RhodeCode

CMAKE_MINIMUM_REQUIRED(VERSION 2.6)

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#  OUTPUT_VARIABLE HG_REVISION

#  OUTPUT_STRIP_TRAILING_WHITESPACE)

SET(PROJECT_VERSION_MAJOR "0")

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SET(PROJECT_VERSION_PATCH "0")

SET(PROJECT_VERSION

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PROJECT(${PROJECT_NAME})

SET(CMAKE_MODULE_PATH ${CMAKE_SOURCE_DIR}/cmake)

INCLUDE(FindDoxygen)

INCLUDE(FindGhostscript)

ENABLE_TESTING()

ADD_SUBDIRECTORY(lemon)

ADD_SUBDIRECTORY(demo)

ADD_SUBDIRECTORY(doc)

ADD_SUBDIRECTORY(test)

IF(WIN32)

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IF(WIN32)

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  # Variables to generate a component-based installer.

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dnl Process this file with autoconf to produce a configure script.

dnl Version information.

m4_define([lemon_version_number], [])

m4_define([lemon_hg_revision], [m4_normalize(esyscmd([hg id -i]))])

m4_define([lemon_version], [ifelse(lemon_version_number(), [], [lemon_hg_revision()], [lemon_version_number()])])

AC_PREREQ([2.59])

AC_CONFIG_AUX_DIR([build-aux])

AC_CONFIG_MACRO_DIR([m4])

AM_INIT_AUTOMAKE([-Wall -Werror foreign subdir-objects nostdinc])

AC_CONFIG_SRCDIR([lemon/list_graph.h])

AC_CONFIG_HEADERS([config.h lemon/config.h])

lx_cmdline_cxxflags_set=${CXXFLAGS+set}

dnl Checks for programs.

AC_PROG_CXX

AC_PROG_CXXCPP

AC_PROG_INSTALL

AC_DISABLE_SHARED

AC_PROG_LIBTOOL

AC_CHECK_PROG([doxygen_found],[doxygen],[yes],[no])

AC_CHECK_PROG([gs_found],[gs],[yes],[no])

dnl Set custom compiler flags when using g++.

if test x"$lx_cmdline_cxxflags_set" != x"set" -a "$GXX" = yes; then

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fi

dnl Checks for libraries.

LX_CHECK_GLPK

LX_CHECK_CPLEX

LX_CHECK_SOPLEX

dnl Disable/enable building the demo programs.

AC_ARG_ENABLE([demo],

AS_HELP_STRING([--enable-demo], [build the demo programs])

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dnl Disable/enable building the binary tools.

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dnl Checks for header files.

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dnl Checks for typedefs, structures, and compiler characteristics.

AC_C_CONST

AC_C_INLINE

AC_TYPE_SIZE_T

AC_HEADER_TIME

AC_STRUCT_TM

dnl Checks for library functions.

AC_HEADER_STDC

AC_CHECK_FUNCS(gettimeofday times ctime_r)

dnl Add dependencies on files generated by configure.

AC_SUBST([CONFIG_STATUS_DEPENDENCIES],

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AC_CONFIG_FILES([

Makefile

doc/Doxyfile

lemon/lemon.pc

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AC_OUTPUT

echo

echo '****************************** SUMMARY ******************************'

echo

echo Package version............... : $PACKAGE-$VERSION

echo

echo C++ compiler.................. : $CXX

echo C++ compiles flags............ : $CXXFLAGS

echo

echo GLPK support.................. : $lx_glpk_found

echo CPLEX support................. : $lx_cplex_found

echo SOPLEX support................ : $lx_soplex_found

echo

echo Build benchmarks.............. : $enable_benchmark

echo Build demo programs........... : $enable_demo

echo Build additional tools........ : $enable_tools

echo

echo The packace will be installed in

echo -n '  '

echo $prefix.

echo

echo '*********************************************************************'

echo

echo Configure complete, now type \'make\' and then \'make install\'.

echo

@ingroup algs

\brief Common graph search algorithms.

This group describes the common graph search algorithms like

Breadth-first search (Bfs) and Depth-first search (Dfs).

*/

/**

@defgroup shortest_path Shortest Path algorithms

@ingroup algs

\brief Algorithms for finding shortest paths.

This group describes the algorithms for finding shortest paths in graphs.

*/

/**

@defgroup max_flow Maximum Flow algorithms

@ingroup algs

\brief Algorithms for finding maximum flows.

This group describes the algorithms for finding maximum flows and

feasible circulations.

The maximum flow problem is to find a flow between a single source and

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

directed graph, an \f$c_a:A\rightarrow\mathbf{R}^+_0\f$ capacity

function and given \f$s, t \in V\f$ source and target node. The

maximum flow is the \f$f_a\f$ solution of the next optimization problem:

\f[ 0 \le f_a \le c_a \f]

\f[ \sum_{v\in\delta^{-}(u)}f_{vu}=\sum_{v\in\delta^{+}(u)}f_{uv}

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

\f[ \max \sum_{v\in\delta^{+}(s)}f_{uv} - \sum_{v\in\delta^{-}(s)}f_{vu}\f]

LEMON contains several algorithms for solving maximum flow problems:

- \ref lemon::EdmondsKarp "Edmonds-Karp"

- \ref lemon::Preflow "Goldberg's Preflow algorithm"

- \ref lemon::DinitzSleatorTarjan "Dinitz's blocking flow algorithm with dynamic trees"

- \ref lemon::GoldbergTarjan "Preflow algorithm with dynamic trees"

In most cases the \ref lemon::Preflow "Preflow" algorithm provides the

fastest method to compute the maximum flow. All impelementations

provides functions to query the minimum cut, which is the dual linear

programming problem of the maximum flow.

*/

/**

@defgroup min_cost_flow Minimum Cost Flow algorithms

@ingroup algs

\brief Algorithms for finding minimum cost flows and circulations.

This group describes the algorithms for finding minimum cost flows and

circulations.

*/

/**

@defgroup min_cut Minimum Cut algorithms

@ingroup algs

\brief Algorithms for finding minimum cut in graphs.

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

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

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

outgoing arcs. Formally, there is \f$G=(V,A)\f$ directed graph, an

\f$c_a:A\rightarrow\mathbf{R}^+_0\f$ 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}c_{uv}\f]

LEMON contains several algorithms related to minimum cut problems:

- \ref lemon::HaoOrlin "Hao-Orlin algorithm" to calculate minimum cut

  in directed graphs

- \ref lemon::NagamochiIbaraki "Nagamochi-Ibaraki algorithm" to

  calculate minimum cut in undirected graphs

- \ref lemon::GomoryHuTree "Gomory-Hu tree computation" to calculate all

  pairs minimum cut in undirected graphs

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

please see the \ref max_flow "Maximum Flow page".

*/

/**

@defgroup graph_prop Connectivity and other graph properties

@ingroup algs

\brief Algorithms for discovering the graph properties

This group describes 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 planar Planarity embedding and drawing

@ingroup algs

\brief Algorithms for planarity checking, embedding and drawing

This group describes the algorithms for planarity checking,

embedding and drawing.

\image html planar.png

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

*/

/**

@defgroup matching Matching algorithms

@ingroup algs

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

This group contains algorithm objects and functions to calculate

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

finding a subset of the arcs which does not shares common endpoints.

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 the finding maximum cardinality, maximum weight or minimum cost

matching. The search can be constrained to find perfect or

maximum cardinality matching.

- \ref lemon::MaxBipartiteMatching "MaxBipartiteMatching" Hopcroft-Karp

  augmenting path algorithm for calculate maximum cardinality matching in

  bipartite graphs

- \ref lemon::PrBipartiteMatching "PrBipartiteMatching" Push-Relabel

  algorithm for calculate maximum cardinality matching in bipartite graphs

- \ref lemon::MaxWeightedBipartiteMatching "MaxWeightedBipartiteMatching"

  Successive shortest path algorithm for calculate maximum weighted matching

  and maximum weighted bipartite matching in bipartite graph

- \ref lemon::MinCostMaxBipartiteMatching "MinCostMaxBipartiteMatching"

  Successive shortest path algorithm for calculate minimum cost maximum

  matching in bipartite graph

- \ref lemon::MaxMatching "MaxMatching" Edmond's blossom shrinking algorithm

  for calculate maximum cardinality matching in general graph

- \ref lemon::MaxWeightedMatching "MaxWeightedMatching" Edmond's blossom

  shrinking algorithm for calculate maximum weighted matching in general

  graph

- \ref lemon::MaxWeightedPerfectMatching "MaxWeightedPerfectMatching"

  Edmond's blossom shrinking algorithm for calculate maximum weighted

  perfect matching in general graph

\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 describes 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 describes some algorithms implemented in LEMON

in order to make it easier to implement complex algorithms.

*/

/**

@defgroup approx Approximation algorithms

\brief Approximation algorithms.

This group describes the approximation and heuristic algorithms

implemented in LEMON.

*/

/**

@defgroup gen_opt_group General Optimization Tools

\brief This group describes some general optimization frameworks

implemented in LEMON.

This group describes 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 describes Lp and Mip solver interfaces for LEMON. The

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

interface.

*/

/**

@defgroup lp_utils Tools for Lp and Mip solvers

@ingroup lp_group

\brief Helper tools to the Lp and Mip solvers.

This group adds some helper tools to general optimization framework

implemented in LEMON.

*/

/**

@defgroup metah Metaheuristics

@ingroup gen_opt_group

\brief Metaheuristics for LEMON library.

This group describes some metaheuristic optimization tools.

*/

/**

@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 describes some simple basic graph utilities.

*/

/**

@defgroup misc Miscellaneous Tools

@ingroup utils

\brief Tools for development, debugging and testing.

This group describes 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 describes simple tools for measuring the performance

of algorithms.

*/

/**

@defgroup graphbits Tools for Graph Implementation

@ingroup utils

\brief Tools to make it easier to create graphs.

This group describes the tools that makes it easier to create graphs and

the maps that dynamically update with the graph changes.

*/

/**

@defgroup exceptions Exceptions

@ingroup utils

\brief Exceptions defined in LEMON.

This group describes the exceptions defined in LEMON.

*/

/**

@defgroup io_group Input-Output

\brief Graph Input-Output methods

This group describes the tools for importing and exporting graphs

and graph related data. Now it supports the LEMON format, the

\c DIMACS format and the encapsulated postscript (EPS) format.

*/

/**

@ingroup io_group

This group describes methods for reading and writing

*/

/**

@defgroup eps_io Postscript exporting

@ingroup io_group

\brief General \c EPS drawer and graph exporter

This group describes general \c EPS drawing methods and special

graph exporting tools.

*/

/**

@defgroup concept Concepts

\brief Skeleton classes and concept checking classes

This group describes 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 describes the skeletons and concept checking classes of LEMON's

graph structures and helper classes used to implement these.

*/

/* --- Unused group

@defgroup experimental Experimental Structures and Algorithms

This group describes some Experimental structures and algorithms.

The stuff here is subject to change.

*/

/**

\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.

It order to compile them, use <tt>--enable-demo</tt> configure option when

build the library.

*/

/**

@defgroup tools Standalone utility applications

Some utility applications are listed here.

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

them, as well.

*/

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

 *

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

 *

 * Copyright (C) 2003-2008

 * 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 {

/*!

The \e LGF is a <em>column oriented</em>

file format for storing graphs and associated data like

node and edge maps.

Each line with \c '#' first non-whitespace

character is considered as a comment line.

Otherwise the file consists of sections starting with

a header line. The header lines starts with an \c '@' character followed by the

type of section. The standard section types are \c \@nodes, \c

\@arcs and \c \@edges

and \@attributes. Each header line may also have an optional

\e name, which can be use to distinguish the sections of the same

type.

The standard sections are column oriented, each line consists of

<em>token</em>s separated by whitespaces. A token can be \e plain or

\e quoted. A plain token is just a sequence of non-whitespace characters,

while a quoted token is a

character sequence surrounded by double quotes, and it can also

contain whitespaces and escape sequences.

The \c \@nodes section describes a set of nodes and associated

maps. The first is a header line, its columns are the names of the

maps appearing in the following lines.

One of the maps must be called \c

"label", which plays special role in the file.

The following

non-empty lines until the next section describes nodes of the

graph. Each line contains the values of the node maps

associated to the current node.

\code

 @nodes

 label  coordinates  size    title

 1      (10,20)      10      "First node"

 2      (80,80)      8       "Second node"

 3      (40,10)      10      "Third node"

\endcode

The \c \@arcs section is very similar to the \c \@nodes section,

it again starts with a header line describing the names of the maps,

but the \c "label" map is not obligatory here. The following lines

describe the arcs. The first two tokens of each line are

the source and the target node of the arc, respectively, then come the map

values. The source and target tokens must be node labels.

\code

 @arcs

         capacity

 1   2   16

 1   3   12

 2   3   18

\endcode

The \c \@edges is just a synonym of \c \@arcs. The @arcs section can

also store the edge set of an undirected graph. In such case there is

a conventional method for store arc maps in the file, if two columns

has the same caption with \c '+' and \c '-' prefix, then these columns

can be regarded as the values of an arc map.

The \c \@attributes section contains key-value pairs, each line

consists of two tokens, an attribute name, and then an attribute

value. The value of the attribute could be also a label value of a

node or an edge, or even an edge label prefixed with \c '+' or \c '-',

which regards to the forward or backward directed arc of the

corresponding edge.

\code

 @attributes

 source 1

 target 3

 caption "LEMON test digraph"

\endcode

The \e LGF can contain extra sections, but there is no restriction on

the format of such sections.

*/

}

//  LocalWords:  whitespace whitespaces

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

 *

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

 *

 * Copyright (C) 2003-2008

 * 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.

 *

 */

#ifndef LEMON_BITS_ALTERATION_NOTIFIER_H

#define LEMON_BITS_ALTERATION_NOTIFIER_H

#include <vector>

#include <list>

#include <lemon/core.h>

///\ingroup graphbits

///\file

///\brief Observer notifier for graph alteration observers.

namespace lemon {

  /// \ingroup graphbits

  ///

  /// \brief Notifier class to notify observes about alterations in

  /// a container.

  ///

  /// The simple graph's can be refered as two containers, one node container

  /// and one edge container. But they are not standard containers they

  /// does not store values directly they are just key continars for more

  /// value containers which are the node and edge maps.

  ///

  /// The graph's node and edge sets can be changed as we add or erase

  /// that the node and edge maps should contain values for all nodes or

  /// edges. If we want to check on every indicing if the map contains

  /// the current indicing key that cause a drawback in the performance

  /// in the library. We use another solution we notify all maps about

  /// an alteration in the graph, which cause only drawback on the

  /// alteration of the graph.

  ///

  /// This class provides an interface to the container. The \e first() and \e

  /// next() member functions make possible to iterate on the keys of the

  /// container. The \e id() function returns an integer id for each key.

  /// The \e maxId() function gives back an upper bound of the ids.

  ///

  /// For the proper functonality of this class, we should notify it

  /// about each alteration in the container. The alterations have four type

  /// as \e add(), \e erase(), \e build() and \e clear(). The \e add() and

  /// \e erase() signals that only one or few items added or erased to or

  /// from the graph. If all items are erased from the graph or from an empty

  /// graph a new graph is builded then it can be signaled with the

  /// clear() and build() members. Important rule that if we erase items

  /// from graph we should first signal the alteration and after that erase

  /// them from the container, on the other way on item addition we should

  /// first extend the container and just after that signal the alteration.

  ///

  /// The alteration can be observed with a class inherited from the

  /// \e ObserverBase nested class. The signals can be handled with

  /// overriding the virtual functions defined in the base class.  The

  /// observer base can be attached to the notifier with the

  /// \e attach() member and can be detached with detach() function. The

  /// alteration handlers should not call any function which signals

  /// an other alteration in the same notifier and should not

  /// detach any observer from the notifier.

  ///

  /// Alteration observers try to be exception safe. If an \e add() or

  /// a \e clear() function throws an exception then the remaining

  /// observeres will not be notified and the fulfilled additions will

  /// be rolled back by calling the \e erase() or \e clear()

  /// functions. Thence the \e erase() and \e clear() should not throw

  /// exception. Actullay, it can be throw only

  /// \ref AlterationObserver::ImmediateDetach ImmediateDetach

  /// exception which detach the observer from the notifier.

  ///

  /// There are some place when the alteration observing is not completly

  /// reliable. If we want to carry out the node degree in the graph

  /// as in the \ref InDegMap and we use the reverseEdge that cause

  /// unreliable functionality. Because the alteration observing signals

  /// only erasing and adding but not the reversing it will stores bad

  /// degrees. The sub graph adaptors cannot signal the alterations because

  /// just a setting in the filter map can modify the graph and this cannot

  /// be watched in any way.

  ///

  /// \param _Container The container which is observed.

  /// \param _Item The item type which is obserbved.

  template <typename _Container, typename _Item>

  class AlterationNotifier {

  public:

    typedef True Notifier;

    typedef _Container Container;

    typedef _Item Item;

    /// \brief Exception which can be called from \e clear() and

    /// \e erase().

    ///

    /// From the \e clear() and \e erase() function only this

    /// exception is allowed to throw. The exception immediatly

    /// detaches the current observer from the notifier. Because the

    /// \e clear() and \e erase() should not throw other exceptions

    /// it can be used to invalidate the observer.

    struct ImmediateDetach {};

    /// \brief ObserverBase is the base class for the observers.

    ///

    /// ObserverBase is the abstract base class for the observers.

    /// It will be notified about an item was inserted into or

    /// erased from the graph.

    ///

    /// The observer interface contains some pure virtual functions

    /// to override. The add() and erase() functions are

    /// to notify the oberver when one item is added or

    /// erased.

    ///

    /// The build() and clear() members are to notify the observer

    /// about the container is built from an empty container or

    /// is cleared to an empty container.

    class ObserverBase {

    protected:

      typedef AlterationNotifier Notifier;

      friend class AlterationNotifier;

      /// \brief Default constructor.

      ///

      /// Default constructor for ObserverBase.

      ///

      ObserverBase() : _notifier(0) {}

      /// \brief Constructor which attach the observer into notifier.

      ///

      /// Constructor which attach the observer into notifier.

      ObserverBase(AlterationNotifier& nf) {

        attach(nf);

      }

      /// \brief Constructor which attach the obserever to the same notifier.

      ///

      /// Constructor which attach the obserever to the same notifier as

      /// the other observer is attached to.

      ObserverBase(const ObserverBase& copy) {

        if (copy.attached()) {

          attach(*copy.notifier());

        }

      }

      /// \brief Destructor

      virtual ~ObserverBase() {

        if (attached()) {

          detach();

        }

      }

      /// \brief Attaches the observer into an AlterationNotifier.

      ///

      /// This member attaches the observer into an AlterationNotifier.

      ///

      void attach(AlterationNotifier& nf) {

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

 *

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

 *

 * Copyright (C) 2003-2008

 * 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.

 *

 */

///\ingroup concept

///\file

///\brief Classes for representing paths in digraphs.

///

///\todo Iterators have obsolete style

#ifndef LEMON_CONCEPT_PATH_H

#define LEMON_CONCEPT_PATH_H

#include <lemon/core.h>

#include <lemon/concept_check.h>

namespace lemon {

  namespace concepts {

    /// \addtogroup concept

    /// @{

    /// \brief A skeleton structure for representing directed paths in

    /// a digraph.

    ///

    /// A skeleton structure for representing directed paths in a

    /// digraph.

    /// \tparam _Digraph The digraph type in which the path is.

    ///

    /// In a sense, the path can be treated as a list of arcs. The

    /// lemon path type stores just this list. As a consequence it

    /// cannot enumerate the nodes in the path and the zero length

    /// paths cannot store the source.

    ///

    template <typename _Digraph>

    class Path {

    public:

      /// Type of the underlying digraph.

      typedef _Digraph Digraph;

      /// Arc type of the underlying digraph.

      typedef typename Digraph::Arc Arc;

      class ArcIt;

      /// \brief Default constructor

      Path() {}

      /// \brief Template constructor

      template <typename CPath>

      Path(const CPath& cpath) {}

      /// \brief Template assigment

      template <typename CPath>

      Path& operator=(const CPath& cpath) {}

      /// Length of the path ie. the number of arcs in the path.

      int length() const { return 0;}

      /// Returns whether the path is empty.

      bool empty() const { return true;}

      /// Resets the path to an empty path.

      void clear() {}

      ///

      /// This class is used to iterate on the arcs of the paths.

      class ArcIt {

      public:

        /// Default constructor

        ArcIt() {}

        /// Invalid constructor

        ArcIt(Invalid) {}

        /// Constructor for first arc

        ArcIt(const Path &) {}

        /// Conversion to Arc

        operator Arc() const { return INVALID; }

        /// Next arc

        ArcIt& operator++() {return *this;}

        /// Comparison operator

        bool operator==(const ArcIt&) const {return true;}

        /// Comparison operator

        bool operator!=(const ArcIt&) const {return true;}

        /// Comparison operator

        bool operator<(const ArcIt&) const {return false;}

      };

      template <typename _Path>

      struct Constraints {

        void constraints() {

          Path<Digraph> pc;

          _Path p, pp(pc);

          int l = p.length();

          int e = p.empty();

          p.clear();

          p = pc;

          typename _Path::ArcIt id, ii(INVALID), i(p);

          ++i;

          typename Digraph::Arc ed = i;

          e = (i == ii);

          e = (i != ii);

          e = (i < ii);

          ignore_unused_variable_warning(l);

          ignore_unused_variable_warning(pp);

          ignore_unused_variable_warning(e);

          ignore_unused_variable_warning(id);

          ignore_unused_variable_warning(ii);

          ignore_unused_variable_warning(ed);

        }

      };

    };

    namespace _path_bits {

      template <typename _Digraph, typename _Path, typename RevPathTag = void>

      struct PathDumperConstraints {

        void constraints() {

          int l = p.length();

          int e = p.empty();

          typename _Path::ArcIt id, i(p);

          ++i;

          typename _Digraph::Arc ed = i;

          e = (i == INVALID);

          e = (i != INVALID);

          ignore_unused_variable_warning(l);

          ignore_unused_variable_warning(e);

          ignore_unused_variable_warning(id);

          ignore_unused_variable_warning(ed);

        }

        _Path& p;

      };

      template <typename _Digraph, typename _Path>

      struct PathDumperConstraints<

        _Digraph, _Path,

        typename enable_if<typename _Path::RevPathTag, void>::type

      > {

        void constraints() {

          int l = p.length();

          int e = p.empty();

          typename _Path::RevArcIt id, i(p);

          ++i;

          typename _Digraph::Arc ed = i;

          e = (i == INVALID);

          e = (i != INVALID);

          ignore_unused_variable_warning(l);

          ignore_unused_variable_warning(e);

          ignore_unused_variable_warning(id);

          ignore_unused_variable_warning(ed);

        }

        _Path& p;

      };

    }

    /// \brief A skeleton structure for path dumpers.

    ///

    /// A skeleton structure for path dumpers. The path dumpers are

    /// the generalization of the paths. The path dumpers can

    /// enumerate the arcs of the path wheter in forward or in

    /// backward order.  In most time these classes are not used

    /// directly rather it used to assign a dumped class to a real

    /// path type.

    ///

    /// The main purpose of this concept is that the shortest path

    /// algorithms can enumerate easily the arcs in reverse order.

    /// If we would like to give back a real path from these

    /// algorithms then we should create a temporarly path object. In