/* -*- C++ -*-

  *

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

  *

  * Copyright (C) 2003-2006

  * 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 Graph utilities.

 ///

 ///

 namespace lemon {

   /// \addtogroup gutils

   /// @{

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

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

   ///of \c Graph: \c Node,  \c NodeIt, \c Edge, \c EdgeIt, \c InEdgeIt,

   ///\c OutEdgeIt

   ///\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 graph maps because of the lack of

   ///template typedefs in C++.

 #define GRAPH_TYPEDEFS(Graph)				\

   typedef Graph::     Node      Node;			\

     typedef Graph::   NodeIt    NodeIt;			\

     typedef Graph::   Edge      Edge;			\

     typedef Graph::   EdgeIt    EdgeIt;			\

     typedef Graph:: InEdgeIt  InEdgeIt;			\

     typedef Graph::OutEdgeIt OutEdgeIt;			

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

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

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

   ///\c UEdge, \c UEdgeIt, \c IncEdgeIt,

   ///

   ///\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 graph maps because of the lack of

   ///template typedefs in C++.

 #define UGRAPH_TYPEDEFS(Graph)				\

   GRAPH_TYPEDEFS(Graph)						\

     typedef Graph:: UEdge   UEdge;			\

     typedef Graph:: UEdgeIt UEdgeIt;			\

     typedef Graph:: IncEdgeIt   IncEdgeIt;		       

   ///\brief Creates convenience typedefs for the bipartite undirected graph 

   ///types and iterators

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

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

   ///\c ANodeIt, \c BNodeIt, 

   ///

   ///\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 graph maps because of the lack of

   ///template typedefs in C++.

 #define BPUGRAPH_TYPEDEFS(Graph)            \

   UGRAPH_TYPEDEFS(Graph)                    \

     typedef Graph::ANode ANode;             \

     typedef Graph::BNode BNode;             \

     typedef Graph::ANodeIt ANodeIt;	    \

     typedef Graph::BNodeIt BNodeIt;

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

   ///

   /// This function counts the items (nodes, edges etc) in the graph.

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

   /// it iterates on all of the items.

   template <typename Graph, typename Item>

   inline int countItems(const Graph& g) {

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

     int num = 0;

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

       ++num;

     }

     return num;

   }

   // Node counting:

   namespace _graph_utils_bits {

     template <typename Graph, typename Enable = void>

     struct CountNodesSelector {

       static int count(const Graph &g) {

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

       }

     };

     template <typename Graph>

     struct CountNodesSelector<

       Graph, typename 

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

     {

       static int count(const Graph &g) {

         return g.nodeNum();

       }

     };    

   }

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

   ///

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

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

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

   ///

   /// \todo refer how to specialize it

   template <typename Graph>

   inline int countNodes(const Graph& g) {

     return _graph_utils_bits::CountNodesSelector<Graph>::count(g);

   }

   namespace _graph_utils_bits {

     template <typename Graph, typename Enable = void>

     struct CountANodesSelector {

       static int count(const Graph &g) {

         return countItems<Graph, typename Graph::ANode>(g);

       }

     };

     template <typename Graph>

     struct CountANodesSelector<

       Graph, typename 

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

     {

       static int count(const Graph &g) {

         return g.aNodeNum();

       }

     };    

   }

   /// \brief Function to count the anodes in the graph.

   ///

   /// This function counts the anodes in the graph.

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

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

   ///

   /// \todo refer how to specialize it

   template <typename Graph>

   inline int countANodes(const Graph& g) {

     return _graph_utils_bits::CountANodesSelector<Graph>::count(g);

   }

   namespace _graph_utils_bits {

     template <typename Graph, typename Enable = void>

     struct CountBNodesSelector {

       static int count(const Graph &g) {

         return countItems<Graph, typename Graph::BNode>(g);

       }

     };

     template <typename Graph>

     struct CountBNodesSelector<

       Graph, typename 

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

     {

       static int count(const Graph &g) {

         return g.bNodeNum();

       }

     };    

   }

   /// \brief Function to count the bnodes in the graph.

   ///

   /// This function counts the bnodes in the graph.

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

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

   ///

   /// \todo refer how to specialize it

   template <typename Graph>

   inline int countBNodes(const Graph& g) {

     return _graph_utils_bits::CountBNodesSelector<Graph>::count(g);

   }

   // Edge counting:

   namespace _graph_utils_bits {

     template <typename Graph, typename Enable = void>

     struct CountEdgesSelector {

       static int count(const Graph &g) {

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

       }

     };

     template <typename Graph>

     struct CountEdgesSelector<

       Graph, 

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

     {

       static int count(const Graph &g) {

         return g.edgeNum();

       }

     };    

   }

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

   ///

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

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

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

   template <typename Graph>

   inline int countEdges(const Graph& g) {

     return _graph_utils_bits::CountEdgesSelector<Graph>::count(g);

   }

   // Undirected edge counting:

   namespace _graph_utils_bits {

     template <typename Graph, typename Enable = void>

     struct CountUEdgesSelector {

       static int count(const Graph &g) {

         return countItems<Graph, typename Graph::UEdge>(g);

       }

     };

     template <typename Graph>

     struct CountUEdgesSelector<

       Graph, 

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

     {

       static int count(const Graph &g) {

         return g.uEdgeNum();

       }

     };    

   }

   /// \brief Function to count the undirected edges in the graph.

   ///

   /// This function counts the undirected edges in the graph.

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

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

   template <typename Graph>

   inline int countUEdges(const Graph& g) {

     return _graph_utils_bits::CountUEdgesSelector<Graph>::count(g);

   }

   template <typename Graph, typename DegIt>

   inline int countNodeDegree(const Graph& _g, const typename Graph::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-edges from node \c n.

   ///

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

   /// in the graph.  

   template <typename Graph>

   inline int countOutEdges(const Graph& _g,  const typename Graph::Node& _n) {

     return countNodeDegree<Graph, typename Graph::OutEdgeIt>(_g, _n);

   }

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

   ///

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

   /// in the graph.  

   template <typename Graph>

   inline int countInEdges(const Graph& _g,  const typename Graph::Node& _n) {

     return countNodeDegree<Graph, typename Graph::InEdgeIt>(_g, _n);

   }

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

   ///

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

   /// in the graph.  

   template <typename Graph>

   inline int countIncEdges(const Graph& _g,  const typename Graph::Node& _n) {

     return countNodeDegree<Graph, typename Graph::IncEdgeIt>(_g, _n);

   }

   namespace _graph_utils_bits {

     template <typename Graph, typename Enable = void>

     struct FindEdgeSelector {

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

       static Edge find(const Graph &g, Node u, Node v, Edge 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 Graph>

     struct FindEdgeSelector<

       Graph, 

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

     {

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

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

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

       }

     };    

   }

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

   ///

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

   ///

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

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

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

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

   ///

   /// Thus you can iterate through each edge 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 EdgeLookUp

   ///\se AllEdgeLookup

   ///\sa ConEdgeIt

   template <typename Graph>

   inline typename Graph::Edge 

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

            typename Graph::Edge prev = INVALID) {

     return _graph_utils_bits::FindEdgeSelector<Graph>::find(g, u, v, prev);

   }

   /// \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<Graph> it(g, src, trg); it != INVALID; ++it) {

   ///   ...

   /// }

   ///\endcode

   /// 

   ///\sa findEdge()

   ///\sa EdgeLookUp

   ///\se AllEdgeLookup

   ///

   /// \author Balazs Dezso 

   template <typename _Graph>

   class ConEdgeIt : public _Graph::Edge {

   public:

     typedef _Graph Graph;

     typedef typename Graph::Edge Parent;

     typedef typename Graph::Edge Edge;

     typedef typename Graph::Node Node;

     /// \brief Constructor.

     ///

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

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

     ConEdgeIt(const Graph& g, Node u, Node v) : graph(g) {

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

     }

     /// \brief Constructor.

     ///

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

     /// the \c e edge.

     ConEdgeIt(const Graph& g, Edge e) : Parent(e), graph(g) {}

     /// \brief Increment operator.

     ///

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

     ConEdgeIt& operator++() {

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

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

       return *this;

     }

   private:

     const Graph& graph;

   };

   namespace _graph_utils_bits {

     template <typename Graph, typename Enable = void>

     struct FindUEdgeSelector {

       typedef typename Graph::Node Node;

       typedef typename Graph::UEdge UEdge;

       static UEdge find(const Graph &g, Node u, Node v, UEdge 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 Graph>

     struct FindUEdgeSelector<

       Graph, 

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

     {

       typedef typename Graph::Node Node;

       typedef typename Graph::UEdge UEdge;

       static UEdge find(const Graph &g, Node u, Node v, UEdge prev) {

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

       }

     };    

   }

   /// \brief Finds an uedge between two nodes of a graph.

   ///

   /// Finds an uedge from node \c u to node \c v in graph \c g.

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

   /// will be enumerated.

   ///

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

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

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

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

   ///

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

   ///\code

   /// for(UEdge e = findUEdge(g,u,v); e != INVALID; 

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

   ///   ...

   /// }

   ///\endcode

   ///

   ///\sa ConEdgeIt

   template <typename Graph>

   inline typename Graph::UEdge 

   findUEdge(const Graph &g, typename Graph::Node u, typename Graph::Node v,

             typename Graph::UEdge p = INVALID) {

     return _graph_utils_bits::FindUEdgeSelector<Graph>::find(g, u, v, p);

   }

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

   ///

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

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

   /// use it the following way:

   ///\code

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

   ///   ...

   /// }

   ///\endcode

   ///

   ///\sa findUEdge()

   ///

   /// \author Balazs Dezso 

   template <typename _Graph>

   class ConUEdgeIt : public _Graph::UEdge {

   public:

     typedef _Graph Graph;

     typedef typename Graph::UEdge Parent;

     typedef typename Graph::UEdge UEdge;

     typedef typename Graph::Node Node;

     /// \brief Constructor.

     ///

     /// Construct a new ConUEdgeIt iterating on the edges which

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

     ConUEdgeIt(const Graph& g, Node u, Node v) : graph(g) {

       Parent::operator=(findUEdge(graph, u, v));

     }

     /// \brief Constructor.

     ///

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

     /// the \c e edge.

     ConUEdgeIt(const Graph& g, UEdge e) : Parent(e), graph(g) {}

     /// \brief Increment operator.

     ///

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

     ConUEdgeIt& operator++() {

       Parent::operator=(findUEdge(graph, graph.source(*this), 

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

       return *this;

     }

   private:

     const Graph& graph;

   };

   /// \brief Copy a map.

   ///

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

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

   /// mapping to convert the source's keys to the target's keys.

   template <typename Target, typename Source, 

 	    typename ItemIt, typename Ref>	    

   void copyMap(Target& target, const Source& source, 

 	       ItemIt it, const Ref& ref) {

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

       target[ref[it]] = source[it];

     }

   }

   /// \brief Copy the source map to the target map.

   ///

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

   /// to iterate on the data structure.

   template <typename Target, typename Source, typename ItemIt>	    

   void copyMap(Target& target, const Source& source, ItemIt it) {

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

       target[it] = source[it];

     }

   }

   namespace _graph_utils_bits {

     template <typename Graph, typename Item, typename RefMap>

     class MapCopyBase {

     public:

       virtual void copy(const Graph& source, const RefMap& refMap) = 0;

       virtual ~MapCopyBase() {}

     };

     template <typename Graph, typename Item, typename RefMap, 

               typename TargetMap, typename SourceMap>

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

     public:

       MapCopy(TargetMap& tmap, const SourceMap& map) 

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

       virtual void copy(const Graph& graph, const RefMap& refMap) {

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

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

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

         }

       }

     private:

       TargetMap& _tmap;

       const SourceMap& _map;

     };

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

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

     public:

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

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

         _it = refMap[_item];

       }

     private:

       It& _it;

       Item _item;

     };

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

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

     public:

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

       virtual void copy(const Graph& graph, const RefMap& refMap) {

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

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

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

         }

       }

     private:

       Ref& _map;

     };

     template <typename Graph, typename Item, typename RefMap, 

               typename CrossRef>

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

     public:

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

       virtual void copy(const Graph& graph, const RefMap& refMap) {

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

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

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

         }

       }

     private:

       CrossRef& _cmap;

     };

     template <typename Graph, typename Enable = void>

     struct GraphCopySelector {

       template <typename Source, typename NodeRefMap, typename EdgeRefMap>

       static void copy(Graph &target, const Source& source,

                        NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {

         for (typename Source::NodeIt it(source); it != INVALID; ++it) {

           nodeRefMap[it] = target.addNode();

         }

         for (typename Source::EdgeIt it(source); it != INVALID; ++it) {

           edgeRefMap[it] = target.addEdge(nodeRefMap[source.source(it)], 

                                           nodeRefMap[source.target(it)]);

         }

       }

     };

     template <typename Graph>

     struct GraphCopySelector<

       Graph, 

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

     {

       template <typename Source, typename NodeRefMap, typename EdgeRefMap>

       static void copy(Graph &target, const Source& source,

                        NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {

         target.clone(source, nodeRefMap, edgeRefMap);

       }

     };

     template <typename UGraph, typename Enable = void>

     struct UGraphCopySelector {

       template <typename Source, typename NodeRefMap, typename UEdgeRefMap>

       static void copy(UGraph &target, const Source& source,

                        NodeRefMap& nodeRefMap, UEdgeRefMap& uEdgeRefMap) {

         for (typename Source::NodeIt it(source); it != INVALID; ++it) {

           nodeRefMap[it] = target.addNode();

         }

         for (typename Source::UEdgeIt it(source); it != INVALID; ++it) {

           uEdgeRefMap[it] = target.addEdge(nodeRefMap[source.source(it)], 

                                           nodeRefMap[source.target(it)]);

         }

       }

     };

     template <typename UGraph>

     struct UGraphCopySelector<

       UGraph, 

       typename enable_if<typename UGraph::CloneableTag, void>::type> 

     {

       template <typename Source, typename NodeRefMap, typename UEdgeRefMap>

       static void copy(UGraph &target, const Source& source,

                        NodeRefMap& nodeRefMap, UEdgeRefMap& uEdgeRefMap) {

         target.clone(source, nodeRefMap, uEdgeRefMap);

       }

     };

     template <typename BpUGraph, typename Enable = void>

     struct BpUGraphCopySelector {

       template <typename Source, typename ANodeRefMap, 

                 typename BNodeRefMap, typename UEdgeRefMap>

       static void copy(BpUGraph &target, const Source& source,

                        ANodeRefMap& aNodeRefMap, BNodeRefMap& bNodeRefMap,

                        UEdgeRefMap& uEdgeRefMap) {

         for (typename Source::ANodeIt it(source); it != INVALID; ++it) {

           aNodeRefMap[it] = target.addANode();

         }

         for (typename Source::BNodeIt it(source); it != INVALID; ++it) {

           bNodeRefMap[it] = target.addBNode();

         }

         for (typename Source::UEdgeIt it(source); it != INVALID; ++it) {

           uEdgeRefMap[it] = target.addEdge(aNodeRefMap[source.aNode(it)], 

                                            bNodeRefMap[source.bNode(it)]);

         }

       }

     };

     template <typename BpUGraph>

     struct BpUGraphCopySelector<

       BpUGraph, 

       typename enable_if<typename BpUGraph::CloneableTag, void>::type> 

     {

       template <typename Source, typename ANodeRefMap, 

                 typename BNodeRefMap, typename UEdgeRefMap>

       static void copy(BpUGraph &target, const Source& source,

                        ANodeRefMap& aNodeRefMap, BNodeRefMap& bNodeRefMap,

                        UEdgeRefMap& uEdgeRefMap) {

         target.clone(source, aNodeRefMap, bNodeRefMap, uEdgeRefMap);

       }

     };

   }

   /// \brief Class to copy a graph.

   ///

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

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

   template <typename Target, typename Source>

   class GraphCopy {

   private:

     typedef typename Source::Node Node;

     typedef typename Source::NodeIt NodeIt;

     typedef typename Source::Edge Edge;

     typedef typename Source::EdgeIt EdgeIt;

     typedef typename Target::Node TNode;

     typedef typename Target::Edge TEdge;

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

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

   public: 

     /// \brief Constructor for the GraphCopy.

     ///

     /// It copies the content of the \c _source graph into the

     /// \c _target graph.

     GraphCopy(Target& _target, const Source& _source) 

       : source(_source), target(_target) {}

     /// \brief Destructor of the GraphCopy

     ///

     /// Destructor of the GraphCopy

     ~GraphCopy() {

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

         delete nodeMapCopies[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 _graph_utils_bits::RefCopy<Source, 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 _graph_utils_bits::CrossRefCopy<Source, Node,

                               NodeRefMap, NodeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's node type,

     /// and the copied map's key type is the source graph's node

     /// type.  

     template <typename TargetMap, typename SourceMap>

     GraphCopy& nodeMap(TargetMap& tmap, const SourceMap& map) {

       nodeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, Node, 

                               NodeRefMap, TargetMap, SourceMap>(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& node) {

       nodeMapCopies.push_back(new _graph_utils_bits::ItemCopy<Source, Node, 

                               NodeRefMap, TNode>(tnode, node));

       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 _graph_utils_bits::RefCopy<Source, 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 _graph_utils_bits::CrossRefCopy<Source, Edge,

                               EdgeRefMap, EdgeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's edge type,

     /// and the copied map's key type is the source graph's edge

     /// type.  

     template <typename TargetMap, typename SourceMap>

     GraphCopy& edgeMap(TargetMap& tmap, const SourceMap& map) {

       edgeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, Edge, 

                               EdgeRefMap, TargetMap, SourceMap>(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& edge) {

       edgeMapCopies.push_back(new _graph_utils_bits::ItemCopy<Source, Edge, 

                               EdgeRefMap, TEdge>(tedge, edge));

       return *this;

     }

     /// \brief Executes the copies.

     ///

     /// Executes the copies.

     void run() {

       NodeRefMap nodeRefMap(source);

       EdgeRefMap edgeRefMap(source);

       _graph_utils_bits::GraphCopySelector<Target>::

         copy(target, source, nodeRefMap, edgeRefMap);

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

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

       }

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

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

       }      

     }

   protected:

     const Source& source;

     Target& target;

     std::vector<_graph_utils_bits::MapCopyBase<Source, Node, NodeRefMap>* > 

     nodeMapCopies;

     std::vector<_graph_utils_bits::MapCopyBase<Source, Edge, EdgeRefMap>* > 

     edgeMapCopies;

   };

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

   ///

   /// Copy a graph to another graph.

   /// The usage of the function:

   /// 

   ///\code

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

   ///\endcode

   /// 

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

   /// source graph's nodes to the target graph's nodes and the \c ecr will

   /// contain the mapping from the target graph's edges to the source's

   /// edges.

   ///

   /// \see GraphCopy 

   template <typename Target, typename Source>

   GraphCopy<Target, Source> copyGraph(Target& target, const Source& source) {

     return GraphCopy<Target, Source>(target, source);

   }

   /// \brief Class to copy an undirected graph.

   ///

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

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

   template <typename Target, typename Source>

   class UGraphCopy {

   private:

     typedef typename Source::Node Node;

     typedef typename Source::NodeIt NodeIt;

     typedef typename Source::Edge Edge;

     typedef typename Source::EdgeIt EdgeIt;

     typedef typename Source::UEdge UEdge;

     typedef typename Source::UEdgeIt UEdgeIt;

     typedef typename Target::Node TNode;

     typedef typename Target::Edge TEdge;

     typedef typename Target::UEdge TUEdge;

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

     typedef typename Source::template UEdgeMap<TUEdge> UEdgeRefMap;

     struct EdgeRefMap {

       EdgeRefMap(const Target& _target, const Source& _source,

                  const UEdgeRefMap& _uedge_ref, const NodeRefMap& _node_ref) 

         : target(_target), source(_source), 

           uedge_ref(_uedge_ref), node_ref(_node_ref) {}

       typedef typename Source::Edge Key;

       typedef typename Target::Edge Value;

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

         bool forward = (source.direction(key) == 

                         (node_ref[source.source((UEdge)key)] == 

                          target.source(uedge_ref[(UEdge)key])));

 	return target.direct(uedge_ref[key], forward); 

       }

       const Target& target;

       const Source& source;

       const UEdgeRefMap& uedge_ref;

       const NodeRefMap& node_ref;

     };

   public: 

     /// \brief Constructor for the GraphCopy.

     ///

     /// It copies the content of the \c _source graph into the

     /// \c _target graph.

     UGraphCopy(Target& _target, const Source& _source) 

       : source(_source), target(_target) {}

     /// \brief Destructor of the GraphCopy

     ///

     /// Destructor of the GraphCopy

     ~UGraphCopy() {

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

         delete nodeMapCopies[i];

       }

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

         delete edgeMapCopies[i];

       }

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

         delete uEdgeMapCopies[i];

       }

     }

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

     ///

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

     template <typename NodeRef>

     UGraphCopy& nodeRef(NodeRef& map) {

       nodeMapCopies.push_back(new _graph_utils_bits::RefCopy<Source, 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>

     UGraphCopy& nodeCrossRef(NodeCrossRef& map) {

       nodeMapCopies.push_back(new _graph_utils_bits::CrossRefCopy<Source, Node,

                               NodeRefMap, NodeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's node type,

     /// and the copied map's key type is the source graph's node

     /// type.  

     template <typename TargetMap, typename SourceMap>

     UGraphCopy& nodeMap(TargetMap& tmap, const SourceMap& map) {

       nodeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, Node, 

                               NodeRefMap, TargetMap, SourceMap>(tmap, map));

       return *this;

     }

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

     ///

     /// Make a copy of the given node.

     UGraphCopy& node(TNode& tnode, const Node& node) {

       nodeMapCopies.push_back(new _graph_utils_bits::ItemCopy<Source, Node, 

                               NodeRefMap, TNode>(tnode, node));

       return *this;

     }

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

     ///

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

     template <typename EdgeRef>

     UGraphCopy& edgeRef(EdgeRef& map) {

       edgeMapCopies.push_back(new _graph_utils_bits::RefCopy<Source, 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>

     UGraphCopy& edgeCrossRef(EdgeCrossRef& map) {

       edgeMapCopies.push_back(new _graph_utils_bits::CrossRefCopy<Source, Edge,

                               EdgeRefMap, EdgeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's edge type,

     /// and the copied map's key type is the source graph's edge

     /// type.  

     template <typename TargetMap, typename SourceMap>

     UGraphCopy& edgeMap(TargetMap& tmap, const SourceMap& map) {

       edgeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, Edge, 

                               EdgeRefMap, TargetMap, SourceMap>(tmap, map));

       return *this;

     }

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

     ///

     /// Make a copy of the given edge.

     UGraphCopy& edge(TEdge& tedge, const Edge& edge) {

       edgeMapCopies.push_back(new _graph_utils_bits::ItemCopy<Source, Edge, 

                               EdgeRefMap, TEdge>(tedge, edge));

       return *this;

     }

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

     ///

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

     template <typename UEdgeRef>

     UGraphCopy& uEdgeRef(UEdgeRef& map) {

       uEdgeMapCopies.push_back(new _graph_utils_bits::RefCopy<Source, UEdge, 

                                UEdgeRefMap, UEdgeRef>(map));

       return *this;

     }

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

     ///

     /// Copies the undirected edge cross references (reverse

     /// references) into the given map.

     template <typename UEdgeCrossRef>

     UGraphCopy& uEdgeCrossRef(UEdgeCrossRef& map) {

       uEdgeMapCopies.push_back(new _graph_utils_bits::CrossRefCopy<Source, 

                                UEdge, UEdgeRefMap, UEdgeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's undirected edge type,

     /// and the copied map's key type is the source graph's undirected edge

     /// type.  

     template <typename TargetMap, typename SourceMap>

     UGraphCopy& uEdgeMap(TargetMap& tmap, const SourceMap& map) {

       uEdgeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, UEdge, 

                                UEdgeRefMap, TargetMap, SourceMap>(tmap, map));

       return *this;

     }

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

     ///

     /// Make a copy of the given undirected edge.

     UGraphCopy& uEdge(TUEdge& tuedge, const UEdge& uedge) {

       uEdgeMapCopies.push_back(new _graph_utils_bits::ItemCopy<Source, UEdge, 

                                UEdgeRefMap, TUEdge>(tuedge, uedge));

       return *this;

     }

     /// \brief Executes the copies.

     ///

     /// Executes the copies.

     void run() {

       NodeRefMap nodeRefMap(source);

       UEdgeRefMap uEdgeRefMap(source);

       EdgeRefMap edgeRefMap(target, source, uEdgeRefMap, nodeRefMap);

       _graph_utils_bits::UGraphCopySelector<Target>::

         copy(target, source, nodeRefMap, uEdgeRefMap);

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

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

       }

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

         uEdgeMapCopies[i]->copy(source, uEdgeRefMap);

       }

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

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

       }

     }

   private:

     const Source& source;

     Target& target;

     std::vector<_graph_utils_bits::MapCopyBase<Source, Node, NodeRefMap>* > 

     nodeMapCopies;

     std::vector<_graph_utils_bits::MapCopyBase<Source, Edge, EdgeRefMap>* > 

     edgeMapCopies;

     std::vector<_graph_utils_bits::MapCopyBase<Source, UEdge, UEdgeRefMap>* > 

     uEdgeMapCopies;

   };

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

   ///

   /// Copy an undirected graph to another graph.

   /// The usage of the function:

   /// 

   ///\code

   /// copyUGraph(trg, src).nodeRef(nr).edgeCrossRef(ecr).run();

   ///\endcode

   /// 

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

   /// source graph's nodes to the target graph's nodes and the \c ecr will

   /// contain the mapping from the target graph's edges to the source's

   /// edges.

   ///

   /// \see UGraphCopy 

   template <typename Target, typename Source>

   UGraphCopy<Target, Source> 

   copyUGraph(Target& target, const Source& source) {

     return UGraphCopy<Target, Source>(target, source);

   }

   /// \brief Class to copy a bipartite undirected graph.

   ///

   /// Class to copy a bipartite undirected graph to another graph

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

   /// the \c copyBpUGraph() function.

   template <typename Target, typename Source>

   class BpUGraphCopy {

   private:

     typedef typename Source::Node Node;

     typedef typename Source::ANode ANode;

     typedef typename Source::BNode BNode;

     typedef typename Source::NodeIt NodeIt;

     typedef typename Source::Edge Edge;

     typedef typename Source::EdgeIt EdgeIt;

     typedef typename Source::UEdge UEdge;

     typedef typename Source::UEdgeIt UEdgeIt;

     typedef typename Target::Node TNode;

     typedef typename Target::Edge TEdge;

     typedef typename Target::UEdge TUEdge;

     typedef typename Source::template ANodeMap<TNode> ANodeRefMap;

     typedef typename Source::template BNodeMap<TNode> BNodeRefMap;

     typedef typename Source::template UEdgeMap<TUEdge> UEdgeRefMap;

     struct NodeRefMap {

       NodeRefMap(const Source& _source, const ANodeRefMap& _anode_ref,

                  const BNodeRefMap& _bnode_ref)

         : source(_source), anode_ref(_anode_ref), bnode_ref(_bnode_ref) {}

       typedef typename Source::Node Key;

       typedef typename Target::Node Value;

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

 	return source.aNode(key) ? anode_ref[key] : bnode_ref[key]; 

       }

       const Source& source;

       const ANodeRefMap& anode_ref;

       const BNodeRefMap& bnode_ref;

     };

     struct EdgeRefMap {

       EdgeRefMap(const Target& _target, const Source& _source,

                  const UEdgeRefMap& _uedge_ref, const NodeRefMap& _node_ref) 

         : target(_target), source(_source), 

           uedge_ref(_uedge_ref), node_ref(_node_ref) {}

       typedef typename Source::Edge Key;

       typedef typename Target::Edge Value;

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

         bool forward = (source.direction(key) == 

                         (node_ref[source.source((UEdge)key)] == 

                          target.source(uedge_ref[(UEdge)key])));

 	return target.direct(uedge_ref[key], forward); 

       }

       const Target& target;

       const Source& source;

       const UEdgeRefMap& uedge_ref;

       const NodeRefMap& node_ref;

     };

   public: 

     /// \brief Constructor for the GraphCopy.

     ///

     /// It copies the content of the \c _source graph into the

     /// \c _target graph.

     BpUGraphCopy(Target& _target, const Source& _source) 

       : source(_source), target(_target) {}

     /// \brief Destructor of the GraphCopy

     ///

     /// Destructor of the GraphCopy

     ~BpUGraphCopy() {

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

         delete aNodeMapCopies[i];

       }

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

         delete bNodeMapCopies[i];

       }

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

         delete nodeMapCopies[i];

       }

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

         delete edgeMapCopies[i];

       }

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

         delete uEdgeMapCopies[i];

       }

     }

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

     ///

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

     template <typename ANodeRef>

     BpUGraphCopy& aNodeRef(ANodeRef& map) {

       aNodeMapCopies.push_back(new _graph_utils_bits::RefCopy<Source, ANode, 

                                ANodeRefMap, ANodeRef>(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 ANodeCrossRef>

     BpUGraphCopy& aNodeCrossRef(ANodeCrossRef& map) {

       aNodeMapCopies.push_back(new _graph_utils_bits::CrossRefCopy<Source, 

                                ANode, ANodeRefMap, ANodeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's node type,

     /// and the copied map's key type is the source graph's node

     /// type.  

     template <typename TargetMap, typename SourceMap>

     BpUGraphCopy& aNodeMap(TargetMap& tmap, const SourceMap& map) {

       aNodeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, ANode, 

                                ANodeRefMap, TargetMap, SourceMap>(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 BNodeRef>

     BpUGraphCopy& bNodeRef(BNodeRef& map) {

       bNodeMapCopies.push_back(new _graph_utils_bits::RefCopy<Source, BNode, 

                                BNodeRefMap, BNodeRef>(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 BNodeCrossRef>

     BpUGraphCopy& bNodeCrossRef(BNodeCrossRef& map) {

       bNodeMapCopies.push_back(new _graph_utils_bits::CrossRefCopy<Source, 

                               BNode, BNodeRefMap, BNodeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's node type,

     /// and the copied map's key type is the source graph's node

     /// type.  

     template <typename TargetMap, typename SourceMap>

     BpUGraphCopy& bNodeMap(TargetMap& tmap, const SourceMap& map) {

       bNodeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, BNode, 

                                BNodeRefMap, TargetMap, SourceMap>(tmap, map));

       return *this;

     }

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

     ///

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

     template <typename NodeRef>

     BpUGraphCopy& nodeRef(NodeRef& map) {

       nodeMapCopies.push_back(new _graph_utils_bits::RefCopy<Source, 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>

     BpUGraphCopy& nodeCrossRef(NodeCrossRef& map) {

       nodeMapCopies.push_back(new _graph_utils_bits::CrossRefCopy<Source, Node,

                               NodeRefMap, NodeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's node type,

     /// and the copied map's key type is the source graph's node

     /// type.  

     template <typename TargetMap, typename SourceMap>

     BpUGraphCopy& nodeMap(TargetMap& tmap, const SourceMap& map) {

       nodeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, Node, 

                               NodeRefMap, TargetMap, SourceMap>(tmap, map));

       return *this;

     }

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

     ///

     /// Make a copy of the given node.

     BpUGraphCopy& node(TNode& tnode, const Node& node) {

       nodeMapCopies.push_back(new _graph_utils_bits::ItemCopy<Source, Node, 

                               NodeRefMap, TNode>(tnode, node));

       return *this;

     }

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

     ///

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

     template <typename EdgeRef>

     BpUGraphCopy& edgeRef(EdgeRef& map) {

       edgeMapCopies.push_back(new _graph_utils_bits::RefCopy<Source, 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>

     BpUGraphCopy& edgeCrossRef(EdgeCrossRef& map) {

       edgeMapCopies.push_back(new _graph_utils_bits::CrossRefCopy<Source, Edge,

                               EdgeRefMap, EdgeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's edge type,

     /// and the copied map's key type is the source graph's edge

     /// type.  

     template <typename TargetMap, typename SourceMap>

     BpUGraphCopy& edgeMap(TargetMap& tmap, const SourceMap& map) {

       edgeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, Edge, 

                               EdgeRefMap, TargetMap, SourceMap>(tmap, map));

       return *this;

     }

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

     ///

     /// Make a copy of the given edge.

     BpUGraphCopy& edge(TEdge& tedge, const Edge& edge) {

       edgeMapCopies.push_back(new _graph_utils_bits::ItemCopy<Source, Edge, 

                               EdgeRefMap, TEdge>(tedge, edge));

       return *this;

     }

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

     ///

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

     template <typename UEdgeRef>

     BpUGraphCopy& uEdgeRef(UEdgeRef& map) {

       uEdgeMapCopies.push_back(new _graph_utils_bits::RefCopy<Source, UEdge, 

                                UEdgeRefMap, UEdgeRef>(map));

       return *this;

     }

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

     ///

     /// Copies the undirected edge cross references (reverse

     /// references) into the given map.

     template <typename UEdgeCrossRef>

     BpUGraphCopy& uEdgeCrossRef(UEdgeCrossRef& map) {

       uEdgeMapCopies.push_back(new _graph_utils_bits::CrossRefCopy<Source, 

                                UEdge, UEdgeRefMap, UEdgeCrossRef>(map));

       return *this;

     }

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

     ///

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

     /// The new map's key type is the target graph's undirected edge type,

     /// and the copied map's key type is the source graph's undirected edge

     /// type.  

     template <typename TargetMap, typename SourceMap>

     BpUGraphCopy& uEdgeMap(TargetMap& tmap, const SourceMap& map) {

       uEdgeMapCopies.push_back(new _graph_utils_bits::MapCopy<Source, UEdge, 

                                UEdgeRefMap, TargetMap, SourceMap>(tmap, map));

       return *this;

     }

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

     ///

     /// Make a copy of the given undirected edge.

     BpUGraphCopy& uEdge(TUEdge& tuedge, const UEdge& uedge) {

       uEdgeMapCopies.push_back(new _graph_utils_bits::ItemCopy<Source, UEdge, 

                                UEdgeRefMap, TUEdge>(tuedge, uedge));

       return *this;

     }

     /// \brief Executes the copies.

     ///

     /// Executes the copies.

     void run() {

       ANodeRefMap aNodeRefMap(source);

       BNodeRefMap bNodeRefMap(source);

       NodeRefMap nodeRefMap(source, aNodeRefMap, bNodeRefMap);

       UEdgeRefMap uEdgeRefMap(source);

       EdgeRefMap edgeRefMap(target, source, uEdgeRefMap, nodeRefMap);

       _graph_utils_bits::BpUGraphCopySelector<Target>::

         copy(target, source, aNodeRefMap, bNodeRefMap, uEdgeRefMap);

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

         aNodeMapCopies[i]->copy(source, aNodeRefMap);

       }

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

         bNodeMapCopies[i]->copy(source, bNodeRefMap);

       }

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

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

       }

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

         uEdgeMapCopies[i]->copy(source, uEdgeRefMap);

       }

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

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

       }

     }

   private:

     const Source& source;

     Target& target;

     std::vector<_graph_utils_bits::MapCopyBase<Source, ANode, ANodeRefMap>* > 

     aNodeMapCopies;

     std::vector<_graph_utils_bits::MapCopyBase<Source, BNode, BNodeRefMap>* > 

     bNodeMapCopies;

     std::vector<_graph_utils_bits::MapCopyBase<Source, Node, NodeRefMap>* > 

     nodeMapCopies;

     std::vector<_graph_utils_bits::MapCopyBase<Source, Edge, EdgeRefMap>* > 

     edgeMapCopies;

     std::vector<_graph_utils_bits::MapCopyBase<Source, UEdge, UEdgeRefMap>* > 

     uEdgeMapCopies;

   };

   /// \brief Copy a bipartite undirected graph to another graph.

   ///

   /// Copy a bipartite undirected graph to another graph.

   /// The usage of the function:

   /// 

   ///\code

   /// copyBpUGraph(trg, src).aNodeRef(anr).edgeCrossRef(ecr).run();

   ///\endcode

   /// 

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

   /// source graph's nodes to the target graph's nodes and the \c ecr will

   /// contain the mapping from the target graph's edges to the source's

   /// edges.

   ///

   /// \see BpUGraphCopy

   template <typename Target, typename Source>

   BpUGraphCopy<Target, Source> 

   copyBpUGraph(Target& target, const Source& source) {

     return BpUGraphCopy<Target, Source>(target, source);

   }

   /// @}

   /// \addtogroup graph_maps

   /// @{

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

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

   /// same type (e.g. node) in the graph. 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 graph. This map can be inverted with its member

   /// class \c InverseMap.

   ///

   template <typename _Graph, typename _Item>

   class IdMap {

   public:

     typedef _Graph Graph;

     typedef int Value;

     typedef _Item Item;

     typedef _Item Key;

     /// \brief Constructor.

     ///

     /// Constructor for creating id map.

     explicit IdMap(const Graph& _graph) : graph(&_graph) {}

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

     ///

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

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

   private:

     const Graph* graph;

   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 Graph& _graph) : graph(&_graph) {}

       /// \brief Constructor.

       ///

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

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

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

       ///

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

       /// 

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

     private:

       const Graph* graph;

     };

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

     ///

     /// Gives back the inverse of the IdMap.

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

   };

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

   /// This type provides simple invertable graph-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 _Graph The graph type.

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

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

   ///

   /// \see IterableValueMap

   template <typename _Graph, typename _Item, typename _Value>

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

   private:

     typedef DefaultMap<_Graph, _Item, _Value> Map;

     typedef _Graph Graph;

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

     Container invMap;    

   public:

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

     typedef typename Map::Key Key;

     /// The value type of the InvertableMap.

     typedef typename Map::Value Value;

     /// \brief Constructor.

     ///

     /// Construct a new InvertableMap for the graph.

     ///

     explicit InvertableMap(const Graph& graph) : Map(graph) {} 

     /// \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);

     }

   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 {

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

 	return it->second;

       }

     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 graph.

   ///

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

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

   /// graph. 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 _Graph The graph class the \c DescriptorMap belongs to.

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

   /// UEdge.

   template <typename _Graph, typename _Item>

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

     typedef _Item Item;

     typedef DefaultMap<_Graph, _Item, int> Map;

   public:

     /// The graph class of DescriptorMap.

     typedef _Graph Graph;

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

     typedef typename Map::Key Key;

     /// The value type of DescriptorMap.

     typedef typename Map::Value Value;

     /// \brief Constructor.

     ///

     /// Constructor for descriptor map.

     explicit DescriptorMap(const Graph& _graph) : Map(_graph) {

       Item it;

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

       for (notifier->first(it); it != INVALID; notifier->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* notifier = Map::getNotifier(); 

       for (notifier->first(it); it != INVALID; notifier->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);

     }

   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.invMap[key];

       }

       /// \brief Size of the map.

       ///

       /// Returns the size of the map.

       unsigned int size() const {

 	return inverted.invMap.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 edge.

   ///

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

   /// \author Balazs Dezso

   template <typename Graph>

   class SourceMap {

   public:

     typedef typename Graph::Node Value;

     typedef typename Graph::Edge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _graph The graph that the map belongs to.

     explicit SourceMap(const Graph& _graph) : graph(_graph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param edge The edge 

     /// \return The source of the edge 

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

       return graph.source(edge);

     }

   private:

     const Graph& graph;

   };

   /// \brief Returns a \ref SourceMap class

   ///

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

   /// \relates SourceMap

   template <typename Graph>

   inline SourceMap<Graph> sourceMap(const Graph& graph) {

     return SourceMap<Graph>(graph);

   } 

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

   ///

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

   /// \author Balazs Dezso

   template <typename Graph>

   class TargetMap {

   public:

     typedef typename Graph::Node Value;

     typedef typename Graph::Edge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _graph The graph that the map belongs to.

     explicit TargetMap(const Graph& _graph) : graph(_graph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param e The edge 

     /// \return The target of the edge 

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

       return graph.target(e);

     }

   private:

     const Graph& graph;

   };

   /// \brief Returns a \ref TargetMap class

   ///

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

   /// \relates TargetMap

   template <typename Graph>

   inline TargetMap<Graph> targetMap(const Graph& graph) {

     return TargetMap<Graph>(graph);

   }

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

   ///

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

   /// \author Balazs Dezso

   template <typename Graph>

   class ForwardMap {

   public:

     typedef typename Graph::Edge Value;

     typedef typename Graph::UEdge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _graph The graph that the map belongs to.

     explicit ForwardMap(const Graph& _graph) : graph(_graph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param key An undirected edge 

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

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

       return graph.direct(key, true);

     }

   private:

     const Graph& graph;

   };

   /// \brief Returns a \ref ForwardMap class

   ///

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

   /// \relates ForwardMap

   template <typename Graph>

   inline ForwardMap<Graph> forwardMap(const Graph& graph) {

     return ForwardMap<Graph>(graph);

   }

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

   ///

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

   /// \author Balazs Dezso

   template <typename Graph>

   class BackwardMap {

   public:

     typedef typename Graph::Edge Value;

     typedef typename Graph::UEdge Key;

     /// \brief Constructor

     ///

     /// Constructor

     /// \param _graph The graph that the map belongs to.

     explicit BackwardMap(const Graph& _graph) : graph(_graph) {}

     /// \brief The subscript operator.

     ///

     /// The subscript operator.

     /// \param key An undirected edge 

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

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

       return graph.direct(key, false);

     }

   private:

     const Graph& graph;

   };

   /// \brief Returns a \ref BackwardMap class

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

   /// \relates BackwardMap

   template <typename Graph>

   inline BackwardMap<Graph> backwardMap(const Graph& graph) {

     return BackwardMap<Graph>(graph);

   }

   /// \brief Potential difference map

   ///

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

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

   /// values of the target and source.

   template <typename Graph, typename NodeMap>

   class PotentialDifferenceMap {

   public:

     typedef typename Graph::Edge Key;

     typedef typename NodeMap::Value Value;

     /// \brief Constructor

     ///

     /// Contructor of the map

     explicit PotentialDifferenceMap(const Graph& _graph, 

                                     const NodeMap& _potential) 

       : graph(_graph), potential(_potential) {}

     /// \brief Const subscription operator

     ///

     /// Const subscription operator

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

       return potential[graph.target(edge)] - potential[graph.source(edge)];

     }

   private:

     const Graph& graph;

     const NodeMap& potential;

   };

   /// \brief Just returns a PotentialDifferenceMap

   ///

   /// Just returns a PotentialDifferenceMap

   /// \relates PotentialDifferenceMap

   template <typename Graph, typename NodeMap>

   PotentialDifferenceMap<Graph, NodeMap> 

   potentialDifferenceMap(const Graph& graph, const NodeMap& potential) {

     return PotentialDifferenceMap<Graph, NodeMap>(graph, 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 graph changes.

   ///

   /// \warning Besides addNode() and addEdge(), a graph structure may provide

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

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

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

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

   /// \ref ListGraph::reverseEdge() "reverseEdge()"

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

   ///

   /// \sa OutDegMap

   template <typename _Graph>

   class InDegMap  

     : protected ItemSetTraits<_Graph, typename _Graph::Edge>

       ::ItemNotifier::ObserverBase {

   public:

     typedef _Graph Graph;

     typedef int Value;

     typedef typename Graph::Node Key;

     typedef typename ItemSetTraits<_Graph, typename _Graph::Edge>

     ::ItemNotifier::ObserverBase Parent;

   private:

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

     public:

       typedef DefaultMap<_Graph, Key, int> Parent;

       typedef typename Parent::Graph Graph;

       AutoNodeMap(const Graph& graph) : Parent(graph, 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);

 	}

       }

     };

   public:

     /// \brief Constructor.

     ///

     /// Constructor for creating in-degree map.

     explicit InDegMap(const Graph& _graph) : graph(_graph), deg(_graph) {

       Parent::attach(graph.getNotifier(typename _Graph::Edge()));

       for(typename _Graph::NodeIt it(graph); it != INVALID; ++it) {

 	deg[it] = countInEdges(graph, it);

       }

     }

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

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

       return deg[key];

     }

   protected:

     typedef typename Graph::Edge Edge;

     virtual void add(const Edge& edge) {

       ++deg[graph.target(edge)];

     }

     virtual void add(const std::vector<Edge>& edges) {

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

         ++deg[graph.target(edges[i])];

       }

     }

     virtual void erase(const Edge& edge) {

       --deg[graph.target(edge)];

     }

     virtual void erase(const std::vector<Edge>& edges) {

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

         --deg[graph.target(edges[i])];

       }

     }

     virtual void build() {

       for(typename _Graph::NodeIt it(graph); it != INVALID; ++it) {

 	deg[it] = countInEdges(graph, it);

       }      

     }

     virtual void clear() {

       for(typename _Graph::NodeIt it(graph); it != INVALID; ++it) {

 	deg[it] = 0;

       }

     }

   private:

     const _Graph& graph;

     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 graph changes.

   ///

   /// \warning Besides addNode() and addEdge(), a graph structure may provide

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

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

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

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

   /// \ref ListGraph::reverseEdge() "reverseEdge()"

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

   ///

   /// \sa InDegMap

   template <typename _Graph>

   class OutDegMap  

     : protected ItemSetTraits<_Graph, typename _Graph::Edge>

       ::ItemNotifier::ObserverBase {

   public:

     typedef typename ItemSetTraits<_Graph, typename _Graph::Edge>

     ::ItemNotifier::ObserverBase Parent;

     typedef _Graph Graph;

     typedef int Value;

     typedef typename Graph::Node Key;

   private:

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

     public:

       typedef DefaultMap<_Graph, Key, int> Parent;

       typedef typename Parent::Graph Graph;

       AutoNodeMap(const Graph& graph) : Parent(graph, 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);

 	}

       }

     };

   public:

     /// \brief Constructor.

     ///

     /// Constructor for creating out-degree map.

     explicit OutDegMap(const Graph& _graph) : graph(_graph), deg(_graph) {

       Parent::attach(graph.getNotifier(typename _Graph::Edge()));

       for(typename _Graph::NodeIt it(graph); it != INVALID; ++it) {

 	deg[it] = countOutEdges(graph, it);

       }

     }

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

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

       return deg[key];

     }

   protected:

     typedef typename Graph::Edge Edge;

     virtual void add(const Edge& edge) {

       ++deg[graph.source(edge)];

     }

     virtual void add(const std::vector<Edge>& edges) {

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

         ++deg[graph.source(edges[i])];

       }

     }

     virtual void erase(const Edge& edge) {

       --deg[graph.source(edge)];

     }

     virtual void erase(const std::vector<Edge>& edges) {

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

         --deg[graph.source(edges[i])];

       }

     }

     virtual void build() {

       for(typename _Graph::NodeIt it(graph); it != INVALID; ++it) {

 	deg[it] = countOutEdges(graph, it);

       }      

     }

     virtual void clear() {

       for(typename _Graph::NodeIt it(graph); it != INVALID; ++it) {

 	deg[it] = 0;

       }

     }

   private:

     const _Graph& graph;

     AutoNodeMap deg;

   };

   ///Fast edge look up between given endpoints.

   ///\ingroup gutils

   ///Using this class, you can find an edge in a graph from a given

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

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

   ///

   ///It is not possible to find \e all parallel edges between two nodes.

   ///Use \ref AllEdgeLookUp for this purpose.

   ///

   ///\warning This class is static, so you should refresh() (or at least

   ///refresh(Node)) this data structure

   ///whenever the graph changes. This is a time consuming (superlinearly

   ///proportional (<em>O(m</em>log<em>m)</em>) to the number of edges).

   ///

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

   ///

   ///\sa AllEdgeLookUp  

   template<class G>

   class EdgeLookUp 

   {

   public:

     GRAPH_TYPEDEFS(typename G)

     typedef G Graph;

   protected:

     const Graph &_g;

     typename Graph::template NodeMap<Edge> _head;

     typename Graph::template EdgeMap<Edge> _left;

     typename Graph::template EdgeMap<Edge> _right;

     class EdgeLess {

       const Graph &g;

     public:

       EdgeLess(const Graph &_g) : g(_g) {}

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

       {

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

       }

     };

   public:

     ///Constructor

     ///Constructor.

     ///

     ///It builds up the search database, which remains valid until the graph

     ///changes.

     EdgeLookUp(const Graph &g) :_g(g),_head(g),_left(g),_right(g) {refresh();}

   private:

     Edge refresh_rec(std::vector<Edge> &v,int a,int b) 

     {

       int m=(a+b)/2;

       Edge me=v[m];

       _left[me] = a<m?refresh_rec(v,a,m-1):INVALID;

       _right[me] = m<b?refresh_rec(v,m+1,b):INVALID;

       return me;

     }

   public:

     ///Refresh the data structure at a node.

     ///Build up the search database of node \c n.

     ///

     ///It runs in time <em>O(d</em>log<em>d)</em>, where <em>d</em> is

     ///the number of the outgoing edges of \c n.

     void refresh(Node n) 

     {

       std::vector<Edge> v;

       for(OutEdgeIt e(_g,n);e!=INVALID;++e) v.push_back(e);

       if(v.size()) {

 	std::sort(v.begin(),v.end(),EdgeLess(_g));

 	_head[n]=refresh_rec(v,0,v.size()-1);

       }

       else _head[n]=INVALID;

     }

     ///Refresh the full data structure.

     ///Build up the full search database. In fact, it simply calls

     ///\ref refresh(Node) "refresh(n)" for each node \c n.

     ///

     ///It runs in time <em>O(m</em>log<em>D)</em>, where <em>m</em> is

     ///the number of the edges of \c n and <em>D</em> is the maximum

     ///out-degree of the graph.

     void refresh() 

     {

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

     }

     ///Find an edge between two nodes.

     ///Find an edge between two nodes in time <em>O(</em>log<em>d)</em>, where

     /// <em>d</em> is the number of outgoing edges of \c s.

     ///\param s The source node

     ///\param t The target node

     ///\return An edge from \c s to \c t if there exists,

     ///\ref INVALID otherwise.

     ///

     ///\warning If you change the graph, refresh() must be called before using

     ///this operator. If you change the outgoing edges of

     ///a single node \c n, then

     ///\ref refresh(Node) "refresh(n)" is enough.

     ///

     Edge operator()(Node s, Node t) const

     {

       Edge e;

       for(e=_head[s];

 	  e!=INVALID&&_g.target(e)!=t;

 	  e = t < _g.target(e)?_left[e]:_right[e]) ;

       return e;

     }

   };

   ///Fast look up of all edges between given endpoints.

   ///\ingroup gutils

   ///This class is the same as \ref EdgeLookUp, with the addition

   ///that it makes it possible to find all edges between given endpoints.

   ///

   ///\warning This class is static, so you should refresh() (or at least

   ///refresh(Node)) this data structure

   ///whenever the graph changes. This is a time consuming (superlinearly

   ///proportional (<em>O(m</em>log<em>m)</em>) to the number of edges).

   ///

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

   ///

   ///\sa EdgeLookUp  

   template<class G>

   class AllEdgeLookUp : public EdgeLookUp<G>

   {

     using EdgeLookUp<G>::_g;

     using EdgeLookUp<G>::_right;

     using EdgeLookUp<G>::_left;

     using EdgeLookUp<G>::_head;

     GRAPH_TYPEDEFS(typename G)

     typedef G Graph;

     typename Graph::template EdgeMap<Edge> _next;

     Edge refreshNext(Edge head,Edge next=INVALID)

     {

       if(head==INVALID) return next;

       else {

 	next=refreshNext(_right[head],next);

 // 	_next[head]=next;

 	_next[head]=( next!=INVALID && _g.target(next)==_g.target(head))

 	  ? next : INVALID;

 	return refreshNext(_left[head],head);

       }

     }

     void refreshNext()

     {

       for(NodeIt n(_g);n!=INVALID;++n) refreshNext(_head[n]);

     }

   public:

     ///Constructor

     ///Constructor.

     ///

     ///It builds up the search database, which remains valid until the graph

     ///changes.

     AllEdgeLookUp(const Graph &g) : EdgeLookUp<G>(g), _next(g) {refreshNext();}

     ///Refresh the data structure at a node.

     ///Build up the search database of node \c n.

     ///

     ///It runs in time <em>O(d</em>log<em>d)</em>, where <em>d</em> is

     ///the number of the outgoing edges of \c n.

     void refresh(Node n) 

     {

       EdgeLookUp<G>::refresh(n);

       refreshNext(_head[n]);

     }

     ///Refresh the full data structure.

     ///Build up the full search database. In fact, it simply calls

     ///\ref refresh(Node) "refresh(n)" for each node \c n.

     ///

     ///It runs in time <em>O(m</em>log<em>D)</em>, where <em>m</em> is

     ///the number of the edges of \c n and <em>D</em> is the maximum

     ///out-degree of the graph.

     void refresh() 

     {

       for(NodeIt n(_g);n!=INVALID;++n) refresh(_head[n]);

     }

     ///Find an edge between two nodes.

     ///Find an edge between two nodes.

     ///\param s The source node

     ///\param t The target node

     ///\param prev The previous edge between \c s and \c t. It it is INVALID or

     ///not given, the operator finds the first appropriate edge.

     ///\return An edge from \c s to \c t after \prev or

     ///\ref INVALID if there is no more.

     ///

     ///For example, you can count the number of edges from \c u to \c v in the

     ///following way.

     ///\code