RhodeCode

/* -*- 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

*/

/**

@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.

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.

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.

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

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 matrices Matrices

@ingroup datas

\brief Two dimensional data storages implemented in LEMON.

*/

/**

@defgroup paths Path Structures

@ingroup datas

\brief %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.

order to make it easier to implement combinatorial algorithms.

*/

/**

@defgroup algs Algorithms

implemented in LEMON.

implemented in LEMON.

*/

/**

@defgroup search Graph Search

@ingroup algs

\brief Common graph search algorithms.

\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.

 - \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 FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms

   for solving the \e all-pairs \e shortest \e paths \e problem 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 max_flow Maximum Flow Algorithms

@ingroup algs

\brief Algorithms for finding maximum flows.

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_{a\in\delta_{out}(s)}f(a) - \sum_{a\in\delta_{in}(s)}f(a) \f]

\f[ \sum_{a\in\delta_{out}(v)} f(a) = \sum_{a\in\delta_{in}(v)} f(a)

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

\f[ 0 \leq f(a) \leq cap(a) \qquad \forall a\in A \f]

LEMON contains several algorithms for solving maximum flow problems:

- \ref EdmondsKarp Edmonds-Karp algorithm.

- \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm.

- \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees.

- \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees.

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

fastest method for computing a maximum flow. All implementations

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

problem of the maximum flow.

*/

/**

@defgroup min_cost_flow Minimum Cost Flow Algorithms

@ingroup algs

\brief Algorithms for finding minimum cost flows and circulations.

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 and arc costs.

Formally, let \f$G=(V,A)\f$ be a digraph,

\f$lower, upper: A\rightarrow\mathbf{Z}^+_0\f$ denote the lower and

upper bounds for the flow values on the arcs,

\f$cost: A\rightarrow\mathbf{Z}^+_0\f$ denotes the cost per unit flow

on the arcs, and

\f$supply: V\rightarrow\mathbf{Z}\f$ denotes the supply/demand values

of the nodes.

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

the following optimization problem.

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

\f[ \sum_{a\in\delta_{out}(v)} f(a) - \sum_{a\in\delta_{in}(v)} f(a) =

    supply(v) \qquad \forall v\in V \f]

\f[ lower(a) \leq f(a) \leq upper(a) \qquad \forall a\in A \f]

LEMON contains several algorithms for solving minimum cost flow problems:

 - \ref CycleCanceling Cycle-canceling algorithms.

 - \ref CapacityScaling Successive shortest path algorithm with optional

   capacity scaling.

 - \ref CostScaling Push-relabel and augment-relabel algorithms based on

   cost scaling.

 - \ref NetworkSimplex Primal network simplex algorithm with various

   pivot strategies.

*/

/**

@defgroup min_cut Minimum Cut Algorithms

@ingroup algs

\brief 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 NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for

  calculating minimum cut in undirected graphs.

  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_prop Connectivity and Other Graph Properties

@ingroup algs

\brief 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

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 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 MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm

  for calculating maximum cardinality matching in bipartite graphs.

- \ref PrBipartiteMatching Push-relabel algorithm

  for calculating maximum cardinality matching in bipartite graphs.

- \ref MaxWeightedBipartiteMatching

  Successive shortest path algorithm for calculating maximum weighted

  matching and maximum weighted bipartite matching in bipartite graphs.

- \ref MinCostMaxBipartiteMatching

  Successive shortest path algorithm for calculating minimum cost maximum

  matching in bipartite graphs.

- \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.

tree in a graph.

*/

/**

@defgroup auxalg Auxiliary Algorithms

@ingroup algs

\brief Auxiliary algorithms implemented in LEMON.

in order to make it easier to implement complex algorithms.

*/

/**

@defgroup approx Approximation Algorithms

@ingroup algs

\brief Approximation algorithms.

implemented in LEMON.

*/

/**

@defgroup gen_opt_group General Optimization Tools

implemented in LEMON.

implemented in LEMON.

*/

/**

@defgroup lp_group Lp and Mip Solvers

@ingroup gen_opt_group

\brief Lp and Mip solver interfaces for LEMON.

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.

*/

/**

@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.

*/

/**

@defgroup misc Miscellaneous Tools

@ingroup utils

\brief Tools for development, debugging and testing.

debugging and testing.

*/

/**

@defgroup timecount Time Measuring and Counting

@ingroup misc

\brief Simple tools for measuring the performance of algorithms.

of algorithms.

*/

/**

@defgroup exceptions Exceptions

@ingroup utils

\brief Exceptions defined in LEMON.

*/

/**

@defgroup io_group Input-Output

\brief Graph Input-Output methods

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.

\ref lgf-format "LEMON Graph Format".

*/

/**

@defgroup eps_io Postscript Exporting

@ingroup io_group

\brief General \c EPS drawer and graph exporter

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

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

graph structures and helper classes used to implement these.

*/

/**

@defgroup map_concepts Map Concepts

@ingroup concept

\brief Skeleton and concept checking classes for 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.

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) 2003-2009

 * 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.

 *

 */

/**

\mainpage LEMON Documentation

\section intro Introduction

\subsection whatis What is LEMON

LEMON stands for

<b>L</b>ibrary of <b>E</b>fficient <b>M</b>odels

and <b>O</b>ptimization in <b>N</b>etworks.

It is a C++ template

library aimed at combinatorial optimization tasks which

often involve in working

with graphs.

<b>

LEMON is an <a class="el" href="http://opensource.org/">open&nbsp;source</a>

project.

You are free to use it in your commercial or

non-commercial applications under very permissive

\ref license "license terms".

</b>

\subsection howtoread How to read the documentation

If you want to get a quick start and see the most important features then

take a look at our \ref quicktour

"Quick Tour to LEMON" which will guide you along.

If you know what you are looking for then try to find it under the

If you are a user of the old (0.x) series of LEMON, please check out the

\ref migration "Migration Guide" for the backward incompatibilities.

*/

    typedef EF EdgeFilterMap;

    typedef GraphAdaptorExtender<

      SubGraphBase<GR, ConstMap<typename GR::Node, Const<bool, true > >, 

                   EF, false> > Parent;

    typedef typename Parent::Edge Edge;

  protected:

    ConstMap<typename GR::Node, Const<bool, true> > const_true_map;

    FilterEdges() : const_true_map(true) {

      Parent::setNodeFilterMap(const_true_map);

    }

  public:

    /// \brief Constructor

    ///

    /// Creates a subgraph for the given graph with the given edge

    /// filter map.

    FilterEdges(GR& graph, EF& edge_filter) 

      : Parent(), const_true_map() {

      Parent::initialize(graph, const_true_map, edge_filter);

    }

    /// \brief Sets the status of the given edge

    ///

    /// This function sets the status of the given edge.

    /// It is done by simply setting the assigned value of \c e

    /// to \c v in the edge filter map.

    void status(const Edge& e, bool v) const { Parent::status(e, v); }

    /// \brief Returns the status of the given edge

    ///

    /// This function returns the status of the given edge.

    /// It is \c true if the given edge is enabled (i.e. not hidden).

    bool status(const Edge& e) const { return Parent::status(e); }

    /// \brief Disables the given edge

    ///

    /// This function disables the given edge in the subgraph,

    /// so the iteration jumps over it.

    /// It is the same as \ref status() "status(e, false)".

    void disable(const Edge& e) const { Parent::status(e, false); }

    /// \brief Enables the given edge

    ///

    /// This function enables the given edge in the subgraph.

    /// It is the same as \ref status() "status(e, true)".

    void enable(const Edge& e) const { Parent::status(e, true); }

  };

  /// \brief Returns a read-only FilterEdges adaptor

  ///

  /// This function just returns a read-only \ref FilterEdges adaptor.

  /// \ingroup graph_adaptors

  /// \relates FilterEdges

  template<typename GR, typename EF>

  FilterEdges<const GR, EF>

  filterEdges(const GR& graph, EF& edge_filter) {

    return FilterEdges<const GR, EF>(graph, edge_filter);

  }

  template<typename GR, typename EF>

  FilterEdges<const GR, const EF>

  filterEdges(const GR& graph, const EF& edge_filter) {

    return FilterEdges<const GR, const EF>(graph, edge_filter);

  }

  template <typename DGR>

  class UndirectorBase {

  public:

    typedef DGR Digraph;

    typedef UndirectorBase Adaptor;

    typedef True UndirectedTag;

    typedef typename Digraph::Arc Edge;

    typedef typename Digraph::Node Node;

    class Arc : public Edge {

      friend class UndirectorBase;

    protected:

      bool _forward;

      Arc(const Edge& edge, bool forward) :

        Edge(edge), _forward(forward) {}

    public:

      Arc() {}

      Arc(Invalid) : Edge(INVALID), _forward(true) {}

      bool operator==(const Arc &other) const {

        return _forward == other._forward &&

          static_cast<const Edge&>(*this) == static_cast<const Edge&>(other);

      }

      bool operator!=(const Arc &other) const {

        return _forward != other._forward ||

          static_cast<const Edge&>(*this) != static_cast<const Edge&>(other);

      }

      bool operator<(const Arc &other) const {

        return _forward < other._forward ||

          (_forward == other._forward &&

           static_cast<const Edge&>(*this) < static_cast<const Edge&>(other));

      }

    };

    void first(Node& n) const {

      _digraph->first(n);

    }

    void next(Node& n) const {

      _digraph->next(n);

    }

    void first(Arc& a) const {

      _digraph->first(a);

      a._forward = true;

    }

    void next(Arc& a) const {

      if (a._forward) {

        a._forward = false;

      } else {

        _digraph->next(a);

        a._forward = true;

      }

    }

    void first(Edge& e) const {

      _digraph->first(e);

    }

    void next(Edge& e) const {

      _digraph->next(e);

    }

    void firstOut(Arc& a, const Node& n) const {

      _digraph->firstIn(a, n);

      if( static_cast<const Edge&>(a) != INVALID ) {

        a._forward = false;

      } else {

        _digraph->firstOut(a, n);

        a._forward = true;

      }

    }

    void nextOut(Arc &a) const {

      if (!a._forward) {

        Node n = _digraph->target(a);

        _digraph->nextIn(a);

        if (static_cast<const Edge&>(a) == INVALID ) {

          _digraph->firstOut(a, n);

          a._forward = true;

        }

      }

      else {

        _digraph->nextOut(a);

      }

    }

    void firstIn(Arc &a, const Node &n) const {

      _digraph->firstOut(a, n);

      if (static_cast<const Edge&>(a) != INVALID ) {

        a._forward = false;

      } else {

        _digraph->firstIn(a, n);

        a._forward = true;

      }

    }

    void nextIn(Arc &a) const {

      if (!a._forward) {

        Node n = _digraph->source(a);

        _digraph->nextOut(a);

        if( static_cast<const Edge&>(a) == INVALID ) {

          _digraph->firstIn(a, n);

          a._forward = true;

        }

      }

      else {

        _digraph->nextIn(a);

      }

    }

    void firstInc(Edge &e, bool &d, const Node &n) const {

      d = true;

      _digraph->firstOut(e, n);

      if (e != INVALID) return;

      d = false;

      _digraph->firstIn(e, n);

    }

    void nextInc(Edge &e, bool &d) const {

      if (d) {

        Node s = _digraph->source(e);

        _digraph->nextOut(e);

        if (e != INVALID) return;

        d = false;

        _digraph->firstIn(e, s);

      } else {

        _digraph->nextIn(e);

      }

    }

    Node u(const Edge& e) const {

      return _digraph->source(e);

    }

    Node v(const Edge& e) const {

      return _digraph->target(e);

    }

    Node source(const Arc &a) const {

      return a._forward ? _digraph->source(a) : _digraph->target(a);

    }

    Node target(const Arc &a) const {

      return a._forward ? _digraph->target(a) : _digraph->source(a);

    }

    static Arc direct(const Edge &e, bool d) {

      return Arc(e, d);

    }

    Arc direct(const Edge &e, const Node& n) const {

      return Arc(e, _digraph->source(e) == n);

    }

    static bool direction(const Arc &a) { return a._forward; }

    Node nodeFromId(int ix) const { return _digraph->nodeFromId(ix); }

    Arc arcFromId(int ix) const {

      return direct(_digraph->arcFromId(ix >> 1), bool(ix & 1));

    }

    Edge edgeFromId(int ix) const { return _digraph->arcFromId(ix); }

    int id(const Node &n) const { return _digraph->id(n); }

    int id(const Arc &a) const {

      return  (_digraph->id(a) << 1) | (a._forward ? 1 : 0);

    }

    int id(const Edge &e) const { return _digraph->id(e); }

    int maxNodeId() const { return _digraph->maxNodeId(); }

    int maxArcId() const { return (_digraph->maxArcId() << 1) | 1; }

    int maxEdgeId() const { return _digraph->maxArcId(); }

    Node addNode() { return _digraph->addNode(); }

    Edge addEdge(const Node& u, const Node& v) {

      return _digraph->addArc(u, v);

    }

    void erase(const Node& i) { _digraph->erase(i); }

    void erase(const Edge& i) { _digraph->erase(i); }

    void clear() { _digraph->clear(); }

    typedef NodeNumTagIndicator<Digraph> NodeNumTag;

    int nodeNum() const { return _digraph->nodeNum(); }

    typedef ArcNumTagIndicator<Digraph> ArcNumTag;

    int arcNum() const { return 2 * _digraph->arcNum(); }

    typedef ArcNumTag EdgeNumTag;

    int edgeNum() const { return _digraph->arcNum(); }

    typedef FindArcTagIndicator<Digraph> FindArcTag;

    Arc findArc(Node s, Node t, Arc p = INVALID) const {

      if (p == INVALID) {

        Edge arc = _digraph->findArc(s, t);

        if (arc != INVALID) return direct(arc, true);

        arc = _digraph->findArc(t, s);

        if (arc != INVALID) return direct(arc, false);

      } else if (direction(p)) {

        Edge arc = _digraph->findArc(s, t, p);

        if (arc != INVALID) return direct(arc, true);

        arc = _digraph->findArc(t, s);

        if (arc != INVALID) return direct(arc, false);

      } else {

        Edge arc = _digraph->findArc(t, s, p);

        if (arc != INVALID) return direct(arc, false);

      }

      return INVALID;

    }

    typedef FindArcTag FindEdgeTag;

    Edge findEdge(Node s, Node t, Edge p = INVALID) const {

      if (s != t) {

        if (p == INVALID) {

          Edge arc = _digraph->findArc(s, t);

          if (arc != INVALID) return arc;

          arc = _digraph->findArc(t, s);

          if (arc != INVALID) return arc;

        } else if (_digraph->source(p) == s) {

          Edge arc = _digraph->findArc(s, t, p);

          if (arc != INVALID) return arc;

          arc = _digraph->findArc(t, s);

          if (arc != INVALID) return arc;

        } else {

          Edge arc = _digraph->findArc(t, s, p);

          if (arc != INVALID) return arc;

        }

      } else {

        return _digraph->findArc(s, t, p);

      }