1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
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2 | * |
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3 | * This file is a part of LEMON, a generic C++ optimization library. |
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4 | * |
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5 | * Copyright (C) 2003-2010 |
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6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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8 | * |
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9 | * Permission to use, modify and distribute this software is granted |
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10 | * provided that this copyright notice appears in all copies. For |
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11 | * precise terms see the accompanying LICENSE file. |
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12 | * |
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13 | * This software is provided "AS IS" with no warranty of any kind, |
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14 | * express or implied, and with no claim as to its suitability for any |
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15 | * purpose. |
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16 | * |
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17 | */ |
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18 | |
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19 | #ifndef LEMON_HOWARD_MMC_H |
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20 | #define LEMON_HOWARD_MMC_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 HowardMmc class. |
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37 | /// |
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38 | /// Default traits class of HowardMmc class. |
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39 | /// \tparam GR The type of the digraph. |
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40 | /// \tparam CM The type of the cost 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 CM> |
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44 | #else |
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45 | template <typename GR, typename CM, |
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46 | bool integer = std::numeric_limits<typename CM::Value>::is_integer> |
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47 | #endif |
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48 | struct HowardMmcDefaultTraits |
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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 cost map |
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53 | typedef CM CostMap; |
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54 | /// The type of the arc costs |
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55 | typedef typename CostMap::Value Cost; |
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56 | |
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57 | /// \brief The large cost type used for internal computations |
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58 | /// |
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59 | /// The large cost type used for internal computations. |
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60 | /// It is \c long \c long if the \c Cost type is integer, |
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61 | /// otherwise it is \c double. |
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62 | /// \c Cost must be convertible to \c LargeCost. |
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63 | typedef double LargeCost; |
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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<LargeCost> 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 cost types |
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77 | template <typename GR, typename CM> |
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78 | struct HowardMmcDefaultTraits<GR, CM, true> |
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79 | { |
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80 | typedef GR Digraph; |
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81 | typedef CM CostMap; |
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82 | typedef typename CostMap::Value Cost; |
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83 | #ifdef LEMON_HAVE_LONG_LONG |
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84 | typedef long long LargeCost; |
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85 | #else |
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86 | typedef long LargeCost; |
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87 | #endif |
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88 | typedef lemon::Tolerance<LargeCost> 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 cost in a digraph |
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101 | /// \cite dasdan98minmeancycle, \cite dasdan04experimental. |
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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 CM The type of the cost 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 HowardMmcDefaultTraits |
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111 | /// "HowardMmcDefaultTraits<GR, CM>". |
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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 CM, typename TR> |
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116 | #else |
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117 | template < typename GR, |
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118 | typename CM = typename GR::template ArcMap<int>, |
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119 | typename TR = HowardMmcDefaultTraits<GR, CM> > |
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120 | #endif |
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121 | class HowardMmc |
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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 cost map |
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128 | typedef typename TR::CostMap CostMap; |
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129 | /// The type of the arc costs |
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130 | typedef typename TR::Cost Cost; |
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131 | |
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132 | /// \brief The large cost type |
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133 | /// |
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134 | /// The large cost type used for internal computations. |
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135 | /// By default, it is \c long \c long if the \c Cost type is integer, |
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136 | /// otherwise it is \c double. |
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137 | typedef typename TR::LargeCost LargeCost; |
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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 HowardMmcDefaultTraits "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 HowardMmcDefaultTraits "traits class" of the algorithm |
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150 | typedef TR Traits; |
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151 | |
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152 | /// \brief Constants for the causes of search termination. |
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153 | /// |
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154 | /// Enum type containing constants for the different causes of search |
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155 | /// termination. The \ref findCycleMean() function returns one of |
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156 | /// these values. |
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157 | enum TerminationCause { |
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158 | |
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159 | /// No directed cycle can be found in the digraph. |
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160 | NO_CYCLE = 0, |
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161 | |
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162 | /// Optimal solution (minimum cycle mean) is found. |
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163 | OPTIMAL = 1, |
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164 | |
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165 | /// The iteration count limit is reached. |
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166 | ITERATION_LIMIT |
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167 | }; |
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168 | |
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169 | private: |
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170 | |
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171 | TEMPLATE_DIGRAPH_TYPEDEFS(Digraph); |
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172 | |
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173 | // The digraph the algorithm runs on |
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174 | const Digraph &_gr; |
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175 | // The cost of the arcs |
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176 | const CostMap &_cost; |
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177 | |
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178 | // Data for the found cycles |
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179 | bool _curr_found, _best_found; |
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180 | LargeCost _curr_cost, _best_cost; |
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181 | int _curr_size, _best_size; |
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182 | Node _curr_node, _best_node; |
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183 | |
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184 | Path *_cycle_path; |
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185 | bool _local_path; |
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186 | |
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187 | // Internal data used by the algorithm |
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188 | typename Digraph::template NodeMap<Arc> _policy; |
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189 | typename Digraph::template NodeMap<bool> _reached; |
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190 | typename Digraph::template NodeMap<int> _level; |
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191 | typename Digraph::template NodeMap<LargeCost> _dist; |
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192 | |
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193 | // Data for storing the strongly connected components |
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194 | int _comp_num; |
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195 | typename Digraph::template NodeMap<int> _comp; |
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196 | std::vector<std::vector<Node> > _comp_nodes; |
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197 | std::vector<Node>* _nodes; |
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198 | typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs; |
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199 | |
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200 | // Queue used for BFS search |
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201 | std::vector<Node> _queue; |
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202 | int _qfront, _qback; |
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203 | |
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204 | Tolerance _tolerance; |
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205 | |
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206 | // Infinite constant |
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207 | const LargeCost INF; |
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208 | |
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209 | public: |
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210 | |
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211 | /// \name Named Template Parameters |
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212 | /// @{ |
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213 | |
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214 | template <typename T> |
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215 | struct SetLargeCostTraits : public Traits { |
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216 | typedef T LargeCost; |
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217 | typedef lemon::Tolerance<T> Tolerance; |
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218 | }; |
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219 | |
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220 | /// \brief \ref named-templ-param "Named parameter" for setting |
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221 | /// \c LargeCost type. |
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222 | /// |
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223 | /// \ref named-templ-param "Named parameter" for setting \c LargeCost |
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224 | /// type. It is used for internal computations in the algorithm. |
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225 | template <typename T> |
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226 | struct SetLargeCost |
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227 | : public HowardMmc<GR, CM, SetLargeCostTraits<T> > { |
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228 | typedef HowardMmc<GR, CM, SetLargeCostTraits<T> > Create; |
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229 | }; |
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230 | |
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231 | template <typename T> |
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232 | struct SetPathTraits : public Traits { |
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233 | typedef T Path; |
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234 | }; |
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235 | |
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236 | /// \brief \ref named-templ-param "Named parameter" for setting |
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237 | /// \c %Path type. |
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238 | /// |
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239 | /// \ref named-templ-param "Named parameter" for setting the \c %Path |
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240 | /// type of the found cycles. |
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241 | /// It must conform to the \ref lemon::concepts::Path "Path" concept |
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242 | /// and it must have an \c addBack() function. |
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243 | template <typename T> |
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244 | struct SetPath |
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245 | : public HowardMmc<GR, CM, SetPathTraits<T> > { |
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246 | typedef HowardMmc<GR, CM, SetPathTraits<T> > Create; |
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247 | }; |
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248 | |
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249 | /// @} |
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250 | |
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251 | protected: |
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252 | |
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253 | HowardMmc() {} |
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254 | |
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255 | public: |
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256 | |
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257 | /// \brief Constructor. |
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258 | /// |
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259 | /// The constructor of the class. |
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260 | /// |
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261 | /// \param digraph The digraph the algorithm runs on. |
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262 | /// \param cost The costs of the arcs. |
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263 | HowardMmc( const Digraph &digraph, |
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264 | const CostMap &cost ) : |
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265 | _gr(digraph), _cost(cost), _best_found(false), |
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266 | _best_cost(0), _best_size(1), _cycle_path(NULL), _local_path(false), |
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267 | _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph), |
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268 | _comp(digraph), _in_arcs(digraph), |
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269 | INF(std::numeric_limits<LargeCost>::has_infinity ? |
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270 | std::numeric_limits<LargeCost>::infinity() : |
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271 | std::numeric_limits<LargeCost>::max()) |
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272 | {} |
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273 | |
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274 | /// Destructor. |
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275 | ~HowardMmc() { |
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276 | if (_local_path) delete _cycle_path; |
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277 | } |
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278 | |
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279 | /// \brief Set the path structure for storing the found cycle. |
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280 | /// |
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281 | /// This function sets an external path structure for storing the |
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282 | /// found cycle. |
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283 | /// |
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284 | /// If you don't call this function before calling \ref run() or |
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285 | /// \ref findCycleMean(), a local \ref Path "path" structure |
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286 | /// will be allocated. The destuctor deallocates this automatically |
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287 | /// allocated object, of course. |
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288 | /// |
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289 | /// \note The algorithm calls only the \ref lemon::Path::addBack() |
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290 | /// "addBack()" function of the given path structure. |
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291 | /// |
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292 | /// \return <tt>(*this)</tt> |
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293 | HowardMmc& cycle(Path &path) { |
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294 | if (_local_path) { |
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295 | delete _cycle_path; |
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296 | _local_path = false; |
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297 | } |
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298 | _cycle_path = &path; |
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299 | return *this; |
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300 | } |
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301 | |
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302 | /// \brief Set the tolerance used by the algorithm. |
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303 | /// |
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304 | /// This function sets the tolerance object used by the algorithm. |
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305 | /// |
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306 | /// \return <tt>(*this)</tt> |
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307 | HowardMmc& tolerance(const Tolerance& tolerance) { |
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308 | _tolerance = tolerance; |
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309 | return *this; |
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310 | } |
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311 | |
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312 | /// \brief Return a const reference to the tolerance. |
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313 | /// |
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314 | /// This function returns a const reference to the tolerance object |
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315 | /// used by the algorithm. |
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316 | const Tolerance& tolerance() const { |
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317 | return _tolerance; |
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318 | } |
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319 | |
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320 | /// \name Execution control |
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321 | /// The simplest way to execute the algorithm is to call the \ref run() |
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322 | /// function.\n |
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323 | /// If you only need the minimum mean cost, you may call |
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324 | /// \ref findCycleMean(). |
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325 | |
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326 | /// @{ |
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327 | |
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328 | /// \brief Run the algorithm. |
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329 | /// |
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330 | /// This function runs the algorithm. |
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331 | /// It can be called more than once (e.g. if the underlying digraph |
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332 | /// and/or the arc costs have been modified). |
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333 | /// |
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334 | /// \return \c true if a directed cycle exists in the digraph. |
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335 | /// |
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336 | /// \note <tt>mmc.run()</tt> is just a shortcut of the following code. |
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337 | /// \code |
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338 | /// return mmc.findCycleMean() && mmc.findCycle(); |
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339 | /// \endcode |
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340 | bool run() { |
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341 | return findCycleMean() && findCycle(); |
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342 | } |
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343 | |
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344 | /// \brief Find the minimum cycle mean (or an upper bound). |
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345 | /// |
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346 | /// This function finds the minimum mean cost of the directed |
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347 | /// cycles in the digraph (or an upper bound for it). |
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348 | /// |
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349 | /// By default, the function finds the exact minimum cycle mean, |
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350 | /// but an optional limit can also be specified for the number of |
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351 | /// iterations performed during the search process. |
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352 | /// The return value indicates if the optimal solution is found |
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353 | /// or the iteration limit is reached. In the latter case, an |
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354 | /// approximate solution is provided, which corresponds to a directed |
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355 | /// cycle whose mean cost is relatively small, but not necessarily |
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356 | /// minimal. |
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357 | /// |
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358 | /// \param limit The maximum allowed number of iterations during |
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359 | /// the search process. Its default value implies that the algorithm |
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360 | /// runs until it finds the exact optimal solution. |
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361 | /// |
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362 | /// \return The termination cause of the search process. |
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363 | /// For more information, see \ref TerminationCause. |
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364 | TerminationCause findCycleMean(int limit = std::numeric_limits<int>::max()) { |
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365 | // Initialize and find strongly connected components |
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366 | init(); |
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367 | findComponents(); |
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368 | |
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369 | // Find the minimum cycle mean in the components |
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370 | int iter_count = 0; |
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371 | bool iter_limit_reached = false; |
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372 | for (int comp = 0; comp < _comp_num; ++comp) { |
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373 | // Find the minimum mean cycle in the current component |
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374 | if (!buildPolicyGraph(comp)) continue; |
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375 | while (true) { |
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376 | if (++iter_count > limit) { |
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377 | iter_limit_reached = true; |
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378 | break; |
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379 | } |
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380 | findPolicyCycle(); |
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381 | if (!computeNodeDistances()) break; |
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382 | } |
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383 | |
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384 | // Update the best cycle (global minimum mean cycle) |
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385 | if ( _curr_found && (!_best_found || |
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386 | _curr_cost * _best_size < _best_cost * _curr_size) ) { |
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387 | _best_found = true; |
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388 | _best_cost = _curr_cost; |
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389 | _best_size = _curr_size; |
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390 | _best_node = _curr_node; |
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391 | } |
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392 | |
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393 | if (iter_limit_reached) break; |
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394 | } |
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395 | |
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396 | if (iter_limit_reached) { |
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397 | return ITERATION_LIMIT; |
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398 | } else { |
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399 | return _best_found ? OPTIMAL : NO_CYCLE; |
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400 | } |
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401 | } |
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402 | |
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403 | /// \brief Find a minimum mean directed cycle. |
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404 | /// |
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405 | /// This function finds a directed cycle of minimum mean cost |
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406 | /// in the digraph using the data computed by findCycleMean(). |
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407 | /// |
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408 | /// \return \c true if a directed cycle exists in the digraph. |
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409 | /// |
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410 | /// \pre \ref findCycleMean() must be called before using this function. |
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411 | bool findCycle() { |
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412 | if (!_best_found) return false; |
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413 | _cycle_path->addBack(_policy[_best_node]); |
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414 | for ( Node v = _best_node; |
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415 | (v = _gr.target(_policy[v])) != _best_node; ) { |
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416 | _cycle_path->addBack(_policy[v]); |
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417 | } |
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418 | return true; |
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419 | } |
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420 | |
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421 | /// @} |
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422 | |
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423 | /// \name Query Functions |
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424 | /// The results of the algorithm can be obtained using these |
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425 | /// functions.\n |
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426 | /// The algorithm should be executed before using them. |
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427 | |
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428 | /// @{ |
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429 | |
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430 | /// \brief Return the total cost of the found cycle. |
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431 | /// |
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432 | /// This function returns the total cost of the found cycle. |
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433 | /// |
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434 | /// \pre \ref run() or \ref findCycleMean() must be called before |
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435 | /// using this function. |
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436 | Cost cycleCost() const { |
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437 | return static_cast<Cost>(_best_cost); |
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438 | } |
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439 | |
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440 | /// \brief Return the number of arcs on the found cycle. |
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441 | /// |
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442 | /// This function returns the number of arcs on the found cycle. |
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443 | /// |
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444 | /// \pre \ref run() or \ref findCycleMean() must be called before |
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445 | /// using this function. |
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446 | int cycleSize() const { |
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447 | return _best_size; |
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448 | } |
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449 | |
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450 | /// \brief Return the mean cost of the found cycle. |
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451 | /// |
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452 | /// This function returns the mean cost of the found cycle. |
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453 | /// |
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454 | /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the |
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455 | /// following code. |
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456 | /// \code |
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457 | /// return static_cast<double>(alg.cycleCost()) / alg.cycleSize(); |
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458 | /// \endcode |
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459 | /// |
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460 | /// \pre \ref run() or \ref findCycleMean() must be called before |
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461 | /// using this function. |
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462 | double cycleMean() const { |
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463 | return static_cast<double>(_best_cost) / _best_size; |
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464 | } |
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465 | |
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466 | /// \brief Return the found cycle. |
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467 | /// |
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468 | /// This function returns a const reference to the path structure |
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469 | /// storing the found cycle. |
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470 | /// |
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471 | /// \pre \ref run() or \ref findCycle() must be called before using |
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472 | /// this function. |
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473 | const Path& cycle() const { |
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474 | return *_cycle_path; |
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475 | } |
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476 | |
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477 | ///@} |
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478 | |
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479 | private: |
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480 | |
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481 | // Initialize |
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482 | void init() { |
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483 | if (!_cycle_path) { |
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484 | _local_path = true; |
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485 | _cycle_path = new Path; |
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486 | } |
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487 | _queue.resize(countNodes(_gr)); |
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488 | _best_found = false; |
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489 | _best_cost = 0; |
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490 | _best_size = 1; |
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491 | _cycle_path->clear(); |
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492 | } |
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493 | |
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494 | // Find strongly connected components and initialize _comp_nodes |
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495 | // and _in_arcs |
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496 | void findComponents() { |
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497 | _comp_num = stronglyConnectedComponents(_gr, _comp); |
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498 | _comp_nodes.resize(_comp_num); |
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499 | if (_comp_num == 1) { |
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500 | _comp_nodes[0].clear(); |
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501 | for (NodeIt n(_gr); n != INVALID; ++n) { |
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502 | _comp_nodes[0].push_back(n); |
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503 | _in_arcs[n].clear(); |
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504 | for (InArcIt a(_gr, n); a != INVALID; ++a) { |
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505 | _in_arcs[n].push_back(a); |
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506 | } |
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507 | } |
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508 | } else { |
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509 | for (int i = 0; i < _comp_num; ++i) |
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510 | _comp_nodes[i].clear(); |
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511 | for (NodeIt n(_gr); n != INVALID; ++n) { |
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512 | int k = _comp[n]; |
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513 | _comp_nodes[k].push_back(n); |
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514 | _in_arcs[n].clear(); |
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515 | for (InArcIt a(_gr, n); a != INVALID; ++a) { |
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516 | if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a); |
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517 | } |
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518 | } |
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519 | } |
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520 | } |
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521 | |
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522 | // Build the policy graph in the given strongly connected component |
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523 | // (the out-degree of every node is 1) |
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524 | bool buildPolicyGraph(int comp) { |
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525 | _nodes = &(_comp_nodes[comp]); |
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526 | if (_nodes->size() < 1 || |
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527 | (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) { |
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528 | return false; |
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529 | } |
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530 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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531 | _dist[(*_nodes)[i]] = INF; |
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532 | } |
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533 | Node u, v; |
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534 | Arc e; |
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535 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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536 | v = (*_nodes)[i]; |
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537 | for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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538 | e = _in_arcs[v][j]; |
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539 | u = _gr.source(e); |
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540 | if (_cost[e] < _dist[u]) { |
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541 | _dist[u] = _cost[e]; |
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542 | _policy[u] = e; |
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543 | } |
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544 | } |
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545 | } |
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546 | return true; |
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547 | } |
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548 | |
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549 | // Find the minimum mean cycle in the policy graph |
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550 | void findPolicyCycle() { |
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551 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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552 | _level[(*_nodes)[i]] = -1; |
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553 | } |
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554 | LargeCost ccost; |
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555 | int csize; |
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556 | Node u, v; |
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557 | _curr_found = false; |
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558 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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559 | u = (*_nodes)[i]; |
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560 | if (_level[u] >= 0) continue; |
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561 | for (; _level[u] < 0; u = _gr.target(_policy[u])) { |
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562 | _level[u] = i; |
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563 | } |
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564 | if (_level[u] == i) { |
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565 | // A cycle is found |
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566 | ccost = _cost[_policy[u]]; |
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567 | csize = 1; |
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568 | for (v = u; (v = _gr.target(_policy[v])) != u; ) { |
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569 | ccost += _cost[_policy[v]]; |
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570 | ++csize; |
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571 | } |
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572 | if ( !_curr_found || |
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573 | (ccost * _curr_size < _curr_cost * csize) ) { |
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574 | _curr_found = true; |
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575 | _curr_cost = ccost; |
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576 | _curr_size = csize; |
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577 | _curr_node = u; |
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578 | } |
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579 | } |
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580 | } |
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581 | } |
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582 | |
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583 | // Contract the policy graph and compute node distances |
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584 | bool computeNodeDistances() { |
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585 | // Find the component of the main cycle and compute node distances |
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586 | // using reverse BFS |
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587 | for (int i = 0; i < int(_nodes->size()); ++i) { |
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588 | _reached[(*_nodes)[i]] = false; |
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589 | } |
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590 | _qfront = _qback = 0; |
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591 | _queue[0] = _curr_node; |
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592 | _reached[_curr_node] = true; |
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593 | _dist[_curr_node] = 0; |
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594 | Node u, v; |
---|
595 | Arc e; |
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596 | while (_qfront <= _qback) { |
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597 | v = _queue[_qfront++]; |
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598 | for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
---|
599 | e = _in_arcs[v][j]; |
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600 | u = _gr.source(e); |
---|
601 | if (_policy[u] == e && !_reached[u]) { |
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602 | _reached[u] = true; |
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603 | _dist[u] = _dist[v] + _cost[e] * _curr_size - _curr_cost; |
---|
604 | _queue[++_qback] = u; |
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605 | } |
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606 | } |
---|
607 | } |
---|
608 | |
---|
609 | // Connect all other nodes to this component and compute node |
---|
610 | // distances using reverse BFS |
---|
611 | _qfront = 0; |
---|
612 | while (_qback < int(_nodes->size())-1) { |
---|
613 | v = _queue[_qfront++]; |
---|
614 | for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
---|
615 | e = _in_arcs[v][j]; |
---|
616 | u = _gr.source(e); |
---|
617 | if (!_reached[u]) { |
---|
618 | _reached[u] = true; |
---|
619 | _policy[u] = e; |
---|
620 | _dist[u] = _dist[v] + _cost[e] * _curr_size - _curr_cost; |
---|
621 | _queue[++_qback] = u; |
---|
622 | } |
---|
623 | } |
---|
624 | } |
---|
625 | |
---|
626 | // Improve node distances |
---|
627 | bool improved = false; |
---|
628 | for (int i = 0; i < int(_nodes->size()); ++i) { |
---|
629 | v = (*_nodes)[i]; |
---|
630 | for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
---|
631 | e = _in_arcs[v][j]; |
---|
632 | u = _gr.source(e); |
---|
633 | LargeCost delta = _dist[v] + _cost[e] * _curr_size - _curr_cost; |
---|
634 | if (_tolerance.less(delta, _dist[u])) { |
---|
635 | _dist[u] = delta; |
---|
636 | _policy[u] = e; |
---|
637 | improved = true; |
---|
638 | } |
---|
639 | } |
---|
640 | } |
---|
641 | return improved; |
---|
642 | } |
---|
643 | |
---|
644 | }; //class HowardMmc |
---|
645 | |
---|
646 | ///@} |
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647 | |
---|
648 | } //namespace lemon |
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649 | |
---|
650 | #endif //LEMON_HOWARD_MMC_H |
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