } |
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250 | } |
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251 | |
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252 | /// \brief Sets the map storing the tree edges. |
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253 | /// |
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254 | /// Sets the map storing the tree edges. |
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255 | /// If you don't use this function before calling \ref run(), |
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256 | /// it will allocate one. The destuctor deallocates this |
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257 | /// automatically allocated map, of course. |
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258 | /// \return \c *this </tt> |
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259 | Kruskal& treeMap(TreeMap &m){ |
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260 | if (local_tree) { |
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261 | delete _tree; |
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262 | local_tree = false; |
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263 | } |
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264 | _tree = &m; |
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265 | return *this; |
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266 | } |
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267 | |
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268 | /// \brief Initialize the algorithm |
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269 | /// |
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270 | /// This member function initializes the unionfind data structure |
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271 | /// and sorts the edges into ascending order |
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272 | void init() { |
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273 | initStructures(); |
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274 | initUnionFind(); |
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275 | for (UEdgeIt e(graph); e != INVALID; ++e) { |
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276 | edges.push_back(e); |
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277 | _tree->set(e, false); |
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278 | } |
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279 | Traits::sort(edges.begin(), edges.end(), cost); |
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280 | index = 0; |
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281 | } |
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282 | |
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283 | |
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284 | /// \brief Initialize the algorithm |
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285 | /// |
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286 | /// This member function initializes the unionfind data structure |
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287 | /// and sets the edge order to the given sequence. The given |
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288 | /// sequence should be a valid STL range of undirected edges. |
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289 | template <typename Iterator> |
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290 | void initPresorted(Iterator begin, Iterator end) { |
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291 | initStructures(); |
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292 | initUnionFind(); |
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293 | edges.clear(); |
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294 | std::copy(begin, end, std::back_inserter(edges)); |
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295 | index = 0; |
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296 | } |
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297 | |
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298 | /// \brief Initialize the algorithm |
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299 | /// |
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300 | /// This member function initializes the unionfind data structure |
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301 | /// and sets the tree to empty. It does not change the order of |
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302 | /// the edges, it uses the order of the previous running. |
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303 | void reinit() { |
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304 | initStructures(); |
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305 | initUnionFind(); |
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306 | for (UEdgeIt e(graph); e != INVALID; ++e) { |
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307 | _tree->set(e, false); |
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308 | } |
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309 | index = 0; |
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310 | } |
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311 | |
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312 | |
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313 | /// \brief Executes the algorithm. |
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314 | /// |
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315 | /// Executes the algorithm. |
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316 | /// |
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317 | /// \pre init() must be called before using this function. |
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318 | /// |
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319 | /// This method runs the %Kruskal algorithm. |
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320 | void start() { |
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321 | while (index < int(edges.size())) { |
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322 | if (uf->join(graph.target(edges[index]), graph.source(edges[index]))) { |
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323 | _tree->set(edges[index], true); |
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324 | } |
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325 | ++index; |
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326 | } |
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327 | } |
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328 | |
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329 | /// \brief Runs the prim algorithm until it find a new tree edge |
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330 | /// |
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331 | /// Runs the prim algorithm until it find a new tree edge. If it |
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332 | /// does not next tree edge in the sequence it gives back \c INVALID. |
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333 | UEdge findNextTreeEdge() { |
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334 | while (index < int(edges.size())) { |
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335 | if (uf->join(graph.target(edges[index]), graph.source(edges[index]))) { |
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336 | _tree->set(edges[index], true); |
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337 | return edges[index++]; |
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338 | } |
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339 | ++index; |
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340 | } |
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341 | return INVALID; |
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342 | } |
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343 | |
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344 | /// \brief Processes the next edge in the sequence |
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345 | /// |
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346 | /// Processes the next edge in the sequence. |
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347 | /// |
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348 | /// \return The prcocessed edge. |
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349 | /// |
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350 | /// \warning The sequence must not be empty! |
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351 | UEdge processNextEdge() { |
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352 | UEdge edge = edges[index++]; |
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353 | processEdge(edge); |
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354 | return edge; |
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355 | } |
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356 | |
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357 | /// \brief Processes an arbitrary edge |
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358 | /// |
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359 | /// Processes the next edge in the sequence. |
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360 | /// |
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361 | /// \return True when the edge is a tree edge. |
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362 | bool processEdge(const UEdge& edge) { |
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363 | if (uf->join(graph.target(edge), graph.source(edge))) { |
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364 | _tree->set(edge, true); |
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365 | return true; |
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366 | } else { |
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367 | return false; |
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368 | } |
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369 | } |
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370 | |
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371 | /// \brief Returns \c false if there are edge to be processed in |
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372 | /// sequence |
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373 | /// |
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374 | /// Returns \c false if there are nodes to be processed in the |
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375 | /// sequence |
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376 | bool emptyQueue() { |
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377 | return index == int(edges.size()); |
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378 | } |
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379 | |
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380 | /// \brief Returns the next edge to be processed |
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381 | /// |
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382 | /// Returns the next edge to be processed |
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383 | /// |
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384 | UEdge nextEdge() const { |
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385 | return edges[index]; |
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386 | } |
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387 | |
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388 | /// \brief Runs %Kruskal algorithm. |
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389 | /// |
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390 | /// This method runs the %Kruskal algorithm in order to compute the |
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391 | /// minimum spanning tree (or minimum spanning forest). The |
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392 | /// method also works on graphs that has more than one components. |
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393 | /// In this case it computes the minimum spanning forest. |
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394 | void run() { |
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395 | init(); |
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396 | start(); |
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397 | } |
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398 | |
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399 | /// \brief Returns a reference to the tree edges map |
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400 | /// |
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401 | /// Returns a reference to the TreeEdgeMap of the edges of the |
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402 | /// minimum spanning tree. The value of the map is \c true only if |
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403 | /// the edge is in the minimum spanning tree. |
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404 | /// |
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405 | const TreeMap &treeMap() const { return *_tree;} |
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406 | |
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407 | /// \brief Returns the total cost of the tree |
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408 | /// |
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409 | /// Returns the total cost of the tree |
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410 | Value treeValue() const { |
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411 | Value value = 0; |
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412 | for (UEdgeIt it(graph); it != INVALID; ++it) { |
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413 | if ((*_tree)[it]) { |
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414 | value += cost[it]; |
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415 | } |
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416 | } |
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417 | return value; |
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418 | } |
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419 | |
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420 | /// \brief Returns true when the given edge is tree edge |
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421 | /// |
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422 | /// Returns true when the given edge is tree edge |
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423 | bool tree(UEdge e) const { |
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424 | return (*_tree)[e]; |
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425 | } |
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426 | |
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427 | |
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428 | }; |
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429 | |
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430 | |
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431 | namespace _kruskal_bits { |
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432 | |
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433 | template <typename Graph, typename In, typename Out> |
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434 | typename In::value_type::second_type |
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435 | kruskal(const Graph& graph, const In& in, Out& out) { |
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436 | typedef typename In::value_type::second_type Value; |
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437 | typedef typename Graph::template NodeMap<int> IndexMap; |
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438 | typedef typename Graph::Node Node; |
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439 | |
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440 | IndexMap index(graph); |
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441 | UnionFind<IndexMap> uf(index); |
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442 | for (typename Graph::NodeIt it(graph); it != INVALID; ++it) { |
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443 | uf.insert(it); |
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444 | } |
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445 | |
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446 | Value tree_value = 0; |
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447 | for (typename In::const_iterator it = in.begin(); it != in.end(); ++it) { |
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448 | if (uf.join(graph.target(it->first),graph.source(it->first))) { |
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449 | out.set(it->first, true); |
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450 | tree_value += it->second; |
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451 | } |
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452 | else { |
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453 | out.set(it->first, false); |
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454 | } |
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455 | } |
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456 | return tree_value; |
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457 | } |
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458 | |
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459 | |
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460 | template <typename Sequence> |
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461 | struct PairComp { |
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462 | typedef typename Sequence::value_type Value; |
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463 | bool operator()(const Value& left, const Value& right) { |
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464 | return left.second < right.second; |
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465 | } |
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466 | }; |
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467 | |
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468 | template <typename In, typename Enable = void> |
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469 | struct SequenceInputIndicator { |
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470 | static const bool value = false; |
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471 | }; |
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472 | |
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473 | template <typename In> |
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474 | struct SequenceInputIndicator<In, |
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475 | typename exists<typename In::value_type::first_type>::type> { |
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476 | static const bool value = true; |
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477 | }; |
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478 | |
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479 | template <typename In, typename Enable = void> |
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480 | struct MapInputIndicator { |
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481 | static const bool value = false; |
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482 | }; |
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483 | |
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484 | template <typename In> |
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485 | struct MapInputIndicator<In, |
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486 | typename exists<typename In::Value>::type> { |
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487 | static const bool value = true; |
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488 | }; |
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489 | |
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490 | template <typename In, typename Enable = void> |
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491 | struct SequenceOutputIndicator { |
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492 | static const bool value = false; |
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493 | }; |
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494 | |
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495 | template <typename Out> |
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496 | struct SequenceOutputIndicator<Out, |
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497 | typename exists<typename Out::value_type>::type> { |
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498 | static const bool value = true; |
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499 | }; |
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500 | |
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501 | template <typename Out, typename Enable = void> |
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502 | struct MapOutputIndicator { |
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503 | static const bool value = false; |
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504 | }; |
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505 | |
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506 | template <typename Out> |
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507 | struct MapOutputIndicator<Out, |
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508 | typename exists<typename Out::Value>::type> { |
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509 | static const bool value = true; |
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510 | }; |
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511 | |
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512 | template <typename In, typename InEnable = void> |
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513 | struct KruskalValueSelector {}; |
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514 | |
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515 | template <typename In> |
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516 | struct KruskalValueSelector<In, |
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517 | typename enable_if<SequenceInputIndicator<In>, void>::type> |
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518 | { |
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519 | typedef typename In::value_type::second_type Value; |
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520 | }; |
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521 | |
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522 | template <typename In> |
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523 | struct KruskalValueSelector<In, |
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524 | typename enable_if<MapInputIndicator<In>, void>::type> |
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525 | { |
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526 | typedef typename In::Value Value; |
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527 | }; |
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528 | |
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529 | template <typename Graph, typename In, typename Out, |
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530 | typename InEnable = void> |
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531 | struct KruskalInputSelector {}; |
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532 | |
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533 | template <typename Graph, typename In, typename Out, |
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534 | typename InEnable = void> |
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535 | struct KruskalOutputSelector {}; |
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536 | |
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537 | template <typename Graph, typename In, typename Out> |
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538 | struct KruskalInputSelector<Graph, In, Out, |
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539 | typename enable_if<SequenceInputIndicator<In>, void>::type > |
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540 | { |
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541 | typedef typename In::value_type::second_type Value; |
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542 | |
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543 | static Value kruskal(const Graph& graph, const In& in, Out& out) { |
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544 | return KruskalOutputSelector<Graph, In, Out>:: |
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545 | kruskal(graph, in, out); |
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546 | } |
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547 | |
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548 | }; |
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549 | |
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550 | template <typename Graph, typename In, typename Out> |
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551 | struct KruskalInputSelector<Graph, In, Out, |
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552 | typename enable_if<MapInputIndicator<In>, void>::type > |
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553 | { |
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554 | typedef typename In::Value Value; |
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555 | static Value kruskal(const Graph& graph, const In& in, Out& out) { |
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556 | typedef typename In::Key MapEdge; |
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557 | typedef typename In::Value Value; |
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558 | typedef typename ItemSetTraits<Graph, MapEdge>::ItemIt MapEdgeIt; |
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559 | typedef std::vector<std::pair<MapEdge, Value> > Sequence; |
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560 | Sequence seq; |
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561 | |
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562 | for (MapEdgeIt it(graph); it != INVALID; ++it) { |
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563 | seq.push_back(std::make_pair(it, in[it])); |
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564 | } |
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565 | |
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566 | std::sort(seq.begin(), seq.end(), PairComp<Sequence>()); |
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567 | return KruskalOutputSelector<Graph, Sequence, Out>:: |
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568 | kruskal(graph, seq, out); |
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569 | } |
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570 | }; |
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571 | |
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572 | template <typename Graph, typename In, typename Out> |
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573 | struct KruskalOutputSelector<Graph, In, Out, |
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574 | typename enable_if<SequenceOutputIndicator<Out>, void>::type > |
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575 | { |
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576 | typedef typename In::value_type::second_type Value; |
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577 | |
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578 | static Value kruskal(const Graph& graph, const In& in, Out& out) { |
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579 | typedef StoreBoolMap<Out> Map; |
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580 | Map map(out); |
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581 | return _kruskal_bits::kruskal(graph, in, map); |
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582 | } |
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583 | |
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584 | }; |
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585 | |
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586 | template <typename Graph, typename In, typename Out> |
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587 | struct KruskalOutputSelector<Graph, In, Out, |
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588 | typename enable_if<MapOutputIndicator<Out>, void>::type > |
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589 | { |
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590 | typedef typename In::value_type::second_type Value; |
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591 | |
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592 | static Value kruskal(const Graph& graph, const In& in, Out& out) { |
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593 | return _kruskal_bits::kruskal(graph, in, out); |
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594 | } |
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595 | }; |
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596 | |
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597 | } |
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598 | |
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599 | /// \ingroup spantree |
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600 | /// |
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601 | /// \brief Kruskal's algorithm to find a minimum cost tree of a graph. |
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602 | /// |
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603 | /// This function runs Kruskal's algorithm to find a minimum cost tree. |
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604 | /// Due to hard C++ hacking, it accepts various input and output types. |
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605 | /// |
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606 | /// \param g The graph the algorithm runs on. |
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607 | /// It can be either \ref concepts::Graph "directed" or |
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608 | /// \ref concepts::UGraph "undirected". |
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609 | /// If the graph is directed, the algorithm consider it to be |
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610 | /// undirected by disregarding the direction of the edges. |
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611 | /// |
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612 | /// \param in This object is used to describe the edge costs. It can be one |
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613 | /// of the following choices. |
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614 | /// |
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615 | /// - An STL compatible 'Forward Container' with |
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616 | /// <tt>std::pair<GR::UEdge,X></tt> or |
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617 | /// <tt>std::pair<GR::Edge,X></tt> as its <tt>value_type</tt>, where |
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618 | /// \c X is the type of the costs. The pairs indicates the edges |
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619 | /// along with the assigned cost. <em>They must be in a |
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620 | /// cost-ascending order.</em> |
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621 | /// - Any readable Edge map. The values of the map indicate the edge costs. |
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622 | /// |
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623 | /// \retval out Here we also have a choise. |
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624 | /// - It can be a writable \c bool edge map. After running the |
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625 | /// algorithm this will contain the found minimum cost spanning |
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626 | /// tree: the value of an edge will be set to \c true if it belongs |
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627 | /// to the tree, otherwise it will be set to \c false. The value of |
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628 | /// each edge will be set exactly once. |
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629 | /// - It can also be an iteraror of an STL Container with |
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630 | /// <tt>GR::UEdge</tt> or <tt>GR::Edge</tt> as its |
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631 | /// <tt>value_type</tt>. The algorithm copies the elements of the |
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632 | /// found tree into this sequence. For example, if we know that the |
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633 | /// spanning tree of the graph \c g has say 53 edges, then we can |
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634 | /// put its edges into an STL vector \c tree with a code like this. |
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635 | ///\code |
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636 | /// std::vector<Edge> tree(53); |
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637 | /// kruskal(g,cost,tree.begin()); |
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638 | ///\endcode |
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639 | /// Or if we don't know in advance the size of the tree, we can |
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640 | /// write this. |
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641 | ///\code std::vector<Edge> tree; |
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642 | /// kruskal(g,cost,std::back_inserter(tree)); |
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643 | ///\endcode |
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644 | /// |
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645 | /// \return The total cost of the found tree. |
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646 | /// |
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647 | /// \warning If kruskal runs on an be consistent of using the same |
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648 | /// Edge type for input and output. |
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649 | /// |
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650 | |
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651 | #ifdef DOXYGEN |
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652 | template <class Graph, class In, class Out> |
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653 | Value kruskal(GR const& g, const In& in, Out& out) |
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654 | #else |
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655 | template <class Graph, class In, class Out> |
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656 | inline typename _kruskal_bits::KruskalValueSelector<In>::Value |
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657 | kruskal(const Graph& graph, const In& in, Out& out) |
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658 | #endif |
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659 | { |
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660 | return _kruskal_bits::KruskalInputSelector<Graph, In, Out>:: |
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661 | kruskal(graph, in, out); |
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662 | } |
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663 | |
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664 | |
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665 | |
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666 | |
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667 | template <class Graph, class In, class Out> |
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668 | inline typename _kruskal_bits::KruskalValueSelector<In>::Value |
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669 | kruskal(const Graph& graph, const In& in, const Out& out) |
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670 | { |
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671 | return _kruskal_bits::KruskalInputSelector<Graph, In, const Out>:: |
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672 | kruskal(graph, in, out); |
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673 | } |
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674 | |
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675 | } //namespace lemon |
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676 | |
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677 | #endif //LEMON_KRUSKAL_H |
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