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

  *

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

  *

  * Copyright (C) 2003-2009

  * 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 semi_adaptors Semi-Adaptor Classes for Graphs

 @ingroup graphs

 \brief Graph types between real graphs and graph adaptors.

 This group contains some graph types between real graphs and graph adaptors.

 These classes wrap graphs to give new functionality as the adaptors do it.

 On the other hand they are not light-weight structures as the adaptors.

 */

 /**

 @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 lemon::concepts::Path

 */

 /**

 @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 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).

 */

 /**

 @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 Dijkstra Dijkstra's algorithm for finding shortest paths from a 

    source node when all arc lengths are non-negative.

  - \ref Suurballe A successive shortest path algorithm for finding

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

 */

 /**

 @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.

 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 implements the preflow push-relabel algorithm of Goldberg and

 Tarjan for solving this problem. 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 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.

 The \e minimum \e cost \e flow \e problem is to find a feasible flow of

 minimum total cost from a set of supply nodes to a set of demand nodes

 in a network with capacity constraints (lower and upper bounds)

 and arc costs.

 Formally, let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{Z}\f$,

 \f$upper: A\rightarrow\mathbf{Z}\cup\{+\infty\}\f$ denote the lower and

 upper bounds for the flow values on the arcs, for which

 \f$lower(uv) \leq upper(uv)\f$ must hold for all \f$uv\in A\f$,

 \f$cost: A\rightarrow\mathbf{Z}\f$ denotes the cost per unit flow

 on the arcs and \f$sup: V\rightarrow\mathbf{Z}\f$ denotes the

 signed supply values of the nodes.

 If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$

 supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with

 \f$-sup(u)\f$ demand.

 A minimum cost flow is an \f$f: A\rightarrow\mathbf{Z}\f$ solution

 of the following optimization problem.

 \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]

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

     sup(u) \quad \forall u\in V \f]

 \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]

 The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be

 zero or negative in order to have a feasible solution (since the sum

 of the expressions on the left-hand side of the inequalities is zero).

 It means that the total demand must be greater or equal to the total

 supply and all the supplies have to be carried out from the supply nodes,

 but there could be demands that are not satisfied.

 If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand

 constraints have to be satisfied with equality, i.e. all demands

 have to be satisfied and all supplies have to be used.

 If you need the opposite inequalities in the supply/demand constraints

 (i.e. the total demand is less than the total supply and all the demands

 have to be satisfied while there could be supplies that are not used),

 then you could easily transform the problem to the above form by reversing

 the direction of the arcs and taking the negative of the supply values

 (e.g. using \ref ReverseDigraph and \ref NegMap adaptors).

 However \ref NetworkSimplex algorithm also supports this form directly

 for the sake of convenience.

 A feasible solution for this problem can be found using \ref Circulation.

 Note that the above formulation is actually more general than the usual

 definition of the minimum cost flow problem, in which strict equalities

 are required in the supply/demand contraints, i.e.

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

     sup(u) \quad \forall u\in V. \f]

 However if the sum of the supply values is zero, then these two problems

 are equivalent. So if you need the equality form, you have to ensure this

 additional contraint for the algorithms.

 The dual solution of the minimum cost flow problem is represented by node 

 potentials \f$\pi: V\rightarrow\mathbf{Z}\f$.

 An \f$f: A\rightarrow\mathbf{Z}\f$ feasible solution of the problem

 is optimal if and only if for some \f$\pi: V\rightarrow\mathbf{Z}\f$

 node potentials the following \e complementary \e slackness optimality

 conditions hold.

  - For all \f$uv\in A\f$ arcs:

    - if \f$cost^\pi(uv)>0\f$, then \f$f(uv)=lower(uv)\f$;

    - if \f$lower(uv)<f(uv)<upper(uv)\f$, then \f$cost^\pi(uv)=0\f$;

    - if \f$cost^\pi(uv)<0\f$, then \f$f(uv)=upper(uv)\f$.

  - For all \f$u\in V\f$ nodes:

    - if \f$\sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \neq sup(u)\f$,

      then \f$\pi(u)=0\f$.

 Here \f$cost^\pi(uv)\f$ denotes the \e reduced \e cost of the arc

 \f$uv\in A\f$ with respect to the potential function \f$\pi\f$, i.e.

 \f[ cost^\pi(uv) = cost(uv) + \pi(u) - \pi(v).\f]

 \ref NetworkSimplex is an efficient implementation of the primal Network

 Simplex algorithm for finding minimum cost flows. It also provides dual

 solution (node potentials), if an optimal flow is found.

 */

 /**

 @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 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 edge_biconnected_components.png

 \image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth

 */

 /**

 @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.

 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 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.

 \image html bipartite_matching.png

 \image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth

 */

 /**

 @defgroup spantree Minimum Spanning Tree Algorithms

 @ingroup algs

 \brief Algorithms for finding a minimum cost spanning tree in a graph.

 This group contains the algorithms for finding a minimum cost spanning

 tree in a graph.

 */

 /**

 @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. The

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

 interface.

 */

 /**

 @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 LEMON's

 graph structures and helper classes used to implement these.

 */

 /**

 @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.

 */

 /**

 \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.

 */

 /**

 @defgroup tools Standalone Utility Applications

 Some utility applications are listed here.

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

 them, as well.

 */

 }