/* -*- C++ -*-

  *

  * 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_GRAPH_UTILS_H

 #define LEMON_GRAPH_UTILS_H

 #include <iterator>

 #include <vector>

 #include <map>

 #include <cmath>

 #include <algorithm>

 #include <lemon/bits/invalid.h>

 #include <lemon/bits/utility.h>

 #include <lemon/maps.h>

 #include <lemon/bits/traits.h>

 #include <lemon/bits/alteration_notifier.h>

 #include <lemon/bits/default_map.h>

 ///\ingroup gutils

 ///\file

 ///\brief Digraph utilities.

 namespace lemon {

   /// \addtogroup gutils

   /// @{

   ///Creates convenience typedefs for the digraph types and iterators

   ///This \c \#define creates convenience typedefs for the following types

   ///of \c Digraph: \c Node,  \c NodeIt, \c Arc, \c ArcIt, \c InArcIt,

   ///\c OutArcIt

   ///\note If \c G it a template parameter, it should be used in this way.

   ///\code

   ///  GRAPH_TYPEDEFS(typename G);

   ///\endcode

   ///

   ///\warning There are no typedefs for the digraph maps because of the lack of

   ///template typedefs in C++.

 #define GRAPH_TYPEDEFS(Digraph)				\

   typedef Digraph::     Node      Node;			\

     typedef Digraph::   NodeIt    NodeIt;			\

     typedef Digraph::   Arc      Arc;			\

     typedef Digraph::   ArcIt    ArcIt;			\

     typedef Digraph:: InArcIt  InArcIt;			\

     typedef Digraph::OutArcIt OutArcIt

   ///Creates convenience typedefs for the graph types and iterators

   ///This \c \#define creates the same convenience typedefs as defined by

   ///\ref GRAPH_TYPEDEFS(Digraph) and three more, namely it creates

   ///\c Edge, \c EdgeIt, \c IncArcIt,

   ///

   ///\note If \c G it a template parameter, it should be used in this way.

   ///\code

   ///  UGRAPH_TYPEDEFS(typename G);

   ///\endcode

   ///

   ///\warning There are no typedefs for the digraph maps because of the lack of

   ///template typedefs in C++.

 #define UGRAPH_TYPEDEFS(Digraph)				\

   GRAPH_TYPEDEFS(Digraph);				\

     typedef Digraph:: Edge   Edge;			\

     typedef Digraph:: EdgeIt EdgeIt;			\

     typedef Digraph:: IncArcIt   IncArcIt

   ///\brief Creates convenience typedefs for the bipartite digraph 

   ///types and iterators

   ///This \c \#define creates the same convenience typedefs as defined by

   ///\ref UGRAPH_TYPEDEFS(Digraph) and two more, namely it creates

   ///\c RedIt, \c BlueIt, 

   ///

   ///\note If \c G it a template parameter, it should be used in this way.

   ///\code

   ///  BPUGRAPH_TYPEDEFS(typename G);

   ///\endcode

   ///

   ///\warning There are no typedefs for the digraph maps because of the lack of

   ///template typedefs in C++.

 #define BPUGRAPH_TYPEDEFS(Digraph)            \

   UGRAPH_TYPEDEFS(Digraph);		    \

     typedef Digraph::Red Red;             \

     typedef Digraph::Blue Blue;             \

     typedef Digraph::RedIt RedIt;	    \

     typedef Digraph::BlueIt BlueIt

   /// \brief Function to count the items in the digraph.

   ///

   /// This function counts the items (nodes, arcs etc) in the digraph.

   /// The complexity of the function is O(n) because

   /// it iterates on all of the items.

   template <typename Digraph, typename Item>

   inline int countItems(const Digraph& g) {

     typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;

     int num = 0;

     for (ItemIt it(g); it != INVALID; ++it) {

       ++num;

     }

     return num;

   }

   // Node counting:

   namespace _digraph_utils_bits {

     template <typename Digraph, typename Enable = void>

     struct CountNodesSelector {

       static int count(const Digraph &g) {

         return countItems<Digraph, typename Digraph::Node>(g);

       }

     };

     template <typename Digraph>

     struct CountNodesSelector<

       Digraph, typename 

       enable_if<typename Digraph::NodeNumTag, void>::type> 

     {

       static int count(const Digraph &g) {

         return g.nodeNum();

       }

     };    

   }

   /// \brief Function to count the nodes in the digraph.

   ///

   /// This function counts the nodes in the digraph.

   /// The complexity of the function is O(n) but for some

   /// digraph structures it is specialized to run in O(1).

   ///

   /// If the digraph contains a \e nodeNum() member function and a 

   /// \e NodeNumTag tag then this function calls directly the member

   /// function to query the cardinality of the node set.

   template <typename Digraph>

   inline int countNodes(const Digraph& g) {

     return _digraph_utils_bits::CountNodesSelector<Digraph>::count(g);

   }

   namespace _digraph_utils_bits {

     template <typename Digraph, typename Enable = void>

     struct CountRedsSelector {

       static int count(const Digraph &g) {

         return countItems<Digraph, typename Digraph::Red>(g);

       }

     };

     template <typename Digraph>

     struct CountRedsSelector<

       Digraph, typename 

       enable_if<typename Digraph::NodeNumTag, void>::type> 

     {

       static int count(const Digraph &g) {

         return g.redNum();

       }

     };    

   }

   /// \brief Function to count the reds in the digraph.

   ///

   /// This function counts the reds in the digraph.

   /// The complexity of the function is O(an) but for some

   /// digraph structures it is specialized to run in O(1).

   ///

   /// If the digraph contains an \e redNum() member function and a 

   /// \e NodeNumTag tag then this function calls directly the member

   /// function to query the cardinality of the A-node set.

   template <typename Digraph>

   inline int countReds(const Digraph& g) {

     return _digraph_utils_bits::CountRedsSelector<Digraph>::count(g);

   }

   namespace _digraph_utils_bits {

     template <typename Digraph, typename Enable = void>

     struct CountBluesSelector {

       static int count(const Digraph &g) {

         return countItems<Digraph, typename Digraph::Blue>(g);

       }

     };

     template <typename Digraph>

     struct CountBluesSelector<

       Digraph, typename 

       enable_if<typename Digraph::NodeNumTag, void>::type> 

     {

       static int count(const Digraph &g) {

         return g.blueNum();

       }

     };    

   }

   /// \brief Function to count the blues in the digraph.

   ///

   /// This function counts the blues in the digraph.

   /// The complexity of the function is O(bn) but for some

   /// digraph structures it is specialized to run in O(1).

   ///

   /// If the digraph contains a \e blueNum() member function and a 

   /// \e NodeNumTag tag then this function calls directly the member

   /// function to query the cardinality of the B-node set.

   template <typename Digraph>

   inline int countBlues(const Digraph& g) {

     return _digraph_utils_bits::CountBluesSelector<Digraph>::count(g);

   }

   // Arc counting:

   namespace _digraph_utils_bits {

     template <typename Digraph, typename Enable = void>

     struct CountArcsSelector {

       static int count(const Digraph &g) {

         return countItems<Digraph, typename Digraph::Arc>(g);

       }

     };

     template <typename Digraph>

     struct CountArcsSelector<

       Digraph, 

       typename enable_if<typename Digraph::ArcNumTag, void>::type> 

     {

       static int count(const Digraph &g) {

         return g.arcNum();

       }

     };    

   }

   /// \brief Function to count the arcs in the digraph.

   ///

   /// This function counts the arcs in the digraph.

   /// The complexity of the function is O(e) but for some

   /// digraph structures it is specialized to run in O(1).

   ///

   /// If the digraph contains a \e arcNum() member function and a 

   /// \e ArcNumTag tag then this function calls directly the member

   /// function to query the cardinality of the arc set.

   template <typename Digraph>

   inline int countArcs(const Digraph& g) {

     return _digraph_utils_bits::CountArcsSelector<Digraph>::count(g);

   }

   // Undirected arc counting:

   namespace _digraph_utils_bits {

     template <typename Digraph, typename Enable = void>

     struct CountEdgesSelector {

       static int count(const Digraph &g) {

         return countItems<Digraph, typename Digraph::Edge>(g);

       }

     };

     template <typename Digraph>

     struct CountEdgesSelector<

       Digraph, 

       typename enable_if<typename Digraph::ArcNumTag, void>::type> 

     {

       static int count(const Digraph &g) {

         return g.edgeNum();

       }

     };    

   }

   /// \brief Function to count the edges in the digraph.

   ///

   /// This function counts the edges in the digraph.

   /// The complexity of the function is O(e) but for some

   /// digraph structures it is specialized to run in O(1).

   ///

   /// If the digraph contains a \e edgeNum() member function and a 

   /// \e ArcNumTag tag then this function calls directly the member

   /// function to query the cardinality of the edge set.

   template <typename Digraph>

   inline int countEdges(const Digraph& g) {

     return _digraph_utils_bits::CountEdgesSelector<Digraph>::count(g);

   }

   template <typename Digraph, typename DegIt>

   inline int countNodeDegree(const Digraph& _g, const typename Digraph::Node& _n) {

     int num = 0;

     for (DegIt it(_g, _n); it != INVALID; ++it) {

       ++num;

     }

     return num;

   }

   /// \brief Function to count the number of the out-arcs from node \c n.

   ///

   /// This function counts the number of the out-arcs from node \c n

   /// in the digraph.  

   template <typename Digraph>

   inline int countOutArcs(const Digraph& _g,  const typename Digraph::Node& _n) {

     return countNodeDegree<Digraph, typename Digraph::OutArcIt>(_g, _n);

   }

   /// \brief Function to count the number of the in-arcs to node \c n.

   ///

   /// This function counts the number of the in-arcs to node \c n

   /// in the digraph.  

   template <typename Digraph>

   inline int countInArcs(const Digraph& _g,  const typename Digraph::Node& _n) {

     return countNodeDegree<Digraph, typename Digraph::InArcIt>(_g, _n);

   }

   /// \brief Function to count the number of the inc-arcs to node \c n.

   ///

   /// This function counts the number of the inc-arcs to node \c n

   /// in the digraph.  

   template <typename Digraph>

   inline int countIncArcs(const Digraph& _g,  const typename Digraph::Node& _n) {

     return countNodeDegree<Digraph, typename Digraph::IncArcIt>(_g, _n);

   }

   namespace _digraph_utils_bits {

     template <typename Digraph, typename Enable = void>

     struct FindArcSelector {

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       static Arc find(const Digraph &g, Node u, Node v, Arc e) {

         if (e == INVALID) {

           g.firstOut(e, u);

         } else {

           g.nextOut(e);

         }

         while (e != INVALID && g.target(e) != v) {

           g.nextOut(e);

         }

         return e;

       }

     };

     template <typename Digraph>

     struct FindArcSelector<

       Digraph, 

       typename enable_if<typename Digraph::FindArcTag, void>::type> 

     {

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Arc Arc;

       static Arc find(const Digraph &g, Node u, Node v, Arc prev) {

         return g.findArc(u, v, prev);

       }

     };    

   }

   /// \brief Finds an arc between two nodes of a digraph.

   ///

   /// Finds an arc from node \c u to node \c v in digraph \c g.

   ///

   /// If \c prev is \ref INVALID (this is the default value), then

   /// it finds the first arc from \c u to \c v. Otherwise it looks for

   /// the next arc from \c u to \c v after \c prev.

   /// \return The found arc or \ref INVALID if there is no such an arc.

   ///

   /// Thus you can iterate through each arc from \c u to \c v as it follows.

   ///\code

   /// for(Arc e=findArc(g,u,v);e!=INVALID;e=findArc(g,u,v,e)) {

   ///   ...

   /// }

   ///\endcode

   ///

   ///\sa ArcLookUp

   ///\sa AllArcLookUp

   ///\sa DynArcLookUp

   ///\sa ConArcIt

   template <typename Digraph>

   inline typename Digraph::Arc 

   findArc(const Digraph &g, typename Digraph::Node u, typename Digraph::Node v,

            typename Digraph::Arc prev = INVALID) {

     return _digraph_utils_bits::FindArcSelector<Digraph>::find(g, u, v, prev);

   }

   /// \brief Iterator for iterating on arcs connected the same nodes.

   ///

   /// Iterator for iterating on arcs connected the same nodes. It is 

   /// higher level interface for the findArc() function. You can

   /// use it the following way:

   ///\code

   /// for (ConArcIt<Digraph> it(g, src, trg); it != INVALID; ++it) {

   ///   ...

   /// }

   ///\endcode

   /// 

   ///\sa findArc()

   ///\sa ArcLookUp

   ///\sa AllArcLookUp

   ///\sa DynArcLookUp

   ///

   /// \author Balazs Dezso 

   template <typename _Digraph>

   class ConArcIt : public _Digraph::Arc {

   public:

     typedef _Digraph Digraph;

     typedef typename Digraph::Arc Parent;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::Node Node;

     /// \brief Constructor.

     ///

     /// Construct a new ConArcIt iterating on the arcs which

     /// connects the \c u and \c v node.

     ConArcIt(const Digraph& g, Node u, Node v) : digraph(g) {

       Parent::operator=(findArc(digraph, u, v));

     }

     /// \brief Constructor.

     ///

     /// Construct a new ConArcIt which continues the iterating from 

     /// the \c e arc.

     ConArcIt(const Digraph& g, Arc e) : Parent(e), digraph(g) {}

     /// \brief Increment operator.

     ///

     /// It increments the iterator and gives back the next arc.

     ConArcIt& operator++() {

       Parent::operator=(findArc(digraph, digraph.source(*this), 

 				 digraph.target(*this), *this));

       return *this;

     }

   private:

     const Digraph& digraph;

   };

   namespace _digraph_utils_bits {

     template <typename Digraph, typename Enable = void>

     struct FindEdgeSelector {

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Edge Edge;

       static Edge find(const Digraph &g, Node u, Node v, Edge e) {

         bool b;

         if (u != v) {

           if (e == INVALID) {

             g.firstInc(e, b, u);

           } else {

             b = g.source(e) == u;

             g.nextInc(e, b);

           }

           while (e != INVALID && (b ? g.target(e) : g.source(e)) != v) {

             g.nextInc(e, b);

           }

         } else {

           if (e == INVALID) {

             g.firstInc(e, b, u);

           } else {

             b = true;

             g.nextInc(e, b);

           }

           while (e != INVALID && (!b || g.target(e) != v)) {

             g.nextInc(e, b);

           }

         }

         return e;

       }

     };

     template <typename Digraph>

     struct FindEdgeSelector<

       Digraph, 

       typename enable_if<typename Digraph::FindArcTag, void>::type> 

     {

       typedef typename Digraph::Node Node;

       typedef typename Digraph::Edge Edge;

       static Edge find(const Digraph &g, Node u, Node v, Edge prev) {

         return g.findEdge(u, v, prev);

       }

     };    

   }

   /// \brief Finds an edge between two nodes of a digraph.

   ///

   /// Finds an edge from node \c u to node \c v in digraph \c g.

   /// If the node \c u and node \c v is equal then each loop arc

   /// will be enumerated.

   ///

   /// If \c prev is \ref INVALID (this is the default value), then

   /// it finds the first arc from \c u to \c v. Otherwise it looks for

   /// the next arc from \c u to \c v after \c prev.

   /// \return The found arc or \ref INVALID if there is no such an arc.

   ///

   /// Thus you can iterate through each arc from \c u to \c v as it follows.

   ///\code

   /// for(Edge e = findEdge(g,u,v); e != INVALID; 

   ///     e = findEdge(g,u,v,e)) {

   ///   ...

   /// }

   ///\endcode

   ///

   ///\sa ConArcIt

   template <typename Digraph>

   inline typename Digraph::Edge 

   findEdge(const Digraph &g, typename Digraph::Node u, typename Digraph::Node v,

             typename Digraph::Edge p = INVALID) {

     return _digraph_utils_bits::FindEdgeSelector<Digraph>::find(g, u, v, p);

   }

   /// \brief Iterator for iterating on edges connected the same nodes.

   ///

   /// Iterator for iterating on edges connected the same nodes. It is 

   /// higher level interface for the findEdge() function. You can

   /// use it the following way:

   ///\code

   /// for (ConEdgeIt<Digraph> it(g, src, trg); it != INVALID; ++it) {

   ///   ...

   /// }

   ///\endcode

   ///

   ///\sa findEdge()

   ///

   /// \author Balazs Dezso 

   template <typename _Digraph>

   class ConEdgeIt : public _Digraph::Edge {

   public:

     typedef _Digraph Digraph;

     typedef typename Digraph::Edge Parent;

     typedef typename Digraph::Edge Edge;

     typedef typename Digraph::Node Node;

     /// \brief Constructor.

     ///

     /// Construct a new ConEdgeIt iterating on the arcs which

     /// connects the \c u and \c v node.

     ConEdgeIt(const Digraph& g, Node u, Node v) : digraph(g) {

       Parent::operator=(findEdge(digraph, u, v));

     }

     /// \brief Constructor.

     ///

     /// Construct a new ConEdgeIt which continues the iterating from 

     /// the \c e arc.

     ConEdgeIt(const Digraph& g, Edge e) : Parent(e), digraph(g) {}

     /// \brief Increment operator.

     ///

     /// It increments the iterator and gives back the next arc.

     ConEdgeIt& operator++() {

       Parent::operator=(findEdge(digraph, digraph.source(*this), 

 				      digraph.target(*this), *this));

       return *this;

     }

   private:

     const Digraph& digraph;

   };

   /// \brief Copy a map.

   ///

   /// This function copies the \c from map to the \c to map. It uses the

   /// given iterator to iterate on the data structure and it uses the \c ref

   /// mapping to convert the from's keys to the to's keys.

   template <typename To, typename From, 

 	    typename ItemIt, typename Ref>	    

   void copyMap(To& to, const From& from, 

 	       ItemIt it, const Ref& ref) {

     for (; it != INVALID; ++it) {

       to[ref[it]] = from[it];

     }

   }

   /// \brief Copy the from map to the to map.

   ///

   /// Copy the \c from map to the \c to map. It uses the given iterator

   /// to iterate on the data structure.

   template <typename To, typename From, typename ItemIt>	    

   void copyMap(To& to, const From& from, ItemIt it) {

     for (; it != INVALID; ++it) {

       to[it] = from[it];

     }

   }

   namespace _digraph_utils_bits {

     template <typename Digraph, typename Item, typename RefMap>

     class MapCopyBase {

     public:

       virtual void copy(const Digraph& from, const RefMap& refMap) = 0;

       virtual ~MapCopyBase() {}

     };

     template <typename Digraph, typename Item, typename RefMap, 

               typename ToMap, typename FromMap>

     class MapCopy : public MapCopyBase<Digraph, Item, RefMap> {

     public:

       MapCopy(ToMap& tmap, const FromMap& map) 

         : _tmap(tmap), _map(map) {}

       virtual void copy(const Digraph& digraph, const RefMap& refMap) {

         typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;

         for (ItemIt it(digraph); it != INVALID; ++it) {

           _tmap.set(refMap[it], _map[it]);

         }

       }

     private:

       ToMap& _tmap;

       const FromMap& _map;

     };

     template <typename Digraph, typename Item, typename RefMap, typename It>

     class ItemCopy : public MapCopyBase<Digraph, Item, RefMap> {

     public:

       ItemCopy(It& it, const Item& item) : _it(it), _item(item) {}

       virtual void copy(const Digraph&, const RefMap& refMap) {

         _it = refMap[_item];

       }

     private:

       It& _it;

       Item _item;

     };

     template <typename Digraph, typename Item, typename RefMap, typename Ref>

     class RefCopy : public MapCopyBase<Digraph, Item, RefMap> {

     public:

       RefCopy(Ref& map) : _map(map) {}

       virtual void copy(const Digraph& digraph, const RefMap& refMap) {

         typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;

         for (ItemIt it(digraph); it != INVALID; ++it) {

           _map.set(it, refMap[it]);

         }

       }

     private:

       Ref& _map;

     };

     template <typename Digraph, typename Item, typename RefMap, 

               typename CrossRef>

     class CrossRefCopy : public MapCopyBase<Digraph, Item, RefMap> {

     public:

       CrossRefCopy(CrossRef& cmap) : _cmap(cmap) {}

       virtual void copy(const Digraph& digraph, const RefMap& refMap) {

         typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;

         for (ItemIt it(digraph); it != INVALID; ++it) {

           _cmap.set(refMap[it], it);

         }

       }

     private:

       CrossRef& _cmap;

     };

     template <typename Digraph, typename Enable = void>

     struct DigraphCopySelector {

       template <typename From, typename NodeRefMap, typename ArcRefMap>

       static void copy(Digraph &to, const From& from,

                        NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {

         for (typename From::NodeIt it(from); it != INVALID; ++it) {

           nodeRefMap[it] = to.addNode();

         }

         for (typename From::ArcIt it(from); it != INVALID; ++it) {

           arcRefMap[it] = to.addArc(nodeRefMap[from.source(it)], 

                                           nodeRefMap[from.target(it)]);

         }

       }

     };

     template <typename Digraph>

     struct DigraphCopySelector<

       Digraph, 

       typename enable_if<typename Digraph::BuildTag, void>::type> 

     {

       template <typename From, typename NodeRefMap, typename ArcRefMap>

       static void copy(Digraph &to, const From& from,

                        NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {

         to.build(from, nodeRefMap, arcRefMap);

       }

     };

     template <typename Graph, typename Enable = void>

     struct GraphCopySelector {

       template <typename From, typename NodeRefMap, typename EdgeRefMap>

       static void copy(Graph &to, const From& from,

                        NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {

         for (typename From::NodeIt it(from); it != INVALID; ++it) {

           nodeRefMap[it] = to.addNode();

         }

         for (typename From::EdgeIt it(from); it != INVALID; ++it) {

           edgeRefMap[it] = to.addArc(nodeRefMap[from.source(it)], 

 				       nodeRefMap[from.target(it)]);

         }

       }

     };

     template <typename Graph>

     struct GraphCopySelector<

       Graph, 

       typename enable_if<typename Graph::BuildTag, void>::type> 

     {

       template <typename From, typename NodeRefMap, typename EdgeRefMap>

       static void copy(Graph &to, const From& from,

                        NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {

         to.build(from, nodeRefMap, edgeRefMap);

       }

     };

     template <typename BpGraph, typename Enable = void>

     struct BpGraphCopySelector {

       template <typename From, typename RedRefMap, 

                 typename BlueRefMap, typename EdgeRefMap>

       static void copy(BpGraph &to, const From& from,

                        RedRefMap& redRefMap, BlueRefMap& blueRefMap,

                        EdgeRefMap& edgeRefMap) {

         for (typename From::RedIt it(from); it != INVALID; ++it) {

           redRefMap[it] = to.addRed();

         }

         for (typename From::BlueIt it(from); it != INVALID; ++it) {

           blueRefMap[it] = to.addBlue();

         }

         for (typename From::EdgeIt it(from); it != INVALID; ++it) {

           edgeRefMap[it] = to.addArc(redRefMap[from.red(it)], 

                                            blueRefMap[from.blue(it)]);

         }

       }

     };

     template <typename BpGraph>

     struct BpGraphCopySelector<

       BpGraph, 

       typename enable_if<typename BpGraph::BuildTag, void>::type> 

     {

       template <typename From, typename RedRefMap, 

                 typename BlueRefMap, typename EdgeRefMap>

       static void copy(BpGraph &to, const From& from,

                        RedRefMap& redRefMap, BlueRefMap& blueRefMap,

                        EdgeRefMap& edgeRefMap) {

         to.build(from, redRefMap, blueRefMap, edgeRefMap);

       }

     };

   }

   /// \brief Class to copy a digraph.

   ///

   /// Class to copy a digraph to another digraph (duplicate a digraph). The

   /// simplest way of using it is through the \c copyDigraph() function.

   template <typename To, typename From>

   class DigraphCopy {

   private:

     typedef typename From::Node Node;

     typedef typename From::NodeIt NodeIt;

     typedef typename From::Arc Arc;

     typedef typename From::ArcIt ArcIt;

     typedef typename To::Node TNode;

     typedef typename To::Arc TArc;

     typedef typename From::template NodeMap<TNode> NodeRefMap;

     typedef typename From::template ArcMap<TArc> ArcRefMap;

   public: 

     /// \brief Constructor for the DigraphCopy.

     ///

     /// It copies the content of the \c _from digraph into the

     /// \c _to digraph.

     DigraphCopy(To& _to, const From& _from) 

       : from(_from), to(_to) {}

     /// \brief Destructor of the DigraphCopy

     ///

     /// Destructor of the DigraphCopy

     ~DigraphCopy() {

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

         delete nodeMapCopies[i];

       }

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

         delete arcMapCopies[i];

       }

     }

     /// \brief Copies the node references into the given map.

     ///

     /// Copies the node references into the given map.

     template <typename NodeRef>

     DigraphCopy& nodeRef(NodeRef& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Node, 

                               NodeRefMap, NodeRef>(map));

       return *this;

     }

     /// \brief Copies the node cross references into the given map.

     ///

     ///  Copies the node cross references (reverse references) into

     ///  the given map.

     template <typename NodeCrossRef>

     DigraphCopy& nodeCrossRef(NodeCrossRef& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Node,

                               NodeRefMap, NodeCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's node type,

     /// and the copied map's key type is the from digraph's node

     /// type.  

     template <typename ToMap, typename FromMap>

     DigraphCopy& nodeMap(ToMap& tmap, const FromMap& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Node, 

                               NodeRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Make a copy of the given node.

     ///

     /// Make a copy of the given node.

     DigraphCopy& node(TNode& tnode, const Node& snode) {

       nodeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Node, 

                               NodeRefMap, TNode>(tnode, snode));

       return *this;

     }

     /// \brief Copies the arc references into the given map.

     ///

     /// Copies the arc references into the given map.

     template <typename ArcRef>

     DigraphCopy& arcRef(ArcRef& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Arc, 

                               ArcRefMap, ArcRef>(map));

       return *this;

     }

     /// \brief Copies the arc cross references into the given map.

     ///

     ///  Copies the arc cross references (reverse references) into

     ///  the given map.

     template <typename ArcCrossRef>

     DigraphCopy& arcCrossRef(ArcCrossRef& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Arc,

                               ArcRefMap, ArcCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's arc type,

     /// and the copied map's key type is the from digraph's arc

     /// type.  

     template <typename ToMap, typename FromMap>

     DigraphCopy& arcMap(ToMap& tmap, const FromMap& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Arc, 

                               ArcRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Make a copy of the given arc.

     ///

     /// Make a copy of the given arc.

     DigraphCopy& arc(TArc& tarc, const Arc& sarc) {

       arcMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Arc, 

                               ArcRefMap, TArc>(tarc, sarc));

       return *this;

     }

     /// \brief Executes the copies.

     ///

     /// Executes the copies.

     void run() {

       NodeRefMap nodeRefMap(from);

       ArcRefMap arcRefMap(from);

       _digraph_utils_bits::DigraphCopySelector<To>::

         copy(to, from, nodeRefMap, arcRefMap);

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

         nodeMapCopies[i]->copy(from, nodeRefMap);

       }

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

         arcMapCopies[i]->copy(from, arcRefMap);

       }      

     }

   protected:

     const From& from;

     To& to;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Node, NodeRefMap>* > 

     nodeMapCopies;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Arc, ArcRefMap>* > 

     arcMapCopies;

   };

   /// \brief Copy a digraph to another digraph.

   ///

   /// Copy a digraph to another digraph.

   /// The usage of the function:

   /// 

   ///\code

   /// copyDigraph(trg, src).nodeRef(nr).arcCrossRef(ecr).run();

   ///\endcode

   /// 

   /// After the copy the \c nr map will contain the mapping from the

   /// nodes of the \c from digraph to the nodes of the \c to digraph and

   /// \c ecr will contain the mapping from the arcs of the \c to digraph

   /// to the arcs of the \c from digraph.

   ///

   /// \see DigraphCopy 

   template <typename To, typename From>

   DigraphCopy<To, From> copyDigraph(To& to, const From& from) {

     return DigraphCopy<To, From>(to, from);

   }

   /// \brief Class to copy an graph.

   ///

   /// Class to copy an graph to another digraph (duplicate a digraph).

   /// The simplest way of using it is through the \c copyGraph() function.

   template <typename To, typename From>

   class GraphCopy {

   private:

     typedef typename From::Node Node;

     typedef typename From::NodeIt NodeIt;

     typedef typename From::Arc Arc;

     typedef typename From::ArcIt ArcIt;

     typedef typename From::Edge Edge;

     typedef typename From::EdgeIt EdgeIt;

     typedef typename To::Node TNode;

     typedef typename To::Arc TArc;

     typedef typename To::Edge TEdge;

     typedef typename From::template NodeMap<TNode> NodeRefMap;

     typedef typename From::template EdgeMap<TEdge> EdgeRefMap;

     struct ArcRefMap {

       ArcRefMap(const To& _to, const From& _from,

                  const EdgeRefMap& _edge_ref, const NodeRefMap& _node_ref) 

         : to(_to), from(_from), 

           edge_ref(_edge_ref), node_ref(_node_ref) {}

       typedef typename From::Arc Key;

       typedef typename To::Arc Value;

       Value operator[](const Key& key) const {

         bool forward = 

           (from.direction(key) == 

            (node_ref[from.source(static_cast<const Edge&>(key))] == 

             to.source(edge_ref[static_cast<const Edge&>(key)])));

 	return to.direct(edge_ref[key], forward); 

       }

       const To& to;

       const From& from;

       const EdgeRefMap& edge_ref;

       const NodeRefMap& node_ref;

     };

   public: 

     /// \brief Constructor for the DigraphCopy.

     ///

     /// It copies the content of the \c _from digraph into the

     /// \c _to digraph.

     GraphCopy(To& _to, const From& _from) 

       : from(_from), to(_to) {}

     /// \brief Destructor of the DigraphCopy

     ///

     /// Destructor of the DigraphCopy

     ~GraphCopy() {

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

         delete nodeMapCopies[i];

       }

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

         delete arcMapCopies[i];

       }

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

         delete edgeMapCopies[i];

       }

     }

     /// \brief Copies the node references into the given map.

     ///

     /// Copies the node references into the given map.

     template <typename NodeRef>

     GraphCopy& nodeRef(NodeRef& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Node, 

                               NodeRefMap, NodeRef>(map));

       return *this;

     }

     /// \brief Copies the node cross references into the given map.

     ///

     ///  Copies the node cross references (reverse references) into

     ///  the given map.

     template <typename NodeCrossRef>

     GraphCopy& nodeCrossRef(NodeCrossRef& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Node,

                               NodeRefMap, NodeCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's node type,

     /// and the copied map's key type is the from digraph's node

     /// type.  

     template <typename ToMap, typename FromMap>

     GraphCopy& nodeMap(ToMap& tmap, const FromMap& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Node, 

                               NodeRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Make a copy of the given node.

     ///

     /// Make a copy of the given node.

     GraphCopy& node(TNode& tnode, const Node& snode) {

       nodeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Node, 

                               NodeRefMap, TNode>(tnode, snode));

       return *this;

     }

     /// \brief Copies the arc references into the given map.

     ///

     /// Copies the arc references into the given map.

     template <typename ArcRef>

     GraphCopy& arcRef(ArcRef& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Arc, 

                               ArcRefMap, ArcRef>(map));

       return *this;

     }

     /// \brief Copies the arc cross references into the given map.

     ///

     ///  Copies the arc cross references (reverse references) into

     ///  the given map.

     template <typename ArcCrossRef>

     GraphCopy& arcCrossRef(ArcCrossRef& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Arc,

                               ArcRefMap, ArcCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's arc type,

     /// and the copied map's key type is the from digraph's arc

     /// type.  

     template <typename ToMap, typename FromMap>

     GraphCopy& arcMap(ToMap& tmap, const FromMap& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Arc, 

                               ArcRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Make a copy of the given arc.

     ///

     /// Make a copy of the given arc.

     GraphCopy& arc(TArc& tarc, const Arc& sarc) {

       arcMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Arc, 

                               ArcRefMap, TArc>(tarc, sarc));

       return *this;

     }

     /// \brief Copies the edge references into the given map.

     ///

     /// Copies the edge references into the given map.

     template <typename EdgeRef>

     GraphCopy& edgeRef(EdgeRef& map) {

       edgeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Edge, 

                                EdgeRefMap, EdgeRef>(map));

       return *this;

     }

     /// \brief Copies the edge cross references into the given map.

     ///

     /// Copies the edge cross references (reverse

     /// references) into the given map.

     template <typename EdgeCrossRef>

     GraphCopy& edgeCrossRef(EdgeCrossRef& map) {

       edgeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, 

                                Edge, EdgeRefMap, EdgeCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's edge type,

     /// and the copied map's key type is the from digraph's edge

     /// type.  

     template <typename ToMap, typename FromMap>

     GraphCopy& edgeMap(ToMap& tmap, const FromMap& map) {

       edgeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Edge, 

                                EdgeRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Make a copy of the given edge.

     ///

     /// Make a copy of the given edge.

     GraphCopy& edge(TEdge& tedge, const Edge& sedge) {

       edgeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Edge, 

                                EdgeRefMap, TEdge>(tedge, sedge));

       return *this;

     }

     /// \brief Executes the copies.

     ///

     /// Executes the copies.

     void run() {

       NodeRefMap nodeRefMap(from);

       EdgeRefMap edgeRefMap(from);

       ArcRefMap arcRefMap(to, from, edgeRefMap, nodeRefMap);

       _digraph_utils_bits::GraphCopySelector<To>::

         copy(to, from, nodeRefMap, edgeRefMap);

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

         nodeMapCopies[i]->copy(from, nodeRefMap);

       }

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

         edgeMapCopies[i]->copy(from, edgeRefMap);

       }

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

         arcMapCopies[i]->copy(from, arcRefMap);

       }

     }

   private:

     const From& from;

     To& to;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Node, NodeRefMap>* > 

     nodeMapCopies;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Arc, ArcRefMap>* > 

     arcMapCopies;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Edge, EdgeRefMap>* > 

     edgeMapCopies;

   };

   /// \brief Copy an graph to another digraph.

   ///

   /// Copy an graph to another digraph.

   /// The usage of the function:

   /// 

   ///\code

   /// copyGraph(trg, src).nodeRef(nr).arcCrossRef(ecr).run();

   ///\endcode

   /// 

   /// After the copy the \c nr map will contain the mapping from the

   /// nodes of the \c from digraph to the nodes of the \c to digraph and

   /// \c ecr will contain the mapping from the arcs of the \c to digraph

   /// to the arcs of the \c from digraph.

   ///

   /// \see GraphCopy 

   template <typename To, typename From>

   GraphCopy<To, From> 

   copyGraph(To& to, const From& from) {

     return GraphCopy<To, From>(to, from);

   }

   /// \brief Class to copy a bipartite digraph.

   ///

   /// Class to copy a bipartite digraph to another digraph

   /// (duplicate a digraph).  The simplest way of using it is through

   /// the \c copyBpGraph() function.

   template <typename To, typename From>

   class BpGraphCopy {

   private:

     typedef typename From::Node Node;

     typedef typename From::Red Red;

     typedef typename From::Blue Blue;

     typedef typename From::NodeIt NodeIt;

     typedef typename From::Arc Arc;

     typedef typename From::ArcIt ArcIt;

     typedef typename From::Edge Edge;

     typedef typename From::EdgeIt EdgeIt;

     typedef typename To::Node TNode;

     typedef typename To::Arc TArc;

     typedef typename To::Edge TEdge;

     typedef typename From::template RedMap<TNode> RedRefMap;

     typedef typename From::template BlueMap<TNode> BlueRefMap;

     typedef typename From::template EdgeMap<TEdge> EdgeRefMap;

     struct NodeRefMap {

       NodeRefMap(const From& _from, const RedRefMap& _red_ref,

                  const BlueRefMap& _blue_ref)

         : from(_from), red_ref(_red_ref), blue_ref(_blue_ref) {}

       typedef typename From::Node Key;

       typedef typename To::Node Value;

       Value operator[](const Key& key) const {

 	return from.red(key) ? red_ref[key] : blue_ref[key]; 

       }

       const From& from;

       const RedRefMap& red_ref;

       const BlueRefMap& blue_ref;

     };

     struct ArcRefMap {

       ArcRefMap(const To& _to, const From& _from,

                  const EdgeRefMap& _edge_ref, const NodeRefMap& _node_ref) 

         : to(_to), from(_from), 

           edge_ref(_edge_ref), node_ref(_node_ref) {}

       typedef typename From::Arc Key;

       typedef typename To::Arc Value;

       Value operator[](const Key& key) const {

         bool forward = 

           (from.direction(key) == 

            (node_ref[from.source(static_cast<const Edge&>(key))] == 

             to.source(edge_ref[static_cast<const Edge&>(key)])));

 	return to.direct(edge_ref[key], forward); 

       }

       const To& to;

       const From& from;

       const EdgeRefMap& edge_ref;

       const NodeRefMap& node_ref;

     };

   public: 

     /// \brief Constructor for the DigraphCopy.

     ///

     /// It copies the content of the \c _from digraph into the

     /// \c _to digraph.

     BpGraphCopy(To& _to, const From& _from) 

       : from(_from), to(_to) {}

     /// \brief Destructor of the DigraphCopy

     ///

     /// Destructor of the DigraphCopy

     ~BpGraphCopy() {

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

         delete redMapCopies[i];

       }

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

         delete blueMapCopies[i];

       }

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

         delete nodeMapCopies[i];

       }

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

         delete arcMapCopies[i];

       }

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

         delete edgeMapCopies[i];

       }

     }

     /// \brief Copies the A-node references into the given map.

     ///

     /// Copies the A-node references into the given map.

     template <typename RedRef>

     BpGraphCopy& redRef(RedRef& map) {

       redMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Red, 

                                RedRefMap, RedRef>(map));

       return *this;

     }

     /// \brief Copies the A-node cross references into the given map.

     ///

     /// Copies the A-node cross references (reverse references) into

     /// the given map.

     template <typename RedCrossRef>

     BpGraphCopy& redCrossRef(RedCrossRef& map) {

       redMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, 

                                Red, RedRefMap, RedCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given A-node map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's node type,

     /// and the copied map's key type is the from digraph's node

     /// type.  

     template <typename ToMap, typename FromMap>

     BpGraphCopy& redMap(ToMap& tmap, const FromMap& map) {

       redMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Red, 

                                RedRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Copies the B-node references into the given map.

     ///

     /// Copies the B-node references into the given map.

     template <typename BlueRef>

     BpGraphCopy& blueRef(BlueRef& map) {

       blueMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Blue, 

                                BlueRefMap, BlueRef>(map));

       return *this;

     }

     /// \brief Copies the B-node cross references into the given map.

     ///

     ///  Copies the B-node cross references (reverse references) into

     ///  the given map.

     template <typename BlueCrossRef>

     BpGraphCopy& blueCrossRef(BlueCrossRef& map) {

       blueMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, 

                               Blue, BlueRefMap, BlueCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given B-node map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's node type,

     /// and the copied map's key type is the from digraph's node

     /// type.  

     template <typename ToMap, typename FromMap>

     BpGraphCopy& blueMap(ToMap& tmap, const FromMap& map) {

       blueMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Blue, 

                                BlueRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Copies the node references into the given map.

     ///

     /// Copies the node references into the given map.

     template <typename NodeRef>

     BpGraphCopy& nodeRef(NodeRef& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Node, 

                               NodeRefMap, NodeRef>(map));

       return *this;

     }

     /// \brief Copies the node cross references into the given map.

     ///

     ///  Copies the node cross references (reverse references) into

     ///  the given map.

     template <typename NodeCrossRef>

     BpGraphCopy& nodeCrossRef(NodeCrossRef& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Node,

                               NodeRefMap, NodeCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's node type,

     /// and the copied map's key type is the from digraph's node

     /// type.  

     template <typename ToMap, typename FromMap>

     BpGraphCopy& nodeMap(ToMap& tmap, const FromMap& map) {

       nodeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Node, 

                               NodeRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Make a copy of the given node.

     ///

     /// Make a copy of the given node.

     BpGraphCopy& node(TNode& tnode, const Node& snode) {

       nodeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Node, 

                               NodeRefMap, TNode>(tnode, snode));

       return *this;

     }

     /// \brief Copies the arc references into the given map.

     ///

     /// Copies the arc references into the given map.

     template <typename ArcRef>

     BpGraphCopy& arcRef(ArcRef& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Arc, 

                               ArcRefMap, ArcRef>(map));

       return *this;

     }

     /// \brief Copies the arc cross references into the given map.

     ///

     ///  Copies the arc cross references (reverse references) into

     ///  the given map.

     template <typename ArcCrossRef>

     BpGraphCopy& arcCrossRef(ArcCrossRef& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Arc,

                               ArcRefMap, ArcCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's arc type,

     /// and the copied map's key type is the from digraph's arc

     /// type.  

     template <typename ToMap, typename FromMap>

     BpGraphCopy& arcMap(ToMap& tmap, const FromMap& map) {

       arcMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Arc, 

                               ArcRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Make a copy of the given arc.

     ///

     /// Make a copy of the given arc.

     BpGraphCopy& arc(TArc& tarc, const Arc& sarc) {

       arcMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Arc, 

                               ArcRefMap, TArc>(tarc, sarc));

       return *this;

     }

     /// \brief Copies the edge references into the given map.

     ///

     /// Copies the edge references into the given map.

     template <typename EdgeRef>

     BpGraphCopy& edgeRef(EdgeRef& map) {

       edgeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Edge, 

                                EdgeRefMap, EdgeRef>(map));

       return *this;

     }

     /// \brief Copies the edge cross references into the given map.

     ///

     /// Copies the edge cross references (reverse

     /// references) into the given map.

     template <typename EdgeCrossRef>

     BpGraphCopy& edgeCrossRef(EdgeCrossRef& map) {

       edgeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, 

                                Edge, EdgeRefMap, EdgeCrossRef>(map));

       return *this;

     }

     /// \brief Make copy of the given map.

     ///

     /// Makes copy of the given map for the newly created digraph. 

     /// The new map's key type is the to digraph's edge type,

     /// and the copied map's key type is the from digraph's edge

     /// type.  

     template <typename ToMap, typename FromMap>

     BpGraphCopy& edgeMap(ToMap& tmap, const FromMap& map) {

       edgeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Edge, 

                                EdgeRefMap, ToMap, FromMap>(tmap, map));

       return *this;

     }

     /// \brief Make a copy of the given edge.

     ///

     /// Make a copy of the given edge.

     BpGraphCopy& edge(TEdge& tedge, const Edge& sedge) {

       edgeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Edge, 

                                EdgeRefMap, TEdge>(tedge, sedge));

       return *this;

     }

     /// \brief Executes the copies.

     ///

     /// Executes the copies.

     void run() {

       RedRefMap redRefMap(from);

       BlueRefMap blueRefMap(from);

       NodeRefMap nodeRefMap(from, redRefMap, blueRefMap);

       EdgeRefMap edgeRefMap(from);

       ArcRefMap arcRefMap(to, from, edgeRefMap, nodeRefMap);

       _digraph_utils_bits::BpGraphCopySelector<To>::

         copy(to, from, redRefMap, blueRefMap, edgeRefMap);

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

         redMapCopies[i]->copy(from, redRefMap);

       }

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

         blueMapCopies[i]->copy(from, blueRefMap);

       }

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

         nodeMapCopies[i]->copy(from, nodeRefMap);

       }

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

         edgeMapCopies[i]->copy(from, edgeRefMap);

       }

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

         arcMapCopies[i]->copy(from, arcRefMap);

       }

     }

   private:

     const From& from;

     To& to;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Red, RedRefMap>* > 

     redMapCopies;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Blue, BlueRefMap>* > 

     blueMapCopies;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Node, NodeRefMap>* > 

     nodeMapCopies;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Arc, ArcRefMap>* > 

     arcMapCopies;

     std::vector<_digraph_utils_bits::MapCopyBase<From, Edge, EdgeRefMap>* > 

     edgeMapCopies;

   };

   /// \brief Copy a bipartite digraph to another digraph.

   ///

   /// Copy a bipartite digraph to another digraph.

   /// The usage of the function:

   /// 

   ///\code

   /// copyBpGraph(trg, src).redRef(anr).arcCrossRef(ecr).run();

   ///\endcode

   /// 

   /// After the copy the \c nr map will contain the mapping from the

   /// nodes of the \c from digraph to the nodes of the \c to digraph and

   /// \c ecr will contain the mapping from the arcs of the \c to digraph

   /// to the arcs of the \c from digraph.

   ///

   /// \see BpGraphCopy

   template <typename To, typename From>

   BpGraphCopy<To, From> 

   copyBpGraph(To& to, const From& from) {

     return BpGraphCopy<To, From>(to, from);

   }

   /// @}

   /// \addtogroup digraph_maps

   /// @{

   /// Provides an immutable and unique id for each item in the digraph.

   /// The IdMap class provides a unique and immutable id for each item of the

   /// same type (e.g. node) in the digraph. This id is <ul><li>\b unique:

   /// different items (nodes) get different ids <li>\b immutable: the id of an

   /// item (node) does not change (even if you delete other nodes).  </ul>

   /// Through this map you get access (i.e. can read) the inner id values of

   /// the items stored in the digraph. This map can be inverted with its member

   /// class \c InverseMap.

   ///

   template <typename _Digraph, typename _Item>

   class IdMap {

   public:

     typedef _Digraph Digraph;

     typedef int Value;

     typedef _Item Item;

     typedef _Item Key;

     /// \brief Constructor.

     ///

     /// Constructor of the map.

     explicit IdMap(const Digraph& _digraph) : digraph(&_digraph) {}

     /// \brief Gives back the \e id of the item.

     ///

     /// Gives back the immutable and unique \e id of the item.

     int operator[](const Item& item) const { return digraph->id(item);}

     /// \brief Gives back the item by its id.

     ///

     /// Gives back the item by its id.

     Item operator()(int id) { return digraph->fromId(id, Item()); }

   private:

     const Digraph* digraph;

   public:

     /// \brief The class represents the inverse of its owner (IdMap).

     ///

     /// The class represents the inverse of its owner (IdMap).

     /// \see inverse()

     class InverseMap {

     public:

       /// \brief Constructor.

       ///

       /// Constructor for creating an id-to-item map.

       explicit InverseMap(const Digraph& _digraph) : digraph(&_digraph) {}

       /// \brief Constructor.

       ///

       /// Constructor for creating an id-to-item map.

       explicit InverseMap(const IdMap& idMap) : digraph(idMap.digraph) {}

       /// \brief Gives back the given item from its id.

       ///

       /// Gives back the given item from its id.

       /// 

       Item operator[](int id) const { return digraph->fromId(id, Item());}

     private:

       const Digraph* digraph;

     };

     /// \brief Gives back the inverse of the map.

     ///

     /// Gives back the inverse of the IdMap.

     InverseMap inverse() const { return InverseMap(*digraph);} 

   };

   /// \brief General invertable digraph-map type.

   /// This type provides simple invertable digraph-maps. 

   /// The InvertableMap wraps an arbitrary ReadWriteMap 

   /// and if a key is set to a new value then store it

   /// in the inverse map.

   ///

   /// The values of the map can be accessed

   /// with stl compatible forward iterator.

   ///

   /// \param _Digraph The digraph type.

   /// \param _Item The item type of the digraph.

   /// \param _Value The value type of the map.

   ///

   /// \see IterableValueMap

   template <typename _Digraph, typename _Item, typename _Value>

   class InvertableMap : protected DefaultMap<_Digraph, _Item, _Value> {

   private:

     typedef DefaultMap<_Digraph, _Item, _Value> Map;

     typedef _Digraph Digraph;

     typedef std::map<_Value, _Item> Container;

     Container invMap;    

   public:

     /// The key type of InvertableMap (Node, Arc, Edge).

     typedef typename Map::Key Key;

     /// The value type of the InvertableMap.

     typedef typename Map::Value Value;

     /// \brief Constructor.

     ///

     /// Construct a new InvertableMap for the digraph.

     ///

     explicit InvertableMap(const Digraph& digraph) : Map(digraph) {} 

     /// \brief Forward iterator for values.

     ///

     /// This iterator is an stl compatible forward

     /// iterator on the values of the map. The values can

     /// be accessed in the [beginValue, endValue) range.

     ///

     class ValueIterator 

       : public std::iterator<std::forward_iterator_tag, Value> {

       friend class InvertableMap;

     private:

       ValueIterator(typename Container::const_iterator _it) 

         : it(_it) {}

     public:

       ValueIterator() {}

       ValueIterator& operator++() { ++it; return *this; }

       ValueIterator operator++(int) { 

         ValueIterator tmp(*this); 

         operator++();

         return tmp; 

       }

       const Value& operator*() const { return it->first; }

       const Value* operator->() const { return &(it->first); }

       bool operator==(ValueIterator jt) const { return it == jt.it; }

       bool operator!=(ValueIterator jt) const { return it != jt.it; }

     private:

       typename Container::const_iterator it;

     };

     /// \brief Returns an iterator to the first value.

     ///

     /// Returns an stl compatible iterator to the 

     /// first value of the map. The values of the

     /// map can be accessed in the [beginValue, endValue)

     /// range.

     ValueIterator beginValue() const {

       return ValueIterator(invMap.begin());

     }

     /// \brief Returns an iterator after the last value.

     ///

     /// Returns an stl compatible iterator after the 

     /// last value of the map. The values of the

     /// map can be accessed in the [beginValue, endValue)

     /// range.

     ValueIterator endValue() const {

       return ValueIterator(invMap.end());

     }

     /// \brief The setter function of the map.

     ///

     /// Sets the mapped value.

     void set(const Key& key, const Value& val) {

       Value oldval = Map::operator[](key);

       typename Container::iterator it = invMap.find(oldval);

       if (it != invMap.end() && it->second == key) {

 	invMap.erase(it);

       }      

       invMap.insert(make_pair(val, key));

       Map::set(key, val);

     }

     /// \brief The getter function of the map.

     ///

     /// It gives back the value associated with the key.

     typename MapTraits<Map>::ConstReturnValue 

     operator[](const Key& key) const {

       return Map::operator[](key);

     }

     /// \brief Gives back the item by its value.

     ///

     /// Gives back the item by its value.

     Key operator()(const Value& key) const {

       typename Container::const_iterator it = invMap.find(key);

       return it != invMap.end() ? it->second : INVALID;

     }

   protected:

     /// \brief Erase the key from the map.

     ///

     /// Erase the key to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const Key& key) {

       Value val = Map::operator[](key);

       typename Container::iterator it = invMap.find(val);

       if (it != invMap.end() && it->second == key) {

 	invMap.erase(it);

       }

       Map::erase(key);

     }

     /// \brief Erase more keys from the map.

     ///

     /// Erase more keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const std::vector<Key>& keys) {

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

 	Value val = Map::operator[](keys[i]);

 	typename Container::iterator it = invMap.find(val);

 	if (it != invMap.end() && it->second == keys[i]) {

 	  invMap.erase(it);

 	}

       }

       Map::erase(keys);

     }

     /// \brief Clear the keys from the map and inverse map.

     ///

     /// Clear the keys from the map and inverse map. It is called by the

     /// \c AlterationNotifier.

     virtual void clear() {

       invMap.clear();

       Map::clear();

     }

   public:

     /// \brief The inverse map type.

     ///

     /// The inverse of this map. The subscript operator of the map

     /// gives back always the item what was last assigned to the value. 

     class InverseMap {

     public:

       /// \brief Constructor of the InverseMap.

       ///

       /// Constructor of the InverseMap.

       explicit InverseMap(const InvertableMap& _inverted) 

         : inverted(_inverted) {}

       /// The value type of the InverseMap.

       typedef typename InvertableMap::Key Value;

       /// The key type of the InverseMap.

       typedef typename InvertableMap::Value Key; 

       /// \brief Subscript operator. 

       ///

       /// Subscript operator. It gives back always the item 

       /// what was last assigned to the value.

       Value operator[](const Key& key) const {

 	return inverted(key);

       }

     private:

       const InvertableMap& inverted;

     };

     /// \brief It gives back the just readable inverse map.

     ///

     /// It gives back the just readable inverse map.

     InverseMap inverse() const {

       return InverseMap(*this);

     } 

   };

   /// \brief Provides a mutable, continuous and unique descriptor for each 

   /// item in the digraph.

   ///

   /// The DescriptorMap class provides a unique and continuous (but mutable)

   /// descriptor (id) for each item of the same type (e.g. node) in the

   /// digraph. This id is <ul><li>\b unique: different items (nodes) get

   /// different ids <li>\b continuous: the range of the ids is the set of

   /// integers between 0 and \c n-1, where \c n is the number of the items of

   /// this type (e.g. nodes) (so the id of a node can change if you delete an

   /// other node, i.e. this id is mutable).  </ul> This map can be inverted

   /// with its member class \c InverseMap.

   ///

   /// \param _Digraph The digraph class the \c DescriptorMap belongs to.

   /// \param _Item The Item is the Key of the Map. It may be Node, Arc or 

   /// Edge.

   template <typename _Digraph, typename _Item>

   class DescriptorMap : protected DefaultMap<_Digraph, _Item, int> {

     typedef _Item Item;

     typedef DefaultMap<_Digraph, _Item, int> Map;

   public:

     /// The digraph class of DescriptorMap.

     typedef _Digraph Digraph;

     /// The key type of DescriptorMap (Node, Arc, Edge).

     typedef typename Map::Key Key;

     /// The value type of DescriptorMap.

     typedef typename Map::Value Value;

     /// \brief Constructor.

     ///

     /// Constructor for descriptor map.

     explicit DescriptorMap(const Digraph& _digraph) : Map(_digraph) {

       Item it;

       const typename Map::Notifier* nf = Map::notifier(); 

       for (nf->first(it); it != INVALID; nf->next(it)) {

 	Map::set(it, invMap.size());

 	invMap.push_back(it);	

       }      

     }

   protected:

     /// \brief Add a new key to the map.

     ///

     /// Add a new key to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void add(const Item& item) {

       Map::add(item);

       Map::set(item, invMap.size());

       invMap.push_back(item);

     }

     /// \brief Add more new keys to the map.

     ///

     /// Add more new keys to the map. It is called by the

     /// \c AlterationNotifier.

     virtual void add(const std::vector<Item>& items) {

       Map::add(items);

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

 	Map::set(items[i], invMap.size());

 	invMap.push_back(items[i]);

       }

     }

     /// \brief Erase the key from the map.

     ///

     /// Erase the key from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const Item& item) {

       Map::set(invMap.back(), Map::operator[](item));

       invMap[Map::operator[](item)] = invMap.back();

       invMap.pop_back();

       Map::erase(item);

     }

     /// \brief Erase more keys from the map.

     ///

     /// Erase more keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void erase(const std::vector<Item>& items) {

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

 	Map::set(invMap.back(), Map::operator[](items[i]));

 	invMap[Map::operator[](items[i])] = invMap.back();

 	invMap.pop_back();

       }

       Map::erase(items);

     }

     /// \brief Build the unique map.

     ///

     /// Build the unique map. It is called by the

     /// \c AlterationNotifier.

     virtual void build() {

       Map::build();

       Item it;

       const typename Map::Notifier* nf = Map::notifier(); 

       for (nf->first(it); it != INVALID; nf->next(it)) {

 	Map::set(it, invMap.size());

 	invMap.push_back(it);	

       }      

     }

     /// \brief Clear the keys from the map.

     ///

     /// Clear the keys from the map. It is called by the

     /// \c AlterationNotifier.

     virtual void clear() {

       invMap.clear();

       Map::clear();

     }

   public:

     /// \brief Returns the maximal value plus one.

     ///

     /// Returns the maximal value plus one in the map.

     unsigned int size() const {

       return invMap.size();

     }

     /// \brief Swaps the position of the two items in the map.

     ///

     /// Swaps the position of the two items in the map.

     void swap(const Item& p, const Item& q) {

       int pi = Map::operator[](p);

       int qi = Map::operator[](q);

       Map::set(p, qi);

       invMap[qi] = p;

       Map::set(q, pi);

       invMap[pi] = q;

     }

     /// \brief Gives back the \e descriptor of the item.

     ///

     /// Gives back the mutable and unique \e descriptor of the map.

     int operator[](const Item& item) const {

       return Map::operator[](item);

     }

     /// \brief Gives back the item by its descriptor.

     ///

     /// Gives back th item by its descriptor.

     Item operator()(int id) const {

       return invMap[id];

     }

   private:

     typedef std::vector<Item> Container;

     Container invMap;

   public:

     /// \brief The inverse map type of DescriptorMap.

     ///

     /// The inverse map type of DescriptorMap.

     class InverseMap {

     public:

       /// \brief Constructor of the InverseMap.

       ///

       /// Constructor of the InverseMap.

       explicit InverseMap(const DescriptorMap& _inverted) 

 	: inverted(_inverted) {}

       /// The value type of the InverseMap.

       typedef typename DescriptorMap::Key Value;

       /// The key type of the InverseMap.

       typedef typename DescriptorMap::Value Key; 

       /// \brief Subscript operator. 

       ///

       /// Subscript operator. It gives back the item 

       /// that the descriptor belongs to currently.

       Value operator[](const Key& key) const {

 	return inverted(key);

       }

       /// \brief Size of the map.

       ///

       /// Returns the size of the map.

       unsigned int size() const {

 	return inverted.size();

       }

     private:

       const DescriptorMap& inverted;

     };

     /// \brief Gives back the inverse of the map.

     ///

     /// Gives back the inverse of the map.

     const InverseMap inverse() const {

       return InverseMap(*this);

     }

   };

   /// \brief Returns the source of the given arc.

   ///

   /// The SourceMap gives back the source Node of the given arc. 

   /// \see TargetMap

   /// \author Balazs Dezso

   template <typename Digraph>

   class SourceMap {

   public:

     typedef typename Digraph::Node Value;

     typedef typename Digraph::Arc Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _digraph The digraph that the map belongs to.

     explicit SourceMap(const Digraph& _digraph) : digraph(_digraph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param arc The arc 

     /// \return The source of the arc 

     Value operator[](const Key& arc) const {

       return digraph.source(arc);

     }

   private:

     const Digraph& digraph;

   };

   /// \brief Returns a \ref SourceMap class.

   ///

   /// This function just returns an \ref SourceMap class.

   /// \relates SourceMap

   template <typename Digraph>

   inline SourceMap<Digraph> sourceMap(const Digraph& digraph) {

     return SourceMap<Digraph>(digraph);

   } 

   /// \brief Returns the target of the given arc.

   ///

   /// The TargetMap gives back the target Node of the given arc. 

   /// \see SourceMap

   /// \author Balazs Dezso

   template <typename Digraph>

   class TargetMap {

   public:

     typedef typename Digraph::Node Value;

     typedef typename Digraph::Arc Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _digraph The digraph that the map belongs to.

     explicit TargetMap(const Digraph& _digraph) : digraph(_digraph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param e The arc 

     /// \return The target of the arc 

     Value operator[](const Key& e) const {

       return digraph.target(e);

     }

   private:

     const Digraph& digraph;

   };

   /// \brief Returns a \ref TargetMap class.

   ///

   /// This function just returns a \ref TargetMap class.

   /// \relates TargetMap

   template <typename Digraph>

   inline TargetMap<Digraph> targetMap(const Digraph& digraph) {

     return TargetMap<Digraph>(digraph);

   }

   /// \brief Returns the "forward" directed arc view of an edge.

   ///

   /// Returns the "forward" directed arc view of an edge.

   /// \see BackwardMap

   /// \author Balazs Dezso

   template <typename Digraph>

   class ForwardMap {

   public:

     typedef typename Digraph::Arc Value;

     typedef typename Digraph::Edge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _digraph The digraph that the map belongs to.

     explicit ForwardMap(const Digraph& _digraph) : digraph(_digraph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param key An edge 

     /// \return The "forward" directed arc view of edge 

     Value operator[](const Key& key) const {

       return digraph.direct(key, true);

     }

   private:

     const Digraph& digraph;

   };

   /// \brief Returns a \ref ForwardMap class.

   ///

   /// This function just returns an \ref ForwardMap class.

   /// \relates ForwardMap

   template <typename Digraph>

   inline ForwardMap<Digraph> forwardMap(const Digraph& digraph) {

     return ForwardMap<Digraph>(digraph);

   }

   /// \brief Returns the "backward" directed arc view of an edge.

   ///

   /// Returns the "backward" directed arc view of an edge.

   /// \see ForwardMap

   /// \author Balazs Dezso

   template <typename Digraph>

   class BackwardMap {

   public:

     typedef typename Digraph::Arc Value;

     typedef typename Digraph::Edge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _digraph The digraph that the map belongs to.

     explicit BackwardMap(const Digraph& _digraph) : digraph(_digraph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param key An edge 

     /// \return The "backward" directed arc view of edge 

     Value operator[](const Key& key) const {

       return digraph.direct(key, false);

     }

   private:

     const Digraph& digraph;

   };

   /// \brief Returns a \ref BackwardMap class

   /// This function just returns a \ref BackwardMap class.

   /// \relates BackwardMap

   template <typename Digraph>

   inline BackwardMap<Digraph> backwardMap(const Digraph& digraph) {

     return BackwardMap<Digraph>(digraph);

   }

   /// \brief Potential difference map

   ///

   /// If there is an potential map on the nodes then we

   /// can get an arc map as we get the substraction of the

   /// values of the target and source.

   template <typename Digraph, typename NodeMap>

   class PotentialDifferenceMap {

   public:

     typedef typename Digraph::Arc Key;

     typedef typename NodeMap::Value Value;

     /// \brief Constructor

     ///

     /// Contructor of the map

     explicit PotentialDifferenceMap(const Digraph& _digraph, 

                                     const NodeMap& _potential) 

       : digraph(_digraph), potential(_potential) {}

     /// \brief Const subscription operator

     ///

     /// Const subscription operator

     Value operator[](const Key& arc) const {

       return potential[digraph.target(arc)] - potential[digraph.source(arc)];

     }

   private:

     const Digraph& digraph;

     const NodeMap& potential;

   };

   /// \brief Returns a PotentialDifferenceMap.

   ///

   /// This function just returns a PotentialDifferenceMap.

   /// \relates PotentialDifferenceMap

   template <typename Digraph, typename NodeMap>

   PotentialDifferenceMap<Digraph, NodeMap> 

   potentialDifferenceMap(const Digraph& digraph, const NodeMap& potential) {

     return PotentialDifferenceMap<Digraph, NodeMap>(digraph, potential);

   }

   /// \brief Map of the node in-degrees.

   ///

   /// This map returns the in-degree of a node. Once it is constructed,

   /// the degrees are stored in a standard NodeMap, so each query is done

   /// in constant time. On the other hand, the values are updated automatically

   /// whenever the digraph changes.

   ///

   /// \warning Besides addNode() and addArc(), a digraph structure may provide

   /// alternative ways to modify the digraph. The correct behavior of InDegMap

   /// is not guarantied if these additional features are used. For example

   /// the functions \ref ListDigraph::changeSource() "changeSource()",

   /// \ref ListDigraph::changeTarget() "changeTarget()" and

   /// \ref ListDigraph::reverseArc() "reverseArc()"

   /// of \ref ListDigraph will \e not update the degree values correctly.

   ///

   /// \sa OutDegMap

   template <typename _Digraph>

   class InDegMap  

     : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>

       ::ItemNotifier::ObserverBase {

   public:

     typedef _Digraph Digraph;

     typedef int Value;

     typedef typename Digraph::Node Key;

     typedef typename ItemSetTraits<_Digraph, typename _Digraph::Arc>

     ::ItemNotifier::ObserverBase Parent;

   private:

     class AutoNodeMap : public DefaultMap<_Digraph, Key, int> {

     public:

       typedef DefaultMap<_Digraph, Key, int> Parent;

       typedef typename Parent::Digraph Digraph;

       AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}

       virtual void add(const Key& key) {

 	Parent::add(key);

 	Parent::set(key, 0);

       }

       virtual void add(const std::vector<Key>& keys) {

 	Parent::add(keys);

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

 	  Parent::set(keys[i], 0);

 	}

       }

       virtual void build() {

 	Parent::build();

 	Key it;

 	typename Parent::Notifier* nf = Parent::notifier();

 	for (nf->first(it); it != INVALID; nf->next(it)) {

 	  Parent::set(it, 0);

 	}

       }

     };

   public:

     /// \brief Constructor.

     ///

     /// Constructor for creating in-degree map.

     explicit InDegMap(const Digraph& _digraph) : digraph(_digraph), deg(_digraph) {

       Parent::attach(digraph.notifier(typename _Digraph::Arc()));

       for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {

 	deg[it] = countInArcs(digraph, it);

       }

     }

     /// Gives back the in-degree of a Node.

     int operator[](const Key& key) const {

       return deg[key];

     }

   protected:

     typedef typename Digraph::Arc Arc;

     virtual void add(const Arc& arc) {

       ++deg[digraph.target(arc)];

     }

     virtual void add(const std::vector<Arc>& arcs) {

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

         ++deg[digraph.target(arcs[i])];

       }

     }

     virtual void erase(const Arc& arc) {

       --deg[digraph.target(arc)];

     }

     virtual void erase(const std::vector<Arc>& arcs) {

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

         --deg[digraph.target(arcs[i])];

       }

     }

     virtual void build() {

       for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {

 	deg[it] = countInArcs(digraph, it);

       }      

     }

     virtual void clear() {

       for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {

 	deg[it] = 0;

       }

     }

   private:

     const _Digraph& digraph;

     AutoNodeMap deg;

   };

   /// \brief Map of the node out-degrees.

   ///

   /// This map returns the out-degree of a node. Once it is constructed,

   /// the degrees are stored in a standard NodeMap, so each query is done

   /// in constant time. On the other hand, the values are updated automatically

   /// whenever the digraph changes.

   ///

   /// \warning Besides addNode() and addArc(), a digraph structure may provide

   /// alternative ways to modify the digraph. The correct behavior of OutDegMap

   /// is not guarantied if these additional features are used. For example

   /// the functions \ref ListDigraph::changeSource() "changeSource()",

   /// \ref ListDigraph::changeTarget() "changeTarget()" and

   /// \ref ListDigraph::reverseArc() "reverseArc()"

   /// of \ref ListDigraph will \e not update the degree values correctly.

   ///

   /// \sa InDegMap

   template <typename _Digraph>

   class OutDegMap  

     : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>

       ::ItemNotifier::ObserverBase {

   public:

     typedef typename ItemSetTraits<_Digraph, typename _Digraph::Arc>

     ::ItemNotifier::ObserverBase Parent;

     typedef _Digraph Digraph;

     typedef int Value;

     typedef typename Digraph::Node Key;

   private:

     class AutoNodeMap : public DefaultMap<_Digraph, Key, int> {

     public:

       typedef DefaultMap<_Digraph, Key, int> Parent;

       typedef typename Parent::Digraph Digraph;

       AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}

       virtual void add(const Key& key) {

 	Parent::add(key);

 	Parent::set(key, 0);

       }

       virtual void add(const std::vector<Key>& keys) {

 	Parent::add(keys);

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

 	  Parent::set(keys[i], 0);

 	}

       }

       virtual void build() {

 	Parent::build();

 	Key it;

 	typename Parent::Notifier* nf = Parent::notifier();

 	for (nf->first(it); it != INVALID; nf->next(it)) {

 	  Parent::set(it, 0);

 	}

       }

     };

   public:

     /// \brief Constructor.

     ///

     /// Constructor for creating out-degree map.

     explicit OutDegMap(const Digraph& _digraph) : digraph(_digraph), deg(_digraph) {

       Parent::attach(digraph.notifier(typename _Digraph::Arc()));

       for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {

 	deg[it] = countOutArcs(digraph, it);

       }

     }

     /// Gives back the out-degree of a Node.

     int operator[](const Key& key) const {

       return deg[key];

     }

   protected:

     typedef typename Digraph::Arc Arc;

     virtual void add(const Arc& arc) {

       ++deg[digraph.source(arc)];

     }

     virtual void add(const std::vector<Arc>& arcs) {

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

         ++deg[digraph.source(arcs[i])];

       }

     }

     virtual void erase(const Arc& arc) {

       --deg[digraph.source(arc)];

     }

     virtual void erase(const std::vector<Arc>& arcs) {

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

         --deg[digraph.source(arcs[i])];

       }

     }

     virtual void build() {

       for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {

 	deg[it] = countOutArcs(digraph, it);

       }      

     }

     virtual void clear() {

       for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {

 	deg[it] = 0;

       }

     }

   private:

     const _Digraph& digraph;

     AutoNodeMap deg;

   };

   ///Dynamic arc look up between given endpoints.

   ///\ingroup gutils

   ///Using this class, you can find an arc in a digraph from a given

   ///source to a given target in amortized time <em>O(log d)</em>,

   ///where <em>d</em> is the out-degree of the source node.

   ///

   ///It is possible to find \e all parallel arcs between two nodes with

   ///the \c findFirst() and \c findNext() members.

   ///

   ///See the \ref ArcLookUp and \ref AllArcLookUp classes if your

   ///digraph do not changed so frequently.

   ///

   ///This class uses a self-adjusting binary search tree, Sleator's

   ///and Tarjan's Splay tree for guarantee the logarithmic amortized

   ///time bound for arc lookups. This class also guarantees the

   ///optimal time bound in a constant factor for any distribution of

   ///queries.

   ///

   ///\param G The type of the underlying digraph.  

   ///

   ///\sa ArcLookUp  

   ///\sa AllArcLookUp  

   template<class G>

   class DynArcLookUp 

     : protected ItemSetTraits<G, typename G::Arc>::ItemNotifier::ObserverBase

   {

   public:

     typedef typename ItemSetTraits<G, typename G::Arc>

     ::ItemNotifier::ObserverBase Parent;

     GRAPH_TYPEDEFS(typename G);

     typedef G Digraph;

   protected:

     class AutoNodeMap : public DefaultMap<G, Node, Arc> {

     public:

       typedef DefaultMap<G, Node, Arc> Parent;

       AutoNodeMap(const G& digraph) : Parent(digraph, INVALID) {}

       virtual void add(const Node& node) {

 	Parent::add(node);

 	Parent::set(node, INVALID);

       }

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

 	Parent::add(nodes);

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

 	  Parent::set(nodes[i], INVALID);

 	}

       }

       virtual void build() {

 	Parent::build();

 	Node it;

 	typename Parent::Notifier* nf = Parent::notifier();

 	for (nf->first(it); it != INVALID; nf->next(it)) {

 	  Parent::set(it, INVALID);

 	}

       }

     };

     const Digraph &_g;

     AutoNodeMap _head;

     typename Digraph::template ArcMap<Arc> _parent;

     typename Digraph::template ArcMap<Arc> _left;

     typename Digraph::template ArcMap<Arc> _right;

     class ArcLess {

       const Digraph &g;

     public:

       ArcLess(const Digraph &_g) : g(_g) {}

       bool operator()(Arc a,Arc b) const 

       {

 	return g.target(a)<g.target(b);

       }

     };

   public:

     ///Constructor

     ///Constructor.

     ///

     ///It builds up the search database.

     DynArcLookUp(const Digraph &g) 

       : _g(g),_head(g),_parent(g),_left(g),_right(g) 

     { 

       Parent::attach(_g.notifier(typename Digraph::Arc()));

       refresh(); 

     }

   protected:

     virtual void add(const Arc& arc) {

       insert(arc);

     }

     virtual void add(const std::vector<Arc>& arcs) {

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

 	insert(arcs[i]);

       }

     }

     virtual void erase(const Arc& arc) {

       remove(arc);

     }

     virtual void erase(const std::vector<Arc>& arcs) {

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

 	remove(arcs[i]);

       }     

     }

     virtual void build() {

       refresh();

     }

     virtual void clear() {

       for(NodeIt n(_g);n!=INVALID;++n) {

 	_head.set(n, INVALID);

       }

     }

     void insert(Arc arc) {

       Node s = _g.source(arc);

       Node t = _g.target(arc);

       _left.set(arc, INVALID);

       _right.set(arc, INVALID);

       Arc e = _head[s];

       if (e == INVALID) {

 	_head.set(s, arc);

 	_parent.set(arc, INVALID);

 	return;

       }

       while (true) {

 	if (t < _g.target(e)) {

 	  if (_left[e] == INVALID) {

 	    _left.set(e, arc);

 	    _parent.set(arc, e);

 	    splay(arc);

 	    return;

 	  } else {

 	    e = _left[e];

 	  }

 	} else {

 	  if (_right[e] == INVALID) {

 	    _right.set(e, arc);

 	    _parent.set(arc, e);

 	    splay(arc);

 	    return;

 	  } else {

 	    e = _right[e];

 	  }

 	}

       }