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_HOWARD_H |
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20 | #define LEMON_HOWARD_H |
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21 | |
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22 | /// \ingroup min_mean_cycle |
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23 | /// |
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24 | /// \file |
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25 | /// \brief Howard's algorithm for finding a minimum mean cycle. |
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26 | |
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27 | #include <vector> |
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28 | #include <limits> |
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29 | #include <lemon/core.h> |
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30 | #include <lemon/path.h> |
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31 | #include <lemon/tolerance.h> |
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32 | #include <lemon/connectivity.h> |
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33 | |
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34 | namespace lemon { |
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35 | |
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36 | /// \brief Default traits class of Howard class. |
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37 | /// |
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38 | /// Default traits class of Howard class. |
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39 | /// \tparam GR The type of the digraph. |
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40 | /// \tparam LEN The type of the length map. |
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41 | /// It must conform to the \ref concepts::ReadMap "ReadMap" concept. |
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42 | #ifdef DOXYGEN |
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43 | template <typename GR, typename LEN> |
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44 | #else |
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45 | template <typename GR, typename LEN, |
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46 | bool integer = std::numeric_limits<typename LEN::Value>::is_integer> |
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47 | #endif |
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48 | struct HowardDefaultTraits |
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49 | { |
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50 | /// The type of the digraph |
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51 | typedef GR Digraph; |
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52 | /// The type of the length map |
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53 | typedef LEN LengthMap; |
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54 | /// The type of the arc lengths |
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55 | typedef typename LengthMap::Value Value; |
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56 | |
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57 | /// \brief The large value type used for internal computations |
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58 | /// |
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59 | /// The large value type used for internal computations. |
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60 | /// It is \c long \c long if the \c Value type is integer, |
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61 | /// otherwise it is \c double. |
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62 | /// \c Value must be convertible to \c LargeValue. |
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63 | typedef double LargeValue; |
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64 | |
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65 | /// The tolerance type used for internal computations |
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66 | typedef lemon::Tolerance<LargeValue> Tolerance; |
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67 | |
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68 | /// \brief The path type of the found cycles |
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69 | /// |
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70 | /// The path type of the found cycles. |
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71 | /// It must conform to the \ref lemon::concepts::Path "Path" concept |
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72 | /// and it must have an \c addBack() function. |
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73 | typedef lemon::Path<Digraph> Path; |
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74 | }; |
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75 | |
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76 | // Default traits class for integer value types |
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77 | template <typename GR, typename LEN> |
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78 | struct HowardDefaultTraits<GR, LEN, true> |
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79 | { |
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80 | typedef GR Digraph; |
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81 | typedef LEN LengthMap; |
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82 | typedef typename LengthMap::Value Value; |
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83 | #ifdef LEMON_HAVE_LONG_LONG |
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84 | typedef long long LargeValue; |
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85 | #else |
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86 | typedef long LargeValue; |
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87 | #endif |
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88 | typedef lemon::Tolerance<LargeValue> Tolerance; |
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89 | typedef lemon::Path<Digraph> Path; |
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90 | }; |
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91 | |
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92 | |
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93 | /// \addtogroup min_mean_cycle |
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94 | /// @{ |
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95 | |
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96 | /// \brief Implementation of Howard's algorithm for finding a minimum |
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97 | /// mean cycle. |
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98 | /// |
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99 | /// This class implements Howard's policy iteration algorithm for finding |
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100 | /// a directed cycle of minimum mean length (cost) in a digraph |
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101 | /// \ref amo93networkflows, \ref dasdan98minmeancycle. |
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102 | /// This class provides the most efficient algorithm for the |
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103 | /// minimum mean cycle problem, though the best known theoretical |
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104 | /// bound on its running time is exponential. |
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105 | /// |
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106 | /// \tparam GR The type of the digraph the algorithm runs on. |
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107 | /// \tparam LEN The type of the length map. The default |
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108 | /// map type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>". |
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109 | /// \tparam TR The traits class that defines various types used by the |
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110 | /// algorithm. By default, it is \ref HowardDefaultTraits |
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111 | /// "HowardDefaultTraits<GR, LEN>". |
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112 | /// In most cases, this parameter should not be set directly, |
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113 | /// consider to use the named template parameters instead. |
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114 | #ifdef DOXYGEN |
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115 | template <typename GR, typename LEN, typename TR> |
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116 | #else |
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117 | template < typename GR, |
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118 | typename LEN = typename GR::template ArcMap<int>, |
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119 | typename TR = HowardDefaultTraits<GR, LEN> > |
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120 | #endif |
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121 | class Howard |
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122 | { |
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123 | public: |
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124 | |
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125 | /// The type of the digraph |
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126 | typedef typename TR::Digraph Digraph; |
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127 | /// The type of the length map |
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128 | typedef typename TR::LengthMap LengthMap; |
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129 | /// The type of the arc lengths |
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130 | typedef typename TR::Value Value; |
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131 | |
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132 | /// \brief The large value type |
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133 | /// |
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134 | /// The large value type used for internal computations. |
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135 | /// By default, it is \c long \c long if the \c Value type is integer, |
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136 | /// otherwise it is \c double. |
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137 | typedef typename TR::LargeValue LargeValue; |
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138 | |
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139 | /// The tolerance type |
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140 | typedef typename TR::Tolerance Tolerance; |
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141 | |
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142 | /// \brief The path type of the found cycles |
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143 | /// |
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144 | /// The path type of the found cycles. |
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145 | /// Using the \ref HowardDefaultTraits "default traits class", |
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146 | /// it is \ref lemon::Path "Path<Digraph>". |
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147 | typedef typename TR::Path Path; |
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148 | |
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149 | /// The \ref HowardDefaultTraits "traits class" of the algorithm |
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150 | typedef TR Traits; |
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151 | |
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152 | private: |
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153 | |
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154 | TEMPLATE_DIGRAPH_TYPEDEFS(Digraph); |
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155 | |
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156 | // The digraph the algorithm runs on |
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157 | const Digraph &_gr; |
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158 | // The length of the arcs |
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159 | const LengthMap &_length; |
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160 | |
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161 | // Data for the found cycles |
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162 | bool _curr_found, _best_found; |
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163 | LargeValue _curr_length, _best_length; |
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164 | int _curr_size, _best_size; |
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165 | Node _curr_node, _best_node; |
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166 | |
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167 | Path *_cycle_path; |
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168 | bool _local_path; |
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169 | |
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170 | // Internal data used by the algorithm |
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171 | typename Digraph::template NodeMap<Arc> _policy; |
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172 | typename Digraph::template NodeMap<bool> _reached; |
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173 | typename Digraph::template NodeMap<int> _level; |
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174 | typename Digraph::template NodeMap<LargeValue> _dist; |
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175 | |
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176 | // Data for storing the strongly connected components |
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177 | int _comp_num; |
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178 | typename Digraph::template NodeMap<int> _comp; |
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179 | std::vector<std::vector<Node> > _comp_nodes; |
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180 | std::vector<Node>* _nodes; |
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181 | typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs; |
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182 | |
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183 | // Queue used for BFS search |
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184 | std::vector<Node> _queue; |
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185 | int _qfront, _qback; |
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186 | |
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187 | Tolerance _tolerance; |
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188 | |
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189 | // Infinite constant |
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190 | const LargeValue INF; |
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191 | |
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192 | public: |
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193 | |
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194 | /// \name Named Template Parameters |
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195 | /// @{ |
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196 | |
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197 | template <typename T> |
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198 | struct SetLargeValueTraits : public Traits { |
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199 | typedef T LargeValue; |
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200 | typedef lemon::Tolerance<T> Tolerance; |
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201 | }; |
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202 | |
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203 | /// \brief \ref named-templ-param "Named parameter" for setting |
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204 | /// \c LargeValue type. |
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205 | /// |
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206 | /// \ref named-templ-param "Named parameter" for setting \c LargeValue |
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207 | /// type. It is used for internal computations in the algorithm. |
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208 | template <typename T> |
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209 | struct SetLargeValue |
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210 | : public Howard<GR, LEN, SetLargeValueTraits<T> > { |
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211 | typedef Howard<GR, LEN, SetLargeValueTraits<T> > Create; |
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212 | }; |
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213 | |
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214 | template <typename T> |
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215 | struct SetPathTraits : public Traits { |
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216 | typedef T Path; |
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217 | }; |
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218 | |
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219 | /// \brief \ref named-templ-param "Named parameter" for setting |
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220 | /// \c %Path type. |
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221 | /// |
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222 | /// \ref named-templ-param "Named parameter" for setting the \c %Path |
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223 | /// type of the found cycles. |
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224 | /// It must conform to the \ref lemon::concepts::Path "Path" concept |
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225 | /// and it must have an \c addBack() function. |
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226 | template <typename T> |
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227 | struct SetPath |
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228 | : public Howard<GR, LEN, SetPathTraits<T> > { |
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229 | typedef Howard<GR, LEN, SetPathTraits<T> > Create; |
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230 | }; |
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231 | |
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232 | /// @} |
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233 | |
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234 | public: |
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235 | |
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236 | /// \brief Constructor. |
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237 | /// |
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238 | /// The constructor of the class. |
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239 | /// |
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240 | /// \param digraph The digraph the algorithm runs on. |
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241 | /// \param length The lengths (costs) of the arcs. |
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242 | Howard( const Digraph &digraph, |
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243 | const LengthMap &length ) : |
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244 | _gr(digraph), _length(length), _best_found(false), |
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245 | _best_length(0), _best_size(1), _cycle_path(NULL), _local_path(false), |
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246 | _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph), |
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247 | _comp(digraph), _in_arcs(digraph), |
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248 | INF(std::numeric_limits<LargeValue>::has_infinity ? |
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249 | std::numeric_limits<LargeValue>::infinity() : |
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250 | std::numeric_limits<LargeValue>::max()) |
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251 | {} |
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252 | |
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253 | /// Destructor. |
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254 | ~Howard() { |
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255 | if (_local_path) delete _cycle_path; |
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256 | } |
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257 | |
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258 | /// \brief Set the path structure for storing the found cycle. |
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259 | /// |
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260 | /// This function sets an external path structure for storing the |
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261 | /// found cycle. |
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262 | /// |
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263 | /// If you don't call this function before calling \ref run() or |
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264 | /// \ref findMinMean(), it will allocate a local \ref Path "path" |
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265 | /// structure. The destuctor deallocates this automatically |
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266 | /// allocated object, of course. |
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267 | /// |
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268 | /// \note The algorithm calls only the \ref lemon::Path::addBack() |
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269 | /// "addBack()" function of the given path structure. |
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270 | /// |
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271 | /// \return <tt>(*this)</tt> |
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272 | Howard& cycle(Path &path) { |
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273 | if (_local_path) { |
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274 | delete _cycle_path; |
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275 | _local_path = false; |
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276 | } |
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277 | _cycle_path = &path; |
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278 | return *this; |
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279 | } |
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280 | |
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281 | /// \brief Set the tolerance used by the algorithm. |
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282 | /// |
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283 | /// This function sets the tolerance object used by the algorithm. |
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284 | /// |
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285 | /// \return <tt>(*this)</tt> |
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286 | Howard& tolerance(const Tolerance& tolerance) { |
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287 | _tolerance = tolerance; |
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288 | return *this; |
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289 | } |
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290 | |
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291 | /// \brief Return a const reference to the tolerance. |
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292 | /// |
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293 | /// This function returns a const reference to the tolerance object |
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294 | /// used by the algorithm. |
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295 | const Tolerance& tolerance() const { |
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296 | return _tolerance; |
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297 | } |
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298 | |
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299 | /// \name Execution control |
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300 | /// The simplest way to execute the algorithm is to call the \ref run() |
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301 | /// function.\n |
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302 | /// If you only need the minimum mean length, you may call |
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303 | /// \ref findMinMean(). |
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304 | |
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305 | /// @{ |
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306 | |
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307 | /// \brief Run the algorithm. |
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308 | /// |
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309 | /// This function runs the algorithm. |
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310 | /// It can be called more than once (e.g. if the underlying digraph |
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311 | /// and/or the arc lengths have been modified). |
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312 | /// |
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313 | /// \return \c true if a directed cycle exists in the digraph. |
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314 | /// |
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315 | /// \note <tt>mmc.run()</tt> is just a shortcut of the following code. |
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316 | /// \code |
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317 | /// return mmc.findMinMean() && mmc.findCycle(); |
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318 | /// \endcode |
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319 | bool run() { |
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320 | return findMinMean() && findCycle(); |
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321 | } |
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322 | |
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323 | /// \brief Find the minimum cycle mean. |
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324 | /// |
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325 | /// This function finds the minimum mean length of the directed |
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326 | /// cycles in the digraph. |
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327 | /// |
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328 | /// \return \c true if a directed cycle exists in the digraph. |
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329 | bool findMinMean() { |
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330 | // Initialize and find strongly connected components |
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331 | init(); |
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332 | findComponents(); |
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333 | |
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334 | // Find the minimum cycle mean in the components |
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335 | for (int comp = 0; comp < _comp_num; ++comp) { |
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336 | // Find the minimum mean cycle in the current component |
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337 | if (!buildPolicyGraph(comp)) continue; |
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338 | while (true) { |
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339 | findPolicyCycle(); |
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340 | if (!computeNodeDistances()) break; |
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341 | } |
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342 | // Update the best cycle (global minimum mean cycle) |
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343 | if ( _curr_found && (!_best_found || |
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344 | _curr_length * _best_size < _best_length * _curr_size) ) { |
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345 | _best_found = true; |
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346 | _best_length = _curr_length; |
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347 | _best_size = _curr_size; |
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348 | _best_node = _curr_node; |
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349 | } |
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350 | } |
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351 | return _best_found; |
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352 | } |
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353 | |
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354 | /// \brief Find a minimum mean directed cycle. |
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355 | /// |
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356 | /// This function finds a directed cycle of minimum mean length |
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357 | /// in the digraph using the data computed by findMinMean(). |
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358 | /// |
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359 | /// \return \c true if a directed cycle exists in the digraph. |
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360 | /// |
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361 | /// \pre \ref findMinMean() must be called before using this function. |
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362 | bool findCycle() { |
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363 | if (!_best_found) return false; |
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364 | _cycle_path->addBack(_policy[_best_node]); |
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365 | for ( Node v = _best_node; |
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366 | (v = _gr.target(_policy[v])) != _best_node; ) { |
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367 | _cycle_path->addBack(_policy[v]); |
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368 | } |
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369 | return true; |
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370 | } |
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371 | |
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372 | /// @} |
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373 | |
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374 | /// \name Query Functions |
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375 | /// The results of the algorithm can be obtained using these |
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376 | /// functions.\n |
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377 | /// The algorithm should be executed before using them. |
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378 | |
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379 | /// @{ |
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380 | |
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381 | /// \brief Return the total length of the found cycle. |
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382 | /// |
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383 | /// This function returns the total length of the found cycle. |
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384 | /// |
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385 | /// \pre \ref run() or \ref findMinMean() must be called before |
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386 | /// using this function. |
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387 | Value cycleLength() const { |
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388 | return static_cast<Value>(_best_length); |
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389 | } |
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390 | |
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391 | /// \brief Return the number of arcs on the found cycle. |
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392 | /// |
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393 | /// This function returns the number of arcs on the found cycle. |
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394 | /// |
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395 | /// \pre \ref run() or \ref findMinMean() must be called before |
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396 | /// using this function. |
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397 | int cycleArcNum() const { |
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398 | return _best_size; |
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399 | } |
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400 | |
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401 | /// \brief Return the mean length of the found cycle. |
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402 | /// |
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403 | /// This function returns the mean length of the found cycle. |
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404 | /// |
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405 | /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the |
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406 | /// following code. |
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407 | /// \code |
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408 | /// return static_cast<double>(alg.cycleLength()) / alg.cycleArcNum(); |
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409 | /// \endcode |
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410 | /// |
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411 | /// \pre \ref run() or \ref findMinMean() must be called before |
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412 | /// using this function. |
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413 | double cycleMean() const { |
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414 | return static_cast<double>(_best_length) / _best_size; |
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415 | } |
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416 | |
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417 | /// \brief Return the found cycle. |
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418 | /// |
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419 | /// This function returns a const reference to the path structure |
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420 | /// storing the found cycle. |
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421 | /// |
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422 | /// \pre \ref run() or \ref findCycle() must be called before using |
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423 | /// this function. |
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424 | const Path& cycle() const { |
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425 | return *_cycle_path; |
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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 | private: |
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431 | |
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432 | // Initialize |
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433 | void init() { |
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434 | if (!_cycle_path) { |
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435 | _local_path = true; |
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436 | _cycle_path = new Path; |
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437 | } |
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438 | _queue.resize(countNodes(_gr)); |
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439 | _best_found = false; |
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440 | _best_length = 0; |
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441 | _best_size = 1; |
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442 | _cycle_path->clear(); |
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443 | } |
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444 | |
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445 | // Find strongly connected components and initialize _comp_nodes |
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446 | // and _in_arcs |
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447 | void findComponents() { |
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448 | _comp_num = stronglyConnectedComponents(_gr, _comp); |
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449 | _comp_nodes.resize(_comp_num); |
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450 | if (_comp_num == 1) { |
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451 | _comp_nodes[0].clear(); |
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452 | for (NodeIt n(_gr); n != INVALID; ++n) { |
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453 | _comp_nodes[0].push_back(n); |
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454 | _in_arcs[n].clear(); |
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455 | for (InArcIt a(_gr, n); a != INVALID; ++a) { |
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456 | _in_arcs[n].push_back(a); |
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457 | } |
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458 | } |
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459 | } else { |
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460 | for (int i = 0; i < _comp_num; ++i) |
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461 | _comp_nodes[i].clear(); |
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462 | for (NodeIt n(_gr); n != INVALID; ++n) { |
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463 | int k = _comp[n]; |
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464 | _comp_nodes[k].push_back(n); |
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465 | _in_arcs[n].clear(); |
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466 | for (InArcIt a(_gr, n); a != INVALID; ++a) { |
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467 | if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a); |
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468 | } |
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469 | } |
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470 | } |
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471 | } |
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472 | |
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473 | // Build the policy graph in the given strongly connected component |
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474 | // (the out-degree of every node is 1) |
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475 | bool buildPolicyGraph(int comp) { |
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476 | _nodes = &(_comp_nodes[comp]); |
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477 | if (_nodes->size() < 1 || |
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478 | (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) { |
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479 | return false; |
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480 | } |
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481 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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482 | _dist[(*_nodes)[i]] = INF; |
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483 | } |
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484 | Node u, v; |
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485 | Arc e; |
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486 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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487 | v = (*_nodes)[i]; |
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488 | for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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489 | e = _in_arcs[v][j]; |
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490 | u = _gr.source(e); |
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491 | if (_length[e] < _dist[u]) { |
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492 | _dist[u] = _length[e]; |
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493 | _policy[u] = e; |
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494 | } |
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495 | } |
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496 | } |
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497 | return true; |
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498 | } |
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499 | |
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500 | // Find the minimum mean cycle in the policy graph |
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501 | void findPolicyCycle() { |
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502 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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503 | _level[(*_nodes)[i]] = -1; |
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504 | } |
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505 | LargeValue clength; |
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506 | int csize; |
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507 | Node u, v; |
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508 | _curr_found = false; |
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509 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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510 | u = (*_nodes)[i]; |
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511 | if (_level[u] >= 0) continue; |
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512 | for (; _level[u] < 0; u = _gr.target(_policy[u])) { |
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513 | _level[u] = i; |
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514 | } |
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515 | if (_level[u] == i) { |
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516 | // A cycle is found |
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517 | clength = _length[_policy[u]]; |
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518 | csize = 1; |
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519 | for (v = u; (v = _gr.target(_policy[v])) != u; ) { |
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520 | clength += _length[_policy[v]]; |
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521 | ++csize; |
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522 | } |
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523 | if ( !_curr_found || |
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524 | (clength * _curr_size < _curr_length * csize) ) { |
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525 | _curr_found = true; |
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526 | _curr_length = clength; |
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527 | _curr_size = csize; |
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528 | _curr_node = u; |
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529 | } |
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530 | } |
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531 | } |
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532 | } |
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533 | |
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534 | // Contract the policy graph and compute node distances |
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535 | bool computeNodeDistances() { |
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536 | // Find the component of the main cycle and compute node distances |
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537 | // using reverse BFS |
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538 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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539 | _reached[(*_nodes)[i]] = false; |
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540 | } |
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541 | _qfront = _qback = 0; |
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542 | _queue[0] = _curr_node; |
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543 | _reached[_curr_node] = true; |
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544 | _dist[_curr_node] = 0; |
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545 | Node u, v; |
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546 | Arc e; |
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547 | while (_qfront <= _qback) { |
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548 | v = _queue[_qfront++]; |
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549 | for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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550 | e = _in_arcs[v][j]; |
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551 | u = _gr.source(e); |
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552 | if (_policy[u] == e && !_reached[u]) { |
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553 | _reached[u] = true; |
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554 | _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length; |
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555 | _queue[++_qback] = u; |
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556 | } |
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557 | } |
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558 | } |
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559 | |
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560 | // Connect all other nodes to this component and compute node |
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561 | // distances using reverse BFS |
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562 | _qfront = 0; |
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563 | while (_qback < int(_nodes->size())-1) { |
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564 | v = _queue[_qfront++]; |
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565 | for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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566 | e = _in_arcs[v][j]; |
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567 | u = _gr.source(e); |
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568 | if (!_reached[u]) { |
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569 | _reached[u] = true; |
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570 | _policy[u] = e; |
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571 | _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length; |
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572 | _queue[++_qback] = u; |
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573 | } |
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574 | } |
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575 | } |
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576 | |
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577 | // Improve node distances |
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578 | bool improved = false; |
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579 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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580 | v = (*_nodes)[i]; |
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581 | for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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582 | e = _in_arcs[v][j]; |
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583 | u = _gr.source(e); |
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584 | LargeValue delta = _dist[v] + _length[e] * _curr_size - _curr_length; |
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585 | if (_tolerance.less(delta, _dist[u])) { |
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586 | _dist[u] = delta; |
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587 | _policy[u] = e; |
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588 | improved = true; |
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589 | } |
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590 | } |
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591 | } |
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592 | return improved; |
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593 | } |
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594 | |
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595 | }; //class Howard |
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596 | |
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597 | ///@} |
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598 | |
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599 | } //namespace lemon |
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600 | |
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601 | #endif //LEMON_HOWARD_H |
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