/* -*- mode: C++; indent-tabs-mode: nil; -*-

  *

  * This file is a part of LEMON, a generic C++ optimization library.

  *

  * Copyright (C) 2003-2010

  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

  * (Egervary Research Group on Combinatorial Optimization, EGRES).

  *

  * Permission to use, modify and distribute this software is granted

  * provided that this copyright notice appears in all copies. For

  * precise terms see the accompanying LICENSE file.

  *

  * This software is provided "AS IS" with no warranty of any kind,

  * express or implied, and with no claim as to its suitability for any

  * purpose.

  *

  */

 namespace lemon {

 /**

 @defgroup datas Data Structures

 This group contains the several data structures implemented in LEMON.

 */

 /**

 @defgroup graphs Graph Structures

 @ingroup datas

 \brief Graph structures implemented in LEMON.

 The implementation of combinatorial algorithms heavily relies on

 efficient graph implementations. LEMON offers data structures which are

 planned to be easily used in an experimental phase of implementation studies,

 and thereafter the program code can be made efficient by small modifications.

 The most efficient implementation of diverse applications require the

 usage of different physical graph implementations. These differences

 appear in the size of graph we require to handle, memory or time usage

 limitations or in the set of operations through which the graph can be

 accessed.  LEMON provides several physical graph structures to meet

 the diverging requirements of the possible users.  In order to save on

 running time or on memory usage, some structures may fail to provide

 some graph features like arc/edge or node deletion.

 Alteration of standard containers need a very limited number of

 operations, these together satisfy the everyday requirements.

 In the case of graph structures, different operations are needed which do

 not alter the physical graph, but gives another view. If some nodes or

 arcs have to be hidden or the reverse oriented graph have to be used, then

 this is the case. It also may happen that in a flow implementation

 the residual graph can be accessed by another algorithm, or a node-set

 is to be shrunk for another algorithm.

 LEMON also provides a variety of graphs for these requirements called

 \ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only

 in conjunction with other graph representations.

 You are free to use the graph structure that fit your requirements

 the best, most graph algorithms and auxiliary data structures can be used

 with any graph structure.

 <b>See also:</b> \ref graph_concepts "Graph Structure Concepts".

 */

 /**

 @defgroup graph_adaptors Adaptor Classes for Graphs

 @ingroup graphs

 \brief Adaptor classes for digraphs and graphs

 This group contains several useful adaptor classes for digraphs and graphs.

 The main parts of LEMON are the different graph structures, generic

 graph algorithms, graph concepts, which couple them, and graph

 adaptors. While the previous notions are more or less clear, the

 latter one needs further explanation. Graph adaptors are graph classes

 which serve for considering graph structures in different ways.

 A short example makes this much clearer.  Suppose that we have an

 instance \c g of a directed graph type, say ListDigraph and an algorithm

 \code

 template <typename Digraph>

 int algorithm(const Digraph&);

 \endcode

 is needed to run on the reverse oriented graph.  It may be expensive

 (in time or in memory usage) to copy \c g with the reversed

 arcs.  In this case, an adaptor class is used, which (according

 to LEMON \ref concepts::Digraph "digraph concepts") works as a digraph.

 The adaptor uses the original digraph structure and digraph operations when

 methods of the reversed oriented graph are called.  This means that the adaptor

 have minor memory usage, and do not perform sophisticated algorithmic

 actions.  The purpose of it is to give a tool for the cases when a

 graph have to be used in a specific alteration.  If this alteration is

 obtained by a usual construction like filtering the node or the arc set or

 considering a new orientation, then an adaptor is worthwhile to use.

 To come back to the reverse oriented graph, in this situation

 \code

 template<typename Digraph> class ReverseDigraph;

 \endcode

 template class can be used. The code looks as follows

 \code

 ListDigraph g;

 ReverseDigraph<ListDigraph> rg(g);

 int result = algorithm(rg);

 \endcode

 During running the algorithm, the original digraph \c g is untouched.

 This techniques give rise to an elegant code, and based on stable

 graph adaptors, complex algorithms can be implemented easily.

 In flow, circulation and matching problems, the residual

 graph is of particular importance. Combining an adaptor implementing

 this with shortest path algorithms or minimum mean cycle algorithms,

 a range of weighted and cardinality optimization algorithms can be

 obtained. For other examples, the interested user is referred to the

 detailed documentation of particular adaptors.

 The behavior of graph adaptors can be very different. Some of them keep

 capabilities of the original graph while in other cases this would be

 meaningless. This means that the concepts that they meet depend

 on the graph adaptor, and the wrapped graph.

 For example, if an arc of a reversed digraph is deleted, this is carried

 out by deleting the corresponding arc of the original digraph, thus the

 adaptor modifies the original digraph.

 However in case of a residual digraph, this operation has no sense.

 Let us stand one more example here to simplify your work.

 ReverseDigraph has constructor

 \code

 ReverseDigraph(Digraph& digraph);

 \endcode

 This means that in a situation, when a <tt>const %ListDigraph&</tt>

 reference to a graph is given, then it have to be instantiated with

 <tt>Digraph=const %ListDigraph</tt>.

 \code

 int algorithm1(const ListDigraph& g) {

   ReverseDigraph<const ListDigraph> rg(g);

   return algorithm2(rg);

 }

 \endcode

 */

 /**

 @defgroup maps Maps

 @ingroup datas

 \brief Map structures implemented in LEMON.

 This group contains the map structures implemented in LEMON.

 LEMON provides several special purpose maps and map adaptors that e.g. combine

 new maps from existing ones.

 <b>See also:</b> \ref map_concepts "Map Concepts".

 */

 /**

 @defgroup graph_maps Graph Maps

 @ingroup maps

 \brief Special graph-related maps.

 This group contains maps that are specifically designed to assign

 values to the nodes and arcs/edges of graphs.

 If you are looking for the standard graph maps (\c NodeMap, \c ArcMap,

 \c EdgeMap), see the \ref graph_concepts "Graph Structure Concepts".

 */

 /**

 \defgroup map_adaptors Map Adaptors

 \ingroup maps

 \brief Tools to create new maps from existing ones

 This group contains map adaptors that are used to create "implicit"

 maps from other maps.

 Most of them are \ref concepts::ReadMap "read-only maps".

 They can make arithmetic and logical operations between one or two maps

 (negation, shifting, addition, multiplication, logical 'and', 'or',

 'not' etc.) or e.g. convert a map to another one of different Value type.

 The typical usage of this classes is passing implicit maps to

 algorithms.  If a function type algorithm is called then the function

 type map adaptors can be used comfortable. For example let's see the

 usage of map adaptors with the \c graphToEps() function.

 \code

   Color nodeColor(int deg) {

     if (deg >= 2) {

       return Color(0.5, 0.0, 0.5);

     } else if (deg == 1) {

       return Color(1.0, 0.5, 1.0);

     } else {

       return Color(0.0, 0.0, 0.0);

     }

   }

   Digraph::NodeMap<int> degree_map(graph);

   graphToEps(graph, "graph.eps")

     .coords(coords).scaleToA4().undirected()

     .nodeColors(composeMap(functorToMap(nodeColor), degree_map))

     .run();

 \endcode

 The \c functorToMap() function makes an \c int to \c Color map from the

 \c nodeColor() function. The \c composeMap() compose the \c degree_map

 and the previously created map. The composed map is a proper function to

 get the color of each node.

 The usage with class type algorithms is little bit harder. In this

 case the function type map adaptors can not be used, because the

 function map adaptors give back temporary objects.

 \code

   Digraph graph;

   typedef Digraph::ArcMap<double> DoubleArcMap;

   DoubleArcMap length(graph);

   DoubleArcMap speed(graph);

   typedef DivMap<DoubleArcMap, DoubleArcMap> TimeMap;

   TimeMap time(length, speed);

   Dijkstra<Digraph, TimeMap> dijkstra(graph, time);

   dijkstra.run(source, target);

 \endcode

 We have a length map and a maximum speed map on the arcs of a digraph.

 The minimum time to pass the arc can be calculated as the division of

 the two maps which can be done implicitly with the \c DivMap template

 class. We use the implicit minimum time map as the length map of the

 \c Dijkstra algorithm.

 */

 /**

 @defgroup paths Path Structures

 @ingroup datas

 \brief %Path structures implemented in LEMON.

 This group contains the path structures implemented in LEMON.

 LEMON provides flexible data structures to work with paths.

 All of them have similar interfaces and they can be copied easily with

 assignment operators and copy constructors. This makes it easy and

 efficient to have e.g. the Dijkstra algorithm to store its result in

 any kind of path structure.

 \sa \ref concepts::Path "Path concept"

 */

 /**

 @defgroup heaps Heap Structures

 @ingroup datas

 \brief %Heap structures implemented in LEMON.

 This group contains the heap structures implemented in LEMON.

 LEMON provides several heap classes. They are efficient implementations

 of the abstract data type \e priority \e queue. They store items with

 specified values called \e priorities in such a way that finding and

 removing the item with minimum priority are efficient.

 The basic operations are adding and erasing items, changing the priority

 of an item, etc.

 Heaps are crucial in several algorithms, such as Dijkstra and Prim.

 The heap implementations have the same interface, thus any of them can be

 used easily in such algorithms.

 \sa \ref concepts::Heap "Heap concept"

 */

 /**

 @defgroup auxdat Auxiliary Data Structures

 @ingroup datas

 \brief Auxiliary data structures implemented in LEMON.

 This group contains some data structures implemented in LEMON in

 order to make it easier to implement combinatorial algorithms.

 */

 /**

 @defgroup geomdat Geometric Data Structures

 @ingroup auxdat

 \brief Geometric data structures implemented in LEMON.

 This group contains geometric data structures implemented in LEMON.

  - \ref lemon::dim2::Point "dim2::Point" implements a two dimensional

    vector with the usual operations.

  - \ref lemon::dim2::Box "dim2::Box" can be used to determine the

    rectangular bounding box of a set of \ref lemon::dim2::Point

    "dim2::Point"'s.

 */

 /**

 @defgroup algs Algorithms

 \brief This group contains the several algorithms

 implemented in LEMON.

 This group contains the several algorithms

 implemented in LEMON.

 */

 /**

 @defgroup search Graph Search

 @ingroup algs

 \brief Common graph search algorithms.

 This group contains the common graph search algorithms, namely

 \e breadth-first \e search (BFS) and \e depth-first \e search (DFS)

 \ref clrs01algorithms.

 */

 /**

 @defgroup shortest_path Shortest Path Algorithms

 @ingroup algs

 \brief Algorithms for finding shortest paths.

 This group contains the algorithms for finding shortest paths in digraphs

 \ref clrs01algorithms.

  - \ref Dijkstra algorithm for finding shortest paths from a source node

    when all arc lengths are non-negative.

  - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths

    from a source node when arc lenghts can be either positive or negative,

    but the digraph should not contain directed cycles with negative total

    length.

  - \ref Suurballe A successive shortest path algorithm for finding

    arc-disjoint paths between two nodes having minimum total length.

 */

 /**

 @defgroup spantree Minimum Spanning Tree Algorithms

 @ingroup algs

 \brief Algorithms for finding minimum cost spanning trees and arborescences.

 This group contains the algorithms for finding minimum cost spanning

 trees and arborescences \ref clrs01algorithms.

 */

 /**

 @defgroup max_flow Maximum Flow Algorithms

 @ingroup algs

 \brief Algorithms for finding maximum flows.

 This group contains the algorithms for finding maximum flows and

 feasible circulations \ref clrs01algorithms, \ref amo93networkflows.

 The \e maximum \e flow \e problem is to find a flow of maximum value between

 a single source and a single target. Formally, there is a \f$G=(V,A)\f$

 digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and

 \f$s, t \in V\f$ source and target nodes.

 A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the

 following optimization problem.

 \f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]

 \f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)

     \quad \forall u\in V\setminus\{s,t\} \f]

 \f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]

 \ref Preflow is an efficient implementation of Goldberg-Tarjan's

 preflow push-relabel algorithm \ref goldberg88newapproach for finding

 maximum flows. It also provides functions to query the minimum cut,

 which is the dual problem of maximum flow.

 \ref Circulation is a preflow push-relabel algorithm implemented directly

 for finding feasible circulations, which is a somewhat different problem,

 but it is strongly related to maximum flow.

 For more information, see \ref Circulation.

 */

 /**

 @defgroup min_cost_flow_algs Minimum Cost Flow Algorithms

 @ingroup algs

 \brief Algorithms for finding minimum cost flows and circulations.

 This group contains the algorithms for finding minimum cost flows and

 circulations \ref amo93networkflows. For more information about this

 problem and its dual solution, see \ref min_cost_flow

 "Minimum Cost Flow Problem".

 LEMON contains several algorithms for this problem.

  - \ref NetworkSimplex Primal Network Simplex algorithm with various

    pivot strategies \ref dantzig63linearprog, \ref kellyoneill91netsimplex.

  - \ref CostScaling Cost Scaling algorithm based on push/augment and

    relabel operations \ref goldberg90approximation, \ref goldberg97efficient,

    \ref bunnagel98efficient.

  - \ref CapacityScaling Capacity Scaling algorithm based on the successive

    shortest path method \ref edmondskarp72theoretical.

  - \ref CycleCanceling Cycle-Canceling algorithms, two of which are

    strongly polynomial \ref klein67primal, \ref goldberg89cyclecanceling.

 In general NetworkSimplex is the most efficient implementation,

 but in special cases other algorithms could be faster.

 For example, if the total supply and/or capacities are rather small,

 CapacityScaling is usually the fastest algorithm (without effective scaling).

 */

 /**

 @defgroup min_cut Minimum Cut Algorithms

 @ingroup algs

 \brief Algorithms for finding minimum cut in graphs.

 This group contains the algorithms for finding minimum cut in graphs.

 The \e minimum \e cut \e problem is to find a non-empty and non-complete

 \f$X\f$ subset of the nodes with minimum overall capacity on

 outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a

 \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum

 cut is the \f$X\f$ solution of the next optimization problem:

 \f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}

     \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f]

 LEMON contains several algorithms related to minimum cut problems:

 - \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut

   in directed graphs.

 - \ref GomoryHu "Gomory-Hu tree computation" for calculating

   all-pairs minimum cut in undirected graphs.

 If you want to find minimum cut just between two distinict nodes,

 see the \ref max_flow "maximum flow problem".

 */

 /**

 @defgroup min_mean_cycle Minimum Mean Cycle Algorithms

 @ingroup algs

 \brief Algorithms for finding minimum mean cycles.

 This group contains the algorithms for finding minimum mean cycles

 \ref clrs01algorithms, \ref amo93networkflows.

 The \e minimum \e mean \e cycle \e problem is to find a directed cycle

 of minimum mean length (cost) in a digraph.

 The mean length of a cycle is the average length of its arcs, i.e. the

 ratio between the total length of the cycle and the number of arcs on it.

 This problem has an important connection to \e conservative \e length

 \e functions, too. A length function on the arcs of a digraph is called

 conservative if and only if there is no directed cycle of negative total

 length. For an arbitrary length function, the negative of the minimum

 cycle mean is the smallest \f$\epsilon\f$ value so that increasing the

 arc lengths uniformly by \f$\epsilon\f$ results in a conservative length

 function.

 LEMON contains three algorithms for solving the minimum mean cycle problem:

 - \ref KarpMmc Karp's original algorithm \ref amo93networkflows,

   \ref dasdan98minmeancycle.

 - \ref HartmannOrlinMmc Hartmann-Orlin's algorithm, which is an improved

   version of Karp's algorithm \ref dasdan98minmeancycle.

 - \ref HowardMmc Howard's policy iteration algorithm

   \ref dasdan98minmeancycle.

 In practice, the \ref HowardMmc "Howard" algorithm proved to be by far the

 most efficient one, though the best known theoretical bound on its running

 time is exponential.

 Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms

 run in time O(ne) and use space O(n<sup>2</sup>+e), but the latter one is

 typically faster due to the applied early termination scheme.

 */

 /**

 @defgroup matching Matching Algorithms

 @ingroup algs

 \brief Algorithms for finding matchings in graphs and bipartite graphs.

 This group contains the algorithms for calculating

 matchings in graphs and bipartite graphs. The general matching problem is

 finding a subset of the edges for which each node has at most one incident

 edge.

 There are several different algorithms for calculate matchings in

 graphs.  The matching problems in bipartite graphs are generally

 easier than in general graphs. The goal of the matching optimization

 can be finding maximum cardinality, maximum weight or minimum cost

 matching. The search can be constrained to find perfect or

 maximum cardinality matching.

 The matching algorithms implemented in LEMON:

 - \ref MaxMatching Edmond's blossom shrinking algorithm for calculating

   maximum cardinality matching in general graphs.

 - \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating

   maximum weighted matching in general graphs.

 - \ref MaxWeightedPerfectMatching

   Edmond's blossom shrinking algorithm for calculating maximum weighted

   perfect matching in general graphs.

 - \ref MaxFractionalMatching Push-relabel algorithm for calculating

   maximum cardinality fractional matching in general graphs.

 - \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating

   maximum weighted fractional matching in general graphs.

 - \ref MaxWeightedPerfectFractionalMatching

   Augmenting path algorithm for calculating maximum weighted

   perfect fractional matching in general graphs.

 \image html matching.png

 \image latex matching.eps "Min Cost Perfect Matching" width=\textwidth

 */

 /**

 @defgroup graph_properties Connectivity and Other Graph Properties

 @ingroup algs

 \brief Algorithms for discovering the graph properties

 This group contains the algorithms for discovering the graph properties

 like connectivity, bipartiteness, euler property, simplicity etc.

 \image html connected_components.png

 \image latex connected_components.eps "Connected components" width=\textwidth

 */

 /**

 @defgroup planar Planarity Embedding and Drawing

 @ingroup algs

 \brief Algorithms for planarity checking, embedding and drawing

 This group contains the algorithms for planarity checking,

 embedding and drawing.

 \image html planar.png

 \image latex planar.eps "Plane graph" width=\textwidth

 */

 /**

 @defgroup auxalg Auxiliary Algorithms

 @ingroup algs

 \brief Auxiliary algorithms implemented in LEMON.

 This group contains some algorithms implemented in LEMON

 in order to make it easier to implement complex algorithms.

 */

 /**

 @defgroup gen_opt_group General Optimization Tools

 \brief This group contains some general optimization frameworks

 implemented in LEMON.

 This group contains some general optimization frameworks

 implemented in LEMON.

 */

 /**

 @defgroup lp_group LP and MIP Solvers

 @ingroup gen_opt_group

 \brief LP and MIP solver interfaces for LEMON.

 This group contains LP and MIP solver interfaces for LEMON.

 Various LP solvers could be used in the same manner with this

 high-level interface.

 The currently supported solvers are \ref glpk, \ref clp, \ref cbc,

 \ref cplex, \ref soplex.

 */

 /**

 @defgroup utils Tools and Utilities

 \brief Tools and utilities for programming in LEMON

 Tools and utilities for programming in LEMON.

 */

 /**

 @defgroup gutils Basic Graph Utilities

 @ingroup utils

 \brief Simple basic graph utilities.

 This group contains some simple basic graph utilities.

 */

 /**

 @defgroup misc Miscellaneous Tools

 @ingroup utils

 \brief Tools for development, debugging and testing.

 This group contains several useful tools for development,

 debugging and testing.

 */

 /**

 @defgroup timecount Time Measuring and Counting

 @ingroup misc

 \brief Simple tools for measuring the performance of algorithms.

 This group contains simple tools for measuring the performance

 of algorithms.

 */

 /**

 @defgroup exceptions Exceptions

 @ingroup utils

 \brief Exceptions defined in LEMON.

 This group contains the exceptions defined in LEMON.

 */

 /**

 @defgroup io_group Input-Output

 \brief Graph Input-Output methods

 This group contains the tools for importing and exporting graphs

 and graph related data. Now it supports the \ref lgf-format

 "LEMON Graph Format", the \c DIMACS format and the encapsulated

 postscript (EPS) format.

 */

 /**

 @defgroup lemon_io LEMON Graph Format

 @ingroup io_group

 \brief Reading and writing LEMON Graph Format.

 This group contains methods for reading and writing

 \ref lgf-format "LEMON Graph Format".

 */

 /**

 @defgroup eps_io Postscript Exporting

 @ingroup io_group

 \brief General \c EPS drawer and graph exporter

 This group contains general \c EPS drawing methods and special

 graph exporting tools.

 */

 /**

 @defgroup dimacs_group DIMACS Format

 @ingroup io_group

 \brief Read and write files in DIMACS format

 Tools to read a digraph from or write it to a file in DIMACS format data.

 */

 /**

 @defgroup nauty_group NAUTY Format

 @ingroup io_group

 \brief Read \e Nauty format

 Tool to read graphs from \e Nauty format data.

 */

 /**

 @defgroup concept Concepts

 \brief Skeleton classes and concept checking classes

 This group contains the data/algorithm skeletons and concept checking

 classes implemented in LEMON.

 The purpose of the classes in this group is fourfold.

 - These classes contain the documentations of the %concepts. In order

   to avoid document multiplications, an implementation of a concept

   simply refers to the corresponding concept class.

 - These classes declare every functions, <tt>typedef</tt>s etc. an

   implementation of the %concepts should provide, however completely

   without implementations and real data structures behind the

   interface. On the other hand they should provide nothing else. All

   the algorithms working on a data structure meeting a certain concept

   should compile with these classes. (Though it will not run properly,

   of course.) In this way it is easily to check if an algorithm

   doesn't use any extra feature of a certain implementation.

 - The concept descriptor classes also provide a <em>checker class</em>

   that makes it possible to check whether a certain implementation of a

   concept indeed provides all the required features.

 - Finally, They can serve as a skeleton of a new implementation of a concept.

 */

 /**

 @defgroup graph_concepts Graph Structure Concepts

 @ingroup concept

 \brief Skeleton and concept checking classes for graph structures

 This group contains the skeletons and concept checking classes of

 graph structures.

 */

 /**

 @defgroup map_concepts Map Concepts

 @ingroup concept

 \brief Skeleton and concept checking classes for maps

 This group contains the skeletons and concept checking classes of maps.

 */

 /**

 @defgroup tools Standalone Utility Applications

 Some utility applications are listed here.

 The standard compilation procedure (<tt>./configure;make</tt>) will compile

 them, as well.

 */

 /**

 \anchor demoprograms

 @defgroup demos Demo Programs

 Some demo programs are listed here. Their full source codes can be found in

 the \c demo subdirectory of the source tree.

 In order to compile them, use the <tt>make demo</tt> or the

 <tt>make check</tt> commands.

 */

 }