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 | namespace lemon { |
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20 | /** |
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21 | [PAGE]sec_graph_adaptors[PAGE] Graph Adaptors |
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22 | |
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23 | In typical algorithms and applications related to graphs and networks, |
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24 | we usually encounter situations in which a specific alteration of a graph |
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25 | has to be considered. |
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26 | If some nodes or arcs have to be hidden (maybe temporarily) or the reverse |
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27 | oriented graph has to be used, then this is the case. |
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28 | However, actually modifing physical storage of the graph or |
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29 | making a copy of the graph structure along with the required maps |
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30 | could be rather expensive (in time or in memory usage) compared to the |
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31 | operations that should be performed on the altered graph. |
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32 | In such cases, the LEMON \e graph \e adaptor \e classes could be used. |
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33 | |
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34 | [SEC]sec_reverse_digraph[SEC] Reverse Oriented Digraph |
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35 | |
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36 | Let us suppose that we have an instance \c g of a directed graph type, say |
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37 | \ref ListDigraph and an algorithm |
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38 | \code |
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39 | template <typename Digraph> |
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40 | int algorithm(const Digraph&); |
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41 | \endcode |
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42 | is needed to run on the reverse oriented digraph. |
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43 | In this situation, a certain adaptor class |
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44 | \code |
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45 | template <typename Digraph> |
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46 | class ReverseDigraph; |
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47 | \endcode |
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48 | can be used. |
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49 | |
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50 | The graph adaptors are special classes that serve for considering other graph |
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51 | structures in different ways. They can be used exactly the same as "real" |
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52 | graphs, i.e. they conform to the \ref graph_concepts "graph concepts", thus all |
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53 | generic algorithms can be performed on them. However, the adaptor classes |
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54 | cannot be used alone but only in conjunction with actual graph representations. |
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55 | They do not alter the physical graph storage, they just give another view of it. |
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56 | When the methods of the adaptors are called, they use the underlying |
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57 | graph structures and their operations, thus these classes have only negligible |
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58 | memory usage and do not perform sophisticated algorithmic actions. |
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59 | |
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60 | This technique yields convenient tools that help writing compact and elegant |
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61 | code, and makes it possible to easily implement complex algorithms based on |
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62 | well tested standard components. |
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63 | |
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64 | For solving the problem introduced above, we could use the follwing code. |
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65 | |
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66 | \code |
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67 | ListDigraph g; |
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68 | ReverseDigraph<ListDigraph> rg(g); |
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69 | int result = algorithm(rg); |
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70 | \endcode |
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71 | |
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72 | Note that the original digraph \c g remains untouched during the whole |
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73 | procedure. |
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74 | |
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75 | LEMON also provides simple "creator functions" for the adaptor |
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76 | classes to make their usage even simpler. |
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77 | For example, \ref reverseDigraph() returns an instance of \ref ReverseDigraph, |
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78 | thus the above code can be written like this. |
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79 | |
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80 | \code |
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81 | ListDigraph g; |
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82 | int result = algorithm(reverseDigraph(g)); |
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83 | \endcode |
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84 | |
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85 | Another essential feature of the adaptors is that their \c Node and \c Arc |
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86 | types convert to the original item types. |
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87 | Therefore, the maps of the original graph can be used in connection with |
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88 | the adaptor. |
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89 | |
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90 | In the following code, Dijksta's algorithm is run on the reverse oriented |
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91 | graph but using the original node and arc maps. |
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92 | |
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93 | \code |
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94 | ListDigraph g; |
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95 | ListDigraph::ArcMap length(g); |
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96 | ListDigraph::NodeMap dist(g); |
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97 | |
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98 | ListDigraph::Node s = g.addNode(); |
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99 | // add more nodes and arcs |
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100 | |
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101 | dijkstra(reverseDigraph(g), length).distMap(dist).run(s); |
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102 | \endcode |
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103 | |
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104 | In the above examples, we used \ref ReverseDigraph in such a way that the |
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105 | underlying digraph was not changed. However, the adaptor class can even be |
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106 | used for modifying the original graph structure. |
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107 | It allows adding and deleting arcs or nodes, and these operations are carried |
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108 | out by calling suitable functions of the underlying digraph (if it supports |
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109 | them). |
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110 | |
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111 | For this, \ref ReverseDigraph "ReverseDigraph<GR>" has a constructor of the |
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112 | following form. |
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113 | \code |
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114 | ReverseDigraph(GR& gr); |
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115 | \endcode |
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116 | |
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117 | This means that in a situation, when the modification of the original graph |
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118 | has to be avoided (e.g. it is given as a const reference), then the adaptor |
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119 | class has to be instantiated with \c GR set to be \c const type |
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120 | (e.g. <tt>GR = const %ListDigraph</tt>), as in the following example. |
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121 | |
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122 | \code |
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123 | int algorithm1(const ListDigraph& g) { |
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124 | ReverseDigraph<const ListDigraph> rg(g); |
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125 | return algorithm2(rg); |
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126 | } |
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127 | \endcode |
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128 | |
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129 | \note Modification capabilities are not supported for all adaptors. |
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130 | E.g. for \ref ResidualDigraph (see \ref sec_other_adaptors "later"), |
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131 | this makes no sense. |
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132 | |
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133 | As a more complex example, let us see how \ref ReverseDigraph can be used |
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134 | together with a graph search algorithm to decide whether a directed graph is |
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135 | strongly connected or not. |
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136 | We exploit the fact the a digraph is strongly connected if and only if |
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137 | for an arbitrarily selected node \c u, each other node is reachable from |
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138 | \c u (along a directed path) and \c u is reachable from each node. |
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139 | The latter condition is the same that each node is reachable from \c u |
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140 | in the reversed digraph. |
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141 | |
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142 | \code |
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143 | template <typename Digraph> |
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144 | bool stronglyConnected(const Digraph& g) { |
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145 | typedef typename Digraph::NodeIt NodeIt; |
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146 | NodeIt u(g); |
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147 | if (u == INVALID) return true; |
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148 | |
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149 | // Run BFS on the original digraph |
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150 | Bfs<Digraph> bfs(g); |
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151 | bfs.run(u); |
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152 | for (NodeIt n(g); n != INVALID; ++n) { |
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153 | if (!bfs.reached(n)) return false; |
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154 | } |
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155 | |
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156 | // Run BFS on the reverse oriented digraph |
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157 | typedef ReverseDigraph<const Digraph> RDigraph; |
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158 | RDigraph rg(g); |
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159 | Bfs<RDigraph> rbfs(rg); |
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160 | rbfs.run(u); |
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161 | for (NodeIt n(g); n != INVALID; ++n) { |
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162 | if (!rbfs.reached(n)) return false; |
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163 | } |
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164 | |
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165 | return true; |
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166 | } |
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167 | \endcode |
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168 | |
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169 | Note that we have to use the adaptor with '<tt>const Digraph</tt>' type, since |
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170 | \c g is a \c const reference to the original graph structure. |
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171 | The \ref stronglyConnected() function provided in LEMON has a quite |
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172 | similar implementation. |
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173 | |
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174 | |
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175 | [SEC]sec_subgraphs[SEC] Subgraph Adaptorts |
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176 | |
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177 | Another typical requirement is the use of certain subgraphs of a graph, |
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178 | or in other words, hiding nodes and/or arcs from a graph. |
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179 | LEMON provides several convenient adaptors for these purposes. |
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180 | In the following picture, a \ref SubDigraph adaptor is applied to an |
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181 | underlying digraph structure to obtain a suitable subgraph. |
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182 | |
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183 | \image html adaptors1.png |
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184 | \image latex adaptors1.eps "SubDigraph adaptor" width=\textwidth |
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185 | |
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186 | \ref FilterArcs can be used when some arcs have to be hidden from a digraph. |
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187 | A \e filter \e map has to be given to the constructor, which assign \c bool |
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188 | values to the arcs specifying whether they have to be shown or not in the |
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189 | subgraph structure. |
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190 | Suppose we have a \ref ListDigraph structure \c g. |
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191 | Then we can construct a subgraph in which some arcs (\c a1, \c a2 etc.) |
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192 | are hidden as follows. |
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193 | |
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194 | \code |
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195 | ListDigraph::ArcMap filter(g, true); |
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196 | filter[a1] = false; |
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197 | filter[a2] = false; |
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198 | // ... |
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199 | FilterArcs<ListDigraph> subgraph(g, filter); |
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200 | \endcode |
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201 | |
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202 | The following more complex code runs Dijkstra's algorithm on a digraph |
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203 | that is obtained from another digraph by hiding all arcs having negative |
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204 | lengths. |
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205 | |
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206 | \code |
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207 | ListDigraph::ArcMap<int> length(g); |
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208 | ListDigraph::NodeMap<int> dist(g); |
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209 | |
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210 | dijkstra(filterArcs( g, lessMap(length, constMap<ListDigraph::Arc>(0)) ), |
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211 | length).distMap(dist).run(s); |
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212 | \endcode |
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213 | |
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214 | Note the extensive use of map adaptors and creator functions, which makes |
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215 | the code really compact and elegant. |
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216 | |
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217 | \note Implicit maps and graphs (e.g. created using functions) can only be |
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218 | used with the function-type interfaces of the algorithms, since they store |
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219 | only references for the used structures. |
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220 | |
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221 | \ref FilterEdges can be used for hiding edges from an undirected graph (like |
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222 | \ref FilterArcs is used for digraphs). \ref FilterNodes serves for filtering |
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223 | nodes along with the incident arcs or edges in a directed or undirected graph. |
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224 | If both arcs/edges and nodes have to be hidden, then you could use |
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225 | \ref SubDigraph or \ref SubGraph adaptors. |
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226 | |
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227 | \code |
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228 | ListGraph ug; |
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229 | ListGraph::NodeMap<bool> node_filter(ug); |
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230 | ListGraph::EdgeMap<bool> edge_filter(ug); |
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231 | |
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232 | SubGraph<ListGraph> sg(ug, node_filter, edge_filter); |
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233 | \endcode |
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234 | |
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235 | As you see, we needed two filter maps in this case: one for the nodes and |
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236 | another for the edges. If a node is hidden, then all of its incident edges |
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237 | are also considered to be hidden independently of their own filter values. |
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238 | |
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239 | The subgraph adaptors also make it possible to modify the filter values |
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240 | even after the construction of the adaptor class, thus the corresponding |
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241 | graph items can be hidden or shown on the fly. |
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242 | The adaptors store references to the filter maps, thus the map values can be |
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243 | set directly and even by using the \c enable(), \c disable() and \c status() |
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244 | functions. |
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245 | |
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246 | \code |
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247 | ListDigraph g; |
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248 | ListDigraph::Node x = g.addNode(); |
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249 | ListDigraph::Node y = g.addNode(); |
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250 | ListDigraph::Node z = g.addNode(); |
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251 | |
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252 | ListDigraph::NodeMap<bool> filter(g, true); |
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253 | FilterNodes<ListDigraph> subgraph(g, filter); |
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254 | std::cout << countNodes(subgraph) << ", "; |
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255 | |
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256 | filter[x] = false; |
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257 | std::cout << countNodes(subgraph) << ", "; |
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258 | |
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259 | subgraph.enable(x); |
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260 | subgraph.disable(y); |
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261 | subgraph.status(z, !subgraph.status(z)); |
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262 | std::cout << countNodes(subgraph) << std::endl; |
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263 | \endcode |
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264 | |
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265 | The above example prints out this line. |
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266 | \code |
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267 | 3, 2, 1 |
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268 | \endcode |
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269 | |
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270 | Similarly to \ref ReverseDigraph, the subgraph adaptors also allow the |
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271 | modification of the underlying graph structures unless the graph template |
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272 | parameter is set to be \c const type. |
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273 | Moreover the item types of the original graphs and the subgraphs are |
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274 | convertible to each other. |
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275 | |
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276 | The iterators of the subgraph adaptors use the iterators of the original |
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277 | graph structures in such a way that each item with \c false filter value |
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278 | is skipped. If both the node and arc sets are filtered, then the arc iterators |
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279 | check for each arc the status of its end nodes in addition to its own assigned |
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280 | filter value. If the arc or one of its end nodes is hidden, then the arc |
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281 | is left out and the next arc is considered. |
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282 | (It is the same for edges in undirected graphs.) |
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283 | Therefore, the iterators of these adaptors are significantly slower than the |
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284 | original iterators. |
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285 | |
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286 | Using adaptors, these efficiency aspects should be kept in mind. |
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287 | For example, if rather complex algorithms have to be performed on a |
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288 | subgraph (e.g. the nodes and arcs need to be traversed several times), |
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289 | then it could worth copying the altered graph into an efficient |
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290 | structure (e.g. \ref StaticDigraph) and run the algorithm on it. |
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291 | Note that the adaptor classes can also be used for doing this easily, |
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292 | without having to copy the graph manually, as shown in the following |
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293 | example. |
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294 | |
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295 | \code |
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296 | ListDigraph g; |
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297 | ListDigraph::NodeMap<bool> filter_map(g); |
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298 | // construct the graph and fill the filter map |
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299 | |
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300 | { |
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301 | StaticDigraph tmp_graph; |
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302 | ListDigraph::NodeMap<StaticDigraph::Node> node_ref(g); |
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303 | digraphCopy(filterNodes(g, filter_map), tmp_graph) |
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304 | .nodeRef(node_ref).run(); |
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305 | |
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306 | // use tmp_graph |
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307 | } |
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308 | \endcode |
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309 | |
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310 | \note Using \ref ReverseDigraph could be as efficient as working with the |
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311 | original graph, but most of the adaptors cannot be so fast, of course. |
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312 | |
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313 | |
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314 | [SEC]sec_other_adaptors[SEC] Other Graph Adaptors |
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315 | |
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316 | Two other practical adaptors are \ref Undirector and \ref Orienter. |
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317 | \ref Undirector makes an undirected graph from a digraph disregarding the |
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318 | orientations of the arcs. More precisely, an arc of the original digraph |
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319 | is considered as an edge (and two arcs, as well) in the adaptor. |
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320 | \ref Orienter can be used for the reverse alteration, it assigns a certain |
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321 | orientation to each edge of an undirected graph to form a directed graph. |
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322 | A \c bool edge map of the underlying graph must be given to the constructor |
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323 | of the class, which define the direction of the arcs in the created adaptor |
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324 | (with respect to the inherent orientation of the original edges). |
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325 | |
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326 | \code |
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327 | ListGraph graph; |
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328 | ListGraph::EdgeMap<bool> dir_map(graph, true); |
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329 | Orienter<ListGraph> directed_graph(graph, dir_map); |
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330 | \endcode |
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331 | |
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332 | Sine the adaptor classes conform to the \ref graph_concepts "graph concepts", |
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333 | we can even apply an adaptor to another one. |
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334 | The following image illustrates a situation when a \ref SubDigraph and an |
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335 | \ref Undirector adaptor is applied on a digraph. |
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336 | |
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337 | \image html adaptors2.png |
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338 | \image latex adaptors2.eps "Arc disjoint paths" width=\textwidth |
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339 | |
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340 | LEMON also provides some more complex adaptors, for |
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341 | instance, \ref SplitNodes, which can be used for splitting each node of a |
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342 | directed graph into an in-node and an out-node. |
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343 | Formally, the adaptor replaces each node u in the graph with two nodes, |
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344 | namely u<sub>in</sub> and u<sub>out</sub>. Each arc (u,v) of the original |
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345 | graph will correspond to an arc (u<sub>out</sub>,v<sub>in</sub>). |
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346 | The adaptor also adds an additional bind arc (u<sub>in</sub>,u<sub>out</sub>) |
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347 | for each node u of the original digraph. |
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348 | |
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349 | The aim of this class is to assign costs or capacities to the nodes when using |
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350 | algorithms which would otherwise consider arc costs or capacities only. |
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351 | For example, let us suppose that we have a digraph \c g with costs assigned to |
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352 | both the nodes and the arcs. Then Dijkstra's algorithm can be used in |
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353 | connection with \ref SplitNodes as follows. |
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354 | |
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355 | \code |
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356 | typedef SplitNodes<ListDigraph> SplitGraph; |
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357 | SplitGraph sg(g); |
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358 | SplitGraph::CombinedArcMap<NodeCostMap, ArcCostMap> |
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359 | combined_cost(node_cost, arc_cost); |
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360 | SplitGraph::NodeMap<double> dist(sg); |
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361 | dijkstra(sg, combined_cost).distMap(dist).run(sg.outNode(u)); |
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362 | \endcode |
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363 | |
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364 | \note This problem can also be solved using map adaptors to create |
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365 | an implicit arc map that assigns for each arc the sum of its cost |
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366 | and the cost of its target node. This map can be used with the original |
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367 | graph more efficiently than using the above solution. |
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368 | |
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369 | Another nice application is the problem of finding disjoint paths in |
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370 | a digraph. |
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371 | The maximum number of \e edge \e disjoint paths from a source node to |
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372 | a sink node in a digraph can be easily computed using a maximum flow |
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373 | algorithm with all arc capacities set to 1. |
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374 | For example, in the following digraph, four arc disjoint paths can be found |
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375 | from the node on the left to the node on the right. |
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376 | |
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377 | \image html splitnodes1.png |
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378 | \image latex splitnodes1.eps "Arc disjoint paths" width=\textwidth |
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379 | |
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380 | On the other hand, \e node \e disjoint paths cannot be found directly |
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381 | using a standard algorithm. |
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382 | However, \ref SplitNodes adaptor makes it really simple. |
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383 | If a maximum flow computation is performed on this adaptor, then the |
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384 | bottleneck of the flow (i.e. the minimum cut) will be formed by bind arcs, |
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385 | thus the found flow will correspond to the union of some node disjoint |
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386 | paths in terms of the original digraph. |
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387 | For example, in the above digraph, there are only three node disjoint paths. |
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388 | |
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389 | \image html splitnodes2.png |
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390 | \image latex splitnodes2.eps "Node disjoint paths" width=\textwidth |
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391 | |
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392 | In flow, circulation and matching problems, the residual network is of |
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393 | particular importance, which is implemented in \ref ResidualDigraph. |
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394 | Combining this adaptor with various algorithms, a range of weighted and |
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395 | cardinality optimization methods can be implemented easily. |
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396 | |
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397 | To construct a residual network, a digraph structure, a flow map and a |
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398 | capacity map have to be given to the constructor of the adaptor as shown |
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399 | in the following code. |
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400 | |
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401 | \code |
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402 | ListDigraph g; |
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403 | ListDigraph::ArcMap<int> flow(g); |
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404 | ListDigraph::ArcMap<int> capacity(g); |
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405 | |
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406 | ResidualDigraph<ListDigraph> res_graph(g, capacity, flow); |
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407 | \endcode |
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408 | |
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409 | \note In fact, this class is implemented using two other adaptors: |
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410 | \ref Undirector and \ref FilterArcs. |
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411 | |
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412 | [TRAILER] |
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413 | */ |
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414 | } |
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