[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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[956] | 5 | * Copyright (C) 2003-2010 |
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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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[209] | 141 | @defgroup maps Maps |
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[40] | 142 | @ingroup datas |
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[50] | 143 | \brief Map structures implemented in LEMON. |
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[40] | 144 | |
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[606] | 145 | This group contains the map structures implemented in LEMON. |
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[50] | 146 | |
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[314] | 147 | LEMON provides several special purpose maps and map adaptors that e.g. combine |
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[40] | 148 | new maps from existing ones. |
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[314] | 149 | |
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| 150 | <b>See also:</b> \ref map_concepts "Map Concepts". |
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[40] | 151 | */ |
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| 152 | |
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| 153 | /** |
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[209] | 154 | @defgroup graph_maps Graph Maps |
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[40] | 155 | @ingroup maps |
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[83] | 156 | \brief Special graph-related maps. |
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[40] | 157 | |
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[606] | 158 | This group contains maps that are specifically designed to assign |
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[422] | 159 | values to the nodes and arcs/edges of graphs. |
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| 160 | |
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| 161 | If you are looking for the standard graph maps (\c NodeMap, \c ArcMap, |
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| 162 | \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts". |
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[40] | 163 | */ |
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| 164 | |
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| 165 | /** |
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| 166 | \defgroup map_adaptors Map Adaptors |
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| 167 | \ingroup maps |
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| 168 | \brief Tools to create new maps from existing ones |
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| 169 | |
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[606] | 170 | This group contains map adaptors that are used to create "implicit" |
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[50] | 171 | maps from other maps. |
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[40] | 172 | |
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[422] | 173 | Most of them are \ref concepts::ReadMap "read-only maps". |
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[83] | 174 | They can make arithmetic and logical operations between one or two maps |
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| 175 | (negation, shifting, addition, multiplication, logical 'and', 'or', |
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| 176 | 'not' etc.) or e.g. convert a map to another one of different Value type. |
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[40] | 177 | |
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[50] | 178 | The typical usage of this classes is passing implicit maps to |
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[40] | 179 | algorithms. If a function type algorithm is called then the function |
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| 180 | type map adaptors can be used comfortable. For example let's see the |
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[314] | 181 | usage of map adaptors with the \c graphToEps() function. |
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[40] | 182 | \code |
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| 183 | Color nodeColor(int deg) { |
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| 184 | if (deg >= 2) { |
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| 185 | return Color(0.5, 0.0, 0.5); |
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| 186 | } else if (deg == 1) { |
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| 187 | return Color(1.0, 0.5, 1.0); |
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| 188 | } else { |
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| 189 | return Color(0.0, 0.0, 0.0); |
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| 190 | } |
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| 191 | } |
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[209] | 192 | |
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[83] | 193 | Digraph::NodeMap<int> degree_map(graph); |
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[209] | 194 | |
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[314] | 195 | graphToEps(graph, "graph.eps") |
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[40] | 196 | .coords(coords).scaleToA4().undirected() |
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[83] | 197 | .nodeColors(composeMap(functorToMap(nodeColor), degree_map)) |
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[40] | 198 | .run(); |
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[209] | 199 | \endcode |
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[83] | 200 | The \c functorToMap() function makes an \c int to \c Color map from the |
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[314] | 201 | \c nodeColor() function. The \c composeMap() compose the \c degree_map |
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[83] | 202 | and the previously created map. The composed map is a proper function to |
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| 203 | get the color of each node. |
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[40] | 204 | |
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| 205 | The usage with class type algorithms is little bit harder. In this |
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| 206 | case the function type map adaptors can not be used, because the |
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[50] | 207 | function map adaptors give back temporary objects. |
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[40] | 208 | \code |
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[83] | 209 | Digraph graph; |
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| 210 | |
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| 211 | typedef Digraph::ArcMap<double> DoubleArcMap; |
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| 212 | DoubleArcMap length(graph); |
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| 213 | DoubleArcMap speed(graph); |
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| 214 | |
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| 215 | typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap; |
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[40] | 216 | TimeMap time(length, speed); |
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[209] | 217 | |
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[83] | 218 | Dijkstra<Digraph, TimeMap> dijkstra(graph, time); |
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[40] | 219 | dijkstra.run(source, target); |
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| 220 | \endcode |
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[83] | 221 | We have a length map and a maximum speed map on the arcs of a digraph. |
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| 222 | The minimum time to pass the arc can be calculated as the division of |
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| 223 | the two maps which can be done implicitly with the \c DivMap template |
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[40] | 224 | class. We use the implicit minimum time map as the length map of the |
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| 225 | \c Dijkstra algorithm. |
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| 226 | */ |
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| 227 | |
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| 228 | /** |
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| 229 | @defgroup paths Path Structures |
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| 230 | @ingroup datas |
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[318] | 231 | \brief %Path structures implemented in LEMON. |
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[40] | 232 | |
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[606] | 233 | This group contains the path structures implemented in LEMON. |
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[40] | 234 | |
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[50] | 235 | LEMON provides flexible data structures to work with paths. |
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| 236 | All of them have similar interfaces and they can be copied easily with |
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| 237 | assignment operators and copy constructors. This makes it easy and |
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[40] | 238 | efficient to have e.g. the Dijkstra algorithm to store its result in |
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| 239 | any kind of path structure. |
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| 240 | |
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[757] | 241 | \sa \ref concepts::Path "Path concept" |
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| 242 | */ |
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| 243 | |
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| 244 | /** |
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| 245 | @defgroup heaps Heap Structures |
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| 246 | @ingroup datas |
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| 247 | \brief %Heap structures implemented in LEMON. |
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| 248 | |
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| 249 | This group contains the heap structures implemented in LEMON. |
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| 250 | |
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| 251 | LEMON provides several heap classes. They are efficient implementations |
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| 252 | of the abstract data type \e priority \e queue. They store items with |
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| 253 | specified values called \e priorities in such a way that finding and |
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| 254 | removing the item with minimum priority are efficient. |
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| 255 | The basic operations are adding and erasing items, changing the priority |
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| 256 | of an item, etc. |
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| 257 | |
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| 258 | Heaps are crucial in several algorithms, such as Dijkstra and Prim. |
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| 259 | The heap implementations have the same interface, thus any of them can be |
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| 260 | used easily in such algorithms. |
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| 261 | |
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| 262 | \sa \ref concepts::Heap "Heap concept" |
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| 263 | */ |
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| 264 | |
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| 265 | /** |
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[40] | 266 | @defgroup auxdat Auxiliary Data Structures |
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| 267 | @ingroup datas |
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[50] | 268 | \brief Auxiliary data structures implemented in LEMON. |
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[40] | 269 | |
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[606] | 270 | This group contains some data structures implemented in LEMON in |
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[40] | 271 | order to make it easier to implement combinatorial algorithms. |
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| 272 | */ |
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| 273 | |
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| 274 | /** |
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[761] | 275 | @defgroup geomdat Geometric Data Structures |
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| 276 | @ingroup auxdat |
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| 277 | \brief Geometric data structures implemented in LEMON. |
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| 278 | |
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| 279 | This group contains geometric data structures implemented in LEMON. |
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| 280 | |
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| 281 | - \ref lemon::dim2::Point "dim2::Point" implements a two dimensional |
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| 282 | vector with the usual operations. |
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| 283 | - \ref lemon::dim2::Box "dim2::Box" can be used to determine the |
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| 284 | rectangular bounding box of a set of \ref lemon::dim2::Point |
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| 285 | "dim2::Point"'s. |
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| 286 | */ |
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| 287 | |
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| 288 | /** |
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| 289 | @defgroup matrices Matrices |
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| 290 | @ingroup auxdat |
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| 291 | \brief Two dimensional data storages implemented in LEMON. |
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| 292 | |
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| 293 | This group contains two dimensional data storages implemented in LEMON. |
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| 294 | */ |
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| 295 | |
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| 296 | /** |
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[40] | 297 | @defgroup algs Algorithms |
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[606] | 298 | \brief This group contains the several algorithms |
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[40] | 299 | implemented in LEMON. |
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| 300 | |
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[606] | 301 | This group contains the several algorithms |
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[40] | 302 | implemented in LEMON. |
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| 303 | */ |
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| 304 | |
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| 305 | /** |
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| 306 | @defgroup search Graph Search |
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| 307 | @ingroup algs |
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[50] | 308 | \brief Common graph search algorithms. |
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[40] | 309 | |
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[606] | 310 | This group contains the common graph search algorithms, namely |
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[802] | 311 | \e breadth-first \e search (BFS) and \e depth-first \e search (DFS) |
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| 312 | \ref clrs01algorithms. |
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[40] | 313 | */ |
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| 314 | |
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| 315 | /** |
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[314] | 316 | @defgroup shortest_path Shortest Path Algorithms |
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[40] | 317 | @ingroup algs |
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[50] | 318 | \brief Algorithms for finding shortest paths. |
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[40] | 319 | |
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[802] | 320 | This group contains the algorithms for finding shortest paths in digraphs |
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| 321 | \ref clrs01algorithms. |
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[422] | 322 | |
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| 323 | - \ref Dijkstra algorithm for finding shortest paths from a source node |
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| 324 | when all arc lengths are non-negative. |
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| 325 | - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths |
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| 326 | from a source node when arc lenghts can be either positive or negative, |
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| 327 | but the digraph should not contain directed cycles with negative total |
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| 328 | length. |
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| 329 | - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms |
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| 330 | for solving the \e all-pairs \e shortest \e paths \e problem when arc |
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| 331 | lenghts can be either positive or negative, but the digraph should |
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| 332 | not contain directed cycles with negative total length. |
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| 333 | - \ref Suurballe A successive shortest path algorithm for finding |
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| 334 | arc-disjoint paths between two nodes having minimum total length. |
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[40] | 335 | */ |
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| 336 | |
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[209] | 337 | /** |
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[761] | 338 | @defgroup spantree Minimum Spanning Tree Algorithms |
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| 339 | @ingroup algs |
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| 340 | \brief Algorithms for finding minimum cost spanning trees and arborescences. |
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| 341 | |
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| 342 | This group contains the algorithms for finding minimum cost spanning |
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[802] | 343 | trees and arborescences \ref clrs01algorithms. |
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[761] | 344 | */ |
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| 345 | |
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| 346 | /** |
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[314] | 347 | @defgroup max_flow Maximum Flow Algorithms |
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[209] | 348 | @ingroup algs |
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[50] | 349 | \brief Algorithms for finding maximum flows. |
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[40] | 350 | |
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[606] | 351 | This group contains the algorithms for finding maximum flows and |
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[802] | 352 | feasible circulations \ref clrs01algorithms, \ref amo93networkflows. |
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[40] | 353 | |
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[422] | 354 | The \e maximum \e flow \e problem is to find a flow of maximum value between |
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| 355 | a single source and a single target. Formally, there is a \f$G=(V,A)\f$ |
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[656] | 356 | digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and |
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[422] | 357 | \f$s, t \in V\f$ source and target nodes. |
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[656] | 358 | A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the |
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[422] | 359 | following optimization problem. |
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[40] | 360 | |
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[656] | 361 | \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f] |
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| 362 | \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu) |
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| 363 | \quad \forall u\in V\setminus\{s,t\} \f] |
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| 364 | \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f] |
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[40] | 365 | |
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[50] | 366 | LEMON contains several algorithms for solving maximum flow problems: |
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[802] | 367 | - \ref EdmondsKarp Edmonds-Karp algorithm |
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| 368 | \ref edmondskarp72theoretical. |
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| 369 | - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm |
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| 370 | \ref goldberg88newapproach. |
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| 371 | - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees |
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| 372 | \ref dinic70algorithm, \ref sleator83dynamic. |
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| 373 | - \ref GoldbergTarjan !Preflow push-relabel algorithm with dynamic trees |
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| 374 | \ref goldberg88newapproach, \ref sleator83dynamic. |
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[40] | 375 | |
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[802] | 376 | In most cases the \ref Preflow algorithm provides the |
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[422] | 377 | fastest method for computing a maximum flow. All implementations |
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[698] | 378 | also provide functions to query the minimum cut, which is the dual |
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| 379 | problem of maximum flow. |
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| 380 | |
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[948] | 381 | \ref Circulation is a preflow push-relabel algorithm implemented directly |
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[698] | 382 | for finding feasible circulations, which is a somewhat different problem, |
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| 383 | but it is strongly related to maximum flow. |
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| 384 | For more information, see \ref Circulation. |
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[40] | 385 | */ |
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| 386 | |
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| 387 | /** |
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[710] | 388 | @defgroup min_cost_flow_algs Minimum Cost Flow Algorithms |
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[40] | 389 | @ingroup algs |
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| 390 | |
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[50] | 391 | \brief Algorithms for finding minimum cost flows and circulations. |
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[40] | 392 | |
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[656] | 393 | This group contains the algorithms for finding minimum cost flows and |
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[802] | 394 | circulations \ref amo93networkflows. For more information about this |
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| 395 | problem and its dual solution, see \ref min_cost_flow |
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| 396 | "Minimum Cost Flow Problem". |
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[422] | 397 | |
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[710] | 398 | LEMON contains several algorithms for this problem. |
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[656] | 399 | - \ref NetworkSimplex Primal Network Simplex algorithm with various |
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[802] | 400 | pivot strategies \ref dantzig63linearprog, \ref kellyoneill91netsimplex. |
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[879] | 401 | - \ref CostScaling Cost Scaling algorithm based on push/augment and |
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| 402 | relabel operations \ref goldberg90approximation, \ref goldberg97efficient, |
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[802] | 403 | \ref bunnagel98efficient. |
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[879] | 404 | - \ref CapacityScaling Capacity Scaling algorithm based on the successive |
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| 405 | shortest path method \ref edmondskarp72theoretical. |
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| 406 | - \ref CycleCanceling Cycle-Canceling algorithms, two of which are |
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| 407 | strongly polynomial \ref klein67primal, \ref goldberg89cyclecanceling. |
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[656] | 408 | |
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[1023] | 409 | In general, \ref NetworkSimplex and \ref CostScaling are the most efficient |
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| 410 | implementations, but the other two algorithms could be faster in special cases. |
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[656] | 411 | For example, if the total supply and/or capacities are rather small, |
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[1023] | 412 | \ref CapacityScaling is usually the fastest algorithm (without effective scaling). |
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[40] | 413 | */ |
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| 414 | |
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| 415 | /** |
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[314] | 416 | @defgroup min_cut Minimum Cut Algorithms |
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[209] | 417 | @ingroup algs |
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[40] | 418 | |
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[50] | 419 | \brief Algorithms for finding minimum cut in graphs. |
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[40] | 420 | |
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[606] | 421 | This group contains the algorithms for finding minimum cut in graphs. |
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[40] | 422 | |
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[422] | 423 | The \e minimum \e cut \e problem is to find a non-empty and non-complete |
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| 424 | \f$X\f$ subset of the nodes with minimum overall capacity on |
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| 425 | outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a |
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| 426 | \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum |
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[50] | 427 | cut is the \f$X\f$ solution of the next optimization problem: |
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[40] | 428 | |
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[210] | 429 | \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}} |
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[760] | 430 | \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f] |
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[40] | 431 | |
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[50] | 432 | LEMON contains several algorithms related to minimum cut problems: |
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[40] | 433 | |
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[422] | 434 | - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut |
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| 435 | in directed graphs. |
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| 436 | - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for |
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| 437 | calculating minimum cut in undirected graphs. |
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[606] | 438 | - \ref GomoryHu "Gomory-Hu tree computation" for calculating |
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[422] | 439 | all-pairs minimum cut in undirected graphs. |
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[40] | 440 | |
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| 441 | If you want to find minimum cut just between two distinict nodes, |
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[422] | 442 | see the \ref max_flow "maximum flow problem". |
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[40] | 443 | */ |
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| 444 | |
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| 445 | /** |
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[815] | 446 | @defgroup min_mean_cycle Minimum Mean Cycle Algorithms |
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[40] | 447 | @ingroup algs |
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[815] | 448 | \brief Algorithms for finding minimum mean cycles. |
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[40] | 449 | |
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[818] | 450 | This group contains the algorithms for finding minimum mean cycles |
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| 451 | \ref clrs01algorithms, \ref amo93networkflows. |
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[40] | 452 | |
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[815] | 453 | The \e minimum \e mean \e cycle \e problem is to find a directed cycle |
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| 454 | of minimum mean length (cost) in a digraph. |
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| 455 | The mean length of a cycle is the average length of its arcs, i.e. the |
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| 456 | ratio between the total length of the cycle and the number of arcs on it. |
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[40] | 457 | |
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[815] | 458 | This problem has an important connection to \e conservative \e length |
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| 459 | \e functions, too. A length function on the arcs of a digraph is called |
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| 460 | conservative if and only if there is no directed cycle of negative total |
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| 461 | length. For an arbitrary length function, the negative of the minimum |
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| 462 | cycle mean is the smallest \f$\epsilon\f$ value so that increasing the |
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| 463 | arc lengths uniformly by \f$\epsilon\f$ results in a conservative length |
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| 464 | function. |
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[40] | 465 | |
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[815] | 466 | LEMON contains three algorithms for solving the minimum mean cycle problem: |
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[959] | 467 | - \ref KarpMmc Karp's original algorithm \ref amo93networkflows, |
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[818] | 468 | \ref dasdan98minmeancycle. |
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[959] | 469 | - \ref HartmannOrlinMmc Hartmann-Orlin's algorithm, which is an improved |
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[818] | 470 | version of Karp's algorithm \ref dasdan98minmeancycle. |
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[959] | 471 | - \ref HowardMmc Howard's policy iteration algorithm |
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[818] | 472 | \ref dasdan98minmeancycle. |
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[40] | 473 | |
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[1023] | 474 | In practice, the \ref HowardMmc "Howard" algorithm turned out to be by far the |
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[959] | 475 | most efficient one, though the best known theoretical bound on its running |
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| 476 | time is exponential. |
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| 477 | Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms |
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| 478 | run in time O(ne) and use space O(n<sup>2</sup>+e), but the latter one is |
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| 479 | typically faster due to the applied early termination scheme. |
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[40] | 480 | */ |
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| 481 | |
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| 482 | /** |
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[314] | 483 | @defgroup matching Matching Algorithms |
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[40] | 484 | @ingroup algs |
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[50] | 485 | \brief Algorithms for finding matchings in graphs and bipartite graphs. |
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[40] | 486 | |
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[637] | 487 | This group contains the algorithms for calculating |
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[40] | 488 | matchings in graphs and bipartite graphs. The general matching problem is |
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[637] | 489 | finding a subset of the edges for which each node has at most one incident |
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| 490 | edge. |
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[209] | 491 | |
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[40] | 492 | There are several different algorithms for calculate matchings in |
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| 493 | graphs. The matching problems in bipartite graphs are generally |
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| 494 | easier than in general graphs. The goal of the matching optimization |
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[422] | 495 | can be finding maximum cardinality, maximum weight or minimum cost |
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[40] | 496 | matching. The search can be constrained to find perfect or |
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| 497 | maximum cardinality matching. |
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| 498 | |
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[422] | 499 | The matching algorithms implemented in LEMON: |
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| 500 | - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm |
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| 501 | for calculating maximum cardinality matching in bipartite graphs. |
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| 502 | - \ref PrBipartiteMatching Push-relabel algorithm |
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| 503 | for calculating maximum cardinality matching in bipartite graphs. |
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| 504 | - \ref MaxWeightedBipartiteMatching |
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| 505 | Successive shortest path algorithm for calculating maximum weighted |
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| 506 | matching and maximum weighted bipartite matching in bipartite graphs. |
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| 507 | - \ref MinCostMaxBipartiteMatching |
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| 508 | Successive shortest path algorithm for calculating minimum cost maximum |
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| 509 | matching in bipartite graphs. |
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| 510 | - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating |
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| 511 | maximum cardinality matching in general graphs. |
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| 512 | - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating |
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| 513 | maximum weighted matching in general graphs. |
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| 514 | - \ref MaxWeightedPerfectMatching |
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| 515 | Edmond's blossom shrinking algorithm for calculating maximum weighted |
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| 516 | perfect matching in general graphs. |
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[948] | 517 | - \ref MaxFractionalMatching Push-relabel algorithm for calculating |
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| 518 | maximum cardinality fractional matching in general graphs. |
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| 519 | - \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating |
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| 520 | maximum weighted fractional matching in general graphs. |
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| 521 | - \ref MaxWeightedPerfectFractionalMatching |
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| 522 | Augmenting path algorithm for calculating maximum weighted |
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| 523 | perfect fractional matching in general graphs. |
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[40] | 524 | |
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[943] | 525 | \image html matching.png |
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[952] | 526 | \image latex matching.eps "Min Cost Perfect Matching" width=\textwidth |
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[40] | 527 | */ |
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| 528 | |
---|
| 529 | /** |
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[761] | 530 | @defgroup graph_properties Connectivity and Other Graph Properties |
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[40] | 531 | @ingroup algs |
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[761] | 532 | \brief Algorithms for discovering the graph properties |
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[40] | 533 | |
---|
[761] | 534 | This group contains the algorithms for discovering the graph properties |
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| 535 | like connectivity, bipartiteness, euler property, simplicity etc. |
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| 536 | |
---|
| 537 | \image html connected_components.png |
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| 538 | \image latex connected_components.eps "Connected components" width=\textwidth |
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| 539 | */ |
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| 540 | |
---|
| 541 | /** |
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[1023] | 542 | @defgroup planar Planar Embedding and Drawing |
---|
[761] | 543 | @ingroup algs |
---|
| 544 | \brief Algorithms for planarity checking, embedding and drawing |
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| 545 | |
---|
| 546 | This group contains the algorithms for planarity checking, |
---|
| 547 | embedding and drawing. |
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| 548 | |
---|
| 549 | \image html planar.png |
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| 550 | \image latex planar.eps "Plane graph" width=\textwidth |
---|
| 551 | */ |
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| 552 | |
---|
| 553 | /** |
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[999] | 554 | @defgroup approx_algs Approximation Algorithms |
---|
[761] | 555 | @ingroup algs |
---|
| 556 | \brief Approximation algorithms. |
---|
| 557 | |
---|
| 558 | This group contains the approximation and heuristic algorithms |
---|
| 559 | implemented in LEMON. |
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[999] | 560 | |
---|
| 561 | <b>Maximum Clique Problem</b> |
---|
| 562 | - \ref GrossoLocatelliPullanMc An efficient heuristic algorithm of |
---|
| 563 | Grosso, Locatelli, and Pullan. |
---|
[40] | 564 | */ |
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| 565 | |
---|
| 566 | /** |
---|
[314] | 567 | @defgroup auxalg Auxiliary Algorithms |
---|
[40] | 568 | @ingroup algs |
---|
[50] | 569 | \brief Auxiliary algorithms implemented in LEMON. |
---|
[40] | 570 | |
---|
[606] | 571 | This group contains some algorithms implemented in LEMON |
---|
[50] | 572 | in order to make it easier to implement complex algorithms. |
---|
[40] | 573 | */ |
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| 574 | |
---|
| 575 | /** |
---|
| 576 | @defgroup gen_opt_group General Optimization Tools |
---|
[606] | 577 | \brief This group contains some general optimization frameworks |
---|
[40] | 578 | implemented in LEMON. |
---|
| 579 | |
---|
[606] | 580 | This group contains some general optimization frameworks |
---|
[40] | 581 | implemented in LEMON. |
---|
| 582 | */ |
---|
| 583 | |
---|
| 584 | /** |
---|
[802] | 585 | @defgroup lp_group LP and MIP Solvers |
---|
[40] | 586 | @ingroup gen_opt_group |
---|
[802] | 587 | \brief LP and MIP solver interfaces for LEMON. |
---|
[40] | 588 | |
---|
[802] | 589 | This group contains LP and MIP solver interfaces for LEMON. |
---|
| 590 | Various LP solvers could be used in the same manner with this |
---|
| 591 | high-level interface. |
---|
| 592 | |
---|
| 593 | The currently supported solvers are \ref glpk, \ref clp, \ref cbc, |
---|
| 594 | \ref cplex, \ref soplex. |
---|
[40] | 595 | */ |
---|
| 596 | |
---|
[209] | 597 | /** |
---|
[314] | 598 | @defgroup lp_utils Tools for Lp and Mip Solvers |
---|
[40] | 599 | @ingroup lp_group |
---|
[50] | 600 | \brief Helper tools to the Lp and Mip solvers. |
---|
[40] | 601 | |
---|
| 602 | This group adds some helper tools to general optimization framework |
---|
| 603 | implemented in LEMON. |
---|
| 604 | */ |
---|
| 605 | |
---|
| 606 | /** |
---|
| 607 | @defgroup metah Metaheuristics |
---|
| 608 | @ingroup gen_opt_group |
---|
| 609 | \brief Metaheuristics for LEMON library. |
---|
| 610 | |
---|
[606] | 611 | This group contains some metaheuristic optimization tools. |
---|
[40] | 612 | */ |
---|
| 613 | |
---|
| 614 | /** |
---|
[209] | 615 | @defgroup utils Tools and Utilities |
---|
[50] | 616 | \brief Tools and utilities for programming in LEMON |
---|
[40] | 617 | |
---|
[50] | 618 | Tools and utilities for programming in LEMON. |
---|
[40] | 619 | */ |
---|
| 620 | |
---|
| 621 | /** |
---|
| 622 | @defgroup gutils Basic Graph Utilities |
---|
| 623 | @ingroup utils |
---|
[50] | 624 | \brief Simple basic graph utilities. |
---|
[40] | 625 | |
---|
[606] | 626 | This group contains some simple basic graph utilities. |
---|
[40] | 627 | */ |
---|
| 628 | |
---|
| 629 | /** |
---|
| 630 | @defgroup misc Miscellaneous Tools |
---|
| 631 | @ingroup utils |
---|
[50] | 632 | \brief Tools for development, debugging and testing. |
---|
| 633 | |
---|
[606] | 634 | This group contains several useful tools for development, |
---|
[40] | 635 | debugging and testing. |
---|
| 636 | */ |
---|
| 637 | |
---|
| 638 | /** |
---|
[314] | 639 | @defgroup timecount Time Measuring and Counting |
---|
[40] | 640 | @ingroup misc |
---|
[50] | 641 | \brief Simple tools for measuring the performance of algorithms. |
---|
| 642 | |
---|
[606] | 643 | This group contains simple tools for measuring the performance |
---|
[40] | 644 | of algorithms. |
---|
| 645 | */ |
---|
| 646 | |
---|
| 647 | /** |
---|
| 648 | @defgroup exceptions Exceptions |
---|
| 649 | @ingroup utils |
---|
[50] | 650 | \brief Exceptions defined in LEMON. |
---|
| 651 | |
---|
[606] | 652 | This group contains the exceptions defined in LEMON. |
---|
[40] | 653 | */ |
---|
| 654 | |
---|
| 655 | /** |
---|
| 656 | @defgroup io_group Input-Output |
---|
[50] | 657 | \brief Graph Input-Output methods |
---|
[40] | 658 | |
---|
[606] | 659 | This group contains the tools for importing and exporting graphs |
---|
[314] | 660 | and graph related data. Now it supports the \ref lgf-format |
---|
| 661 | "LEMON Graph Format", the \c DIMACS format and the encapsulated |
---|
| 662 | postscript (EPS) format. |
---|
[40] | 663 | */ |
---|
| 664 | |
---|
| 665 | /** |
---|
[363] | 666 | @defgroup lemon_io LEMON Graph Format |
---|
[40] | 667 | @ingroup io_group |
---|
[314] | 668 | \brief Reading and writing LEMON Graph Format. |
---|
[40] | 669 | |
---|
[606] | 670 | This group contains methods for reading and writing |
---|
[236] | 671 | \ref lgf-format "LEMON Graph Format". |
---|
[40] | 672 | */ |
---|
| 673 | |
---|
| 674 | /** |
---|
[314] | 675 | @defgroup eps_io Postscript Exporting |
---|
[40] | 676 | @ingroup io_group |
---|
| 677 | \brief General \c EPS drawer and graph exporter |
---|
| 678 | |
---|
[606] | 679 | This group contains general \c EPS drawing methods and special |
---|
[209] | 680 | graph exporting tools. |
---|
[40] | 681 | */ |
---|
| 682 | |
---|
| 683 | /** |
---|
[761] | 684 | @defgroup dimacs_group DIMACS Format |
---|
[403] | 685 | @ingroup io_group |
---|
| 686 | \brief Read and write files in DIMACS format |
---|
| 687 | |
---|
| 688 | Tools to read a digraph from or write it to a file in DIMACS format data. |
---|
| 689 | */ |
---|
| 690 | |
---|
| 691 | /** |
---|
[363] | 692 | @defgroup nauty_group NAUTY Format |
---|
| 693 | @ingroup io_group |
---|
| 694 | \brief Read \e Nauty format |
---|
[403] | 695 | |
---|
[363] | 696 | Tool to read graphs from \e Nauty format data. |
---|
| 697 | */ |
---|
| 698 | |
---|
| 699 | /** |
---|
[40] | 700 | @defgroup concept Concepts |
---|
| 701 | \brief Skeleton classes and concept checking classes |
---|
| 702 | |
---|
[606] | 703 | This group contains the data/algorithm skeletons and concept checking |
---|
[40] | 704 | classes implemented in LEMON. |
---|
| 705 | |
---|
| 706 | The purpose of the classes in this group is fourfold. |
---|
[209] | 707 | |
---|
[318] | 708 | - These classes contain the documentations of the %concepts. In order |
---|
[40] | 709 | to avoid document multiplications, an implementation of a concept |
---|
| 710 | simply refers to the corresponding concept class. |
---|
| 711 | |
---|
| 712 | - These classes declare every functions, <tt>typedef</tt>s etc. an |
---|
[318] | 713 | implementation of the %concepts should provide, however completely |
---|
[40] | 714 | without implementations and real data structures behind the |
---|
| 715 | interface. On the other hand they should provide nothing else. All |
---|
| 716 | the algorithms working on a data structure meeting a certain concept |
---|
| 717 | should compile with these classes. (Though it will not run properly, |
---|
| 718 | of course.) In this way it is easily to check if an algorithm |
---|
| 719 | doesn't use any extra feature of a certain implementation. |
---|
| 720 | |
---|
| 721 | - The concept descriptor classes also provide a <em>checker class</em> |
---|
[50] | 722 | that makes it possible to check whether a certain implementation of a |
---|
[40] | 723 | concept indeed provides all the required features. |
---|
| 724 | |
---|
| 725 | - Finally, They can serve as a skeleton of a new implementation of a concept. |
---|
| 726 | */ |
---|
| 727 | |
---|
| 728 | /** |
---|
| 729 | @defgroup graph_concepts Graph Structure Concepts |
---|
| 730 | @ingroup concept |
---|
| 731 | \brief Skeleton and concept checking classes for graph structures |
---|
| 732 | |
---|
[782] | 733 | This group contains the skeletons and concept checking classes of |
---|
| 734 | graph structures. |
---|
[40] | 735 | */ |
---|
| 736 | |
---|
[314] | 737 | /** |
---|
| 738 | @defgroup map_concepts Map Concepts |
---|
| 739 | @ingroup concept |
---|
| 740 | \brief Skeleton and concept checking classes for maps |
---|
| 741 | |
---|
[606] | 742 | This group contains the skeletons and concept checking classes of maps. |
---|
[40] | 743 | */ |
---|
| 744 | |
---|
| 745 | /** |
---|
[761] | 746 | @defgroup tools Standalone Utility Applications |
---|
| 747 | |
---|
| 748 | Some utility applications are listed here. |
---|
| 749 | |
---|
| 750 | The standard compilation procedure (<tt>./configure;make</tt>) will compile |
---|
| 751 | them, as well. |
---|
| 752 | */ |
---|
| 753 | |
---|
| 754 | /** |
---|
[40] | 755 | \anchor demoprograms |
---|
| 756 | |
---|
[422] | 757 | @defgroup demos Demo Programs |
---|
[40] | 758 | |
---|
| 759 | Some demo programs are listed here. Their full source codes can be found in |
---|
| 760 | the \c demo subdirectory of the source tree. |
---|
| 761 | |
---|
[611] | 762 | In order to compile them, use the <tt>make demo</tt> or the |
---|
| 763 | <tt>make check</tt> commands. |
---|
[40] | 764 | */ |
---|
| 765 | |
---|
[422] | 766 | } |
---|