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

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 matrices Matrices

@ingroup datas

\brief Two dimensional data storages implemented in LEMON.

This group contains two dimensional data storages implemented in LEMON.

*/

/**

@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 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.

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]

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

also provide 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,

/* -*- 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_EDGE_SET_H

#define LEMON_EDGE_SET_H

#include <lemon/core.h>

#include <lemon/bits/edge_set_extender.h>

/// \file

/// \brief ArcSet and EdgeSet classes.

///

/// Graphs which use another graph's node-set as own.

namespace lemon {

  template <typename GR>

  class ListArcSetBase {

  public:

    typedef typename GR::Node Node;

    typedef typename GR::NodeIt NodeIt;

  protected:

    struct NodeT {

      int first_out, first_in;

      NodeT() : first_out(-1), first_in(-1) {}

    };

    typedef typename ItemSetTraits<GR, Node>::

    template Map<NodeT>::Type NodesImplBase;

    NodesImplBase* _nodes;

    struct ArcT {

      Node source, target;

      int next_out, next_in;

      int prev_out, prev_in;

      ArcT() : prev_out(-1), prev_in(-1) {}

    };

    std::vector<ArcT> arcs;

    int first_arc;

    int first_free_arc;

    const GR* _graph;

    void initalize(const GR& graph, NodesImplBase& nodes) {

      _graph = &graph;

      _nodes = &nodes;

    }

  public:

    class Arc {

      friend class ListArcSetBase<GR>;

    protected:

      Arc(int _id) : id(_id) {}

      int id;

    public:

      Arc() {}

      Arc(Invalid) : id(-1) {}

      bool operator==(const Arc& arc) const { return id == arc.id; }

      bool operator!=(const Arc& arc) const { return id != arc.id; }

      bool operator<(const Arc& arc) const { return id < arc.id; }

    };

    ListArcSetBase() : first_arc(-1), first_free_arc(-1) {}

    Arc addArc(const Node& u, const Node& v) {

      int n;

      if (first_free_arc == -1) {

        n = arcs.size();

        arcs.push_back(ArcT());

      } else {

        n = first_free_arc;

        first_free_arc = arcs[first_free_arc].next_in;

      }

      arcs[n].next_in = (*_nodes)[v].first_in;

      if ((*_nodes)[v].first_in != -1) {

        arcs[(*_nodes)[v].first_in].prev_in = n;

      }

      (*_nodes)[v].first_in = n;

      arcs[n].next_out = (*_nodes)[u].first_out;

      if ((*_nodes)[u].first_out != -1) {

        arcs[(*_nodes)[u].first_out].prev_out = n;

      }

      (*_nodes)[u].first_out = n;

      arcs[n].source = u;

      arcs[n].target = v;

      return Arc(n);

    }

    void erase(const Arc& arc) {

      int n = arc.id;

      if (arcs[n].prev_in != -1) {

        arcs[arcs[n].prev_in].next_in = arcs[n].next_in;

      } else {

        (*_nodes)[arcs[n].target].first_in = arcs[n].next_in;

      }

      if (arcs[n].next_in != -1) {

        arcs[arcs[n].next_in].prev_in = arcs[n].prev_in;

      }

      if (arcs[n].prev_out != -1) {

        arcs[arcs[n].prev_out].next_out = arcs[n].next_out;

      } else {

        (*_nodes)[arcs[n].source].first_out = arcs[n].next_out;

      }

      if (arcs[n].next_out != -1) {

        arcs[arcs[n].next_out].prev_out = arcs[n].prev_out;

      }

    }

    void clear() {

      Node node;

      for (first(node); node != INVALID; next(node)) {

        (*_nodes)[node].first_in = -1;

        (*_nodes)[node].first_out = -1;

      }

      arcs.clear();

      first_arc = -1;

      first_free_arc = -1;

    }

    void first(Node& node) const {

      _graph->first(node);

    }

    void next(Node& node) const {

      _graph->next(node);

    }

    void first(Arc& arc) const {

      Node node;

      first(node);

      while (node != INVALID && (*_nodes)[node].first_in == -1) {

        next(node);

      }

      arc.id = (node == INVALID) ? -1 : (*_nodes)[node].first_in;

    }

    void next(Arc& arc) const {

      if (arcs[arc.id].next_in != -1) {

        arc.id = arcs[arc.id].next_in;

      } else {

        Node node = arcs[arc.id].target;

        next(node);

        while (node != INVALID && (*_nodes)[node].first_in == -1) {

          next(node);

        }

        arc.id = (node == INVALID) ? -1 : (*_nodes)[node].first_in;

      }

    }

    void firstOut(Arc& arc, const Node& node) const {

      arc.id = (*_nodes)[node].first_out;

    }

    void nextOut(Arc& arc) const {

      arc.id = arcs[arc.id].next_out;

    }

    void firstIn(Arc& arc, const Node& node) const {

      arc.id = (*_nodes)[node].first_in;

    }

    void nextIn(Arc& arc) const {

      arc.id = arcs[arc.id].next_in;

    }

    int id(const Node& node) const { return _graph->id(node); }

    int id(const Arc& arc) const { return arc.id; }

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

    Arc arcFromId(int ix) const { return Arc(ix); }

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

    int maxArcId() const { return arcs.size() - 1; }

    Node source(const Arc& arc) const { return arcs[arc.id].source;}

    Node target(const Arc& arc) const { return arcs[arc.id].target;}

    typedef typename ItemSetTraits<GR, Node>::ItemNotifier NodeNotifier;

    NodeNotifier& notifier(Node) const {

      return _graph->notifier(Node());

    }

    template <typename V>

    class NodeMap : public GR::template NodeMap<V> {

      typedef typename GR::template NodeMap<V> Parent;

    public:

      explicit NodeMap(const ListArcSetBase<GR>& arcset)

        : Parent(*arcset._graph) {}

      NodeMap(const ListArcSetBase<GR>& arcset, const V& value)

        : Parent(*arcset._graph, value) {}

      NodeMap& operator=(const NodeMap& cmap) {

        return operator=<NodeMap>(cmap);

      }

      template <typename CMap>

      NodeMap& operator=(const CMap& cmap) {

        Parent::operator=(cmap);

        return *this;

      }

    };

  };

  ///

  /// \brief Digraph using a node set of another digraph or graph and

  /// an own arc set.

  ///

  /// This structure can be used to establish another directed graph

  /// over a node set of an existing one. This class uses the same

  /// Node type as the underlying graph, and each valid node of the

  /// original graph is valid in this arc set, therefore the node

  /// objects of the original graph can be used directly with this

  /// class. The node handling functions (id handling, observing, and

  /// iterators) works equivalently as in the original graph.

  ///

  /// This implementation is based on doubly-linked lists, from each

  /// node the outgoing and the incoming arcs make up lists, therefore

  /// one arc can be erased in constant time. It also makes possible,

  /// that node can be removed from the underlying graph, in this case

  /// all arcs incident to the given node is erased from the arc set.

  ///

  /// \param GR The type of the graph which shares its node set with

  /// this class. Its interface must conform to the

  /// \ref concepts::Digraph "Digraph" or \ref concepts::Graph "Graph"

  /// concept.

  ///

  /// This class fully conforms to the \ref concepts::Digraph

  /// "Digraph" concept.

  template <typename GR>

  class ListArcSet : public ArcSetExtender<ListArcSetBase<GR> > {

    typedef ArcSetExtender<ListArcSetBase<GR> > Parent;

  public:

    typedef typename Parent::Node Node;

    typedef typename Parent::Arc Arc;

    typedef typename Parent::NodesImplBase NodesImplBase;

    void eraseNode(const Node& node) {

      Arc arc;

      Parent::firstOut(arc, node);

      while (arc != INVALID ) {

        erase(arc);

        Parent::firstOut(arc, node);

      }

      Parent::firstIn(arc, node);

      while (arc != INVALID ) {

        erase(arc);

        Parent::firstIn(arc, node);

      }

    }

    void clearNodes() {

      Parent::clear();

    }

    class NodesImpl : public NodesImplBase {

      typedef NodesImplBase Parent;

    public:

      NodesImpl(const GR& graph, ListArcSet& arcset)

        : Parent(graph), _arcset(arcset) {}

      virtual ~NodesImpl() {}

    protected:

      virtual void erase(const Node& node) {

        _arcset.eraseNode(node);

        Parent::erase(node);

      }

      virtual void erase(const std::vector<Node>& nodes) {

        for (int i = 0; i < int(nodes.size()); ++i) {

          _arcset.eraseNode(nodes[i]);

        }

        Parent::erase(nodes);

      }

      virtual void clear() {

        _arcset.clearNodes();

        Parent::clear();

      }

    private:

      ListArcSet& _arcset;

    };

    NodesImpl _nodes;

  public:

    /// \brief Constructor of the ArcSet.

    ///

    /// Constructor of the ArcSet.

    ListArcSet(const GR& graph) : _nodes(graph, *this) {

      Parent::initalize(graph, _nodes);

    }

    /// \brief Add a new arc to the digraph.

    ///

    /// Add a new arc to the digraph with source node \c s

    /// and target node \c t.

    /// \return The new arc.

    Arc addArc(const Node& s, const Node& t) {

      return Parent::addArc(s, t);

    }

    /// \brief Erase an arc from the digraph.

    ///

    /// Erase an arc \c a from the digraph.

    void erase(const Arc& a) {

      return Parent::erase(a);

    }

  };

  template <typename GR>

  class ListEdgeSetBase {

  public:

    typedef typename GR::Node Node;

    typedef typename GR::NodeIt NodeIt;

  protected:

    struct NodeT {

      int first_out;

      NodeT() : first_out(-1) {}

    };

    typedef typename ItemSetTraits<GR, Node>::

    template Map<NodeT>::Type NodesImplBase;

    NodesImplBase* _nodes;

    struct ArcT {

      Node target;

      int prev_out, next_out;

      ArcT() : prev_out(-1), next_out(-1) {}

    };

    std::vector<ArcT> arcs;

    int first_arc;

    int first_free_arc;

    const GR* _graph;

    void initalize(const GR& graph, NodesImplBase& nodes) {

      _graph = &graph;

      _nodes = &nodes;

    }

  public:

    class Edge {

      friend class ListEdgeSetBase;

    protected:

      int id;

      explicit Edge(int _id) { id = _id;}

    public:

      Edge() {}

      Edge (Invalid) { id = -1; }

      bool operator==(const Edge& arc) const {return id == arc.id;}

      bool operator!=(const Edge& arc) const {return id != arc.id;}

      bool operator<(const Edge& arc) const {return id < arc.id;}

    };

    class Arc {

      friend class ListEdgeSetBase;

    protected:

      Arc(int _id) : id(_id) {}

      int id;

    public:

      operator Edge() const { return edgeFromId(id / 2); }

      Arc() {}

      Arc(Invalid) : id(-1) {}

      bool operator==(const Arc& arc) const { return id == arc.id; }

      bool operator!=(const Arc& arc) const { return id != arc.id; }

      bool operator<(const Arc& arc) const { return id < arc.id; }

    };

    ListEdgeSetBase() : first_arc(-1), first_free_arc(-1) {}

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

      int n;

      if (first_free_arc == -1) {

        n = arcs.size();

        arcs.push_back(ArcT());

        arcs.push_back(ArcT());

        arcs[arcs[n | 1].next_out].prev_out = arcs[n | 1].prev_out;

      }

      if (arcs[n | 1].prev_out != -1) {

        arcs[arcs[n | 1].prev_out].next_out = arcs[n | 1].next_out;

      } else {

        (*_nodes)[arcs[n].target].first_out = arcs[n | 1].next_out;

      }

      arcs[n].next_out = first_free_arc;

      first_free_arc = n;

    }

    void clear() {

      Node node;

      for (first(node); node != INVALID; next(node)) {

        (*_nodes)[node].first_out = -1;

      }

      arcs.clear();

      first_arc = -1;

      first_free_arc = -1;

    }

    void first(Node& node) const {

      _graph->first(node);

    }

    void next(Node& node) const {

      _graph->next(node);

    }

    void first(Arc& arc) const {

      Node node;

      first(node);

      while (node != INVALID && (*_nodes)[node].first_out == -1) {

        next(node);

      }

      arc.id = (node == INVALID) ? -1 : (*_nodes)[node].first_out;

    }

    void next(Arc& arc) const {

      if (arcs[arc.id].next_out != -1) {

        arc.id = arcs[arc.id].next_out;

      } else {

        Node node = arcs[arc.id ^ 1].target;

        next(node);

        while(node != INVALID && (*_nodes)[node].first_out == -1) {

          next(node);

        }

        arc.id = (node == INVALID) ? -1 : (*_nodes)[node].first_out;

      }

    }

    void first(Edge& edge) const {

      Node node;

      first(node);

      while (node != INVALID) {

        edge.id = (*_nodes)[node].first_out;

        while ((edge.id & 1) != 1) {

          edge.id = arcs[edge.id].next_out;

        }

        if (edge.id != -1) {

          edge.id /= 2;

          return;

        }

        next(node);

      }

      edge.id = -1;

    }

    void next(Edge& edge) const {

      Node node = arcs[edge.id * 2].target;

      edge.id = arcs[(edge.id * 2) | 1].next_out;

      while ((edge.id & 1) != 1) {

        edge.id = arcs[edge.id].next_out;

      }

      if (edge.id != -1) {

        edge.id /= 2;

        return;

      }

      next(node);

      while (node != INVALID) {

        edge.id = (*_nodes)[node].first_out;

        while ((edge.id & 1) != 1) {

          edge.id = arcs[edge.id].next_out;

        }

        if (edge.id != -1) {

          edge.id /= 2;

          return;

        }

        next(node);

      }

      edge.id = -1;

    }

    void firstOut(Arc& arc, const Node& node) const {

      arc.id = (*_nodes)[node].first_out;

    }

    void nextOut(Arc& arc) const {

      arc.id = arcs[arc.id].next_out;

    }

    void firstIn(Arc& arc, const Node& node) const {

      arc.id = (((*_nodes)[node].first_out) ^ 1);

      if (arc.id == -2) arc.id = -1;

    }

    void nextIn(Arc& arc) const {

      arc.id = ((arcs[arc.id ^ 1].next_out) ^ 1);

      if (arc.id == -2) arc.id = -1;

    }

    void firstInc(Edge &arc, bool& dir, const Node& node) const {

      int de = (*_nodes)[node].first_out;

      if (de != -1 ) {

        arc.id = de / 2;

        dir = ((de & 1) == 1);

      } else {

        arc.id = -1;

        dir = true;

      }

    }

    void nextInc(Edge &arc, bool& dir) const {

      int de = (arcs[(arc.id * 2) | (dir ? 1 : 0)].next_out);

      if (de != -1 ) {

        arc.id = de / 2;

        dir = ((de & 1) == 1);

      } else {

        arc.id = -1;

        dir = true;

      }

    }

    static bool direction(Arc arc) {

      return (arc.id & 1) == 1;

    }

    static Arc direct(Edge edge, bool dir) {

      return Arc(edge.id * 2 + (dir ? 1 : 0));

    }

    int id(const Node& node) const { return _graph->id(node); }

    static int id(Arc e) { return e.id; }

    static int id(Edge e) { return e.id; }

    Node nodeFromId(int id) const { return _graph->nodeFromId(id); }

    static Arc arcFromId(int id) { return Arc(id);}

    static Edge edgeFromId(int id) { return Edge(id);}

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

    int maxEdgeId() const { return arcs.size() / 2 - 1; }

    int maxArcId() const { return arcs.size()-1; }

    Node source(Arc e) const { return arcs[e.id ^ 1].target; }

    Node target(Arc e) const { return arcs[e.id].target; }

    Node u(Edge e) const { return arcs[2 * e.id].target; }

    Node v(Edge e) const { return arcs[2 * e.id + 1].target; }

    typedef typename ItemSetTraits<GR, Node>::ItemNotifier NodeNotifier;

    NodeNotifier& notifier(Node) const {

      return _graph->notifier(Node());

    }

    template <typename V>

    class NodeMap : public GR::template NodeMap<V> {

      typedef typename GR::template NodeMap<V> Parent;

    public:

      explicit NodeMap(const ListEdgeSetBase<GR>& arcset)

        : Parent(*arcset._graph) {}

      NodeMap(const ListEdgeSetBase<GR>& arcset, const V& value)

        : Parent(*arcset._graph, value) {}

      NodeMap& operator=(const NodeMap& cmap) {

        return operator=<NodeMap>(cmap);

      }

      template <typename CMap>

      NodeMap& operator=(const CMap& cmap) {

        Parent::operator=(cmap);

        return *this;

      }

    };

  };

  ///

  /// \brief Graph using a node set of another digraph or graph and an

  /// own edge set.

  ///

  /// This structure can be used to establish another graph over a

  /// node set of an existing one. This class uses the same Node type

  /// as the underlying graph, and each valid node of the original

  /// graph is valid in this arc set, therefore the node objects of

  /// the original graph can be used directly with this class. The

  /// node handling functions (id handling, observing, and iterators)

  /// works equivalently as in the original graph.

  ///

  /// This implementation is based on doubly-linked lists, from each

  /// node the incident edges make up lists, therefore one edge can be

  /// erased in constant time. It also makes possible, that node can

  /// be removed from the underlying graph, in this case all edges

  /// incident to the given node is erased from the arc set.

  ///

  /// \param GR The type of the graph which shares its node set

  /// with this class. Its interface must conform to the

  /// \ref concepts::Digraph "Digraph" or \ref concepts::Graph "Graph"

  /// concept.

  ///

  /// This class fully conforms to the \ref concepts::Graph "Graph"

  /// concept.

  template <typename GR>

  class ListEdgeSet : public EdgeSetExtender<ListEdgeSetBase<GR> > {

    typedef EdgeSetExtender<ListEdgeSetBase<GR> > Parent;

  public:

    typedef typename Parent::Node Node;

    typedef typename Parent::Arc Arc;

    typedef typename Parent::Edge Edge;

    typedef typename Parent::NodesImplBase NodesImplBase;

    void eraseNode(const Node& node) {

      Arc arc;

      Parent::firstOut(arc, node);

      while (arc != INVALID ) {

        erase(arc);

        Parent::firstOut(arc, node);

      }

    }

    void clearNodes() {

      Parent::clear();

    }

    class NodesImpl : public NodesImplBase {

      typedef NodesImplBase Parent;

    public:

      NodesImpl(const GR& graph, ListEdgeSet& arcset)

        : Parent(graph), _arcset(arcset) {}

      virtual ~NodesImpl() {}

    protected:

      virtual void erase(const Node& node) {

        _arcset.eraseNode(node);

        Parent::erase(node);

      }

      virtual void erase(const std::vector<Node>& nodes) {

        for (int i = 0; i < int(nodes.size()); ++i) {

          _arcset.eraseNode(nodes[i]);

        }

        Parent::erase(nodes);

      }

      virtual void clear() {

        _arcset.clearNodes();

        Parent::clear();

      }

    private:

      ListEdgeSet& _arcset;

    };

    NodesImpl _nodes;

  public:

    /// \brief Constructor of the EdgeSet.

    ///

    /// Constructor of the EdgeSet.

    ListEdgeSet(const GR& graph) : _nodes(graph, *this) {

      Parent::initalize(graph, _nodes);

    }

    /// \brief Add a new edge to the graph.

    ///

    /// Add a new edge to the graph with node \c u

    /// and node \c v endpoints.

    /// \return The new edge.

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

      return Parent::addEdge(u, v);

    }

    /// \brief Erase an edge from the graph.

    ///

    /// Erase the edge \c e from the graph.

    void erase(const Edge& e) {

      return Parent::erase(e);

    }

  };

  template <typename GR>

  class SmartArcSetBase {

  public:

    typedef typename GR::Node Node;

    typedef typename GR::NodeIt NodeIt;

  protected:

    struct NodeT {

      int first_out, first_in;

      NodeT() : first_out(-1), first_in(-1) {}

    };

    typedef typename ItemSetTraits<GR, Node>::

    template Map<NodeT>::Type NodesImplBase;

    NodesImplBase* _nodes;

    struct ArcT {

      Node source, target;

      int next_out, next_in;

      ArcT() {}

    };

    std::vector<ArcT> arcs;

    const GR* _graph;

    void initalize(const GR& graph, NodesImplBase& nodes) {

      _graph = &graph;

      _nodes = &nodes;

    }

  public:

    class Arc {

      friend class SmartArcSetBase<GR>;

    protected:

      Arc(int _id) : id(_id) {}