/* -*- 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 Graph 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, \c BoolNodeMap, \c IntNodeMap, \c DoubleNodeMap, 

   ///\c BoolArcMap, \c IntArcMap, \c DoubleArcMap.

   ///

   ///\note If the graph type is a dependent type, ie. the graph type depend

   ///on a template parameter, then use \c TEMPLATE_DIGRAPH_TYPEDEFS()

   ///macro.

 #define DIGRAPH_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;					\

   typedef Digraph::NodeMap<bool> BoolNodeMap;				\

   typedef Digraph::NodeMap<int> IntNodeMap;				\

   typedef Digraph::NodeMap<double> DoubleNodeMap;			\

   typedef Digraph::ArcMap<bool> BoolArcMap;				\

   typedef Digraph::ArcMap<int> IntArcMap;				\

   typedef Digraph::ArcMap<double> DoubleArcMap

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

   ///\see DIGRAPH_TYPEDEFS

   ///

   ///\note Use this macro, if the graph type is a dependent type,

   ///ie. the graph type depend on a template parameter.

 #define TEMPLATE_DIGRAPH_TYPEDEFS(Digraph)				\

   typedef typename Digraph::Node Node;					\

   typedef typename Digraph::NodeIt NodeIt;				\

   typedef typename Digraph::Arc Arc;					\

   typedef typename Digraph::ArcIt ArcIt;				\

   typedef typename Digraph::InArcIt InArcIt;				\

   typedef typename Digraph::OutArcIt OutArcIt;				\

   typedef typename Digraph::template NodeMap<bool> BoolNodeMap;		\

   typedef typename Digraph::template NodeMap<int> IntNodeMap;		\

   typedef typename Digraph::template NodeMap<double> DoubleNodeMap;	\

   typedef typename Digraph::template ArcMap<bool> BoolArcMap;		\

   typedef typename Digraph::template ArcMap<int> IntArcMap;		\

   typedef typename Digraph::template ArcMap<double> DoubleArcMap

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

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

   ///by \ref DIGRAPH_TYPEDEFS(Graph) and six more, namely it creates

   ///\c Edge, \c EdgeIt, \c IncEdgeIt, \c BoolEdgeMap, \c IntEdgeMap,

   ///\c DoubleEdgeMap.

   ///

   ///\note If the graph type is a dependent type, ie. the graph type depend

   ///on a template parameter, then use \c TEMPLATE_DIGRAPH_TYPEDEFS()

   ///macro.

 #define GRAPH_TYPEDEFS(Graph)						\

   DIGRAPH_TYPEDEFS(Graph);						\

   typedef Graph::Edge Edge;						\

   typedef Graph::EdgeIt EdgeIt;						\

   typedef Graph::IncEdgeIt IncEdgeIt;					\

   typedef Graph::EdgeMap<bool> BoolEdgeMap;				\

   typedef Graph::EdgeMap<int> IntEdgeMap;				\

   typedef Graph::EdgeMap<double> DoubleEdgeMap

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

   ///\see GRAPH_TYPEDEFS

   ///

   ///\note Use this macro, if the graph type is a dependent type,

   ///ie. the graph type depend on a template parameter.

 #define TEMPLATE_GRAPH_TYPEDEFS(Graph)					\

   TEMPLATE_DIGRAPH_TYPEDEFS(Graph);					\

   typedef typename Graph::Edge Edge;					\

   typedef typename Graph::EdgeIt EdgeIt;				\

   typedef typename Graph::IncEdgeIt IncEdgeIt;				\

   typedef typename Graph::template EdgeMap<bool> BoolEdgeMap;		\

   typedef typename Graph::template EdgeMap<int> IntEdgeMap;		\

   typedef typename Graph::template EdgeMap<double> DoubleEdgeMap

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

   ///

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

   ///

   /// If the graph 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 Graph>

   inline int countNodes(const Graph& g) {

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

   }

   // Arc counting:

   namespace _graph_utils_bits {

     template <typename Graph, typename Enable = void>

     struct CountArcsSelector {

       static int count(const Graph &g) {

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

       }

     };

     template <typename Graph>

     struct CountArcsSelector<

       Graph, 

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

     {

       static int count(const Graph &g) {

         return g.arcNum();

       }

     };    

   }

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

   ///

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

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

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

   ///

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

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

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

   template <typename Graph>

   inline int countArcs(const Graph& g) {

     return _graph_utils_bits::CountArcsSelector<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(m) but for some

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

   ///

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

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

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

   template <typename Graph>

   inline int countEdges(const Graph& g) {

     return _graph_utils_bits::CountEdgesSelector<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-arcs from node \c n.

   ///

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

   /// in the graph.  

   template <typename Graph>

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

     return countNodeDegree<Graph, typename Graph::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 graph.  

   template <typename Graph>

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

     return countNodeDegree<Graph, typename Graph::InArcIt>(_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 FindArcSelector {

       typedef typename Graph::Node Node;

       typedef typename Graph::Arc Arc;

       static Arc find(const Graph &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 Graph>

     struct FindArcSelector<

       Graph, 

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

     {

       typedef typename Graph::Node Node;

       typedef typename Graph::Arc Arc;

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

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

       }

     };    

   }

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

   ///

   /// Finds an arc 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 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 Graph>

   inline typename Graph::Arc 

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

            typename Graph::Arc prev = INVALID) {

     return _graph_utils_bits::FindArcSelector<Graph>::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<Graph> it(g, src, trg); it != INVALID; ++it) {

   ///   ...

   /// }

   ///\endcode

   /// 

   ///\sa findArc()

   ///\sa ArcLookUp

   ///\sa AllArcLookUp

   ///\sa DynArcLookUp

   template <typename _Graph>

   class ConArcIt : public _Graph::Arc {

   public:

     typedef _Graph Graph;

     typedef typename Graph::Arc Parent;

     typedef typename Graph::Arc Arc;

     typedef typename Graph::Node Node;

     /// \brief Constructor.

     ///

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

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

     ConArcIt(const Graph& g, Node u, Node v) : _graph(g) {

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

     }

     /// \brief Constructor.

     ///

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

     /// the \c e arc.

     ConArcIt(const Graph& g, Arc a) : Parent(a), _graph(g) {}

     /// \brief Increment operator.

     ///

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

     ConArcIt& operator++() {

       Parent::operator=(findArc(_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 FindEdgeSelector {

       typedef typename Graph::Node Node;

       typedef typename Graph::Edge Edge;

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

         bool b;

         if (u != v) {

           if (e == INVALID) {

             g.firstInc(e, b, u);

           } else {

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

             g.nextInc(e, b);

           }

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

             g.nextInc(e, b);

           }

         }

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

   /// will be enumerated once.

   ///

   /// 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 ConEdgeIt

   template <typename Graph>

   inline typename Graph::Edge 

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

             typename Graph::Edge p = INVALID) {

     return _graph_utils_bits::FindEdgeSelector<Graph>::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<Graph> it(g, src, trg); it != INVALID; ++it) {

   ///   ...

   /// }

   ///\endcode

   ///

   ///\sa findEdge()

   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.u(*this), 

 				 _graph.v(*this), *this));

       return *this;

     }

   private:

     const Graph& _graph;

   };

   namespace _graph_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.addEdge(nodeRefMap[from.u(it)], 

 				      nodeRefMap[from.v(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);

       }

     };

   }

   /// \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.

   ///

   /// This class not just make a copy of a graph, but it can create

   /// references and cross references between the nodes and arcs of

   /// the two graphs, it can copy maps for use with the newly created

   /// graph and copy nodes and arcs.

   ///

   /// To make a copy from a graph, first an instance of DigraphCopy

   /// should be created, then the data belongs to the graph should

   /// assigned to copy. In the end, the \c run() member should be

   /// called.

   ///

   /// The next code copies a graph with several data:

   ///\code

   ///  DigraphCopy<NewGraph, OrigGraph> dc(new_graph, orig_graph);

   ///  // create a reference for the nodes

   ///  OrigGraph::NodeMap<NewGraph::Node> nr(orig_graph);

   ///  dc.nodeRef(nr);

   ///  // create a cross reference (inverse) for the arcs

   ///  NewGraph::ArcMap<OrigGraph::Arc> acr(new_graph);

   ///  dc.arcCrossRef(acr);

   ///  // copy an arc map

   ///  OrigGraph::ArcMap<double> oamap(orig_graph);

   ///  NewGraph::ArcMap<double> namap(new_graph);

   ///  dc.arcMap(namap, oamap);

   ///  // copy a node

   ///  OrigGraph::Node on;

   ///  NewGraph::Node nn;

   ///  dc.node(nn, on);

   ///  // Executions of copy

   ///  dc.run();

   ///\endcode

   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(_node_maps.size()); ++i) {

         delete _node_maps[i];

       }

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

         delete _arc_maps[i];

       }

     }

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

     ///

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

     /// should be a map, which key type is the Node type of the source

     /// graph, while the value type is the Node type of the

     /// destination graph.

     template <typename NodeRef>

     DigraphCopy& nodeRef(NodeRef& map) {

       _node_maps.push_back(new _graph_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. The parameter should be a map, which key type

     ///  is the Node type of the destination graph, while the value type is

     ///  the Node type of the source graph.

     template <typename NodeCrossRef>

     DigraphCopy& nodeCrossRef(NodeCrossRef& map) {

       _node_maps.push_back(new _graph_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 destination graph's node type,

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

     template <typename ToMap, typename FromMap>

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

       _node_maps.push_back(new _graph_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) {

       _node_maps.push_back(new _graph_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) {

       _arc_maps.push_back(new _graph_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) {

       _arc_maps.push_back(new _graph_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) {

       _arc_maps.push_back(new _graph_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) {

       _arc_maps.push_back(new _graph_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);

       _graph_utils_bits::DigraphCopySelector<To>::

         copy(_to, _from, nodeRefMap, arcRefMap);

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

         _node_maps[i]->copy(_from, nodeRefMap);

       }

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

         _arc_maps[i]->copy(_from, arcRefMap);

       }      

     }

   protected:

     const From& _from;

     To& _to;

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

     _node_maps;

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

     _arc_maps;

   };

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

   ///

   /// Copy a digraph to another digraph. The complete usage of the

   /// function is detailed in the DigraphCopy class, but a short

   /// example shows a basic work:

   ///\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 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.

   ///

   /// This class not just make a copy of a graph, but it can create

   /// references and cross references between the nodes, edges and arcs of

   /// the two graphs, it can copy maps for use with the newly created

   /// graph and copy nodes, edges and arcs.

   ///

   /// To make a copy from a graph, first an instance of GraphCopy

   /// should be created, then the data belongs to the graph should

   /// assigned to copy. In the end, the \c run() member should be

   /// called.

   ///

   /// The next code copies a graph with several data:

   ///\code

   ///  GraphCopy<NewGraph, OrigGraph> dc(new_graph, orig_graph);

   ///  // create a reference for the nodes

   ///  OrigGraph::NodeMap<NewGraph::Node> nr(orig_graph);

   ///  dc.nodeRef(nr);

   ///  // create a cross reference (inverse) for the edges

   ///  NewGraph::EdgeMap<OrigGraph::Arc> ecr(new_graph);

   ///  dc.edgeCrossRef(ecr);

   ///  // copy an arc map

   ///  OrigGraph::ArcMap<double> oamap(orig_graph);

   ///  NewGraph::ArcMap<double> namap(new_graph);

   ///  dc.arcMap(namap, oamap);

   ///  // copy a node

   ///  OrigGraph::Node on;

   ///  NewGraph::Node nn;

   ///  dc.node(nn, on);

   ///  // Executions of copy

   ///  dc.run();

   ///\endcode

   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.u(key) != _from.v(key) ?

 	  _node_ref[_from.source(key)] == 

 	  _to.source(_to.direct(_edge_ref[key], true)) :

 	  _from.direction(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 GraphCopy.

     ///

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

     /// \c _to graph.

     GraphCopy(To& to, const From& from) 

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

     /// \brief Destructor of the GraphCopy

     ///

     /// Destructor of the GraphCopy

     ~GraphCopy() {

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

         delete _node_maps[i];

       }

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

         delete _arc_maps[i];

       }

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

         delete _edge_maps[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) {

       _node_maps.push_back(new _graph_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) {

       _node_maps.push_back(new _graph_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 graph. 

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

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

     /// type.  

     template <typename ToMap, typename FromMap>

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

       _node_maps.push_back(new _graph_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) {

       _node_maps.push_back(new _graph_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) {

       _arc_maps.push_back(new _graph_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) {

       _arc_maps.push_back(new _graph_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 graph. 

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

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

     /// type.  

     template <typename ToMap, typename FromMap>

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

       _arc_maps.push_back(new _graph_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) {

       _arc_maps.push_back(new _graph_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) {

       _edge_maps.push_back(new _graph_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) {

       _edge_maps.push_back(new _graph_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 graph. 

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

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

     /// type.  

     template <typename ToMap, typename FromMap>

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

       _edge_maps.push_back(new _graph_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) {

       _edge_maps.push_back(new _graph_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);

       _graph_utils_bits::GraphCopySelector<To>::

         copy(_to, _from, nodeRefMap, edgeRefMap);

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

         _node_maps[i]->copy(_from, nodeRefMap);

       }

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

         _edge_maps[i]->copy(_from, edgeRefMap);

       }

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

         _arc_maps[i]->copy(_from, arcRefMap);

       }

     }

   private:

     const From& _from;

     To& _to;

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

     _node_maps;

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

     _arc_maps;

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

     _edge_maps;

   };

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

   ///

   /// Copy a graph to another graph. The complete usage of the

   /// function is detailed in the GraphCopy class, but a short

   /// example shows a basic work:

   ///\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 graph to the nodes of the \c to graph and

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

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

   ///

   /// \see GraphCopy 

   template <typename To, typename From>

   GraphCopy<To, From> 

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

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

   }

   /// @}

   /// \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 or with the \c operator() member.

   ///

   template <typename _Graph, typename _Item>

   class IdMap {

   public:

     typedef _Graph Graph;

     typedef int Value;

     typedef _Item Item;

     typedef _Item Key;

     /// \brief Constructor.

     ///

     /// Constructor of the 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 item.

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

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

     ///

     /// Gives back the item by its id.

     Item operator()(int id) { return _graph->fromId(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& map) : _graph(map._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.

   ///

   /// \tparam _Graph The graph type.

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

   /// \tparam _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 _inv_map;    

   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 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(_inv_map.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(_inv_map.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 = _inv_map.find(oldval);

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

 	_inv_map.erase(it);

       }      

       _inv_map.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 = _inv_map.find(key);

       return it != _inv_map.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 = _inv_map.find(val);

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

 	_inv_map.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 = _inv_map.find(val);

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

 	  _inv_map.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() {

       _inv_map.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 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, or with the \c operator() member.

   ///

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

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

   /// Edge.

   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, 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 Graph& _graph) : Map(_graph) {

       Item it;

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

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

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

 	_inv_map.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, _inv_map.size());

       _inv_map.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], _inv_map.size());

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

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

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

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

 	_inv_map.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, _inv_map.size());

 	_inv_map.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() {

       _inv_map.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 _inv_map.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);

       _inv_map[qi] = p;

       Map::set(q, pi);

       _inv_map[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 _inv_map[id];

     }

   private:

     typedef std::vector<Item> Container;

     Container _inv_map;

   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

   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

   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

   template <typename Graph>

   class ForwardMap {

   public:

     typedef typename Graph::Arc Value;

     typedef typename Graph::Edge 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 edge 

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

   ///

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

   /// \see ForwardMap

   template <typename Graph>

   class BackwardMap {

   public:

     typedef typename Graph::Arc Value;

     typedef typename Graph::Edge 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 edge 

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

       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 _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;

       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 is 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.

   ///

   ///\tparam 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;

     TEMPLATE_DIGRAPH_TYPEDEFS(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];

 	  }

 	}

       }

     }

     void remove(Arc arc) {

       if (_left[arc] == INVALID) {

 	if (_right[arc] != INVALID) {

 	  _parent.set(_right[arc], _parent[arc]);

 	}

 	if (_parent[arc] != INVALID) {

 	  if (_left[_parent[arc]] == arc) {

 	    _left.set(_parent[arc], _right[arc]);

 	  } else {

 	    _right.set(_parent[arc], _right[arc]);

 	  }

 	} else {

 	  _head.set(_g.source(arc), _right[arc]);

 	}

       } else if (_right[arc] == INVALID) {

 	_parent.set(_left[arc], _parent[arc]);

 	if (_parent[arc] != INVALID) {

 	  if (_left[_parent[arc]] == arc) {

 	    _left.set(_parent[arc], _left[arc]);

 	  } else {

 	    _right.set(_parent[arc], _left[arc]);

 	  }

 	} else {

 	  _head.set(_g.source(arc), _left[arc]);

 	}

       } else {

 	Arc e = _left[arc];

 	if (_right[e] != INVALID) {

 	  e = _right[e];	  

 	  while (_right[e] != INVALID) {

 	    e = _right[e];

 	  }

 	  Arc s = _parent[e];

 	  _right.set(_parent[e], _left[e]);

 	  if (_left[e] != INVALID) {

 	    _parent.set(_left[e], _parent[e]);

 	  }

 	  _left.set(e, _left[arc]);

 	  _parent.set(_left[arc], e);

 	  _right.set(e, _right[arc]);

 	  _parent.set(_right[arc], e);

 	  _parent.set(e, _parent[arc]);

 	  if (_parent[arc] != INVALID) {

 	    if (_left[_parent[arc]] == arc) {

 	      _left.set(_parent[arc], e);

 	    } else {

 	      _right.set(_parent[arc], e);

 	    }

 	  }

 	  splay(s);

 	} else {

 	  _right.set(e, _right[arc]);

 	  _parent.set(_right[arc], e);

 	  if (_parent[arc] != INVALID) {

 	    if (_left[_parent[arc]] == arc) {

 	      _left.set(_parent[arc], e);

 	    } else {

 	      _right.set(_parent[arc], e);

 	    }

 	  } else {

 	    _head.set(_g.source(arc), e);

 	  }

 	}

       }

     }

     Arc refreshRec(std::vector<Arc> &v,int a,int b) 

     {

       int m=(a+b)/2;

       Arc me=v[m];

       if (a < m) {

 	Arc left = refreshRec(v,a,m-1);

 	_left.set(me, left);

 	_parent.set(left, me);

       } else {

 	_left.set(me, INVALID);

       }

       if (m < b) {

 	Arc right = refreshRec(v,m+1,b);

 	_right.set(me, right);

 	_parent.set(right, me);

       } else {

 	_right.set(me, INVALID);

       }

       return me;

     }

     void refresh() {

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

 	std::vector<Arc> v;

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

 	if(v.size()) {

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

 	  Arc head = refreshRec(v,0,v.size()-1);

 	  _head.set(n, head);

 	  _parent.set(head, INVALID);

 	}

 	else _head.set(n, INVALID);

       }

     }

     void zig(Arc v) {        

       Arc w = _parent[v];

       _parent.set(v, _parent[w]);

       _parent.set(w, v);

       _left.set(w, _right[v]);

       _right.set(v, w);

       if (_parent[v] != INVALID) {

 	if (_right[_parent[v]] == w) {

 	  _right.set(_parent[v], v);

 	} else {

 	  _left.set(_parent[v], v);

 	}

       }

       if (_left[w] != INVALID){

 	_parent.set(_left[w], w);

       }

     }

     void zag(Arc v) {        

       Arc w = _parent[v];

       _parent.set(v, _parent[w]);

       _parent.set(w, v);

       _right.set(w, _left[v]);

       _left.set(v, w);

       if (_parent[v] != INVALID){

 	if (_left[_parent[v]] == w) {

 	  _left.set(_parent[v], v);

 	} else {

 	  _right.set(_parent[v], v);

 	}

       }

       if (_right[w] != INVALID){

 	_parent.set(_right[w], w);

       }

     }

     void splay(Arc v) {

       while (_parent[v] != INVALID) {

 	if (v == _left[_parent[v]]) {

 	  if (_parent[_parent[v]] == INVALID) {

 	    zig(v);

 	  } else {

 	    if (_parent[v] == _left[_parent[_parent[v]]]) {

 	      zig(_parent[v]);

 	      zig(v);

 	    } else {

 	      zig(v);

 	      zag(v);

 	    }

 	  }

 	} else {

 	  if (_parent[_parent[v]] == INVALID) {

 	    zag(v);

 	  } else {

 	    if (_parent[v] == _left[_parent[_parent[v]]]) {

 	      zag(v);

 	      zig(v);

 	    } else {

 	      zag(_parent[v]);

 	      zag(v);

 	    }

 	  }

 	}

       }

       _head[_g.source(v)] = v;

     }

   public:

     ///Find an arc between two nodes.

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

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

     ///\param s The source node

     ///\param t The target node

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

     ///\ref INVALID otherwise.

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

     {

       Arc a = _head[s];

       while (true) {

 	if (_g.target(a) == t) {

 	  const_cast<DynArcLookUp&>(*this).splay(a);

 	  return a;

 	} else if (t < _g.target(a)) {

 	  if (_left[a] == INVALID) {

 	    const_cast<DynArcLookUp&>(*this).splay(a);

 	    return INVALID;

 	  } else {

 	    a = _left[a];

 	  }

 	} else  {

 	  if (_right[a] == INVALID) {

 	    const_cast<DynArcLookUp&>(*this).splay(a);

 	    return INVALID;

 	  } else {

 	    a = _right[a];

 	  }

 	}

       }

     }

     ///Find the first arc between two nodes.

     ///Find the first arc between two nodes in time

     /// <em>O(</em>log<em>d)</em>, where <em>d</em> is the number of

     /// outgoing arcs of \c s.  

     ///\param s The source node 

     ///\param t The target node

     ///\return An arc from \c s to \c t if there exists, \ref INVALID

     /// otherwise.

     Arc findFirst(Node s, Node t) const

     {

       Arc a = _head[s];

       Arc r = INVALID;

       while (true) {

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

 	  if (_right[a] == INVALID) {

 	    const_cast<DynArcLookUp&>(*this).splay(a);

 	    return r;

 	  } else {

 	    a = _right[a];

 	  }

 	} else {

 	  if (_g.target(a) == t) {

 	    r = a;

 	  }

 	  if (_left[a] == INVALID) {

 	    const_cast<DynArcLookUp&>(*this).splay(a);

 	    return r;

 	  } else {

 	    a = _left[a];

 	  }

 	}

       }

     }

     ///Find the next arc between two nodes.

     ///Find the next arc between two nodes in time

     /// <em>O(</em>log<em>d)</em>, where <em>d</em> is the number of

     /// outgoing arcs of \c s.  

     ///\param s The source node 

     ///\param t The target node

     ///\return An arc from \c s to \c t if there exists, \ref INVALID

     /// otherwise.

     ///\note If \c e is not the result of the previous \c findFirst()

     ///operation then the amorized time bound can not be guaranteed.

 #ifdef DOXYGEN

     Arc findNext(Node s, Node t, Arc a) const

 #else

     Arc findNext(Node, Node t, Arc a) const

 #endif

     {

       if (_right[a] != INVALID) {

 	a = _right[a];

 	while (_left[a] != INVALID) {

 	  a = _left[a];

 	}

 	const_cast<DynArcLookUp&>(*this).splay(a);

       } else {

 	while (_parent[a] != INVALID && _right[_parent[a]] ==  a) {

 	  a = _parent[a];

 	}

 	if (_parent[a] == INVALID) {

 	  return INVALID;

 	} else {

 	  a = _parent[a];

 	  const_cast<DynArcLookUp&>(*this).splay(a);

 	}

       }

       if (_g.target(a) == t) return a;

       else return INVALID;    

     }

   };

   ///Fast 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 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 arcs between two nodes.

   ///Use \ref AllArcLookUp for this purpose.

   ///

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

   ///refresh(Node)) this data structure

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

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

   ///

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

   ///

   ///\sa DynArcLookUp

   ///\sa AllArcLookUp  

   template<class G>

   class ArcLookUp 

   {

   public:

     TEMPLATE_DIGRAPH_TYPEDEFS(G);

     typedef G Digraph;

   protected:

     const Digraph &_g;

     typename Digraph::template NodeMap<Arc> _head;

     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, which remains valid until the digraph

     ///changes.

     ArcLookUp(const Digraph &g) :_g(g),_head(g),_left(g),_right(g) {refresh();}

   private:

     Arc refreshRec(std::vector<Arc> &v,int a,int b) 

     {

       int m=(a+b)/2;

       Arc me=v[m];

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

       _right[me] = m<b?refreshRec(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 arcs of \c n.

     void refresh(Node n) 

     {

       std::vector<Arc> v;

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

       if(v.size()) {

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

 	_head[n]=refreshRec(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 arcs of \c n and <em>D</em> is the maximum

     ///out-degree of the digraph.

     void refresh() 

     {

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

     }

     ///Find an arc between two nodes.

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

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

     ///\param s The source node

     ///\param t The target node

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

     ///\ref INVALID otherwise.

     ///

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

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

     ///a single node \c n, then

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

     ///

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

     {

       Arc 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 arcs between given endpoints.

   ///\ingroup gutils

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

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

   ///

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

   ///refresh(Node)) this data structure

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

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

   ///

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

   ///

   ///\sa DynArcLookUp

   ///\sa ArcLookUp  

   template<class G>