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 protected: |
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235 |
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236 Howard() {} |
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237 |
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238 public: |
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239 |
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240 /// \brief Constructor. |
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241 /// |
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242 /// The constructor of the class. |
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243 /// |
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244 /// \param digraph The digraph the algorithm runs on. |
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245 /// \param length The lengths (costs) of the arcs. |
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246 Howard( const Digraph &digraph, |
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247 const LengthMap &length ) : |
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248 _gr(digraph), _length(length), _best_found(false), |
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249 _best_length(0), _best_size(1), _cycle_path(NULL), _local_path(false), |
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250 _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph), |
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251 _comp(digraph), _in_arcs(digraph), |
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252 INF(std::numeric_limits<LargeValue>::has_infinity ? |
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253 std::numeric_limits<LargeValue>::infinity() : |
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254 std::numeric_limits<LargeValue>::max()) |
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255 {} |
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256 |
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257 /// Destructor. |
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258 ~Howard() { |
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259 if (_local_path) delete _cycle_path; |
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260 } |
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261 |
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262 /// \brief Set the path structure for storing the found cycle. |
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263 /// |
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264 /// This function sets an external path structure for storing the |
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265 /// found cycle. |
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266 /// |
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267 /// If you don't call this function before calling \ref run() or |
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268 /// \ref findMinMean(), it will allocate a local \ref Path "path" |
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269 /// structure. The destuctor deallocates this automatically |
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270 /// allocated object, of course. |
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271 /// |
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272 /// \note The algorithm calls only the \ref lemon::Path::addBack() |
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273 /// "addBack()" function of the given path structure. |
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274 /// |
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275 /// \return <tt>(*this)</tt> |
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276 Howard& cycle(Path &path) { |
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277 if (_local_path) { |
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278 delete _cycle_path; |
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279 _local_path = false; |
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280 } |
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281 _cycle_path = &path; |
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282 return *this; |
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283 } |
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284 |
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285 /// \brief Set the tolerance used by the algorithm. |
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286 /// |
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287 /// This function sets the tolerance object used by the algorithm. |
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288 /// |
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289 /// \return <tt>(*this)</tt> |
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290 Howard& tolerance(const Tolerance& tolerance) { |
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291 _tolerance = tolerance; |
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292 return *this; |
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293 } |
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294 |
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295 /// \brief Return a const reference to the tolerance. |
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296 /// |
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297 /// This function returns a const reference to the tolerance object |
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298 /// used by the algorithm. |
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299 const Tolerance& tolerance() const { |
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300 return _tolerance; |
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301 } |
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302 |
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303 /// \name Execution control |
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304 /// The simplest way to execute the algorithm is to call the \ref run() |
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305 /// function.\n |
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306 /// If you only need the minimum mean length, you may call |
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307 /// \ref findMinMean(). |
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308 |
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309 /// @{ |
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310 |
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311 /// \brief Run the algorithm. |
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312 /// |
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313 /// This function runs the algorithm. |
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314 /// It can be called more than once (e.g. if the underlying digraph |
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315 /// and/or the arc lengths have been modified). |
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316 /// |
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317 /// \return \c true if a directed cycle exists in the digraph. |
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318 /// |
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319 /// \note <tt>mmc.run()</tt> is just a shortcut of the following code. |
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320 /// \code |
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321 /// return mmc.findMinMean() && mmc.findCycle(); |
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322 /// \endcode |
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323 bool run() { |
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324 return findMinMean() && findCycle(); |
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325 } |
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326 |
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327 /// \brief Find the minimum cycle mean. |
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328 /// |
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329 /// This function finds the minimum mean length of the directed |
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330 /// cycles in the digraph. |
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331 /// |
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332 /// \return \c true if a directed cycle exists in the digraph. |
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333 bool findMinMean() { |
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334 // Initialize and find strongly connected components |
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335 init(); |
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336 findComponents(); |
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337 |
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338 // Find the minimum cycle mean in the components |
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339 for (int comp = 0; comp < _comp_num; ++comp) { |
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340 // Find the minimum mean cycle in the current component |
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341 if (!buildPolicyGraph(comp)) continue; |
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342 while (true) { |
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343 findPolicyCycle(); |
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344 if (!computeNodeDistances()) break; |
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345 } |
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346 // Update the best cycle (global minimum mean cycle) |
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347 if ( _curr_found && (!_best_found || |
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348 _curr_length * _best_size < _best_length * _curr_size) ) { |
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349 _best_found = true; |
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350 _best_length = _curr_length; |
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351 _best_size = _curr_size; |
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352 _best_node = _curr_node; |
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353 } |
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354 } |
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355 return _best_found; |
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356 } |
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357 |
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358 /// \brief Find a minimum mean directed cycle. |
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359 /// |
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360 /// This function finds a directed cycle of minimum mean length |
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361 /// in the digraph using the data computed by findMinMean(). |
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362 /// |
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363 /// \return \c true if a directed cycle exists in the digraph. |
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364 /// |
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365 /// \pre \ref findMinMean() must be called before using this function. |
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366 bool findCycle() { |
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367 if (!_best_found) return false; |
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368 _cycle_path->addBack(_policy[_best_node]); |
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369 for ( Node v = _best_node; |
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370 (v = _gr.target(_policy[v])) != _best_node; ) { |
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371 _cycle_path->addBack(_policy[v]); |
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372 } |
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373 return true; |
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374 } |
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375 |
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376 /// @} |
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377 |
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378 /// \name Query Functions |
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379 /// The results of the algorithm can be obtained using these |
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380 /// functions.\n |
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381 /// The algorithm should be executed before using them. |
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382 |
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383 /// @{ |
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384 |
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385 /// \brief Return the total length of the found cycle. |
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386 /// |
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387 /// This function returns the total length of the found cycle. |
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388 /// |
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389 /// \pre \ref run() or \ref findMinMean() must be called before |
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390 /// using this function. |
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391 Value cycleLength() const { |
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392 return static_cast<Value>(_best_length); |
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393 } |
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394 |
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395 /// \brief Return the number of arcs on the found cycle. |
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396 /// |
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397 /// This function returns the number of arcs on the found cycle. |
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398 /// |
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399 /// \pre \ref run() or \ref findMinMean() must be called before |
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400 /// using this function. |
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401 int cycleArcNum() const { |
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402 return _best_size; |
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403 } |
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404 |
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405 /// \brief Return the mean length of the found cycle. |
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406 /// |
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407 /// This function returns the mean length of the found cycle. |
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408 /// |
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409 /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the |
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410 /// following code. |
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411 /// \code |
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412 /// return static_cast<double>(alg.cycleLength()) / alg.cycleArcNum(); |
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413 /// \endcode |
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414 /// |
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415 /// \pre \ref run() or \ref findMinMean() must be called before |
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416 /// using this function. |
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417 double cycleMean() const { |
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418 return static_cast<double>(_best_length) / _best_size; |
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419 } |
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420 |
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421 /// \brief Return the found cycle. |
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422 /// |
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423 /// This function returns a const reference to the path structure |
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424 /// storing the found cycle. |
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425 /// |
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426 /// \pre \ref run() or \ref findCycle() must be called before using |
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427 /// this function. |
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428 const Path& cycle() const { |
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429 return *_cycle_path; |
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430 } |
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431 |
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432 ///@} |
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433 |
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434 private: |
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435 |
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436 // Initialize |
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437 void init() { |
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438 if (!_cycle_path) { |
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439 _local_path = true; |
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440 _cycle_path = new Path; |
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441 } |
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442 _queue.resize(countNodes(_gr)); |
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443 _best_found = false; |
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444 _best_length = 0; |
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445 _best_size = 1; |
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446 _cycle_path->clear(); |
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447 } |
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448 |
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449 // Find strongly connected components and initialize _comp_nodes |
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450 // and _in_arcs |
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451 void findComponents() { |
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452 _comp_num = stronglyConnectedComponents(_gr, _comp); |
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453 _comp_nodes.resize(_comp_num); |
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454 if (_comp_num == 1) { |
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455 _comp_nodes[0].clear(); |
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456 for (NodeIt n(_gr); n != INVALID; ++n) { |
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457 _comp_nodes[0].push_back(n); |
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458 _in_arcs[n].clear(); |
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459 for (InArcIt a(_gr, n); a != INVALID; ++a) { |
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460 _in_arcs[n].push_back(a); |
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461 } |
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462 } |
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463 } else { |
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464 for (int i = 0; i < _comp_num; ++i) |
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465 _comp_nodes[i].clear(); |
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466 for (NodeIt n(_gr); n != INVALID; ++n) { |
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467 int k = _comp[n]; |
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468 _comp_nodes[k].push_back(n); |
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469 _in_arcs[n].clear(); |
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470 for (InArcIt a(_gr, n); a != INVALID; ++a) { |
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471 if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a); |
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472 } |
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473 } |
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474 } |
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475 } |
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476 |
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477 // Build the policy graph in the given strongly connected component |
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478 // (the out-degree of every node is 1) |
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479 bool buildPolicyGraph(int comp) { |
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480 _nodes = &(_comp_nodes[comp]); |
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481 if (_nodes->size() < 1 || |
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482 (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) { |
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483 return false; |
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484 } |
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485 for (int i = 0; i < int(_nodes->size()); ++i) { |
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486 _dist[(*_nodes)[i]] = INF; |
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487 } |
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488 Node u, v; |
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489 Arc e; |
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490 for (int i = 0; i < int(_nodes->size()); ++i) { |
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491 v = (*_nodes)[i]; |
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492 for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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493 e = _in_arcs[v][j]; |
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494 u = _gr.source(e); |
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495 if (_length[e] < _dist[u]) { |
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496 _dist[u] = _length[e]; |
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497 _policy[u] = e; |
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498 } |
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499 } |
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500 } |
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501 return true; |
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502 } |
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503 |
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504 // Find the minimum mean cycle in the policy graph |
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505 void findPolicyCycle() { |
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506 for (int i = 0; i < int(_nodes->size()); ++i) { |
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507 _level[(*_nodes)[i]] = -1; |
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508 } |
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509 LargeValue clength; |
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510 int csize; |
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511 Node u, v; |
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512 _curr_found = false; |
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513 for (int i = 0; i < int(_nodes->size()); ++i) { |
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514 u = (*_nodes)[i]; |
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515 if (_level[u] >= 0) continue; |
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516 for (; _level[u] < 0; u = _gr.target(_policy[u])) { |
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517 _level[u] = i; |
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518 } |
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519 if (_level[u] == i) { |
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520 // A cycle is found |
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521 clength = _length[_policy[u]]; |
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522 csize = 1; |
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523 for (v = u; (v = _gr.target(_policy[v])) != u; ) { |
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524 clength += _length[_policy[v]]; |
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525 ++csize; |
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526 } |
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527 if ( !_curr_found || |
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528 (clength * _curr_size < _curr_length * csize) ) { |
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529 _curr_found = true; |
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530 _curr_length = clength; |
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531 _curr_size = csize; |
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532 _curr_node = u; |
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533 } |
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534 } |
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535 } |
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536 } |
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537 |
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538 // Contract the policy graph and compute node distances |
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539 bool computeNodeDistances() { |
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540 // Find the component of the main cycle and compute node distances |
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541 // using reverse BFS |
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542 for (int i = 0; i < int(_nodes->size()); ++i) { |
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543 _reached[(*_nodes)[i]] = false; |
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544 } |
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545 _qfront = _qback = 0; |
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546 _queue[0] = _curr_node; |
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547 _reached[_curr_node] = true; |
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548 _dist[_curr_node] = 0; |
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549 Node u, v; |
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550 Arc e; |
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551 while (_qfront <= _qback) { |
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552 v = _queue[_qfront++]; |
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553 for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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554 e = _in_arcs[v][j]; |
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555 u = _gr.source(e); |
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556 if (_policy[u] == e && !_reached[u]) { |
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557 _reached[u] = true; |
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558 _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length; |
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559 _queue[++_qback] = u; |
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560 } |
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561 } |
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562 } |
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563 |
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564 // Connect all other nodes to this component and compute node |
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565 // distances using reverse BFS |
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566 _qfront = 0; |
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567 while (_qback < int(_nodes->size())-1) { |
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568 v = _queue[_qfront++]; |
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569 for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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570 e = _in_arcs[v][j]; |
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571 u = _gr.source(e); |
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572 if (!_reached[u]) { |
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573 _reached[u] = true; |
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574 _policy[u] = e; |
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575 _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length; |
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576 _queue[++_qback] = u; |
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577 } |
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578 } |
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579 } |
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580 |
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581 // Improve node distances |
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582 bool improved = false; |
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583 for (int i = 0; i < int(_nodes->size()); ++i) { |
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584 v = (*_nodes)[i]; |
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585 for (int j = 0; j < int(_in_arcs[v].size()); ++j) { |
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586 e = _in_arcs[v][j]; |
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587 u = _gr.source(e); |
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588 LargeValue delta = _dist[v] + _length[e] * _curr_size - _curr_length; |
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589 if (_tolerance.less(delta, _dist[u])) { |
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590 _dist[u] = delta; |
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591 _policy[u] = e; |
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592 improved = true; |
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593 } |
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594 } |
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595 } |
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596 return improved; |
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597 } |
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598 |
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599 }; //class Howard |
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600 |
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601 ///@} |
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602 |
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603 } //namespace lemon |
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604 |
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605 #endif //LEMON_HOWARD_H |
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