1 | /** |
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2 | |
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3 | \page quicktour Quick Tour to LEMON |
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4 | |
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5 | Let us first answer the question <b>"What do I want to use LEMON for?" |
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6 | </b>. |
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7 | LEMON is a C++ library, so you can use it if you want to write C++ |
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8 | programs. What kind of tasks does the library LEMON help to solve? |
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9 | It helps to write programs that solve optimization problems that arise |
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10 | frequently when <b>designing and testing certain networks</b>, for example |
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11 | in telecommunication, computer networks, and other areas that I cannot |
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12 | think of now. A very natural way of modelling these networks is by means |
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13 | of a <b> graph</b> (we will always mean a directed graph by that and say |
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14 | <b> undirected graph </b> otherwise). |
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15 | So if you want to write a program that works with |
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16 | graphs then you might find it useful to use our library LEMON. LEMON |
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17 | defines various graph concepts depending on what you want to do with the |
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18 | graph: a very good description can be found in the page |
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19 | about \ref graphs "graphs". |
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20 | |
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21 | You will also want to assign data to the edges or nodes of the graph, for |
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22 | example a length or capacity function defined on the edges. You can do this in |
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23 | LEMON using so called \b maps. You can define a map on the nodes or on the edges of the graph and the value of the map (the range of the function) can be practically almost of any type. Read more about maps \ref maps-page "here". |
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24 | |
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25 | Some examples are the following (you will find links next to the code fragments that help to download full demo programs: save them on your computer and compile them according to the description in the page about \ref getsart How to start using LEMON): |
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26 | |
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27 | <ul> <li> The first thing to discuss is the way one can create data structures |
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28 | like graphs and maps in a program using LEMON. |
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29 | //There are more graph types |
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30 | //implemented in LEMON and you can implement your own graph type just as well: |
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31 | //read more about this in the already mentioned page on \ref graphs "graphs". |
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32 | |
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33 | First we show how to add nodes and edges to a graph manually. We will also |
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34 | define a map on the edges of the graph. After this we show the way one can |
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35 | read a graph (and perhaps maps on it) from a stream (e.g. a file). Of course |
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36 | we also have routines that write a graph (and perhaps maps) to a stream |
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37 | (file): this will also be shown. LEMON supports the DIMACS file formats to |
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38 | store network optimization problems, but more importantly we also have our own |
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39 | file format that gives a more flexible way to store data related to network |
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40 | optimization. |
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41 | |
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42 | <ol> <li>The following code fragment shows how to fill a graph with |
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43 | data. It creates a complete graph on 4 nodes. The type Listgraph is one of the |
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44 | LEMON graph types: the typedefs in the beginning are for convenience and we |
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45 | will suppose them later as well. |
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46 | |
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47 | \code |
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48 | |
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49 | typedef ListGraph Graph; |
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50 | typedef Graph::NodeIt NodeIt; |
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51 | |
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52 | Graph g; |
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53 | |
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54 | for (int i = 0; i < 3; i++) |
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55 | g.addNode(); |
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56 | |
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57 | for (NodeIt i(g); i!=INVALID; ++i) |
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58 | for (NodeIt j(g); j!=INVALID; ++j) |
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59 | if (i != j) g.addEdge(i, j); |
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60 | |
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61 | \endcode |
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62 | |
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63 | See the whole program in file \ref helloworld.cc. |
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64 | |
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65 | If you want to read more on the LEMON graph structures and concepts, read the page about \ref graphs "graphs". |
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66 | |
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67 | <li> The following code shows how to read a graph from a stream (e.g. a file) |
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68 | in the DIMACS file format (find the documentation of the DIMACS file formats on the web). |
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69 | |
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70 | \code |
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71 | Graph g; |
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72 | std::ifstream f("graph.dim"); |
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73 | readDimacs(f, g); |
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74 | \endcode |
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75 | |
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76 | One can also store network (graph+capacity on the edges) instances and other |
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77 | things (minimum cost flow instances etc.) in DIMACS format and use these in LEMON: to see the details read the |
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78 | documentation of the \ref dimacs.h "Dimacs file format reader". There you will |
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79 | also find the details about the output routines into files of the DIMACS |
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80 | format. |
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81 | |
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82 | <li>We needed much greater flexibility than the DIMACS formats could give us, |
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83 | so we worked out our own file format. Instead of any explanation let us give a |
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84 | short example file in this format: read the detailed description of the LEMON |
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85 | graph file format and input-output routines \ref graph-io-page here. |
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86 | |
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87 | So here is a file describing a graph of 10 nodes (0 to 9), two nodemaps |
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88 | (called \c coordinates_x and \c coordinates_y), several edges, an edge map |
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89 | called \c length and two designated nodes (called \c source and \c target). |
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90 | |
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91 | \todo Maybe another example would be better here. |
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92 | |
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93 | \code |
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94 | @nodeset |
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95 | id coordinates_x coordinates_y |
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96 | 9 447.907 578.328 |
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97 | 8 79.2573 909.464 |
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98 | 7 878.677 960.04 |
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99 | 6 11.5504 938.413 |
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100 | 5 327.398 815.035 |
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101 | 4 427.002 954.002 |
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102 | 3 148.549 753.748 |
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103 | 2 903.889 326.476 |
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104 | 1 408.248 577.327 |
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105 | 0 189.239 92.5316 |
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106 | @edgeset |
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107 | length |
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108 | 2 3 901.074 |
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109 | 8 5 270.85 |
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110 | 6 9 601.553 |
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111 | 5 9 285.022 |
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112 | 9 4 408.091 |
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113 | 3 0 719.712 |
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114 | 7 5 612.836 |
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115 | 0 4 933.353 |
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116 | 5 0 778.871 |
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117 | 5 5 0 |
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118 | 7 1 664.049 |
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119 | 5 5 0 |
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120 | 0 9 560.464 |
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121 | 4 8 352.36 |
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122 | 4 9 399.625 |
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123 | 4 1 402.171 |
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124 | 1 2 591.688 |
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125 | 3 8 182.376 |
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126 | 4 5 180.254 |
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127 | 3 1 345.283 |
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128 | 5 4 184.511 |
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129 | 6 2 1112.45 |
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130 | 0 1 556.624 |
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131 | @nodes |
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132 | source 1 |
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133 | target 8 |
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134 | @end |
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135 | \endcode |
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136 | |
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137 | Finally let us give a simple example that reads a graph from a file and writes |
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138 | it to another. |
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139 | |
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140 | \todo This is to be done! |
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141 | |
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142 | </ol> |
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143 | <li> If you want to solve some transportation problems in a network then |
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144 | you will want to find shortest paths between nodes of a graph. This is |
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145 | usually solved using Dijkstra's algorithm. A utility |
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146 | that solves this is the \ref lemon::Dijkstra "LEMON Dijkstra class". |
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147 | The following code is a simple program using the |
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148 | \ref lemon::Dijkstra "LEMON Dijkstra class" and it also shows how to define a map on the edges (the length |
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149 | function): |
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150 | |
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151 | \code |
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152 | |
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153 | typedef ListGraph Graph; |
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154 | typedef Graph::Node Node; |
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155 | typedef Graph::Edge Edge; |
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156 | typedef Graph::EdgeMap<int> LengthMap; |
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157 | |
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158 | Graph g; |
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159 | |
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160 | //An example from Ahuja's book |
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161 | |
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162 | Node s=g.addNode(); |
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163 | Node v2=g.addNode(); |
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164 | Node v3=g.addNode(); |
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165 | Node v4=g.addNode(); |
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166 | Node v5=g.addNode(); |
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167 | Node t=g.addNode(); |
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168 | |
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169 | Edge s_v2=g.addEdge(s, v2); |
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170 | Edge s_v3=g.addEdge(s, v3); |
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171 | Edge v2_v4=g.addEdge(v2, v4); |
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172 | Edge v2_v5=g.addEdge(v2, v5); |
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173 | Edge v3_v5=g.addEdge(v3, v5); |
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174 | Edge v4_t=g.addEdge(v4, t); |
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175 | Edge v5_t=g.addEdge(v5, t); |
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176 | |
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177 | LengthMap len(g); |
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178 | |
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179 | len.set(s_v2, 10); |
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180 | len.set(s_v3, 10); |
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181 | len.set(v2_v4, 5); |
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182 | len.set(v2_v5, 8); |
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183 | len.set(v3_v5, 5); |
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184 | len.set(v4_t, 8); |
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185 | len.set(v5_t, 8); |
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186 | |
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187 | std::cout << "The id of s is " << g.id(s)<< std::endl; |
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188 | std::cout <<"The id of t is " << g.id(t)<<"."<<std::endl; |
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189 | |
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190 | std::cout << "Dijkstra algorithm test..." << std::endl; |
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191 | |
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192 | Dijkstra<Graph, LengthMap> dijkstra_test(g,len); |
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193 | |
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194 | dijkstra_test.run(s); |
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195 | |
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196 | |
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197 | std::cout << "The distance of node t from node s: " << dijkstra_test.dist(t)<<std::endl; |
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198 | |
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199 | std::cout << "The shortest path from s to t goes through the following nodes" <<std::endl; |
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200 | std::cout << " (the first one is t, the last one is s): "<<std::endl; |
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201 | |
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202 | for (Node v=t;v != s; v=dijkstra_test.predNode(v)){ |
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203 | std::cout << g.id(v) << "<-"; |
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204 | } |
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205 | std::cout << g.id(s) << std::endl; |
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206 | \endcode |
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207 | |
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208 | See the whole program in \ref dijkstra_demo.cc. |
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209 | |
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210 | The first part of the code is self-explanatory: we build the graph and set the |
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211 | length values of the edges. Then we instantiate a member of the Dijkstra class |
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212 | and run the Dijkstra algorithm from node \c s. After this we read some of the |
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213 | results. |
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214 | You can do much more with the Dijkstra class, for example you can run it step |
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215 | by step and gain full control of the execution. For a detailed description, see the documentation of the \ref lemon::Dijkstra "LEMON Dijkstra class". |
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216 | |
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217 | |
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218 | <li> If you want to design a network and want to minimize the total length |
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219 | of wires then you might be looking for a <b>minimum spanning tree</b> in |
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220 | an undirected graph. This can be found using the Kruskal algorithm: the |
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221 | class \ref lemon::Kruskal "LEMON Kruskal class" does this job for you. |
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222 | The following code fragment shows an example: |
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223 | |
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224 | Ide Zsuzska fog irni! |
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225 | |
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226 | <li>Many problems in network optimization can be formalized by means |
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227 | of a linear programming problem (LP problem, for short). In our |
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228 | library we decided not to write an LP solver, since such packages are |
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229 | available in the commercial world just as well as in the open source |
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230 | world, and it is also a difficult task to compete these. Instead we |
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231 | decided to develop an interface that makes it easier to use these |
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232 | solvers together with LEMON. The advantage of this approach is |
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233 | twofold. Firstly our C++ interface is more comfortable than the |
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234 | solvers' native interface. Secondly, changing the underlying solver in |
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235 | a certain software using LEMON's LP interface needs zero effort. So, |
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236 | for example, one may try his idea using a free solver, demonstrate its |
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237 | usability for a customer and if it works well, but the performance |
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238 | should be improved, then one may decide to purchase and use a better |
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239 | commercial solver. |
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240 | |
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241 | So far we have an |
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242 | interface for the commercial LP solver software \b CLPLEX (developed by ILOG) |
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243 | and for the open source solver \b GLPK (a shorthand for Gnu Linear Programming |
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244 | Toolkit). |
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245 | |
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246 | We will show two examples, the first one shows how simple it is to formalize |
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247 | and solve an LP problem in LEMON, while the second one shows how LEMON |
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248 | facilitates solving network optimization problems using LP solvers. |
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249 | |
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250 | <ol> |
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251 | <li>The following code shows how to solve an LP problem using the LEMON lp |
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252 | interface. The code together with the comments is self-explanatory. |
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253 | |
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254 | \code |
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255 | |
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256 | //A default solver is taken |
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257 | LpDefault lp; |
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258 | typedef LpDefault::Row Row; |
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259 | typedef LpDefault::Col Col; |
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260 | |
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261 | |
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262 | //This will be a maximization |
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263 | lp.max(); |
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264 | |
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265 | //We add coloumns (variables) to our problem |
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266 | Col x1 = lp.addCol(); |
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267 | Col x2 = lp.addCol(); |
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268 | Col x3 = lp.addCol(); |
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269 | |
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270 | //Constraints |
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271 | lp.addRow(x1+x2+x3 <=100); |
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272 | lp.addRow(10*x1+4*x2+5*x3<=600); |
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273 | lp.addRow(2*x1+2*x2+6*x3<=300); |
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274 | //Nonnegativity of the variables |
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275 | lp.colLowerBound(x1, 0); |
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276 | lp.colLowerBound(x2, 0); |
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277 | lp.colLowerBound(x3, 0); |
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278 | //Objective function |
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279 | lp.setObj(10*x1+6*x2+4*x3); |
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280 | |
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281 | //Call the routine of the underlying LP solver |
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282 | lp.solve(); |
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283 | |
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284 | //Print results |
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285 | if (lp.primalStatus()==LpSolverBase::OPTIMAL){ |
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286 | printf("Z = %g; x1 = %g; x2 = %g; x3 = %g\n", |
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287 | lp.primalValue(), |
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288 | lp.primal(x1), lp.primal(x2), lp.primal(x3)); |
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289 | } |
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290 | else{ |
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291 | std::cout<<"Optimal solution not found!"<<std::endl; |
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292 | } |
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293 | |
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294 | |
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295 | \endcode |
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296 | |
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297 | See the whole code in \ref lp_demo.cc. |
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298 | |
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299 | <li>The second example shows how easy it is to formalize a max-flow |
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300 | problem as an LP problem using the LEMON LP interface: we are looking |
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301 | for a real valued function defined on the edges of the digraph |
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302 | satisfying the nonnegativity-, the capacity constraints and the |
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303 | flow-conservation constraints and giving the largest flow value |
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304 | between to designated nodes. |
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305 | |
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306 | In the following code we suppose that we already have the graph \c g, |
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307 | the capacity map \c cap, the source node \c s and the target node \c t |
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308 | in the memory. We will also omit the typedefs. |
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309 | |
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310 | \code |
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311 | //Define a map on the edges for the variables of the LP problem |
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312 | typename G::template EdgeMap<LpDefault::Col> x(g); |
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313 | lp.addColSet(x); |
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314 | |
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315 | //Nonnegativity and capacity constraints |
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316 | for(EdgeIt e(g);e!=INVALID;++e) { |
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317 | lp.colUpperBound(x[e],cap[e]); |
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318 | lp.colLowerBound(x[e],0); |
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319 | } |
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320 | |
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321 | |
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322 | //Flow conservation constraints for the nodes (except for 's' and 't') |
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323 | for(NodeIt n(g);n!=INVALID;++n) if(n!=s&&n!=t) { |
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324 | LpDefault::Expr ex; |
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325 | for(InEdgeIt e(g,n);e!=INVALID;++e) ex+=x[e]; |
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326 | for(OutEdgeIt e(g,n);e!=INVALID;++e) ex-=x[e]; |
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327 | lp.addRow(ex==0); |
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328 | } |
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329 | |
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330 | //Objective function: the flow value entering 't' |
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331 | { |
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332 | LpDefault::Expr ex; |
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333 | for(InEdgeIt e(g,t);e!=INVALID;++e) ex+=x[e]; |
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334 | for(OutEdgeIt e(g,t);e!=INVALID;++e) ex-=x[e]; |
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335 | lp.setObj(ex); |
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336 | } |
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337 | |
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338 | //Maximization |
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339 | lp.max(); |
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340 | |
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341 | //Solve with the underlying solver |
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342 | lp.solve(); |
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343 | |
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344 | \endcode |
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345 | |
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346 | The complete program can be found in file \ref lp_maxflow_demo.cc. After compiling run it in the form: |
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347 | |
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348 | <tt>./lp_maxflow_demo < ?????????.lgf</tt> |
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349 | |
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350 | where ?????????.lgf is a file in the lemon format containing a maxflow instance (designated "source", "target" nodes and "capacity" map on the edges). |
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351 | |
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352 | |
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353 | |
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354 | </ol> |
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355 | </ul> |
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356 | |
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357 | */ |
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