[209] | 1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
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[40] | 2 | * |
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[209] | 3 | * This file is a part of LEMON, a generic C++ optimization library. |
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[40] | 4 | * |
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[463] | 5 | * Copyright (C) 2003-2009 |
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[40] | 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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[422] | 19 | namespace lemon { |
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| 20 | |
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[40] | 21 | /** |
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| 22 | @defgroup datas Data Structures |
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[606] | 23 | This group contains the several data structures implemented in LEMON. |
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[40] | 24 | */ |
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| 25 | |
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| 26 | /** |
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| 27 | @defgroup graphs Graph Structures |
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| 28 | @ingroup datas |
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| 29 | \brief Graph structures implemented in LEMON. |
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| 30 | |
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[209] | 31 | The implementation of combinatorial algorithms heavily relies on |
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| 32 | efficient graph implementations. LEMON offers data structures which are |
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| 33 | planned to be easily used in an experimental phase of implementation studies, |
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| 34 | and thereafter the program code can be made efficient by small modifications. |
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[40] | 35 | |
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| 36 | The most efficient implementation of diverse applications require the |
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| 37 | usage of different physical graph implementations. These differences |
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| 38 | appear in the size of graph we require to handle, memory or time usage |
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| 39 | limitations or in the set of operations through which the graph can be |
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| 40 | accessed. LEMON provides several physical graph structures to meet |
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| 41 | the diverging requirements of the possible users. In order to save on |
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| 42 | running time or on memory usage, some structures may fail to provide |
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[83] | 43 | some graph features like arc/edge or node deletion. |
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[40] | 44 | |
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[209] | 45 | Alteration of standard containers need a very limited number of |
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| 46 | operations, these together satisfy the everyday requirements. |
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| 47 | In the case of graph structures, different operations are needed which do |
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| 48 | not alter the physical graph, but gives another view. If some nodes or |
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[83] | 49 | arcs have to be hidden or the reverse oriented graph have to be used, then |
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[209] | 50 | this is the case. It also may happen that in a flow implementation |
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| 51 | the residual graph can be accessed by another algorithm, or a node-set |
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| 52 | is to be shrunk for another algorithm. |
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| 53 | LEMON also provides a variety of graphs for these requirements called |
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| 54 | \ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only |
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| 55 | in conjunction with other graph representations. |
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[40] | 56 | |
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| 57 | You are free to use the graph structure that fit your requirements |
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| 58 | the best, most graph algorithms and auxiliary data structures can be used |
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[314] | 59 | with any graph structure. |
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| 60 | |
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| 61 | <b>See also:</b> \ref graph_concepts "Graph Structure Concepts". |
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[40] | 62 | */ |
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| 63 | |
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| 64 | /** |
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[474] | 65 | @defgroup graph_adaptors Adaptor Classes for Graphs |
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[432] | 66 | @ingroup graphs |
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[474] | 67 | \brief Adaptor classes for digraphs and graphs |
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| 68 | |
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| 69 | This group contains several useful adaptor classes for digraphs and graphs. |
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[432] | 70 | |
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| 71 | The main parts of LEMON are the different graph structures, generic |
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[474] | 72 | graph algorithms, graph concepts, which couple them, and graph |
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[432] | 73 | adaptors. While the previous notions are more or less clear, the |
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| 74 | latter one needs further explanation. Graph adaptors are graph classes |
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| 75 | which serve for considering graph structures in different ways. |
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| 76 | |
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| 77 | A short example makes this much clearer. Suppose that we have an |
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[474] | 78 | instance \c g of a directed graph type, say ListDigraph and an algorithm |
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[432] | 79 | \code |
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| 80 | template <typename Digraph> |
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| 81 | int algorithm(const Digraph&); |
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| 82 | \endcode |
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| 83 | is needed to run on the reverse oriented graph. It may be expensive |
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| 84 | (in time or in memory usage) to copy \c g with the reversed |
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| 85 | arcs. In this case, an adaptor class is used, which (according |
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[474] | 86 | to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph. |
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| 87 | The adaptor uses the original digraph structure and digraph operations when |
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| 88 | methods of the reversed oriented graph are called. This means that the adaptor |
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| 89 | have minor memory usage, and do not perform sophisticated algorithmic |
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[432] | 90 | actions. The purpose of it is to give a tool for the cases when a |
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| 91 | graph have to be used in a specific alteration. If this alteration is |
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[474] | 92 | obtained by a usual construction like filtering the node or the arc set or |
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[432] | 93 | considering a new orientation, then an adaptor is worthwhile to use. |
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| 94 | To come back to the reverse oriented graph, in this situation |
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| 95 | \code |
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| 96 | template<typename Digraph> class ReverseDigraph; |
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| 97 | \endcode |
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| 98 | template class can be used. The code looks as follows |
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| 99 | \code |
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| 100 | ListDigraph g; |
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[474] | 101 | ReverseDigraph<ListDigraph> rg(g); |
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[432] | 102 | int result = algorithm(rg); |
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| 103 | \endcode |
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[474] | 104 | During running the algorithm, the original digraph \c g is untouched. |
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| 105 | This techniques give rise to an elegant code, and based on stable |
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[432] | 106 | graph adaptors, complex algorithms can be implemented easily. |
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| 107 | |
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[474] | 108 | In flow, circulation and matching problems, the residual |
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[432] | 109 | graph is of particular importance. Combining an adaptor implementing |
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[474] | 110 | this with shortest path algorithms or minimum mean cycle algorithms, |
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[432] | 111 | a range of weighted and cardinality optimization algorithms can be |
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| 112 | obtained. For other examples, the interested user is referred to the |
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| 113 | detailed documentation of particular adaptors. |
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| 114 | |
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| 115 | The behavior of graph adaptors can be very different. Some of them keep |
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| 116 | capabilities of the original graph while in other cases this would be |
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[474] | 117 | meaningless. This means that the concepts that they meet depend |
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| 118 | on the graph adaptor, and the wrapped graph. |
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| 119 | For example, if an arc of a reversed digraph is deleted, this is carried |
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| 120 | out by deleting the corresponding arc of the original digraph, thus the |
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| 121 | adaptor modifies the original digraph. |
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| 122 | However in case of a residual digraph, this operation has no sense. |
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[432] | 123 | |
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| 124 | Let us stand one more example here to simplify your work. |
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[474] | 125 | ReverseDigraph has constructor |
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[432] | 126 | \code |
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| 127 | ReverseDigraph(Digraph& digraph); |
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| 128 | \endcode |
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[474] | 129 | This means that in a situation, when a <tt>const %ListDigraph&</tt> |
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[432] | 130 | reference to a graph is given, then it have to be instantiated with |
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[474] | 131 | <tt>Digraph=const %ListDigraph</tt>. |
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[432] | 132 | \code |
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| 133 | int algorithm1(const ListDigraph& g) { |
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[474] | 134 | ReverseDigraph<const ListDigraph> rg(g); |
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[432] | 135 | return algorithm2(rg); |
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| 136 | } |
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| 137 | \endcode |
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| 138 | */ |
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| 139 | |
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| 140 | /** |
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[50] | 141 | @defgroup semi_adaptors Semi-Adaptor Classes for Graphs |
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[40] | 142 | @ingroup graphs |
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| 143 | \brief Graph types between real graphs and graph adaptors. |
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| 144 | |
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[606] | 145 | This group contains some graph types between real graphs and graph adaptors. |
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[209] | 146 | These classes wrap graphs to give new functionality as the adaptors do it. |
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[50] | 147 | On the other hand they are not light-weight structures as the adaptors. |
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[40] | 148 | */ |
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| 149 | |
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| 150 | /** |
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[209] | 151 | @defgroup maps Maps |
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[40] | 152 | @ingroup datas |
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[50] | 153 | \brief Map structures implemented in LEMON. |
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[40] | 154 | |
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[606] | 155 | This group contains the map structures implemented in LEMON. |
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[50] | 156 | |
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[314] | 157 | LEMON provides several special purpose maps and map adaptors that e.g. combine |
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[40] | 158 | new maps from existing ones. |
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[314] | 159 | |
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| 160 | <b>See also:</b> \ref map_concepts "Map Concepts". |
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[40] | 161 | */ |
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| 162 | |
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| 163 | /** |
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[209] | 164 | @defgroup graph_maps Graph Maps |
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[40] | 165 | @ingroup maps |
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[83] | 166 | \brief Special graph-related maps. |
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[40] | 167 | |
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[606] | 168 | This group contains maps that are specifically designed to assign |
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[422] | 169 | values to the nodes and arcs/edges of graphs. |
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| 170 | |
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| 171 | If you are looking for the standard graph maps (\c NodeMap, \c ArcMap, |
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| 172 | \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts". |
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[40] | 173 | */ |
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| 174 | |
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| 175 | /** |
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| 176 | \defgroup map_adaptors Map Adaptors |
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| 177 | \ingroup maps |
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| 178 | \brief Tools to create new maps from existing ones |
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| 179 | |
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[606] | 180 | This group contains map adaptors that are used to create "implicit" |
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[50] | 181 | maps from other maps. |
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[40] | 182 | |
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[422] | 183 | Most of them are \ref concepts::ReadMap "read-only maps". |
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[83] | 184 | They can make arithmetic and logical operations between one or two maps |
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| 185 | (negation, shifting, addition, multiplication, logical 'and', 'or', |
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| 186 | 'not' etc.) or e.g. convert a map to another one of different Value type. |
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[40] | 187 | |
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[50] | 188 | The typical usage of this classes is passing implicit maps to |
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[40] | 189 | algorithms. If a function type algorithm is called then the function |
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| 190 | type map adaptors can be used comfortable. For example let's see the |
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[314] | 191 | usage of map adaptors with the \c graphToEps() function. |
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[40] | 192 | \code |
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| 193 | Color nodeColor(int deg) { |
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| 194 | if (deg >= 2) { |
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| 195 | return Color(0.5, 0.0, 0.5); |
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| 196 | } else if (deg == 1) { |
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| 197 | return Color(1.0, 0.5, 1.0); |
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| 198 | } else { |
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| 199 | return Color(0.0, 0.0, 0.0); |
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| 200 | } |
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| 201 | } |
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[209] | 202 | |
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[83] | 203 | Digraph::NodeMap<int> degree_map(graph); |
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[209] | 204 | |
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[314] | 205 | graphToEps(graph, "graph.eps") |
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[40] | 206 | .coords(coords).scaleToA4().undirected() |
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[83] | 207 | .nodeColors(composeMap(functorToMap(nodeColor), degree_map)) |
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[40] | 208 | .run(); |
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[209] | 209 | \endcode |
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[83] | 210 | The \c functorToMap() function makes an \c int to \c Color map from the |
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[314] | 211 | \c nodeColor() function. The \c composeMap() compose the \c degree_map |
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[83] | 212 | and the previously created map. The composed map is a proper function to |
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| 213 | get the color of each node. |
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[40] | 214 | |
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| 215 | The usage with class type algorithms is little bit harder. In this |
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| 216 | case the function type map adaptors can not be used, because the |
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[50] | 217 | function map adaptors give back temporary objects. |
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[40] | 218 | \code |
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[83] | 219 | Digraph graph; |
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| 220 | |
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| 221 | typedef Digraph::ArcMap<double> DoubleArcMap; |
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| 222 | DoubleArcMap length(graph); |
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| 223 | DoubleArcMap speed(graph); |
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| 224 | |
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| 225 | typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap; |
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[40] | 226 | TimeMap time(length, speed); |
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[209] | 227 | |
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[83] | 228 | Dijkstra<Digraph, TimeMap> dijkstra(graph, time); |
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[40] | 229 | dijkstra.run(source, target); |
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| 230 | \endcode |
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[83] | 231 | We have a length map and a maximum speed map on the arcs of a digraph. |
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| 232 | The minimum time to pass the arc can be calculated as the division of |
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| 233 | the two maps which can be done implicitly with the \c DivMap template |
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[40] | 234 | class. We use the implicit minimum time map as the length map of the |
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| 235 | \c Dijkstra algorithm. |
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| 236 | */ |
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| 237 | |
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| 238 | /** |
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[209] | 239 | @defgroup matrices Matrices |
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[40] | 240 | @ingroup datas |
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[50] | 241 | \brief Two dimensional data storages implemented in LEMON. |
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[40] | 242 | |
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[606] | 243 | This group contains two dimensional data storages implemented in LEMON. |
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[40] | 244 | */ |
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| 245 | |
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| 246 | /** |
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| 247 | @defgroup paths Path Structures |
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| 248 | @ingroup datas |
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[318] | 249 | \brief %Path structures implemented in LEMON. |
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[40] | 250 | |
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[606] | 251 | This group contains the path structures implemented in LEMON. |
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[40] | 252 | |
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[50] | 253 | LEMON provides flexible data structures to work with paths. |
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| 254 | All of them have similar interfaces and they can be copied easily with |
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| 255 | assignment operators and copy constructors. This makes it easy and |
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[40] | 256 | efficient to have e.g. the Dijkstra algorithm to store its result in |
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| 257 | any kind of path structure. |
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| 258 | |
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| 259 | \sa lemon::concepts::Path |
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| 260 | */ |
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| 261 | |
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| 262 | /** |
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| 263 | @defgroup auxdat Auxiliary Data Structures |
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| 264 | @ingroup datas |
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[50] | 265 | \brief Auxiliary data structures implemented in LEMON. |
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[40] | 266 | |
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[606] | 267 | This group contains some data structures implemented in LEMON in |
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[40] | 268 | order to make it easier to implement combinatorial algorithms. |
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| 269 | */ |
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| 270 | |
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| 271 | /** |
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| 272 | @defgroup algs Algorithms |
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[606] | 273 | \brief This group contains the several algorithms |
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[40] | 274 | implemented in LEMON. |
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| 275 | |
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[606] | 276 | This group contains the several algorithms |
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[40] | 277 | implemented in LEMON. |
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| 278 | */ |
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| 279 | |
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| 280 | /** |
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| 281 | @defgroup search Graph Search |
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| 282 | @ingroup algs |
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[50] | 283 | \brief Common graph search algorithms. |
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[40] | 284 | |
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[606] | 285 | This group contains the common graph search algorithms, namely |
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[422] | 286 | \e breadth-first \e search (BFS) and \e depth-first \e search (DFS). |
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[40] | 287 | */ |
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| 288 | |
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| 289 | /** |
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[314] | 290 | @defgroup shortest_path Shortest Path Algorithms |
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[40] | 291 | @ingroup algs |
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[50] | 292 | \brief Algorithms for finding shortest paths. |
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[40] | 293 | |
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[606] | 294 | This group contains the algorithms for finding shortest paths in digraphs. |
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[422] | 295 | |
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| 296 | - \ref Dijkstra algorithm for finding shortest paths from a source node |
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| 297 | when all arc lengths are non-negative. |
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| 298 | - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths |
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| 299 | from a source node when arc lenghts can be either positive or negative, |
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| 300 | but the digraph should not contain directed cycles with negative total |
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| 301 | length. |
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| 302 | - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms |
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| 303 | for solving the \e all-pairs \e shortest \e paths \e problem when arc |
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| 304 | lenghts can be either positive or negative, but the digraph should |
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| 305 | not contain directed cycles with negative total length. |
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| 306 | - \ref Suurballe A successive shortest path algorithm for finding |
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| 307 | arc-disjoint paths between two nodes having minimum total length. |
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[40] | 308 | */ |
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| 309 | |
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[209] | 310 | /** |
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[314] | 311 | @defgroup max_flow Maximum Flow Algorithms |
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[209] | 312 | @ingroup algs |
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[50] | 313 | \brief Algorithms for finding maximum flows. |
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[40] | 314 | |
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[606] | 315 | This group contains the algorithms for finding maximum flows and |
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[40] | 316 | feasible circulations. |
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| 317 | |
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[422] | 318 | The \e maximum \e flow \e problem is to find a flow of maximum value between |
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| 319 | a single source and a single target. Formally, there is a \f$G=(V,A)\f$ |
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[656] | 320 | digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and |
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[422] | 321 | \f$s, t \in V\f$ source and target nodes. |
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[656] | 322 | A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the |
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[422] | 323 | following optimization problem. |
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[40] | 324 | |
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[656] | 325 | \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f] |
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| 326 | \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu) |
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| 327 | \quad \forall u\in V\setminus\{s,t\} \f] |
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| 328 | \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f] |
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[40] | 329 | |
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[50] | 330 | LEMON contains several algorithms for solving maximum flow problems: |
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[422] | 331 | - \ref EdmondsKarp Edmonds-Karp algorithm. |
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| 332 | - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm. |
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| 333 | - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees. |
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| 334 | - \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees. |
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[40] | 335 | |
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[422] | 336 | In most cases the \ref Preflow "Preflow" algorithm provides the |
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| 337 | fastest method for computing a maximum flow. All implementations |
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| 338 | provides functions to also query the minimum cut, which is the dual |
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| 339 | problem of the maximum flow. |
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[40] | 340 | */ |
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| 341 | |
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| 342 | /** |
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[314] | 343 | @defgroup min_cost_flow Minimum Cost Flow Algorithms |
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[40] | 344 | @ingroup algs |
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| 345 | |
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[50] | 346 | \brief Algorithms for finding minimum cost flows and circulations. |
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[40] | 347 | |
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[656] | 348 | This group contains the algorithms for finding minimum cost flows and |
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[209] | 349 | circulations. |
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[422] | 350 | |
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| 351 | The \e minimum \e cost \e flow \e problem is to find a feasible flow of |
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| 352 | minimum total cost from a set of supply nodes to a set of demand nodes |
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[656] | 353 | in a network with capacity constraints (lower and upper bounds) |
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| 354 | and arc costs. |
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[687] | 355 | Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{Z}\f$, |
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| 356 | \f$upper: A\rightarrow\mathbf{Z}\cup\{+\infty\}\f$ denote the lower and |
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[656] | 357 | upper bounds for the flow values on the arcs, for which |
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[687] | 358 | \f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$, |
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| 359 | \f$cost: A\rightarrow\mathbf{Z}\f$ denotes the cost per unit flow |
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| 360 | on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the |
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[656] | 361 | signed supply values of the nodes. |
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| 362 | If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$ |
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| 363 | supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with |
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| 364 | \f$-sup(u)\f$ demand. |
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[687] | 365 | A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution |
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[656] | 366 | of the following optimization problem. |
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[422] | 367 | |
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[656] | 368 | \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f] |
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| 369 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq |
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| 370 | sup(u) \quad \forall u\in V \f] |
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| 371 | \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f] |
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[422] | 372 | |
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[656] | 373 | The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be |
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| 374 | zero or negative in order to have a feasible solution (since the sum |
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| 375 | of the expressions on the left-hand side of the inequalities is zero). |
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| 376 | It means that the total demand must be greater or equal to the total |
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| 377 | supply and all the supplies have to be carried out from the supply nodes, |
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| 378 | but there could be demands that are not satisfied. |
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| 379 | If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand |
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| 380 | constraints have to be satisfied with equality, i.e. all demands |
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| 381 | have to be satisfied and all supplies have to be used. |
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| 382 | |
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| 383 | If you need the opposite inequalities in the supply/demand constraints |
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| 384 | (i.e. the total demand is less than the total supply and all the demands |
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| 385 | have to be satisfied while there could be supplies that are not used), |
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| 386 | then you could easily transform the problem to the above form by reversing |
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| 387 | the direction of the arcs and taking the negative of the supply values |
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| 388 | (e.g. using \ref ReverseDigraph and \ref NegMap adaptors). |
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| 389 | However \ref NetworkSimplex algorithm also supports this form directly |
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| 390 | for the sake of convenience. |
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| 391 | |
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| 392 | A feasible solution for this problem can be found using \ref Circulation. |
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| 393 | |
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| 394 | Note that the above formulation is actually more general than the usual |
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| 395 | definition of the minimum cost flow problem, in which strict equalities |
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| 396 | are required in the supply/demand contraints, i.e. |
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| 397 | |
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| 398 | \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) = |
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| 399 | sup(u) \quad \forall u\in V. \f] |
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| 400 | |
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| 401 | However if the sum of the supply values is zero, then these two problems |
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| 402 | are equivalent. So if you need the equality form, you have to ensure this |
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| 403 | additional contraint for the algorithms. |
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| 404 | |
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| 405 | The dual solution of the minimum cost flow problem is represented by node |
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| 406 | potentials \f$\pi: V\rightarrow\mathbf{Z}\f$. |
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[687] | 407 | An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem |
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[656] | 408 | is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$ |
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| 409 | node potentials the following \e complementary \e slackness optimality |
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| 410 | conditions hold. |
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| 411 | |
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| 412 | - For all \f$uv\in A\f$ arcs: |
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| 413 | - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$; |
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| 414 | - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$; |
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| 415 | - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$. |
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[687] | 416 | - For all \f$u\in V\f$ nodes: |
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[656] | 417 | - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$, |
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| 418 | then \f$\pi(u)=0\f$. |
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| 419 | |
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| 420 | Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc |
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[687] | 421 | \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e. |
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[656] | 422 | \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f] |
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| 423 | |
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[687] | 424 | All algorithms provide dual solution (node potentials) as well, |
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[656] | 425 | if an optimal flow is found. |
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| 426 | |
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| 427 | LEMON contains several algorithms for solving minimum cost flow problems. |
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| 428 | - \ref NetworkSimplex Primal Network Simplex algorithm with various |
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| 429 | pivot strategies. |
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| 430 | - \ref CostScaling Push-Relabel and Augment-Relabel algorithms based on |
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| 431 | cost scaling. |
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| 432 | - \ref CapacityScaling Successive Shortest %Path algorithm with optional |
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[422] | 433 | capacity scaling. |
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[656] | 434 | - \ref CancelAndTighten The Cancel and Tighten algorithm. |
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| 435 | - \ref CycleCanceling Cycle-Canceling algorithms. |
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| 436 | |
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| 437 | Most of these implementations support the general inequality form of the |
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| 438 | minimum cost flow problem, but CancelAndTighten and CycleCanceling |
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| 439 | only support the equality form due to the primal method they use. |
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| 440 | |
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| 441 | In general NetworkSimplex is the most efficient implementation, |
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| 442 | but in special cases other algorithms could be faster. |
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| 443 | For example, if the total supply and/or capacities are rather small, |
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| 444 | CapacityScaling is usually the fastest algorithm (without effective scaling). |
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[40] | 445 | */ |
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| 446 | |
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| 447 | /** |
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[314] | 448 | @defgroup min_cut Minimum Cut Algorithms |
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[209] | 449 | @ingroup algs |
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[40] | 450 | |
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[50] | 451 | \brief Algorithms for finding minimum cut in graphs. |
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[40] | 452 | |
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[606] | 453 | This group contains the algorithms for finding minimum cut in graphs. |
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[40] | 454 | |
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[422] | 455 | The \e minimum \e cut \e problem is to find a non-empty and non-complete |
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| 456 | \f$X\f$ subset of the nodes with minimum overall capacity on |
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| 457 | outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a |
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| 458 | \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum |
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[50] | 459 | cut is the \f$X\f$ solution of the next optimization problem: |
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[40] | 460 | |
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[210] | 461 | \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}} |
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[422] | 462 | \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f] |
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[40] | 463 | |
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[50] | 464 | LEMON contains several algorithms related to minimum cut problems: |
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[40] | 465 | |
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[422] | 466 | - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut |
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| 467 | in directed graphs. |
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| 468 | - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for |
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| 469 | calculating minimum cut in undirected graphs. |
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[606] | 470 | - \ref GomoryHu "Gomory-Hu tree computation" for calculating |
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[422] | 471 | all-pairs minimum cut in undirected graphs. |
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[40] | 472 | |
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| 473 | If you want to find minimum cut just between two distinict nodes, |
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[422] | 474 | see the \ref max_flow "maximum flow problem". |
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[40] | 475 | */ |
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| 476 | |
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| 477 | /** |
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[633] | 478 | @defgroup graph_properties Connectivity and Other Graph Properties |
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[40] | 479 | @ingroup algs |
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[50] | 480 | \brief Algorithms for discovering the graph properties |
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[40] | 481 | |
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[606] | 482 | This group contains the algorithms for discovering the graph properties |
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[50] | 483 | like connectivity, bipartiteness, euler property, simplicity etc. |
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[40] | 484 | |
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| 485 | \image html edge_biconnected_components.png |
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| 486 | \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth |
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| 487 | */ |
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| 488 | |
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| 489 | /** |
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[314] | 490 | @defgroup planar Planarity Embedding and Drawing |
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[40] | 491 | @ingroup algs |
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[50] | 492 | \brief Algorithms for planarity checking, embedding and drawing |
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[40] | 493 | |
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[606] | 494 | This group contains the algorithms for planarity checking, |
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[210] | 495 | embedding and drawing. |
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[40] | 496 | |
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| 497 | \image html planar.png |
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| 498 | \image latex planar.eps "Plane graph" width=\textwidth |
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| 499 | */ |
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| 500 | |
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| 501 | /** |
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[314] | 502 | @defgroup matching Matching Algorithms |
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[40] | 503 | @ingroup algs |
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[50] | 504 | \brief Algorithms for finding matchings in graphs and bipartite graphs. |
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[40] | 505 | |
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[637] | 506 | This group contains the algorithms for calculating |
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[40] | 507 | matchings in graphs and bipartite graphs. The general matching problem is |
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[637] | 508 | finding a subset of the edges for which each node has at most one incident |
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| 509 | edge. |
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[209] | 510 | |
---|
[40] | 511 | There are several different algorithms for calculate matchings in |
---|
| 512 | graphs. The matching problems in bipartite graphs are generally |
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| 513 | easier than in general graphs. The goal of the matching optimization |
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[422] | 514 | can be finding maximum cardinality, maximum weight or minimum cost |
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[40] | 515 | matching. The search can be constrained to find perfect or |
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| 516 | maximum cardinality matching. |
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| 517 | |
---|
[422] | 518 | The matching algorithms implemented in LEMON: |
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| 519 | - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm |
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| 520 | for calculating maximum cardinality matching in bipartite graphs. |
---|
| 521 | - \ref PrBipartiteMatching Push-relabel algorithm |
---|
| 522 | for calculating maximum cardinality matching in bipartite graphs. |
---|
| 523 | - \ref MaxWeightedBipartiteMatching |
---|
| 524 | Successive shortest path algorithm for calculating maximum weighted |
---|
| 525 | matching and maximum weighted bipartite matching in bipartite graphs. |
---|
| 526 | - \ref MinCostMaxBipartiteMatching |
---|
| 527 | Successive shortest path algorithm for calculating minimum cost maximum |
---|
| 528 | matching in bipartite graphs. |
---|
| 529 | - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating |
---|
| 530 | maximum cardinality matching in general graphs. |
---|
| 531 | - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating |
---|
| 532 | maximum weighted matching in general graphs. |
---|
| 533 | - \ref MaxWeightedPerfectMatching |
---|
| 534 | Edmond's blossom shrinking algorithm for calculating maximum weighted |
---|
| 535 | perfect matching in general graphs. |
---|
[40] | 536 | |
---|
| 537 | \image html bipartite_matching.png |
---|
| 538 | \image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth |
---|
| 539 | */ |
---|
| 540 | |
---|
| 541 | /** |
---|
[314] | 542 | @defgroup spantree Minimum Spanning Tree Algorithms |
---|
[40] | 543 | @ingroup algs |
---|
[50] | 544 | \brief Algorithms for finding a minimum cost spanning tree in a graph. |
---|
[40] | 545 | |
---|
[606] | 546 | This group contains the algorithms for finding a minimum cost spanning |
---|
[422] | 547 | tree in a graph. |
---|
[40] | 548 | */ |
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| 549 | |
---|
| 550 | /** |
---|
[314] | 551 | @defgroup auxalg Auxiliary Algorithms |
---|
[40] | 552 | @ingroup algs |
---|
[50] | 553 | \brief Auxiliary algorithms implemented in LEMON. |
---|
[40] | 554 | |
---|
[606] | 555 | This group contains some algorithms implemented in LEMON |
---|
[50] | 556 | in order to make it easier to implement complex algorithms. |
---|
[40] | 557 | */ |
---|
| 558 | |
---|
| 559 | /** |
---|
[314] | 560 | @defgroup approx Approximation Algorithms |
---|
| 561 | @ingroup algs |
---|
[50] | 562 | \brief Approximation algorithms. |
---|
[40] | 563 | |
---|
[606] | 564 | This group contains the approximation and heuristic algorithms |
---|
[50] | 565 | implemented in LEMON. |
---|
[40] | 566 | */ |
---|
| 567 | |
---|
| 568 | /** |
---|
| 569 | @defgroup gen_opt_group General Optimization Tools |
---|
[606] | 570 | \brief This group contains some general optimization frameworks |
---|
[40] | 571 | implemented in LEMON. |
---|
| 572 | |
---|
[606] | 573 | This group contains some general optimization frameworks |
---|
[40] | 574 | implemented in LEMON. |
---|
| 575 | */ |
---|
| 576 | |
---|
| 577 | /** |
---|
[314] | 578 | @defgroup lp_group Lp and Mip Solvers |
---|
[40] | 579 | @ingroup gen_opt_group |
---|
| 580 | \brief Lp and Mip solver interfaces for LEMON. |
---|
| 581 | |
---|
[606] | 582 | This group contains Lp and Mip solver interfaces for LEMON. The |
---|
[40] | 583 | various LP solvers could be used in the same manner with this |
---|
| 584 | interface. |
---|
| 585 | */ |
---|
| 586 | |
---|
[209] | 587 | /** |
---|
[314] | 588 | @defgroup lp_utils Tools for Lp and Mip Solvers |
---|
[40] | 589 | @ingroup lp_group |
---|
[50] | 590 | \brief Helper tools to the Lp and Mip solvers. |
---|
[40] | 591 | |
---|
| 592 | This group adds some helper tools to general optimization framework |
---|
| 593 | implemented in LEMON. |
---|
| 594 | */ |
---|
| 595 | |
---|
| 596 | /** |
---|
| 597 | @defgroup metah Metaheuristics |
---|
| 598 | @ingroup gen_opt_group |
---|
| 599 | \brief Metaheuristics for LEMON library. |
---|
| 600 | |
---|
[606] | 601 | This group contains some metaheuristic optimization tools. |
---|
[40] | 602 | */ |
---|
| 603 | |
---|
| 604 | /** |
---|
[209] | 605 | @defgroup utils Tools and Utilities |
---|
[50] | 606 | \brief Tools and utilities for programming in LEMON |
---|
[40] | 607 | |
---|
[50] | 608 | Tools and utilities for programming in LEMON. |
---|
[40] | 609 | */ |
---|
| 610 | |
---|
| 611 | /** |
---|
| 612 | @defgroup gutils Basic Graph Utilities |
---|
| 613 | @ingroup utils |
---|
[50] | 614 | \brief Simple basic graph utilities. |
---|
[40] | 615 | |
---|
[606] | 616 | This group contains some simple basic graph utilities. |
---|
[40] | 617 | */ |
---|
| 618 | |
---|
| 619 | /** |
---|
| 620 | @defgroup misc Miscellaneous Tools |
---|
| 621 | @ingroup utils |
---|
[50] | 622 | \brief Tools for development, debugging and testing. |
---|
| 623 | |
---|
[606] | 624 | This group contains several useful tools for development, |
---|
[40] | 625 | debugging and testing. |
---|
| 626 | */ |
---|
| 627 | |
---|
| 628 | /** |
---|
[314] | 629 | @defgroup timecount Time Measuring and Counting |
---|
[40] | 630 | @ingroup misc |
---|
[50] | 631 | \brief Simple tools for measuring the performance of algorithms. |
---|
| 632 | |
---|
[606] | 633 | This group contains simple tools for measuring the performance |
---|
[40] | 634 | of algorithms. |
---|
| 635 | */ |
---|
| 636 | |
---|
| 637 | /** |
---|
| 638 | @defgroup exceptions Exceptions |
---|
| 639 | @ingroup utils |
---|
[50] | 640 | \brief Exceptions defined in LEMON. |
---|
| 641 | |
---|
[606] | 642 | This group contains the exceptions defined in LEMON. |
---|
[40] | 643 | */ |
---|
| 644 | |
---|
| 645 | /** |
---|
| 646 | @defgroup io_group Input-Output |
---|
[50] | 647 | \brief Graph Input-Output methods |
---|
[40] | 648 | |
---|
[606] | 649 | This group contains the tools for importing and exporting graphs |
---|
[314] | 650 | and graph related data. Now it supports the \ref lgf-format |
---|
| 651 | "LEMON Graph Format", the \c DIMACS format and the encapsulated |
---|
| 652 | postscript (EPS) format. |
---|
[40] | 653 | */ |
---|
| 654 | |
---|
| 655 | /** |
---|
[363] | 656 | @defgroup lemon_io LEMON Graph Format |
---|
[40] | 657 | @ingroup io_group |
---|
[314] | 658 | \brief Reading and writing LEMON Graph Format. |
---|
[40] | 659 | |
---|
[606] | 660 | This group contains methods for reading and writing |
---|
[236] | 661 | \ref lgf-format "LEMON Graph Format". |
---|
[40] | 662 | */ |
---|
| 663 | |
---|
| 664 | /** |
---|
[314] | 665 | @defgroup eps_io Postscript Exporting |
---|
[40] | 666 | @ingroup io_group |
---|
| 667 | \brief General \c EPS drawer and graph exporter |
---|
| 668 | |
---|
[606] | 669 | This group contains general \c EPS drawing methods and special |
---|
[209] | 670 | graph exporting tools. |
---|
[40] | 671 | */ |
---|
| 672 | |
---|
| 673 | /** |
---|
[403] | 674 | @defgroup dimacs_group DIMACS format |
---|
| 675 | @ingroup io_group |
---|
| 676 | \brief Read and write files in DIMACS format |
---|
| 677 | |
---|
| 678 | Tools to read a digraph from or write it to a file in DIMACS format data. |
---|
| 679 | */ |
---|
| 680 | |
---|
| 681 | /** |
---|
[363] | 682 | @defgroup nauty_group NAUTY Format |
---|
| 683 | @ingroup io_group |
---|
| 684 | \brief Read \e Nauty format |
---|
[403] | 685 | |
---|
[363] | 686 | Tool to read graphs from \e Nauty format data. |
---|
| 687 | */ |
---|
| 688 | |
---|
| 689 | /** |
---|
[40] | 690 | @defgroup concept Concepts |
---|
| 691 | \brief Skeleton classes and concept checking classes |
---|
| 692 | |
---|
[606] | 693 | This group contains the data/algorithm skeletons and concept checking |
---|
[40] | 694 | classes implemented in LEMON. |
---|
| 695 | |
---|
| 696 | The purpose of the classes in this group is fourfold. |
---|
[209] | 697 | |
---|
[318] | 698 | - These classes contain the documentations of the %concepts. In order |
---|
[40] | 699 | to avoid document multiplications, an implementation of a concept |
---|
| 700 | simply refers to the corresponding concept class. |
---|
| 701 | |
---|
| 702 | - These classes declare every functions, <tt>typedef</tt>s etc. an |
---|
[318] | 703 | implementation of the %concepts should provide, however completely |
---|
[40] | 704 | without implementations and real data structures behind the |
---|
| 705 | interface. On the other hand they should provide nothing else. All |
---|
| 706 | the algorithms working on a data structure meeting a certain concept |
---|
| 707 | should compile with these classes. (Though it will not run properly, |
---|
| 708 | of course.) In this way it is easily to check if an algorithm |
---|
| 709 | doesn't use any extra feature of a certain implementation. |
---|
| 710 | |
---|
| 711 | - The concept descriptor classes also provide a <em>checker class</em> |
---|
[50] | 712 | that makes it possible to check whether a certain implementation of a |
---|
[40] | 713 | concept indeed provides all the required features. |
---|
| 714 | |
---|
| 715 | - Finally, They can serve as a skeleton of a new implementation of a concept. |
---|
| 716 | */ |
---|
| 717 | |
---|
| 718 | /** |
---|
| 719 | @defgroup graph_concepts Graph Structure Concepts |
---|
| 720 | @ingroup concept |
---|
| 721 | \brief Skeleton and concept checking classes for graph structures |
---|
| 722 | |
---|
[606] | 723 | This group contains the skeletons and concept checking classes of LEMON's |
---|
[40] | 724 | graph structures and helper classes used to implement these. |
---|
| 725 | */ |
---|
| 726 | |
---|
[314] | 727 | /** |
---|
| 728 | @defgroup map_concepts Map Concepts |
---|
| 729 | @ingroup concept |
---|
| 730 | \brief Skeleton and concept checking classes for maps |
---|
| 731 | |
---|
[606] | 732 | This group contains the skeletons and concept checking classes of maps. |
---|
[40] | 733 | */ |
---|
| 734 | |
---|
| 735 | /** |
---|
| 736 | \anchor demoprograms |
---|
| 737 | |
---|
[422] | 738 | @defgroup demos Demo Programs |
---|
[40] | 739 | |
---|
| 740 | Some demo programs are listed here. Their full source codes can be found in |
---|
| 741 | the \c demo subdirectory of the source tree. |
---|
| 742 | |
---|
[611] | 743 | In order to compile them, use the <tt>make demo</tt> or the |
---|
| 744 | <tt>make check</tt> commands. |
---|
[40] | 745 | */ |
---|
| 746 | |
---|
| 747 | /** |
---|
[422] | 748 | @defgroup tools Standalone Utility Applications |
---|
[40] | 749 | |
---|
[209] | 750 | Some utility applications are listed here. |
---|
[40] | 751 | |
---|
| 752 | The standard compilation procedure (<tt>./configure;make</tt>) will compile |
---|
[209] | 753 | them, as well. |
---|
[40] | 754 | */ |
---|
| 755 | |
---|
[422] | 756 | } |
---|