1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
3 * This file is a part of LEMON, a generic C++ optimization library.
5 * Copyright (C) 2003-2009
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
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.
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
22 @defgroup datas Data Structures
23 This group contains the several data structures implemented in LEMON.
27 @defgroup graphs Graph Structures
29 \brief Graph structures implemented in LEMON.
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.
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
43 some graph features like arc/edge or node deletion.
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
49 arcs have to be hidden or the reverse oriented graph have to be used, then
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.
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
59 with any graph structure.
61 <b>See also:</b> \ref graph_concepts "Graph Structure Concepts".
65 @defgroup graph_adaptors Adaptor Classes for Graphs
67 \brief Adaptor classes for digraphs and graphs
69 This group contains several useful adaptor classes for digraphs and graphs.
71 The main parts of LEMON are the different graph structures, generic
72 graph algorithms, graph concepts, which couple them, and graph
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.
77 A short example makes this much clearer. Suppose that we have an
78 instance \c g of a directed graph type, say ListDigraph and an algorithm
80 template <typename Digraph>
81 int algorithm(const Digraph&);
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
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
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
92 obtained by a usual construction like filtering the node or the arc set or
93 considering a new orientation, then an adaptor is worthwhile to use.
94 To come back to the reverse oriented graph, in this situation
96 template<typename Digraph> class ReverseDigraph;
98 template class can be used. The code looks as follows
101 ReverseDigraph<ListDigraph> rg(g);
102 int result = algorithm(rg);
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
106 graph adaptors, complex algorithms can be implemented easily.
108 In flow, circulation and matching problems, the residual
109 graph is of particular importance. Combining an adaptor implementing
110 this with shortest path algorithms or minimum mean cycle algorithms,
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.
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
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.
124 Let us stand one more example here to simplify your work.
125 ReverseDigraph has constructor
127 ReverseDigraph(Digraph& digraph);
129 This means that in a situation, when a <tt>const %ListDigraph&</tt>
130 reference to a graph is given, then it have to be instantiated with
131 <tt>Digraph=const %ListDigraph</tt>.
133 int algorithm1(const ListDigraph& g) {
134 ReverseDigraph<const ListDigraph> rg(g);
135 return algorithm2(rg);
143 \brief Map structures implemented in LEMON.
145 This group contains the map structures implemented in LEMON.
147 LEMON provides several special purpose maps and map adaptors that e.g. combine
148 new maps from existing ones.
150 <b>See also:</b> \ref map_concepts "Map Concepts".
154 @defgroup graph_maps Graph Maps
156 \brief Special graph-related maps.
158 This group contains maps that are specifically designed to assign
159 values to the nodes and arcs/edges of graphs.
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".
166 \defgroup map_adaptors Map Adaptors
168 \brief Tools to create new maps from existing ones
170 This group contains map adaptors that are used to create "implicit"
171 maps from other maps.
173 Most of them are \ref concepts::ReadMap "read-only maps".
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.
178 The typical usage of this classes is passing implicit maps to
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
181 usage of map adaptors with the \c graphToEps() function.
183 Color nodeColor(int deg) {
185 return Color(0.5, 0.0, 0.5);
186 } else if (deg == 1) {
187 return Color(1.0, 0.5, 1.0);
189 return Color(0.0, 0.0, 0.0);
193 Digraph::NodeMap<int> degree_map(graph);
195 graphToEps(graph, "graph.eps")
196 .coords(coords).scaleToA4().undirected()
197 .nodeColors(composeMap(functorToMap(nodeColor), degree_map))
200 The \c functorToMap() function makes an \c int to \c Color map from the
201 \c nodeColor() function. The \c composeMap() compose the \c degree_map
202 and the previously created map. The composed map is a proper function to
203 get the color of each node.
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
207 function map adaptors give back temporary objects.
211 typedef Digraph::ArcMap<double> DoubleArcMap;
212 DoubleArcMap length(graph);
213 DoubleArcMap speed(graph);
215 typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;
216 TimeMap time(length, speed);
218 Dijkstra<Digraph, TimeMap> dijkstra(graph, time);
219 dijkstra.run(source, target);
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
224 class. We use the implicit minimum time map as the length map of the
225 \c Dijkstra algorithm.
229 @defgroup matrices Matrices
231 \brief Two dimensional data storages implemented in LEMON.
233 This group contains two dimensional data storages implemented in LEMON.
237 @defgroup paths Path Structures
239 \brief %Path structures implemented in LEMON.
241 This group contains the path structures implemented in LEMON.
243 LEMON provides flexible data structures to work with paths.
244 All of them have similar interfaces and they can be copied easily with
245 assignment operators and copy constructors. This makes it easy and
246 efficient to have e.g. the Dijkstra algorithm to store its result in
247 any kind of path structure.
249 \sa lemon::concepts::Path
253 @defgroup auxdat Auxiliary Data Structures
255 \brief Auxiliary data structures implemented in LEMON.
257 This group contains some data structures implemented in LEMON in
258 order to make it easier to implement combinatorial algorithms.
262 @defgroup algs Algorithms
263 \brief This group contains the several algorithms
264 implemented in LEMON.
266 This group contains the several algorithms
267 implemented in LEMON.
271 @defgroup search Graph Search
273 \brief Common graph search algorithms.
275 This group contains the common graph search algorithms, namely
276 \e breadth-first \e search (BFS) and \e depth-first \e search (DFS).
280 @defgroup shortest_path Shortest Path Algorithms
282 \brief Algorithms for finding shortest paths.
284 This group contains the algorithms for finding shortest paths in digraphs.
286 - \ref Dijkstra algorithm for finding shortest paths from a source node
287 when all arc lengths are non-negative.
288 - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths
289 from a source node when arc lenghts can be either positive or negative,
290 but the digraph should not contain directed cycles with negative total
292 - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms
293 for solving the \e all-pairs \e shortest \e paths \e problem when arc
294 lenghts can be either positive or negative, but the digraph should
295 not contain directed cycles with negative total length.
296 - \ref Suurballe A successive shortest path algorithm for finding
297 arc-disjoint paths between two nodes having minimum total length.
301 @defgroup max_flow Maximum Flow Algorithms
303 \brief Algorithms for finding maximum flows.
305 This group contains the algorithms for finding maximum flows and
306 feasible circulations.
308 The \e maximum \e flow \e problem is to find a flow of maximum value between
309 a single source and a single target. Formally, there is a \f$G=(V,A)\f$
310 digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and
311 \f$s, t \in V\f$ source and target nodes.
312 A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the
313 following optimization problem.
315 \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]
316 \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)
317 \quad \forall u\in V\setminus\{s,t\} \f]
318 \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]
320 LEMON contains several algorithms for solving maximum flow problems:
321 - \ref EdmondsKarp Edmonds-Karp algorithm.
322 - \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm.
323 - \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees.
324 - \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees.
326 In most cases the \ref Preflow "Preflow" algorithm provides the
327 fastest method for computing a maximum flow. All implementations
328 also provide functions to query the minimum cut, which is the dual
329 problem of maximum flow.
331 \ref Circulation is a preflow push-relabel algorithm implemented directly
332 for finding feasible circulations, which is a somewhat different problem,
333 but it is strongly related to maximum flow.
334 For more information, see \ref Circulation.
338 @defgroup min_cost_flow Minimum Cost Flow Algorithms
341 \brief Algorithms for finding minimum cost flows and circulations.
343 This group contains the algorithms for finding minimum cost flows and
346 The \e minimum \e cost \e flow \e problem is to find a feasible flow of
347 minimum total cost from a set of supply nodes to a set of demand nodes
348 in a network with capacity constraints (lower and upper bounds)
350 Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{Z}\f$,
351 \f$upper: A\rightarrow\mathbf{Z}\cup\{+\infty\}\f$ denote the lower and
352 upper bounds for the flow values on the arcs, for which
353 \f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$,
354 \f$cost: A\rightarrow\mathbf{Z}\f$ denotes the cost per unit flow
355 on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the
356 signed supply values of the nodes.
357 If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$
358 supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with
359 \f$-sup(u)\f$ demand.
360 A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution
361 of the following optimization problem.
363 \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]
364 \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq
365 sup(u) \quad \forall u\in V \f]
366 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]
368 The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be
369 zero or negative in order to have a feasible solution (since the sum
370 of the expressions on the left-hand side of the inequalities is zero).
371 It means that the total demand must be greater or equal to the total
372 supply and all the supplies have to be carried out from the supply nodes,
373 but there could be demands that are not satisfied.
374 If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand
375 constraints have to be satisfied with equality, i.e. all demands
376 have to be satisfied and all supplies have to be used.
378 If you need the opposite inequalities in the supply/demand constraints
379 (i.e. the total demand is less than the total supply and all the demands
380 have to be satisfied while there could be supplies that are not used),
381 then you could easily transform the problem to the above form by reversing
382 the direction of the arcs and taking the negative of the supply values
383 (e.g. using \ref ReverseDigraph and \ref NegMap adaptors).
384 However \ref NetworkSimplex algorithm also supports this form directly
385 for the sake of convenience.
387 A feasible solution for this problem can be found using \ref Circulation.
389 Note that the above formulation is actually more general than the usual
390 definition of the minimum cost flow problem, in which strict equalities
391 are required in the supply/demand contraints, i.e.
393 \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) =
394 sup(u) \quad \forall u\in V. \f]
396 However if the sum of the supply values is zero, then these two problems
397 are equivalent. So if you need the equality form, you have to ensure this
398 additional contraint for the algorithms.
400 The dual solution of the minimum cost flow problem is represented by node
401 potentials \f$\pi: V\rightarrow\mathbf{Z}\f$.
402 An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem
403 is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$
404 node potentials the following \e complementary \e slackness optimality
407 - For all \f$uv\in A\f$ arcs:
408 - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$;
409 - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$;
410 - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.
411 - For all \f$u\in V\f$ nodes:
412 - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,
415 Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc
416 \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.
417 \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f]
419 All algorithms provide dual solution (node potentials) as well,
420 if an optimal flow is found.
422 LEMON contains several algorithms for solving minimum cost flow problems.
423 - \ref NetworkSimplex Primal Network Simplex algorithm with various
425 - \ref CostScaling Push-Relabel and Augment-Relabel algorithms based on
427 - \ref CapacityScaling Successive Shortest %Path algorithm with optional
429 - \ref CancelAndTighten The Cancel and Tighten algorithm.
430 - \ref CycleCanceling Cycle-Canceling algorithms.
432 Most of these implementations support the general inequality form of the
433 minimum cost flow problem, but CancelAndTighten and CycleCanceling
434 only support the equality form due to the primal method they use.
436 In general NetworkSimplex is the most efficient implementation,
437 but in special cases other algorithms could be faster.
438 For example, if the total supply and/or capacities are rather small,
439 CapacityScaling is usually the fastest algorithm (without effective scaling).
443 @defgroup min_cut Minimum Cut Algorithms
446 \brief Algorithms for finding minimum cut in graphs.
448 This group contains the algorithms for finding minimum cut in graphs.
450 The \e minimum \e cut \e problem is to find a non-empty and non-complete
451 \f$X\f$ subset of the nodes with minimum overall capacity on
452 outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a
453 \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum
454 cut is the \f$X\f$ solution of the next optimization problem:
456 \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}
457 \sum_{uv\in A, u\in X, v\not\in X}cap(uv) \f]
459 LEMON contains several algorithms related to minimum cut problems:
461 - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut
463 - \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for
464 calculating minimum cut in undirected graphs.
465 - \ref GomoryHu "Gomory-Hu tree computation" for calculating
466 all-pairs minimum cut in undirected graphs.
468 If you want to find minimum cut just between two distinict nodes,
469 see the \ref max_flow "maximum flow problem".
473 @defgroup graph_properties Connectivity and Other Graph Properties
475 \brief Algorithms for discovering the graph properties
477 This group contains the algorithms for discovering the graph properties
478 like connectivity, bipartiteness, euler property, simplicity etc.
480 \image html edge_biconnected_components.png
481 \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth
485 @defgroup planar Planarity Embedding and Drawing
487 \brief Algorithms for planarity checking, embedding and drawing
489 This group contains the algorithms for planarity checking,
490 embedding and drawing.
492 \image html planar.png
493 \image latex planar.eps "Plane graph" width=\textwidth
497 @defgroup matching Matching Algorithms
499 \brief Algorithms for finding matchings in graphs and bipartite graphs.
501 This group contains the algorithms for calculating
502 matchings in graphs and bipartite graphs. The general matching problem is
503 finding a subset of the edges for which each node has at most one incident
506 There are several different algorithms for calculate matchings in
507 graphs. The matching problems in bipartite graphs are generally
508 easier than in general graphs. The goal of the matching optimization
509 can be finding maximum cardinality, maximum weight or minimum cost
510 matching. The search can be constrained to find perfect or
511 maximum cardinality matching.
513 The matching algorithms implemented in LEMON:
514 - \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm
515 for calculating maximum cardinality matching in bipartite graphs.
516 - \ref PrBipartiteMatching Push-relabel algorithm
517 for calculating maximum cardinality matching in bipartite graphs.
518 - \ref MaxWeightedBipartiteMatching
519 Successive shortest path algorithm for calculating maximum weighted
520 matching and maximum weighted bipartite matching in bipartite graphs.
521 - \ref MinCostMaxBipartiteMatching
522 Successive shortest path algorithm for calculating minimum cost maximum
523 matching in bipartite graphs.
524 - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating
525 maximum cardinality matching in general graphs.
526 - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating
527 maximum weighted matching in general graphs.
528 - \ref MaxWeightedPerfectMatching
529 Edmond's blossom shrinking algorithm for calculating maximum weighted
530 perfect matching in general graphs.
532 \image html bipartite_matching.png
533 \image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth
537 @defgroup spantree Minimum Spanning Tree Algorithms
539 \brief Algorithms for finding minimum cost spanning trees and arborescences.
541 This group contains the algorithms for finding minimum cost spanning
542 trees and arborescences.
546 @defgroup auxalg Auxiliary Algorithms
548 \brief Auxiliary algorithms implemented in LEMON.
550 This group contains some algorithms implemented in LEMON
551 in order to make it easier to implement complex algorithms.
555 @defgroup approx Approximation Algorithms
557 \brief Approximation algorithms.
559 This group contains the approximation and heuristic algorithms
560 implemented in LEMON.
564 @defgroup gen_opt_group General Optimization Tools
565 \brief This group contains some general optimization frameworks
566 implemented in LEMON.
568 This group contains some general optimization frameworks
569 implemented in LEMON.
573 @defgroup lp_group Lp and Mip Solvers
574 @ingroup gen_opt_group
575 \brief Lp and Mip solver interfaces for LEMON.
577 This group contains Lp and Mip solver interfaces for LEMON. The
578 various LP solvers could be used in the same manner with this
583 @defgroup lp_utils Tools for Lp and Mip Solvers
585 \brief Helper tools to the Lp and Mip solvers.
587 This group adds some helper tools to general optimization framework
588 implemented in LEMON.
592 @defgroup metah Metaheuristics
593 @ingroup gen_opt_group
594 \brief Metaheuristics for LEMON library.
596 This group contains some metaheuristic optimization tools.
600 @defgroup utils Tools and Utilities
601 \brief Tools and utilities for programming in LEMON
603 Tools and utilities for programming in LEMON.
607 @defgroup gutils Basic Graph Utilities
609 \brief Simple basic graph utilities.
611 This group contains some simple basic graph utilities.
615 @defgroup misc Miscellaneous Tools
617 \brief Tools for development, debugging and testing.
619 This group contains several useful tools for development,
620 debugging and testing.
624 @defgroup timecount Time Measuring and Counting
626 \brief Simple tools for measuring the performance of algorithms.
628 This group contains simple tools for measuring the performance
633 @defgroup exceptions Exceptions
635 \brief Exceptions defined in LEMON.
637 This group contains the exceptions defined in LEMON.
641 @defgroup io_group Input-Output
642 \brief Graph Input-Output methods
644 This group contains the tools for importing and exporting graphs
645 and graph related data. Now it supports the \ref lgf-format
646 "LEMON Graph Format", the \c DIMACS format and the encapsulated
647 postscript (EPS) format.
651 @defgroup lemon_io LEMON Graph Format
653 \brief Reading and writing LEMON Graph Format.
655 This group contains methods for reading and writing
656 \ref lgf-format "LEMON Graph Format".
660 @defgroup eps_io Postscript Exporting
662 \brief General \c EPS drawer and graph exporter
664 This group contains general \c EPS drawing methods and special
665 graph exporting tools.
669 @defgroup dimacs_group DIMACS format
671 \brief Read and write files in DIMACS format
673 Tools to read a digraph from or write it to a file in DIMACS format data.
677 @defgroup nauty_group NAUTY Format
679 \brief Read \e Nauty format
681 Tool to read graphs from \e Nauty format data.
685 @defgroup concept Concepts
686 \brief Skeleton classes and concept checking classes
688 This group contains the data/algorithm skeletons and concept checking
689 classes implemented in LEMON.
691 The purpose of the classes in this group is fourfold.
693 - These classes contain the documentations of the %concepts. In order
694 to avoid document multiplications, an implementation of a concept
695 simply refers to the corresponding concept class.
697 - These classes declare every functions, <tt>typedef</tt>s etc. an
698 implementation of the %concepts should provide, however completely
699 without implementations and real data structures behind the
700 interface. On the other hand they should provide nothing else. All
701 the algorithms working on a data structure meeting a certain concept
702 should compile with these classes. (Though it will not run properly,
703 of course.) In this way it is easily to check if an algorithm
704 doesn't use any extra feature of a certain implementation.
706 - The concept descriptor classes also provide a <em>checker class</em>
707 that makes it possible to check whether a certain implementation of a
708 concept indeed provides all the required features.
710 - Finally, They can serve as a skeleton of a new implementation of a concept.
714 @defgroup graph_concepts Graph Structure Concepts
716 \brief Skeleton and concept checking classes for graph structures
718 This group contains the skeletons and concept checking classes of LEMON's
719 graph structures and helper classes used to implement these.
723 @defgroup map_concepts Map Concepts
725 \brief Skeleton and concept checking classes for maps
727 This group contains the skeletons and concept checking classes of maps.
733 @defgroup demos Demo Programs
735 Some demo programs are listed here. Their full source codes can be found in
736 the \c demo subdirectory of the source tree.
738 In order to compile them, use the <tt>make demo</tt> or the
739 <tt>make check</tt> commands.
743 @defgroup tools Standalone Utility Applications
745 Some utility applications are listed here.
747 The standard compilation procedure (<tt>./configure;make</tt>) will compile