[209] | 1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
---|
[40] | 2 | * |
---|
[209] | 3 | * This file is a part of LEMON, a generic C++ optimization library. |
---|
[40] | 4 | * |
---|
[877] | 5 | * Copyright (C) 2003-2010 |
---|
[40] | 6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
---|
| 7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
---|
| 8 | * |
---|
| 9 | * Permission to use, modify and distribute this software is granted |
---|
| 10 | * provided that this copyright notice appears in all copies. For |
---|
| 11 | * precise terms see the accompanying LICENSE file. |
---|
| 12 | * |
---|
| 13 | * This software is provided "AS IS" with no warranty of any kind, |
---|
| 14 | * express or implied, and with no claim as to its suitability for any |
---|
| 15 | * purpose. |
---|
| 16 | * |
---|
| 17 | */ |
---|
| 18 | |
---|
[406] | 19 | namespace lemon { |
---|
| 20 | |
---|
[40] | 21 | /** |
---|
| 22 | @defgroup datas Data Structures |
---|
[559] | 23 | This group contains the several data structures implemented in LEMON. |
---|
[40] | 24 | */ |
---|
| 25 | |
---|
| 26 | /** |
---|
| 27 | @defgroup graphs Graph Structures |
---|
| 28 | @ingroup datas |
---|
| 29 | \brief Graph structures implemented in LEMON. |
---|
| 30 | |
---|
[209] | 31 | The implementation of combinatorial algorithms heavily relies on |
---|
| 32 | efficient graph implementations. LEMON offers data structures which are |
---|
| 33 | planned to be easily used in an experimental phase of implementation studies, |
---|
| 34 | and thereafter the program code can be made efficient by small modifications. |
---|
[40] | 35 | |
---|
| 36 | The most efficient implementation of diverse applications require the |
---|
| 37 | usage of different physical graph implementations. These differences |
---|
| 38 | appear in the size of graph we require to handle, memory or time usage |
---|
| 39 | limitations or in the set of operations through which the graph can be |
---|
| 40 | accessed. LEMON provides several physical graph structures to meet |
---|
| 41 | the diverging requirements of the possible users. In order to save on |
---|
| 42 | running time or on memory usage, some structures may fail to provide |
---|
[83] | 43 | some graph features like arc/edge or node deletion. |
---|
[40] | 44 | |
---|
[209] | 45 | Alteration of standard containers need a very limited number of |
---|
| 46 | operations, these together satisfy the everyday requirements. |
---|
| 47 | In the case of graph structures, different operations are needed which do |
---|
| 48 | not alter the physical graph, but gives another view. If some nodes or |
---|
[83] | 49 | arcs have to be hidden or the reverse oriented graph have to be used, then |
---|
[209] | 50 | this is the case. It also may happen that in a flow implementation |
---|
| 51 | the residual graph can be accessed by another algorithm, or a node-set |
---|
| 52 | is to be shrunk for another algorithm. |
---|
| 53 | LEMON also provides a variety of graphs for these requirements called |
---|
| 54 | \ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only |
---|
| 55 | in conjunction with other graph representations. |
---|
[40] | 56 | |
---|
| 57 | You are free to use the graph structure that fit your requirements |
---|
| 58 | the best, most graph algorithms and auxiliary data structures can be used |
---|
[314] | 59 | with any graph structure. |
---|
| 60 | |
---|
| 61 | <b>See also:</b> \ref graph_concepts "Graph Structure Concepts". |
---|
[40] | 62 | */ |
---|
| 63 | |
---|
| 64 | /** |
---|
[451] | 65 | @defgroup graph_adaptors Adaptor Classes for Graphs |
---|
[416] | 66 | @ingroup graphs |
---|
[451] | 67 | \brief Adaptor classes for digraphs and graphs |
---|
| 68 | |
---|
| 69 | This group contains several useful adaptor classes for digraphs and graphs. |
---|
[416] | 70 | |
---|
| 71 | The main parts of LEMON are the different graph structures, generic |
---|
[451] | 72 | graph algorithms, graph concepts, which couple them, and graph |
---|
[416] | 73 | adaptors. While the previous notions are more or less clear, the |
---|
| 74 | latter one needs further explanation. Graph adaptors are graph classes |
---|
| 75 | which serve for considering graph structures in different ways. |
---|
| 76 | |
---|
| 77 | A short example makes this much clearer. Suppose that we have an |
---|
[451] | 78 | instance \c g of a directed graph type, say ListDigraph and an algorithm |
---|
[416] | 79 | \code |
---|
| 80 | template <typename Digraph> |
---|
| 81 | int algorithm(const Digraph&); |
---|
| 82 | \endcode |
---|
| 83 | is needed to run on the reverse oriented graph. It may be expensive |
---|
| 84 | (in time or in memory usage) to copy \c g with the reversed |
---|
| 85 | arcs. In this case, an adaptor class is used, which (according |
---|
[451] | 86 | to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph. |
---|
| 87 | The adaptor uses the original digraph structure and digraph operations when |
---|
| 88 | methods of the reversed oriented graph are called. This means that the adaptor |
---|
| 89 | have minor memory usage, and do not perform sophisticated algorithmic |
---|
[416] | 90 | actions. The purpose of it is to give a tool for the cases when a |
---|
| 91 | graph have to be used in a specific alteration. If this alteration is |
---|
[451] | 92 | obtained by a usual construction like filtering the node or the arc set or |
---|
[416] | 93 | considering a new orientation, then an adaptor is worthwhile to use. |
---|
| 94 | To come back to the reverse oriented graph, in this situation |
---|
| 95 | \code |
---|
| 96 | template<typename Digraph> class ReverseDigraph; |
---|
| 97 | \endcode |
---|
| 98 | template class can be used. The code looks as follows |
---|
| 99 | \code |
---|
| 100 | ListDigraph g; |
---|
[451] | 101 | ReverseDigraph<ListDigraph> rg(g); |
---|
[416] | 102 | int result = algorithm(rg); |
---|
| 103 | \endcode |
---|
[451] | 104 | During running the algorithm, the original digraph \c g is untouched. |
---|
| 105 | This techniques give rise to an elegant code, and based on stable |
---|
[416] | 106 | graph adaptors, complex algorithms can be implemented easily. |
---|
| 107 | |
---|
[451] | 108 | In flow, circulation and matching problems, the residual |
---|
[416] | 109 | graph is of particular importance. Combining an adaptor implementing |
---|
[451] | 110 | this with shortest path algorithms or minimum mean cycle algorithms, |
---|
[416] | 111 | a range of weighted and cardinality optimization algorithms can be |
---|
| 112 | obtained. For other examples, the interested user is referred to the |
---|
| 113 | detailed documentation of particular adaptors. |
---|
| 114 | |
---|
| 115 | The behavior of graph adaptors can be very different. Some of them keep |
---|
| 116 | capabilities of the original graph while in other cases this would be |
---|
[451] | 117 | meaningless. This means that the concepts that they meet depend |
---|
| 118 | on the graph adaptor, and the wrapped graph. |
---|
| 119 | For example, if an arc of a reversed digraph is deleted, this is carried |
---|
| 120 | out by deleting the corresponding arc of the original digraph, thus the |
---|
| 121 | adaptor modifies the original digraph. |
---|
| 122 | However in case of a residual digraph, this operation has no sense. |
---|
[416] | 123 | |
---|
| 124 | Let us stand one more example here to simplify your work. |
---|
[451] | 125 | ReverseDigraph has constructor |
---|
[416] | 126 | \code |
---|
| 127 | ReverseDigraph(Digraph& digraph); |
---|
| 128 | \endcode |
---|
[451] | 129 | This means that in a situation, when a <tt>const %ListDigraph&</tt> |
---|
[416] | 130 | reference to a graph is given, then it have to be instantiated with |
---|
[451] | 131 | <tt>Digraph=const %ListDigraph</tt>. |
---|
[416] | 132 | \code |
---|
| 133 | int algorithm1(const ListDigraph& g) { |
---|
[451] | 134 | ReverseDigraph<const ListDigraph> rg(g); |
---|
[416] | 135 | return algorithm2(rg); |
---|
| 136 | } |
---|
| 137 | \endcode |
---|
| 138 | */ |
---|
| 139 | |
---|
| 140 | /** |
---|
[209] | 141 | @defgroup maps Maps |
---|
[40] | 142 | @ingroup datas |
---|
[50] | 143 | \brief Map structures implemented in LEMON. |
---|
[40] | 144 | |
---|
[559] | 145 | This group contains the map structures implemented in LEMON. |
---|
[50] | 146 | |
---|
[314] | 147 | LEMON provides several special purpose maps and map adaptors that e.g. combine |
---|
[40] | 148 | new maps from existing ones. |
---|
[314] | 149 | |
---|
| 150 | <b>See also:</b> \ref map_concepts "Map Concepts". |
---|
[40] | 151 | */ |
---|
| 152 | |
---|
| 153 | /** |
---|
[209] | 154 | @defgroup graph_maps Graph Maps |
---|
[40] | 155 | @ingroup maps |
---|
[83] | 156 | \brief Special graph-related maps. |
---|
[40] | 157 | |
---|
[559] | 158 | This group contains maps that are specifically designed to assign |
---|
[406] | 159 | values to the nodes and arcs/edges of graphs. |
---|
| 160 | |
---|
| 161 | If you are looking for the standard graph maps (\c NodeMap, \c ArcMap, |
---|
| 162 | \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts". |
---|
[40] | 163 | */ |
---|
| 164 | |
---|
| 165 | /** |
---|
| 166 | \defgroup map_adaptors Map Adaptors |
---|
| 167 | \ingroup maps |
---|
| 168 | \brief Tools to create new maps from existing ones |
---|
| 169 | |
---|
[559] | 170 | This group contains map adaptors that are used to create "implicit" |
---|
[50] | 171 | maps from other maps. |
---|
[40] | 172 | |
---|
[406] | 173 | Most of them are \ref concepts::ReadMap "read-only maps". |
---|
[83] | 174 | They can make arithmetic and logical operations between one or two maps |
---|
| 175 | (negation, shifting, addition, multiplication, logical 'and', 'or', |
---|
| 176 | 'not' etc.) or e.g. convert a map to another one of different Value type. |
---|
[40] | 177 | |
---|
[50] | 178 | The typical usage of this classes is passing implicit maps to |
---|
[40] | 179 | algorithms. If a function type algorithm is called then the function |
---|
| 180 | type map adaptors can be used comfortable. For example let's see the |
---|
[314] | 181 | usage of map adaptors with the \c graphToEps() function. |
---|
[40] | 182 | \code |
---|
| 183 | Color nodeColor(int deg) { |
---|
| 184 | if (deg >= 2) { |
---|
| 185 | return Color(0.5, 0.0, 0.5); |
---|
| 186 | } else if (deg == 1) { |
---|
| 187 | return Color(1.0, 0.5, 1.0); |
---|
| 188 | } else { |
---|
| 189 | return Color(0.0, 0.0, 0.0); |
---|
| 190 | } |
---|
| 191 | } |
---|
[209] | 192 | |
---|
[83] | 193 | Digraph::NodeMap<int> degree_map(graph); |
---|
[209] | 194 | |
---|
[314] | 195 | graphToEps(graph, "graph.eps") |
---|
[40] | 196 | .coords(coords).scaleToA4().undirected() |
---|
[83] | 197 | .nodeColors(composeMap(functorToMap(nodeColor), degree_map)) |
---|
[40] | 198 | .run(); |
---|
[209] | 199 | \endcode |
---|
[83] | 200 | The \c functorToMap() function makes an \c int to \c Color map from the |
---|
[314] | 201 | \c nodeColor() function. The \c composeMap() compose the \c degree_map |
---|
[83] | 202 | and the previously created map. The composed map is a proper function to |
---|
| 203 | get the color of each node. |
---|
[40] | 204 | |
---|
| 205 | The usage with class type algorithms is little bit harder. In this |
---|
| 206 | case the function type map adaptors can not be used, because the |
---|
[50] | 207 | function map adaptors give back temporary objects. |
---|
[40] | 208 | \code |
---|
[83] | 209 | Digraph graph; |
---|
| 210 | |
---|
| 211 | typedef Digraph::ArcMap<double> DoubleArcMap; |
---|
| 212 | DoubleArcMap length(graph); |
---|
| 213 | DoubleArcMap speed(graph); |
---|
| 214 | |
---|
| 215 | typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap; |
---|
[40] | 216 | TimeMap time(length, speed); |
---|
[209] | 217 | |
---|
[83] | 218 | Dijkstra<Digraph, TimeMap> dijkstra(graph, time); |
---|
[40] | 219 | dijkstra.run(source, target); |
---|
| 220 | \endcode |
---|
[83] | 221 | We have a length map and a maximum speed map on the arcs of a digraph. |
---|
| 222 | The minimum time to pass the arc can be calculated as the division of |
---|
| 223 | the two maps which can be done implicitly with the \c DivMap template |
---|
[40] | 224 | class. We use the implicit minimum time map as the length map of the |
---|
| 225 | \c Dijkstra algorithm. |
---|
| 226 | */ |
---|
| 227 | |
---|
| 228 | /** |
---|
| 229 | @defgroup paths Path Structures |
---|
| 230 | @ingroup datas |
---|
[318] | 231 | \brief %Path structures implemented in LEMON. |
---|
[40] | 232 | |
---|
[559] | 233 | This group contains the path structures implemented in LEMON. |
---|
[40] | 234 | |
---|
[50] | 235 | LEMON provides flexible data structures to work with paths. |
---|
| 236 | All of them have similar interfaces and they can be copied easily with |
---|
| 237 | assignment operators and copy constructors. This makes it easy and |
---|
[40] | 238 | efficient to have e.g. the Dijkstra algorithm to store its result in |
---|
| 239 | any kind of path structure. |
---|
| 240 | |
---|
[710] | 241 | \sa \ref concepts::Path "Path concept" |
---|
| 242 | */ |
---|
| 243 | |
---|
| 244 | /** |
---|
| 245 | @defgroup heaps Heap Structures |
---|
| 246 | @ingroup datas |
---|
| 247 | \brief %Heap structures implemented in LEMON. |
---|
| 248 | |
---|
| 249 | This group contains the heap structures implemented in LEMON. |
---|
| 250 | |
---|
| 251 | LEMON provides several heap classes. They are efficient implementations |
---|
| 252 | of the abstract data type \e priority \e queue. They store items with |
---|
| 253 | specified values called \e priorities in such a way that finding and |
---|
| 254 | removing the item with minimum priority are efficient. |
---|
| 255 | The basic operations are adding and erasing items, changing the priority |
---|
| 256 | of an item, etc. |
---|
| 257 | |
---|
| 258 | Heaps are crucial in several algorithms, such as Dijkstra and Prim. |
---|
| 259 | The heap implementations have the same interface, thus any of them can be |
---|
| 260 | used easily in such algorithms. |
---|
| 261 | |
---|
| 262 | \sa \ref concepts::Heap "Heap concept" |
---|
| 263 | */ |
---|
| 264 | |
---|
| 265 | /** |
---|
[40] | 266 | @defgroup auxdat Auxiliary Data Structures |
---|
| 267 | @ingroup datas |
---|
[50] | 268 | \brief Auxiliary data structures implemented in LEMON. |
---|
[40] | 269 | |
---|
[559] | 270 | This group contains some data structures implemented in LEMON in |
---|
[40] | 271 | order to make it easier to implement combinatorial algorithms. |
---|
| 272 | */ |
---|
| 273 | |
---|
| 274 | /** |
---|
[714] | 275 | @defgroup geomdat Geometric Data Structures |
---|
| 276 | @ingroup auxdat |
---|
| 277 | \brief Geometric data structures implemented in LEMON. |
---|
| 278 | |
---|
| 279 | This group contains geometric data structures implemented in LEMON. |
---|
| 280 | |
---|
| 281 | - \ref lemon::dim2::Point "dim2::Point" implements a two dimensional |
---|
| 282 | vector with the usual operations. |
---|
| 283 | - \ref lemon::dim2::Box "dim2::Box" can be used to determine the |
---|
| 284 | rectangular bounding box of a set of \ref lemon::dim2::Point |
---|
| 285 | "dim2::Point"'s. |
---|
| 286 | */ |
---|
| 287 | |
---|
| 288 | /** |
---|
| 289 | @defgroup matrices Matrices |
---|
| 290 | @ingroup auxdat |
---|
| 291 | \brief Two dimensional data storages implemented in LEMON. |
---|
| 292 | |
---|
| 293 | This group contains two dimensional data storages implemented in LEMON. |
---|
| 294 | */ |
---|
| 295 | |
---|
| 296 | /** |
---|
[40] | 297 | @defgroup algs Algorithms |
---|
[559] | 298 | \brief This group contains the several algorithms |
---|
[40] | 299 | implemented in LEMON. |
---|
| 300 | |
---|
[559] | 301 | This group contains the several algorithms |
---|
[40] | 302 | implemented in LEMON. |
---|
| 303 | */ |
---|
| 304 | |
---|
| 305 | /** |
---|
| 306 | @defgroup search Graph Search |
---|
| 307 | @ingroup algs |
---|
[50] | 308 | \brief Common graph search algorithms. |
---|
[40] | 309 | |
---|
[559] | 310 | This group contains the common graph search algorithms, namely |
---|
[755] | 311 | \e breadth-first \e search (BFS) and \e depth-first \e search (DFS) |
---|
| 312 | \ref clrs01algorithms. |
---|
[40] | 313 | */ |
---|
| 314 | |
---|
| 315 | /** |
---|
[314] | 316 | @defgroup shortest_path Shortest Path Algorithms |
---|
[40] | 317 | @ingroup algs |
---|
[50] | 318 | \brief Algorithms for finding shortest paths. |
---|
[40] | 319 | |
---|
[755] | 320 | This group contains the algorithms for finding shortest paths in digraphs |
---|
| 321 | \ref clrs01algorithms. |
---|
[406] | 322 | |
---|
| 323 | - \ref Dijkstra algorithm for finding shortest paths from a source node |
---|
| 324 | when all arc lengths are non-negative. |
---|
| 325 | - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths |
---|
| 326 | from a source node when arc lenghts can be either positive or negative, |
---|
| 327 | but the digraph should not contain directed cycles with negative total |
---|
| 328 | length. |
---|
| 329 | - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms |
---|
| 330 | for solving the \e all-pairs \e shortest \e paths \e problem when arc |
---|
| 331 | lenghts can be either positive or negative, but the digraph should |
---|
| 332 | not contain directed cycles with negative total length. |
---|
| 333 | - \ref Suurballe A successive shortest path algorithm for finding |
---|
| 334 | arc-disjoint paths between two nodes having minimum total length. |
---|
[40] | 335 | */ |
---|
| 336 | |
---|
[209] | 337 | /** |
---|
[714] | 338 | @defgroup spantree Minimum Spanning Tree Algorithms |
---|
| 339 | @ingroup algs |
---|
| 340 | \brief Algorithms for finding minimum cost spanning trees and arborescences. |
---|
| 341 | |
---|
| 342 | This group contains the algorithms for finding minimum cost spanning |
---|
[755] | 343 | trees and arborescences \ref clrs01algorithms. |
---|
[714] | 344 | */ |
---|
| 345 | |
---|
| 346 | /** |
---|
[314] | 347 | @defgroup max_flow Maximum Flow Algorithms |
---|
[209] | 348 | @ingroup algs |
---|
[50] | 349 | \brief Algorithms for finding maximum flows. |
---|
[40] | 350 | |
---|
[559] | 351 | This group contains the algorithms for finding maximum flows and |
---|
[755] | 352 | feasible circulations \ref clrs01algorithms, \ref amo93networkflows. |
---|
[40] | 353 | |
---|
[406] | 354 | The \e maximum \e flow \e problem is to find a flow of maximum value between |
---|
| 355 | a single source and a single target. Formally, there is a \f$G=(V,A)\f$ |
---|
[609] | 356 | digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and |
---|
[406] | 357 | \f$s, t \in V\f$ source and target nodes. |
---|
[609] | 358 | A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the |
---|
[406] | 359 | following optimization problem. |
---|
[40] | 360 | |
---|
[609] | 361 | \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f] |
---|
| 362 | \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu) |
---|
| 363 | \quad \forall u\in V\setminus\{s,t\} \f] |
---|
| 364 | \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f] |
---|
[40] | 365 | |
---|
[50] | 366 | LEMON contains several algorithms for solving maximum flow problems: |
---|
[755] | 367 | - \ref EdmondsKarp Edmonds-Karp algorithm |
---|
| 368 | \ref edmondskarp72theoretical. |
---|
| 369 | - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm |
---|
| 370 | \ref goldberg88newapproach. |
---|
| 371 | - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees |
---|
| 372 | \ref dinic70algorithm, \ref sleator83dynamic. |
---|
| 373 | - \ref GoldbergTarjan !Preflow push-relabel algorithm with dynamic trees |
---|
| 374 | \ref goldberg88newapproach, \ref sleator83dynamic. |
---|
[40] | 375 | |
---|
[755] | 376 | In most cases the \ref Preflow algorithm provides the |
---|
[406] | 377 | fastest method for computing a maximum flow. All implementations |
---|
[651] | 378 | also provide functions to query the minimum cut, which is the dual |
---|
| 379 | problem of maximum flow. |
---|
| 380 | |
---|
[869] | 381 | \ref Circulation is a preflow push-relabel algorithm implemented directly |
---|
[651] | 382 | for finding feasible circulations, which is a somewhat different problem, |
---|
| 383 | but it is strongly related to maximum flow. |
---|
| 384 | For more information, see \ref Circulation. |
---|
[40] | 385 | */ |
---|
| 386 | |
---|
| 387 | /** |
---|
[663] | 388 | @defgroup min_cost_flow_algs Minimum Cost Flow Algorithms |
---|
[40] | 389 | @ingroup algs |
---|
| 390 | |
---|
[50] | 391 | \brief Algorithms for finding minimum cost flows and circulations. |
---|
[40] | 392 | |
---|
[609] | 393 | This group contains the algorithms for finding minimum cost flows and |
---|
[755] | 394 | circulations \ref amo93networkflows. For more information about this |
---|
[1049] | 395 | problem and its dual solution, see: \ref min_cost_flow |
---|
[755] | 396 | "Minimum Cost Flow Problem". |
---|
[406] | 397 | |
---|
[663] | 398 | LEMON contains several algorithms for this problem. |
---|
[609] | 399 | - \ref NetworkSimplex Primal Network Simplex algorithm with various |
---|
[755] | 400 | pivot strategies \ref dantzig63linearprog, \ref kellyoneill91netsimplex. |
---|
[813] | 401 | - \ref CostScaling Cost Scaling algorithm based on push/augment and |
---|
| 402 | relabel operations \ref goldberg90approximation, \ref goldberg97efficient, |
---|
[755] | 403 | \ref bunnagel98efficient. |
---|
[813] | 404 | - \ref CapacityScaling Capacity Scaling algorithm based on the successive |
---|
| 405 | shortest path method \ref edmondskarp72theoretical. |
---|
| 406 | - \ref CycleCanceling Cycle-Canceling algorithms, two of which are |
---|
| 407 | strongly polynomial \ref klein67primal, \ref goldberg89cyclecanceling. |
---|
[609] | 408 | |
---|
[919] | 409 | In general, \ref NetworkSimplex and \ref CostScaling are the most efficient |
---|
[1003] | 410 | implementations. |
---|
| 411 | \ref NetworkSimplex is usually the fastest on relatively small graphs (up to |
---|
| 412 | several thousands of nodes) and on dense graphs, while \ref CostScaling is |
---|
| 413 | typically more efficient on large graphs (e.g. hundreds of thousands of |
---|
| 414 | nodes or above), especially if they are sparse. |
---|
| 415 | However, other algorithms could be faster in special cases. |
---|
[609] | 416 | For example, if the total supply and/or capacities are rather small, |
---|
[919] | 417 | \ref CapacityScaling is usually the fastest algorithm (without effective scaling). |
---|
[1002] | 418 | |
---|
| 419 | These classes are intended to be used with integer-valued input data |
---|
| 420 | (capacities, supply values, and costs), except for \ref CapacityScaling, |
---|
| 421 | which is capable of handling real-valued arc costs (other numerical |
---|
| 422 | data are required to be integer). |
---|
[40] | 423 | */ |
---|
| 424 | |
---|
| 425 | /** |
---|
[314] | 426 | @defgroup min_cut Minimum Cut Algorithms |
---|
[209] | 427 | @ingroup algs |
---|
[40] | 428 | |
---|
[50] | 429 | \brief Algorithms for finding minimum cut in graphs. |
---|
[40] | 430 | |
---|
[559] | 431 | This group contains the algorithms for finding minimum cut in graphs. |
---|
[40] | 432 | |
---|
[406] | 433 | The \e minimum \e cut \e problem is to find a non-empty and non-complete |
---|
| 434 | \f$X\f$ subset of the nodes with minimum overall capacity on |
---|
| 435 | outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a |
---|
| 436 | \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum |
---|
[50] | 437 | cut is the \f$X\f$ solution of the next optimization problem: |
---|
[40] | 438 | |
---|
[210] | 439 | \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}} |
---|
[713] | 440 | \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f] |
---|
[40] | 441 | |
---|
[50] | 442 | LEMON contains several algorithms related to minimum cut problems: |
---|
[40] | 443 | |
---|
[406] | 444 | - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut |
---|
| 445 | in directed graphs. |
---|
| 446 | - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for |
---|
| 447 | calculating minimum cut in undirected graphs. |
---|
[559] | 448 | - \ref GomoryHu "Gomory-Hu tree computation" for calculating |
---|
[406] | 449 | all-pairs minimum cut in undirected graphs. |
---|
[40] | 450 | |
---|
| 451 | If you want to find minimum cut just between two distinict nodes, |
---|
[406] | 452 | see the \ref max_flow "maximum flow problem". |
---|
[40] | 453 | */ |
---|
| 454 | |
---|
| 455 | /** |
---|
[768] | 456 | @defgroup min_mean_cycle Minimum Mean Cycle Algorithms |
---|
[40] | 457 | @ingroup algs |
---|
[768] | 458 | \brief Algorithms for finding minimum mean cycles. |
---|
[40] | 459 | |
---|
[771] | 460 | This group contains the algorithms for finding minimum mean cycles |
---|
[1002] | 461 | \ref amo93networkflows, \ref karp78characterization. |
---|
[40] | 462 | |
---|
[768] | 463 | The \e minimum \e mean \e cycle \e problem is to find a directed cycle |
---|
| 464 | of minimum mean length (cost) in a digraph. |
---|
| 465 | The mean length of a cycle is the average length of its arcs, i.e. the |
---|
| 466 | ratio between the total length of the cycle and the number of arcs on it. |
---|
[40] | 467 | |
---|
[768] | 468 | This problem has an important connection to \e conservative \e length |
---|
| 469 | \e functions, too. A length function on the arcs of a digraph is called |
---|
| 470 | conservative if and only if there is no directed cycle of negative total |
---|
| 471 | length. For an arbitrary length function, the negative of the minimum |
---|
| 472 | cycle mean is the smallest \f$\epsilon\f$ value so that increasing the |
---|
| 473 | arc lengths uniformly by \f$\epsilon\f$ results in a conservative length |
---|
| 474 | function. |
---|
[40] | 475 | |
---|
[768] | 476 | LEMON contains three algorithms for solving the minimum mean cycle problem: |
---|
[1002] | 477 | - \ref KarpMmc Karp's original algorithm \ref karp78characterization. |
---|
[879] | 478 | - \ref HartmannOrlinMmc Hartmann-Orlin's algorithm, which is an improved |
---|
[1002] | 479 | version of Karp's algorithm \ref hartmann93finding. |
---|
[879] | 480 | - \ref HowardMmc Howard's policy iteration algorithm |
---|
[1002] | 481 | \ref dasdan98minmeancycle, \ref dasdan04experimental. |
---|
[40] | 482 | |
---|
[919] | 483 | In practice, the \ref HowardMmc "Howard" algorithm turned out to be by far the |
---|
[879] | 484 | most efficient one, though the best known theoretical bound on its running |
---|
| 485 | time is exponential. |
---|
| 486 | Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms |
---|
[1049] | 487 | run in time O(ne) and use space O(n<sup>2</sup>+e). |
---|
[40] | 488 | */ |
---|
| 489 | |
---|
| 490 | /** |
---|
[314] | 491 | @defgroup matching Matching Algorithms |
---|
[40] | 492 | @ingroup algs |
---|
[50] | 493 | \brief Algorithms for finding matchings in graphs and bipartite graphs. |
---|
[40] | 494 | |
---|
[590] | 495 | This group contains the algorithms for calculating |
---|
[40] | 496 | matchings in graphs and bipartite graphs. The general matching problem is |
---|
[590] | 497 | finding a subset of the edges for which each node has at most one incident |
---|
| 498 | edge. |
---|
[209] | 499 | |
---|
[40] | 500 | There are several different algorithms for calculate matchings in |
---|
| 501 | graphs. The matching problems in bipartite graphs are generally |
---|
| 502 | easier than in general graphs. The goal of the matching optimization |
---|
[406] | 503 | can be finding maximum cardinality, maximum weight or minimum cost |
---|
[40] | 504 | matching. The search can be constrained to find perfect or |
---|
| 505 | maximum cardinality matching. |
---|
| 506 | |
---|
[406] | 507 | The matching algorithms implemented in LEMON: |
---|
| 508 | - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm |
---|
| 509 | for calculating maximum cardinality matching in bipartite graphs. |
---|
| 510 | - \ref PrBipartiteMatching Push-relabel algorithm |
---|
| 511 | for calculating maximum cardinality matching in bipartite graphs. |
---|
| 512 | - \ref MaxWeightedBipartiteMatching |
---|
| 513 | Successive shortest path algorithm for calculating maximum weighted |
---|
| 514 | matching and maximum weighted bipartite matching in bipartite graphs. |
---|
| 515 | - \ref MinCostMaxBipartiteMatching |
---|
| 516 | Successive shortest path algorithm for calculating minimum cost maximum |
---|
| 517 | matching in bipartite graphs. |
---|
| 518 | - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating |
---|
| 519 | maximum cardinality matching in general graphs. |
---|
| 520 | - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating |
---|
| 521 | maximum weighted matching in general graphs. |
---|
| 522 | - \ref MaxWeightedPerfectMatching |
---|
| 523 | Edmond's blossom shrinking algorithm for calculating maximum weighted |
---|
| 524 | perfect matching in general graphs. |
---|
[869] | 525 | - \ref MaxFractionalMatching Push-relabel algorithm for calculating |
---|
| 526 | maximum cardinality fractional matching in general graphs. |
---|
| 527 | - \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating |
---|
| 528 | maximum weighted fractional matching in general graphs. |
---|
| 529 | - \ref MaxWeightedPerfectFractionalMatching |
---|
| 530 | Augmenting path algorithm for calculating maximum weighted |
---|
| 531 | perfect fractional matching in general graphs. |
---|
[40] | 532 | |
---|
[865] | 533 | \image html matching.png |
---|
[873] | 534 | \image latex matching.eps "Min Cost Perfect Matching" width=\textwidth |
---|
[40] | 535 | */ |
---|
| 536 | |
---|
| 537 | /** |
---|
[714] | 538 | @defgroup graph_properties Connectivity and Other Graph Properties |
---|
[40] | 539 | @ingroup algs |
---|
[714] | 540 | \brief Algorithms for discovering the graph properties |
---|
[40] | 541 | |
---|
[714] | 542 | This group contains the algorithms for discovering the graph properties |
---|
| 543 | like connectivity, bipartiteness, euler property, simplicity etc. |
---|
| 544 | |
---|
| 545 | \image html connected_components.png |
---|
| 546 | \image latex connected_components.eps "Connected components" width=\textwidth |
---|
| 547 | */ |
---|
| 548 | |
---|
| 549 | /** |
---|
[919] | 550 | @defgroup planar Planar Embedding and Drawing |
---|
[714] | 551 | @ingroup algs |
---|
| 552 | \brief Algorithms for planarity checking, embedding and drawing |
---|
| 553 | |
---|
| 554 | This group contains the algorithms for planarity checking, |
---|
| 555 | embedding and drawing. |
---|
| 556 | |
---|
| 557 | \image html planar.png |
---|
| 558 | \image latex planar.eps "Plane graph" width=\textwidth |
---|
| 559 | */ |
---|
[1032] | 560 | |
---|
| 561 | /** |
---|
| 562 | @defgroup tsp Traveling Salesman Problem |
---|
| 563 | @ingroup algs |
---|
| 564 | \brief Algorithms for the symmetric traveling salesman problem |
---|
| 565 | |
---|
| 566 | This group contains basic heuristic algorithms for the the symmetric |
---|
| 567 | \e traveling \e salesman \e problem (TSP). |
---|
| 568 | Given an \ref FullGraph "undirected full graph" with a cost map on its edges, |
---|
| 569 | the problem is to find a shortest possible tour that visits each node exactly |
---|
| 570 | once (i.e. the minimum cost Hamiltonian cycle). |
---|
| 571 | |
---|
[1034] | 572 | These TSP algorithms are intended to be used with a \e metric \e cost |
---|
| 573 | \e function, i.e. the edge costs should satisfy the triangle inequality. |
---|
| 574 | Otherwise the algorithms could yield worse results. |
---|
[1032] | 575 | |
---|
| 576 | LEMON provides five well-known heuristics for solving symmetric TSP: |
---|
| 577 | - \ref NearestNeighborTsp Neareast neighbor algorithm |
---|
| 578 | - \ref GreedyTsp Greedy algorithm |
---|
| 579 | - \ref InsertionTsp Insertion heuristic (with four selection methods) |
---|
| 580 | - \ref ChristofidesTsp Christofides algorithm |
---|
| 581 | - \ref Opt2Tsp 2-opt algorithm |
---|
| 582 | |
---|
[1036] | 583 | \ref NearestNeighborTsp, \ref GreedyTsp, and \ref InsertionTsp are the fastest |
---|
| 584 | solution methods. Furthermore, \ref InsertionTsp is usually quite effective. |
---|
| 585 | |
---|
| 586 | \ref ChristofidesTsp is somewhat slower, but it has the best guaranteed |
---|
| 587 | approximation factor: 3/2. |
---|
| 588 | |
---|
| 589 | \ref Opt2Tsp usually provides the best results in practice, but |
---|
| 590 | it is the slowest method. It can also be used to improve given tours, |
---|
| 591 | for example, the results of other algorithms. |
---|
| 592 | |
---|
[1032] | 593 | \image html tsp.png |
---|
| 594 | \image latex tsp.eps "Traveling salesman problem" width=\textwidth |
---|
| 595 | */ |
---|
[714] | 596 | |
---|
| 597 | /** |
---|
[904] | 598 | @defgroup approx_algs Approximation Algorithms |
---|
[714] | 599 | @ingroup algs |
---|
| 600 | \brief Approximation algorithms. |
---|
| 601 | |
---|
| 602 | This group contains the approximation and heuristic algorithms |
---|
| 603 | implemented in LEMON. |
---|
[904] | 604 | |
---|
| 605 | <b>Maximum Clique Problem</b> |
---|
| 606 | - \ref GrossoLocatelliPullanMc An efficient heuristic algorithm of |
---|
| 607 | Grosso, Locatelli, and Pullan. |
---|
[40] | 608 | */ |
---|
| 609 | |
---|
| 610 | /** |
---|
[314] | 611 | @defgroup auxalg Auxiliary Algorithms |
---|
[40] | 612 | @ingroup algs |
---|
[50] | 613 | \brief Auxiliary algorithms implemented in LEMON. |
---|
[40] | 614 | |
---|
[559] | 615 | This group contains some algorithms implemented in LEMON |
---|
[50] | 616 | in order to make it easier to implement complex algorithms. |
---|
[40] | 617 | */ |
---|
| 618 | |
---|
| 619 | /** |
---|
| 620 | @defgroup gen_opt_group General Optimization Tools |
---|
[559] | 621 | \brief This group contains some general optimization frameworks |
---|
[40] | 622 | implemented in LEMON. |
---|
| 623 | |
---|
[559] | 624 | This group contains some general optimization frameworks |
---|
[40] | 625 | implemented in LEMON. |
---|
| 626 | */ |
---|
| 627 | |
---|
| 628 | /** |
---|
[755] | 629 | @defgroup lp_group LP and MIP Solvers |
---|
[40] | 630 | @ingroup gen_opt_group |
---|
[755] | 631 | \brief LP and MIP solver interfaces for LEMON. |
---|
[40] | 632 | |
---|
[755] | 633 | This group contains LP and MIP solver interfaces for LEMON. |
---|
| 634 | Various LP solvers could be used in the same manner with this |
---|
| 635 | high-level interface. |
---|
| 636 | |
---|
| 637 | The currently supported solvers are \ref glpk, \ref clp, \ref cbc, |
---|
| 638 | \ref cplex, \ref soplex. |
---|
[40] | 639 | */ |
---|
| 640 | |
---|
[209] | 641 | /** |
---|
[314] | 642 | @defgroup lp_utils Tools for Lp and Mip Solvers |
---|
[40] | 643 | @ingroup lp_group |
---|
[50] | 644 | \brief Helper tools to the Lp and Mip solvers. |
---|
[40] | 645 | |
---|
| 646 | This group adds some helper tools to general optimization framework |
---|
| 647 | implemented in LEMON. |
---|
| 648 | */ |
---|
| 649 | |
---|
| 650 | /** |
---|
| 651 | @defgroup metah Metaheuristics |
---|
| 652 | @ingroup gen_opt_group |
---|
| 653 | \brief Metaheuristics for LEMON library. |
---|
| 654 | |
---|
[559] | 655 | This group contains some metaheuristic optimization tools. |
---|
[40] | 656 | */ |
---|
| 657 | |
---|
| 658 | /** |
---|
[209] | 659 | @defgroup utils Tools and Utilities |
---|
[50] | 660 | \brief Tools and utilities for programming in LEMON |
---|
[40] | 661 | |
---|
[50] | 662 | Tools and utilities for programming in LEMON. |
---|
[40] | 663 | */ |
---|
| 664 | |
---|
| 665 | /** |
---|
| 666 | @defgroup gutils Basic Graph Utilities |
---|
| 667 | @ingroup utils |
---|
[50] | 668 | \brief Simple basic graph utilities. |
---|
[40] | 669 | |
---|
[559] | 670 | This group contains some simple basic graph utilities. |
---|
[40] | 671 | */ |
---|
| 672 | |
---|
| 673 | /** |
---|
| 674 | @defgroup misc Miscellaneous Tools |
---|
| 675 | @ingroup utils |
---|
[50] | 676 | \brief Tools for development, debugging and testing. |
---|
| 677 | |
---|
[559] | 678 | This group contains several useful tools for development, |
---|
[40] | 679 | debugging and testing. |
---|
| 680 | */ |
---|
| 681 | |
---|
| 682 | /** |
---|
[314] | 683 | @defgroup timecount Time Measuring and Counting |
---|
[40] | 684 | @ingroup misc |
---|
[50] | 685 | \brief Simple tools for measuring the performance of algorithms. |
---|
| 686 | |
---|
[559] | 687 | This group contains simple tools for measuring the performance |
---|
[40] | 688 | of algorithms. |
---|
| 689 | */ |
---|
| 690 | |
---|
| 691 | /** |
---|
| 692 | @defgroup exceptions Exceptions |
---|
| 693 | @ingroup utils |
---|
[50] | 694 | \brief Exceptions defined in LEMON. |
---|
| 695 | |
---|
[559] | 696 | This group contains the exceptions defined in LEMON. |
---|
[40] | 697 | */ |
---|
| 698 | |
---|
| 699 | /** |
---|
| 700 | @defgroup io_group Input-Output |
---|
[50] | 701 | \brief Graph Input-Output methods |
---|
[40] | 702 | |
---|
[559] | 703 | This group contains the tools for importing and exporting graphs |
---|
[314] | 704 | and graph related data. Now it supports the \ref lgf-format |
---|
| 705 | "LEMON Graph Format", the \c DIMACS format and the encapsulated |
---|
| 706 | postscript (EPS) format. |
---|
[40] | 707 | */ |
---|
| 708 | |
---|
| 709 | /** |
---|
[351] | 710 | @defgroup lemon_io LEMON Graph Format |
---|
[40] | 711 | @ingroup io_group |
---|
[314] | 712 | \brief Reading and writing LEMON Graph Format. |
---|
[40] | 713 | |
---|
[559] | 714 | This group contains methods for reading and writing |
---|
[236] | 715 | \ref lgf-format "LEMON Graph Format". |
---|
[40] | 716 | */ |
---|
| 717 | |
---|
| 718 | /** |
---|
[314] | 719 | @defgroup eps_io Postscript Exporting |
---|
[40] | 720 | @ingroup io_group |
---|
| 721 | \brief General \c EPS drawer and graph exporter |
---|
| 722 | |
---|
[559] | 723 | This group contains general \c EPS drawing methods and special |
---|
[209] | 724 | graph exporting tools. |
---|
[40] | 725 | */ |
---|
| 726 | |
---|
| 727 | /** |
---|
[714] | 728 | @defgroup dimacs_group DIMACS Format |
---|
[388] | 729 | @ingroup io_group |
---|
| 730 | \brief Read and write files in DIMACS format |
---|
| 731 | |
---|
| 732 | Tools to read a digraph from or write it to a file in DIMACS format data. |
---|
| 733 | */ |
---|
| 734 | |
---|
| 735 | /** |
---|
[351] | 736 | @defgroup nauty_group NAUTY Format |
---|
| 737 | @ingroup io_group |
---|
| 738 | \brief Read \e Nauty format |
---|
[388] | 739 | |
---|
[351] | 740 | Tool to read graphs from \e Nauty format data. |
---|
| 741 | */ |
---|
| 742 | |
---|
| 743 | /** |
---|
[40] | 744 | @defgroup concept Concepts |
---|
| 745 | \brief Skeleton classes and concept checking classes |
---|
| 746 | |
---|
[559] | 747 | This group contains the data/algorithm skeletons and concept checking |
---|
[40] | 748 | classes implemented in LEMON. |
---|
| 749 | |
---|
| 750 | The purpose of the classes in this group is fourfold. |
---|
[209] | 751 | |
---|
[318] | 752 | - These classes contain the documentations of the %concepts. In order |
---|
[40] | 753 | to avoid document multiplications, an implementation of a concept |
---|
| 754 | simply refers to the corresponding concept class. |
---|
| 755 | |
---|
| 756 | - These classes declare every functions, <tt>typedef</tt>s etc. an |
---|
[318] | 757 | implementation of the %concepts should provide, however completely |
---|
[40] | 758 | without implementations and real data structures behind the |
---|
| 759 | interface. On the other hand they should provide nothing else. All |
---|
| 760 | the algorithms working on a data structure meeting a certain concept |
---|
| 761 | should compile with these classes. (Though it will not run properly, |
---|
| 762 | of course.) In this way it is easily to check if an algorithm |
---|
| 763 | doesn't use any extra feature of a certain implementation. |
---|
| 764 | |
---|
| 765 | - The concept descriptor classes also provide a <em>checker class</em> |
---|
[50] | 766 | that makes it possible to check whether a certain implementation of a |
---|
[40] | 767 | concept indeed provides all the required features. |
---|
| 768 | |
---|
| 769 | - Finally, They can serve as a skeleton of a new implementation of a concept. |
---|
| 770 | */ |
---|
| 771 | |
---|
| 772 | /** |
---|
| 773 | @defgroup graph_concepts Graph Structure Concepts |
---|
| 774 | @ingroup concept |
---|
| 775 | \brief Skeleton and concept checking classes for graph structures |
---|
| 776 | |
---|
[735] | 777 | This group contains the skeletons and concept checking classes of |
---|
| 778 | graph structures. |
---|
[40] | 779 | */ |
---|
| 780 | |
---|
[314] | 781 | /** |
---|
| 782 | @defgroup map_concepts Map Concepts |
---|
| 783 | @ingroup concept |
---|
| 784 | \brief Skeleton and concept checking classes for maps |
---|
| 785 | |
---|
[559] | 786 | This group contains the skeletons and concept checking classes of maps. |
---|
[40] | 787 | */ |
---|
| 788 | |
---|
| 789 | /** |
---|
[714] | 790 | @defgroup tools Standalone Utility Applications |
---|
| 791 | |
---|
| 792 | Some utility applications are listed here. |
---|
| 793 | |
---|
| 794 | The standard compilation procedure (<tt>./configure;make</tt>) will compile |
---|
| 795 | them, as well. |
---|
| 796 | */ |
---|
| 797 | |
---|
| 798 | /** |
---|
[40] | 799 | \anchor demoprograms |
---|
| 800 | |
---|
[406] | 801 | @defgroup demos Demo Programs |
---|
[40] | 802 | |
---|
| 803 | Some demo programs are listed here. Their full source codes can be found in |
---|
| 804 | the \c demo subdirectory of the source tree. |
---|
| 805 | |
---|
[564] | 806 | In order to compile them, use the <tt>make demo</tt> or the |
---|
| 807 | <tt>make check</tt> commands. |
---|
[40] | 808 | */ |
---|
| 809 | |
---|
[406] | 810 | } |
---|