diff -r cc7e6b8b59bf -r 70f3967ca6eb lemon/graph_utils.h
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/lemon/graph_utils.h	Thu Mar 20 12:12:24 2008 +0000
@@ -0,0 +1,3179 @@
+/* -*- C++ -*-
+ *
+ * This file is a part of LEMON, a generic C++ optimization library
+ *
+ * Copyright (C) 2003-2008
+ * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
+ * (Egervary Research Group on Combinatorial Optimization, EGRES).
+ *
+ * Permission to use, modify and distribute this software is granted
+ * provided that this copyright notice appears in all copies. For
+ * precise terms see the accompanying LICENSE file.
+ *
+ * This software is provided "AS IS" with no warranty of any kind,
+ * express or implied, and with no claim as to its suitability for any
+ * purpose.
+ *
+ */
+
+#ifndef LEMON_GRAPH_UTILS_H
+#define LEMON_GRAPH_UTILS_H
+
+#include <iterator>
+#include <vector>
+#include <map>
+#include <cmath>
+#include <algorithm>
+
+#include <lemon/bits/invalid.h>
+#include <lemon/bits/utility.h>
+#include <lemon/maps.h>
+#include <lemon/bits/traits.h>
+
+#include <lemon/bits/alteration_notifier.h>
+#include <lemon/bits/default_map.h>
+
+///\ingroup gutils
+///\file
+///\brief Digraph utilities.
+
+namespace lemon {
+
+  /// \addtogroup gutils
+  /// @{
+
+  ///Creates convenience typedefs for the digraph types and iterators
+
+  ///This \c \#define creates convenience typedefs for the following types
+  ///of \c Digraph: \c Node,  \c NodeIt, \c Arc, \c ArcIt, \c InArcIt,
+  ///\c OutArcIt
+  ///\note If \c G it a template parameter, it should be used in this way.
+  ///\code
+  ///  GRAPH_TYPEDEFS(typename G);
+  ///\endcode
+  ///
+  ///\warning There are no typedefs for the digraph maps because of the lack of
+  ///template typedefs in C++.
+#define GRAPH_TYPEDEFS(Digraph)				\
+  typedef Digraph::     Node      Node;			\
+    typedef Digraph::   NodeIt    NodeIt;			\
+    typedef Digraph::   Arc      Arc;			\
+    typedef Digraph::   ArcIt    ArcIt;			\
+    typedef Digraph:: InArcIt  InArcIt;			\
+    typedef Digraph::OutArcIt OutArcIt
+
+  ///Creates convenience typedefs for the graph types and iterators
+
+  ///This \c \#define creates the same convenience typedefs as defined by
+  ///\ref GRAPH_TYPEDEFS(Digraph) and three more, namely it creates
+  ///\c Edge, \c EdgeIt, \c IncArcIt,
+  ///
+  ///\note If \c G it a template parameter, it should be used in this way.
+  ///\code
+  ///  UGRAPH_TYPEDEFS(typename G);
+  ///\endcode
+  ///
+  ///\warning There are no typedefs for the digraph maps because of the lack of
+  ///template typedefs in C++.
+#define UGRAPH_TYPEDEFS(Digraph)				\
+  GRAPH_TYPEDEFS(Digraph);				\
+    typedef Digraph:: Edge   Edge;			\
+    typedef Digraph:: EdgeIt EdgeIt;			\
+    typedef Digraph:: IncArcIt   IncArcIt
+
+  ///\brief Creates convenience typedefs for the bipartite digraph 
+  ///types and iterators
+
+  ///This \c \#define creates the same convenience typedefs as defined by
+  ///\ref UGRAPH_TYPEDEFS(Digraph) and two more, namely it creates
+  ///\c RedIt, \c BlueIt, 
+  ///
+  ///\note If \c G it a template parameter, it should be used in this way.
+  ///\code
+  ///  BPUGRAPH_TYPEDEFS(typename G);
+  ///\endcode
+  ///
+  ///\warning There are no typedefs for the digraph maps because of the lack of
+  ///template typedefs in C++.
+#define BPUGRAPH_TYPEDEFS(Digraph)            \
+  UGRAPH_TYPEDEFS(Digraph);		    \
+    typedef Digraph::Red Red;             \
+    typedef Digraph::Blue Blue;             \
+    typedef Digraph::RedIt RedIt;	    \
+    typedef Digraph::BlueIt BlueIt
+
+  /// \brief Function to count the items in the digraph.
+  ///
+  /// This function counts the items (nodes, arcs etc) in the digraph.
+  /// The complexity of the function is O(n) because
+  /// it iterates on all of the items.
+
+  template <typename Digraph, typename Item>
+  inline int countItems(const Digraph& g) {
+    typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
+    int num = 0;
+    for (ItemIt it(g); it != INVALID; ++it) {
+      ++num;
+    }
+    return num;
+  }
+
+  // Node counting:
+
+  namespace _digraph_utils_bits {
+    
+    template <typename Digraph, typename Enable = void>
+    struct CountNodesSelector {
+      static int count(const Digraph &g) {
+        return countItems<Digraph, typename Digraph::Node>(g);
+      }
+    };
+
+    template <typename Digraph>
+    struct CountNodesSelector<
+      Digraph, typename 
+      enable_if<typename Digraph::NodeNumTag, void>::type> 
+    {
+      static int count(const Digraph &g) {
+        return g.nodeNum();
+      }
+    };    
+  }
+
+  /// \brief Function to count the nodes in the digraph.
+  ///
+  /// This function counts the nodes in the digraph.
+  /// The complexity of the function is O(n) but for some
+  /// digraph structures it is specialized to run in O(1).
+  ///
+  /// If the digraph contains a \e nodeNum() member function and a 
+  /// \e NodeNumTag tag then this function calls directly the member
+  /// function to query the cardinality of the node set.
+  template <typename Digraph>
+  inline int countNodes(const Digraph& g) {
+    return _digraph_utils_bits::CountNodesSelector<Digraph>::count(g);
+  }
+
+  namespace _digraph_utils_bits {
+    
+    template <typename Digraph, typename Enable = void>
+    struct CountRedsSelector {
+      static int count(const Digraph &g) {
+        return countItems<Digraph, typename Digraph::Red>(g);
+      }
+    };
+
+    template <typename Digraph>
+    struct CountRedsSelector<
+      Digraph, typename 
+      enable_if<typename Digraph::NodeNumTag, void>::type> 
+    {
+      static int count(const Digraph &g) {
+        return g.redNum();
+      }
+    };    
+  }
+
+  /// \brief Function to count the reds in the digraph.
+  ///
+  /// This function counts the reds in the digraph.
+  /// The complexity of the function is O(an) but for some
+  /// digraph structures it is specialized to run in O(1).
+  ///
+  /// If the digraph contains an \e redNum() member function and a 
+  /// \e NodeNumTag tag then this function calls directly the member
+  /// function to query the cardinality of the A-node set.
+  template <typename Digraph>
+  inline int countReds(const Digraph& g) {
+    return _digraph_utils_bits::CountRedsSelector<Digraph>::count(g);
+  }
+
+  namespace _digraph_utils_bits {
+    
+    template <typename Digraph, typename Enable = void>
+    struct CountBluesSelector {
+      static int count(const Digraph &g) {
+        return countItems<Digraph, typename Digraph::Blue>(g);
+      }
+    };
+
+    template <typename Digraph>
+    struct CountBluesSelector<
+      Digraph, typename 
+      enable_if<typename Digraph::NodeNumTag, void>::type> 
+    {
+      static int count(const Digraph &g) {
+        return g.blueNum();
+      }
+    };    
+  }
+
+  /// \brief Function to count the blues in the digraph.
+  ///
+  /// This function counts the blues in the digraph.
+  /// The complexity of the function is O(bn) but for some
+  /// digraph structures it is specialized to run in O(1).
+  ///
+  /// If the digraph contains a \e blueNum() member function and a 
+  /// \e NodeNumTag tag then this function calls directly the member
+  /// function to query the cardinality of the B-node set.
+  template <typename Digraph>
+  inline int countBlues(const Digraph& g) {
+    return _digraph_utils_bits::CountBluesSelector<Digraph>::count(g);
+  }
+
+
+  // Arc counting:
+
+  namespace _digraph_utils_bits {
+    
+    template <typename Digraph, typename Enable = void>
+    struct CountArcsSelector {
+      static int count(const Digraph &g) {
+        return countItems<Digraph, typename Digraph::Arc>(g);
+      }
+    };
+
+    template <typename Digraph>
+    struct CountArcsSelector<
+      Digraph, 
+      typename enable_if<typename Digraph::ArcNumTag, void>::type> 
+    {
+      static int count(const Digraph &g) {
+        return g.arcNum();
+      }
+    };    
+  }
+
+  /// \brief Function to count the arcs in the digraph.
+  ///
+  /// This function counts the arcs in the digraph.
+  /// The complexity of the function is O(e) but for some
+  /// digraph structures it is specialized to run in O(1).
+  ///
+  /// If the digraph contains a \e arcNum() member function and a 
+  /// \e ArcNumTag tag then this function calls directly the member
+  /// function to query the cardinality of the arc set.
+  template <typename Digraph>
+  inline int countArcs(const Digraph& g) {
+    return _digraph_utils_bits::CountArcsSelector<Digraph>::count(g);
+  }
+
+  // Undirected arc counting:
+  namespace _digraph_utils_bits {
+    
+    template <typename Digraph, typename Enable = void>
+    struct CountEdgesSelector {
+      static int count(const Digraph &g) {
+        return countItems<Digraph, typename Digraph::Edge>(g);
+      }
+    };
+
+    template <typename Digraph>
+    struct CountEdgesSelector<
+      Digraph, 
+      typename enable_if<typename Digraph::ArcNumTag, void>::type> 
+    {
+      static int count(const Digraph &g) {
+        return g.edgeNum();
+      }
+    };    
+  }
+
+  /// \brief Function to count the edges in the digraph.
+  ///
+  /// This function counts the edges in the digraph.
+  /// The complexity of the function is O(e) but for some
+  /// digraph structures it is specialized to run in O(1).
+  ///
+  /// If the digraph contains a \e edgeNum() member function and a 
+  /// \e ArcNumTag tag then this function calls directly the member
+  /// function to query the cardinality of the edge set.
+  template <typename Digraph>
+  inline int countEdges(const Digraph& g) {
+    return _digraph_utils_bits::CountEdgesSelector<Digraph>::count(g);
+
+  }
+
+
+  template <typename Digraph, typename DegIt>
+  inline int countNodeDegree(const Digraph& _g, const typename Digraph::Node& _n) {
+    int num = 0;
+    for (DegIt it(_g, _n); it != INVALID; ++it) {
+      ++num;
+    }
+    return num;
+  }
+
+  /// \brief Function to count the number of the out-arcs from node \c n.
+  ///
+  /// This function counts the number of the out-arcs from node \c n
+  /// in the digraph.  
+  template <typename Digraph>
+  inline int countOutArcs(const Digraph& _g,  const typename Digraph::Node& _n) {
+    return countNodeDegree<Digraph, typename Digraph::OutArcIt>(_g, _n);
+  }
+
+  /// \brief Function to count the number of the in-arcs to node \c n.
+  ///
+  /// This function counts the number of the in-arcs to node \c n
+  /// in the digraph.  
+  template <typename Digraph>
+  inline int countInArcs(const Digraph& _g,  const typename Digraph::Node& _n) {
+    return countNodeDegree<Digraph, typename Digraph::InArcIt>(_g, _n);
+  }
+
+  /// \brief Function to count the number of the inc-arcs to node \c n.
+  ///
+  /// This function counts the number of the inc-arcs to node \c n
+  /// in the digraph.  
+  template <typename Digraph>
+  inline int countIncArcs(const Digraph& _g,  const typename Digraph::Node& _n) {
+    return countNodeDegree<Digraph, typename Digraph::IncArcIt>(_g, _n);
+  }
+
+  namespace _digraph_utils_bits {
+    
+    template <typename Digraph, typename Enable = void>
+    struct FindArcSelector {
+      typedef typename Digraph::Node Node;
+      typedef typename Digraph::Arc Arc;
+      static Arc find(const Digraph &g, Node u, Node v, Arc e) {
+        if (e == INVALID) {
+          g.firstOut(e, u);
+        } else {
+          g.nextOut(e);
+        }
+        while (e != INVALID && g.target(e) != v) {
+          g.nextOut(e);
+        }
+        return e;
+      }
+    };
+
+    template <typename Digraph>
+    struct FindArcSelector<
+      Digraph, 
+      typename enable_if<typename Digraph::FindArcTag, void>::type> 
+    {
+      typedef typename Digraph::Node Node;
+      typedef typename Digraph::Arc Arc;
+      static Arc find(const Digraph &g, Node u, Node v, Arc prev) {
+        return g.findArc(u, v, prev);
+      }
+    };    
+  }
+
+  /// \brief Finds an arc between two nodes of a digraph.
+  ///
+  /// Finds an arc from node \c u to node \c v in digraph \c g.
+  ///
+  /// If \c prev is \ref INVALID (this is the default value), then
+  /// it finds the first arc from \c u to \c v. Otherwise it looks for
+  /// the next arc from \c u to \c v after \c prev.
+  /// \return The found arc or \ref INVALID if there is no such an arc.
+  ///
+  /// Thus you can iterate through each arc from \c u to \c v as it follows.
+  ///\code
+  /// for(Arc e=findArc(g,u,v);e!=INVALID;e=findArc(g,u,v,e)) {
+  ///   ...
+  /// }
+  ///\endcode
+  ///
+  ///\sa ArcLookUp
+  ///\sa AllArcLookUp
+  ///\sa DynArcLookUp
+  ///\sa ConArcIt
+  template <typename Digraph>
+  inline typename Digraph::Arc 
+  findArc(const Digraph &g, typename Digraph::Node u, typename Digraph::Node v,
+           typename Digraph::Arc prev = INVALID) {
+    return _digraph_utils_bits::FindArcSelector<Digraph>::find(g, u, v, prev);
+  }
+
+  /// \brief Iterator for iterating on arcs connected the same nodes.
+  ///
+  /// Iterator for iterating on arcs connected the same nodes. It is 
+  /// higher level interface for the findArc() function. You can
+  /// use it the following way:
+  ///\code
+  /// for (ConArcIt<Digraph> it(g, src, trg); it != INVALID; ++it) {
+  ///   ...
+  /// }
+  ///\endcode
+  /// 
+  ///\sa findArc()
+  ///\sa ArcLookUp
+  ///\sa AllArcLookUp
+  ///\sa DynArcLookUp
+  ///
+  /// \author Balazs Dezso 
+  template <typename _Digraph>
+  class ConArcIt : public _Digraph::Arc {
+  public:
+
+    typedef _Digraph Digraph;
+    typedef typename Digraph::Arc Parent;
+
+    typedef typename Digraph::Arc Arc;
+    typedef typename Digraph::Node Node;
+
+    /// \brief Constructor.
+    ///
+    /// Construct a new ConArcIt iterating on the arcs which
+    /// connects the \c u and \c v node.
+    ConArcIt(const Digraph& g, Node u, Node v) : digraph(g) {
+      Parent::operator=(findArc(digraph, u, v));
+    }
+
+    /// \brief Constructor.
+    ///
+    /// Construct a new ConArcIt which continues the iterating from 
+    /// the \c e arc.
+    ConArcIt(const Digraph& g, Arc e) : Parent(e), digraph(g) {}
+    
+    /// \brief Increment operator.
+    ///
+    /// It increments the iterator and gives back the next arc.
+    ConArcIt& operator++() {
+      Parent::operator=(findArc(digraph, digraph.source(*this), 
+				 digraph.target(*this), *this));
+      return *this;
+    }
+  private:
+    const Digraph& digraph;
+  };
+
+  namespace _digraph_utils_bits {
+    
+    template <typename Digraph, typename Enable = void>
+    struct FindEdgeSelector {
+      typedef typename Digraph::Node Node;
+      typedef typename Digraph::Edge Edge;
+      static Edge find(const Digraph &g, Node u, Node v, Edge e) {
+        bool b;
+        if (u != v) {
+          if (e == INVALID) {
+            g.firstInc(e, b, u);
+          } else {
+            b = g.source(e) == u;
+            g.nextInc(e, b);
+          }
+          while (e != INVALID && (b ? g.target(e) : g.source(e)) != v) {
+            g.nextInc(e, b);
+          }
+        } else {
+          if (e == INVALID) {
+            g.firstInc(e, b, u);
+          } else {
+            b = true;
+            g.nextInc(e, b);
+          }
+          while (e != INVALID && (!b || g.target(e) != v)) {
+            g.nextInc(e, b);
+          }
+        }
+        return e;
+      }
+    };
+
+    template <typename Digraph>
+    struct FindEdgeSelector<
+      Digraph, 
+      typename enable_if<typename Digraph::FindArcTag, void>::type> 
+    {
+      typedef typename Digraph::Node Node;
+      typedef typename Digraph::Edge Edge;
+      static Edge find(const Digraph &g, Node u, Node v, Edge prev) {
+        return g.findEdge(u, v, prev);
+      }
+    };    
+  }
+
+  /// \brief Finds an edge between two nodes of a digraph.
+  ///
+  /// Finds an edge from node \c u to node \c v in digraph \c g.
+  /// If the node \c u and node \c v is equal then each loop arc
+  /// will be enumerated.
+  ///
+  /// If \c prev is \ref INVALID (this is the default value), then
+  /// it finds the first arc from \c u to \c v. Otherwise it looks for
+  /// the next arc from \c u to \c v after \c prev.
+  /// \return The found arc or \ref INVALID if there is no such an arc.
+  ///
+  /// Thus you can iterate through each arc from \c u to \c v as it follows.
+  ///\code
+  /// for(Edge e = findEdge(g,u,v); e != INVALID; 
+  ///     e = findEdge(g,u,v,e)) {
+  ///   ...
+  /// }
+  ///\endcode
+  ///
+  ///\sa ConArcIt
+
+  template <typename Digraph>
+  inline typename Digraph::Edge 
+  findEdge(const Digraph &g, typename Digraph::Node u, typename Digraph::Node v,
+            typename Digraph::Edge p = INVALID) {
+    return _digraph_utils_bits::FindEdgeSelector<Digraph>::find(g, u, v, p);
+  }
+
+  /// \brief Iterator for iterating on edges connected the same nodes.
+  ///
+  /// Iterator for iterating on edges connected the same nodes. It is 
+  /// higher level interface for the findEdge() function. You can
+  /// use it the following way:
+  ///\code
+  /// for (ConEdgeIt<Digraph> it(g, src, trg); it != INVALID; ++it) {
+  ///   ...
+  /// }
+  ///\endcode
+  ///
+  ///\sa findEdge()
+  ///
+  /// \author Balazs Dezso 
+  template <typename _Digraph>
+  class ConEdgeIt : public _Digraph::Edge {
+  public:
+
+    typedef _Digraph Digraph;
+    typedef typename Digraph::Edge Parent;
+
+    typedef typename Digraph::Edge Edge;
+    typedef typename Digraph::Node Node;
+
+    /// \brief Constructor.
+    ///
+    /// Construct a new ConEdgeIt iterating on the arcs which
+    /// connects the \c u and \c v node.
+    ConEdgeIt(const Digraph& g, Node u, Node v) : digraph(g) {
+      Parent::operator=(findEdge(digraph, u, v));
+    }
+
+    /// \brief Constructor.
+    ///
+    /// Construct a new ConEdgeIt which continues the iterating from 
+    /// the \c e arc.
+    ConEdgeIt(const Digraph& g, Edge e) : Parent(e), digraph(g) {}
+    
+    /// \brief Increment operator.
+    ///
+    /// It increments the iterator and gives back the next arc.
+    ConEdgeIt& operator++() {
+      Parent::operator=(findEdge(digraph, digraph.source(*this), 
+				      digraph.target(*this), *this));
+      return *this;
+    }
+  private:
+    const Digraph& digraph;
+  };
+
+  /// \brief Copy a map.
+  ///
+  /// This function copies the \c from map to the \c to map. It uses the
+  /// given iterator to iterate on the data structure and it uses the \c ref
+  /// mapping to convert the from's keys to the to's keys.
+  template <typename To, typename From, 
+	    typename ItemIt, typename Ref>	    
+  void copyMap(To& to, const From& from, 
+	       ItemIt it, const Ref& ref) {
+    for (; it != INVALID; ++it) {
+      to[ref[it]] = from[it];
+    }
+  }
+
+  /// \brief Copy the from map to the to map.
+  ///
+  /// Copy the \c from map to the \c to map. It uses the given iterator
+  /// to iterate on the data structure.
+  template <typename To, typename From, typename ItemIt>	    
+  void copyMap(To& to, const From& from, ItemIt it) {
+    for (; it != INVALID; ++it) {
+      to[it] = from[it];
+    }
+  }
+
+  namespace _digraph_utils_bits {
+
+    template <typename Digraph, typename Item, typename RefMap>
+    class MapCopyBase {
+    public:
+      virtual void copy(const Digraph& from, const RefMap& refMap) = 0;
+      
+      virtual ~MapCopyBase() {}
+    };
+
+    template <typename Digraph, typename Item, typename RefMap, 
+              typename ToMap, typename FromMap>
+    class MapCopy : public MapCopyBase<Digraph, Item, RefMap> {
+    public:
+
+      MapCopy(ToMap& tmap, const FromMap& map) 
+        : _tmap(tmap), _map(map) {}
+      
+      virtual void copy(const Digraph& digraph, const RefMap& refMap) {
+        typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
+        for (ItemIt it(digraph); it != INVALID; ++it) {
+          _tmap.set(refMap[it], _map[it]);
+        }
+      }
+
+    private:
+      ToMap& _tmap;
+      const FromMap& _map;
+    };
+
+    template <typename Digraph, typename Item, typename RefMap, typename It>
+    class ItemCopy : public MapCopyBase<Digraph, Item, RefMap> {
+    public:
+
+      ItemCopy(It& it, const Item& item) : _it(it), _item(item) {}
+      
+      virtual void copy(const Digraph&, const RefMap& refMap) {
+        _it = refMap[_item];
+      }
+
+    private:
+      It& _it;
+      Item _item;
+    };
+
+    template <typename Digraph, typename Item, typename RefMap, typename Ref>
+    class RefCopy : public MapCopyBase<Digraph, Item, RefMap> {
+    public:
+
+      RefCopy(Ref& map) : _map(map) {}
+      
+      virtual void copy(const Digraph& digraph, const RefMap& refMap) {
+        typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
+        for (ItemIt it(digraph); it != INVALID; ++it) {
+          _map.set(it, refMap[it]);
+        }
+      }
+
+    private:
+      Ref& _map;
+    };
+
+    template <typename Digraph, typename Item, typename RefMap, 
+              typename CrossRef>
+    class CrossRefCopy : public MapCopyBase<Digraph, Item, RefMap> {
+    public:
+
+      CrossRefCopy(CrossRef& cmap) : _cmap(cmap) {}
+      
+      virtual void copy(const Digraph& digraph, const RefMap& refMap) {
+        typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
+        for (ItemIt it(digraph); it != INVALID; ++it) {
+          _cmap.set(refMap[it], it);
+        }
+      }
+
+    private:
+      CrossRef& _cmap;
+    };
+
+    template <typename Digraph, typename Enable = void>
+    struct DigraphCopySelector {
+      template <typename From, typename NodeRefMap, typename ArcRefMap>
+      static void copy(Digraph &to, const From& from,
+                       NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {
+        for (typename From::NodeIt it(from); it != INVALID; ++it) {
+          nodeRefMap[it] = to.addNode();
+        }
+        for (typename From::ArcIt it(from); it != INVALID; ++it) {
+          arcRefMap[it] = to.addArc(nodeRefMap[from.source(it)], 
+                                          nodeRefMap[from.target(it)]);
+        }
+      }
+    };
+
+    template <typename Digraph>
+    struct DigraphCopySelector<
+      Digraph, 
+      typename enable_if<typename Digraph::BuildTag, void>::type> 
+    {
+      template <typename From, typename NodeRefMap, typename ArcRefMap>
+      static void copy(Digraph &to, const From& from,
+                       NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {
+        to.build(from, nodeRefMap, arcRefMap);
+      }
+    };
+
+    template <typename Graph, typename Enable = void>
+    struct GraphCopySelector {
+      template <typename From, typename NodeRefMap, typename EdgeRefMap>
+      static void copy(Graph &to, const From& from,
+                       NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {
+        for (typename From::NodeIt it(from); it != INVALID; ++it) {
+          nodeRefMap[it] = to.addNode();
+        }
+        for (typename From::EdgeIt it(from); it != INVALID; ++it) {
+          edgeRefMap[it] = to.addArc(nodeRefMap[from.source(it)], 
+				       nodeRefMap[from.target(it)]);
+        }
+      }
+    };
+
+    template <typename Graph>
+    struct GraphCopySelector<
+      Graph, 
+      typename enable_if<typename Graph::BuildTag, void>::type> 
+    {
+      template <typename From, typename NodeRefMap, typename EdgeRefMap>
+      static void copy(Graph &to, const From& from,
+                       NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {
+        to.build(from, nodeRefMap, edgeRefMap);
+      }
+    };
+
+    template <typename BpGraph, typename Enable = void>
+    struct BpGraphCopySelector {
+      template <typename From, typename RedRefMap, 
+                typename BlueRefMap, typename EdgeRefMap>
+      static void copy(BpGraph &to, const From& from,
+                       RedRefMap& redRefMap, BlueRefMap& blueRefMap,
+                       EdgeRefMap& edgeRefMap) {
+        for (typename From::RedIt it(from); it != INVALID; ++it) {
+          redRefMap[it] = to.addRed();
+        }
+        for (typename From::BlueIt it(from); it != INVALID; ++it) {
+          blueRefMap[it] = to.addBlue();
+        }
+        for (typename From::EdgeIt it(from); it != INVALID; ++it) {
+          edgeRefMap[it] = to.addArc(redRefMap[from.red(it)], 
+                                           blueRefMap[from.blue(it)]);
+        }
+      }
+    };
+
+    template <typename BpGraph>
+    struct BpGraphCopySelector<
+      BpGraph, 
+      typename enable_if<typename BpGraph::BuildTag, void>::type> 
+    {
+      template <typename From, typename RedRefMap, 
+                typename BlueRefMap, typename EdgeRefMap>
+      static void copy(BpGraph &to, const From& from,
+                       RedRefMap& redRefMap, BlueRefMap& blueRefMap,
+                       EdgeRefMap& edgeRefMap) {
+        to.build(from, redRefMap, blueRefMap, edgeRefMap);
+      }
+    };
+    
+
+  }
+
+  /// \brief Class to copy a digraph.
+  ///
+  /// Class to copy a digraph to another digraph (duplicate a digraph). The
+  /// simplest way of using it is through the \c copyDigraph() function.
+  template <typename To, typename From>
+  class DigraphCopy {
+  private:
+
+    typedef typename From::Node Node;
+    typedef typename From::NodeIt NodeIt;
+    typedef typename From::Arc Arc;
+    typedef typename From::ArcIt ArcIt;
+
+    typedef typename To::Node TNode;
+    typedef typename To::Arc TArc;
+
+    typedef typename From::template NodeMap<TNode> NodeRefMap;
+    typedef typename From::template ArcMap<TArc> ArcRefMap;
+    
+    
+  public: 
+
+
+    /// \brief Constructor for the DigraphCopy.
+    ///
+    /// It copies the content of the \c _from digraph into the
+    /// \c _to digraph.
+    DigraphCopy(To& _to, const From& _from) 
+      : from(_from), to(_to) {}
+
+    /// \brief Destructor of the DigraphCopy
+    ///
+    /// Destructor of the DigraphCopy
+    ~DigraphCopy() {
+      for (int i = 0; i < int(nodeMapCopies.size()); ++i) {
+        delete nodeMapCopies[i];
+      }
+      for (int i = 0; i < int(arcMapCopies.size()); ++i) {
+        delete arcMapCopies[i];
+      }
+
+    }
+
+    /// \brief Copies the node references into the given map.
+    ///
+    /// Copies the node references into the given map.
+    template <typename NodeRef>
+    DigraphCopy& nodeRef(NodeRef& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Node, 
+                              NodeRefMap, NodeRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the node cross references into the given map.
+    ///
+    ///  Copies the node cross references (reverse references) into
+    ///  the given map.
+    template <typename NodeCrossRef>
+    DigraphCopy& nodeCrossRef(NodeCrossRef& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Node,
+                              NodeRefMap, NodeCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's node type,
+    /// and the copied map's key type is the from digraph's node
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    DigraphCopy& nodeMap(ToMap& tmap, const FromMap& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Node, 
+                              NodeRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Make a copy of the given node.
+    ///
+    /// Make a copy of the given node.
+    DigraphCopy& node(TNode& tnode, const Node& snode) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Node, 
+                              NodeRefMap, TNode>(tnode, snode));
+      return *this;
+    }
+
+    /// \brief Copies the arc references into the given map.
+    ///
+    /// Copies the arc references into the given map.
+    template <typename ArcRef>
+    DigraphCopy& arcRef(ArcRef& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Arc, 
+                              ArcRefMap, ArcRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the arc cross references into the given map.
+    ///
+    ///  Copies the arc cross references (reverse references) into
+    ///  the given map.
+    template <typename ArcCrossRef>
+    DigraphCopy& arcCrossRef(ArcCrossRef& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Arc,
+                              ArcRefMap, ArcCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's arc type,
+    /// and the copied map's key type is the from digraph's arc
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    DigraphCopy& arcMap(ToMap& tmap, const FromMap& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Arc, 
+                              ArcRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Make a copy of the given arc.
+    ///
+    /// Make a copy of the given arc.
+    DigraphCopy& arc(TArc& tarc, const Arc& sarc) {
+      arcMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Arc, 
+                              ArcRefMap, TArc>(tarc, sarc));
+      return *this;
+    }
+
+    /// \brief Executes the copies.
+    ///
+    /// Executes the copies.
+    void run() {
+      NodeRefMap nodeRefMap(from);
+      ArcRefMap arcRefMap(from);
+      _digraph_utils_bits::DigraphCopySelector<To>::
+        copy(to, from, nodeRefMap, arcRefMap);
+      for (int i = 0; i < int(nodeMapCopies.size()); ++i) {
+        nodeMapCopies[i]->copy(from, nodeRefMap);
+      }
+      for (int i = 0; i < int(arcMapCopies.size()); ++i) {
+        arcMapCopies[i]->copy(from, arcRefMap);
+      }      
+    }
+
+  protected:
+
+
+    const From& from;
+    To& to;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Node, NodeRefMap>* > 
+    nodeMapCopies;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Arc, ArcRefMap>* > 
+    arcMapCopies;
+
+  };
+
+  /// \brief Copy a digraph to another digraph.
+  ///
+  /// Copy a digraph to another digraph.
+  /// The usage of the function:
+  /// 
+  ///\code
+  /// copyDigraph(trg, src).nodeRef(nr).arcCrossRef(ecr).run();
+  ///\endcode
+  /// 
+  /// After the copy the \c nr map will contain the mapping from the
+  /// nodes of the \c from digraph to the nodes of the \c to digraph and
+  /// \c ecr will contain the mapping from the arcs of the \c to digraph
+  /// to the arcs of the \c from digraph.
+  ///
+  /// \see DigraphCopy 
+  template <typename To, typename From>
+  DigraphCopy<To, From> copyDigraph(To& to, const From& from) {
+    return DigraphCopy<To, From>(to, from);
+  }
+
+  /// \brief Class to copy an graph.
+  ///
+  /// Class to copy an graph to another digraph (duplicate a digraph).
+  /// The simplest way of using it is through the \c copyGraph() function.
+  template <typename To, typename From>
+  class GraphCopy {
+  private:
+
+    typedef typename From::Node Node;
+    typedef typename From::NodeIt NodeIt;
+    typedef typename From::Arc Arc;
+    typedef typename From::ArcIt ArcIt;
+    typedef typename From::Edge Edge;
+    typedef typename From::EdgeIt EdgeIt;
+
+    typedef typename To::Node TNode;
+    typedef typename To::Arc TArc;
+    typedef typename To::Edge TEdge;
+
+    typedef typename From::template NodeMap<TNode> NodeRefMap;
+    typedef typename From::template EdgeMap<TEdge> EdgeRefMap;
+
+    struct ArcRefMap {
+      ArcRefMap(const To& _to, const From& _from,
+                 const EdgeRefMap& _edge_ref, const NodeRefMap& _node_ref) 
+        : to(_to), from(_from), 
+          edge_ref(_edge_ref), node_ref(_node_ref) {}
+
+      typedef typename From::Arc Key;
+      typedef typename To::Arc Value;
+
+      Value operator[](const Key& key) const {
+        bool forward = 
+          (from.direction(key) == 
+           (node_ref[from.source(static_cast<const Edge&>(key))] == 
+            to.source(edge_ref[static_cast<const Edge&>(key)])));
+	return to.direct(edge_ref[key], forward); 
+      }
+      
+      const To& to;
+      const From& from;
+      const EdgeRefMap& edge_ref;
+      const NodeRefMap& node_ref;
+    };
+
+    
+  public: 
+
+
+    /// \brief Constructor for the DigraphCopy.
+    ///
+    /// It copies the content of the \c _from digraph into the
+    /// \c _to digraph.
+    GraphCopy(To& _to, const From& _from) 
+      : from(_from), to(_to) {}
+
+    /// \brief Destructor of the DigraphCopy
+    ///
+    /// Destructor of the DigraphCopy
+    ~GraphCopy() {
+      for (int i = 0; i < int(nodeMapCopies.size()); ++i) {
+        delete nodeMapCopies[i];
+      }
+      for (int i = 0; i < int(arcMapCopies.size()); ++i) {
+        delete arcMapCopies[i];
+      }
+      for (int i = 0; i < int(edgeMapCopies.size()); ++i) {
+        delete edgeMapCopies[i];
+      }
+
+    }
+
+    /// \brief Copies the node references into the given map.
+    ///
+    /// Copies the node references into the given map.
+    template <typename NodeRef>
+    GraphCopy& nodeRef(NodeRef& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Node, 
+                              NodeRefMap, NodeRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the node cross references into the given map.
+    ///
+    ///  Copies the node cross references (reverse references) into
+    ///  the given map.
+    template <typename NodeCrossRef>
+    GraphCopy& nodeCrossRef(NodeCrossRef& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Node,
+                              NodeRefMap, NodeCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's node type,
+    /// and the copied map's key type is the from digraph's node
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    GraphCopy& nodeMap(ToMap& tmap, const FromMap& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Node, 
+                              NodeRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Make a copy of the given node.
+    ///
+    /// Make a copy of the given node.
+    GraphCopy& node(TNode& tnode, const Node& snode) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Node, 
+                              NodeRefMap, TNode>(tnode, snode));
+      return *this;
+    }
+
+    /// \brief Copies the arc references into the given map.
+    ///
+    /// Copies the arc references into the given map.
+    template <typename ArcRef>
+    GraphCopy& arcRef(ArcRef& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Arc, 
+                              ArcRefMap, ArcRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the arc cross references into the given map.
+    ///
+    ///  Copies the arc cross references (reverse references) into
+    ///  the given map.
+    template <typename ArcCrossRef>
+    GraphCopy& arcCrossRef(ArcCrossRef& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Arc,
+                              ArcRefMap, ArcCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's arc type,
+    /// and the copied map's key type is the from digraph's arc
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    GraphCopy& arcMap(ToMap& tmap, const FromMap& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Arc, 
+                              ArcRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Make a copy of the given arc.
+    ///
+    /// Make a copy of the given arc.
+    GraphCopy& arc(TArc& tarc, const Arc& sarc) {
+      arcMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Arc, 
+                              ArcRefMap, TArc>(tarc, sarc));
+      return *this;
+    }
+
+    /// \brief Copies the edge references into the given map.
+    ///
+    /// Copies the edge references into the given map.
+    template <typename EdgeRef>
+    GraphCopy& edgeRef(EdgeRef& map) {
+      edgeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Edge, 
+                               EdgeRefMap, EdgeRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the edge cross references into the given map.
+    ///
+    /// Copies the edge cross references (reverse
+    /// references) into the given map.
+    template <typename EdgeCrossRef>
+    GraphCopy& edgeCrossRef(EdgeCrossRef& map) {
+      edgeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, 
+                               Edge, EdgeRefMap, EdgeCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's edge type,
+    /// and the copied map's key type is the from digraph's edge
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    GraphCopy& edgeMap(ToMap& tmap, const FromMap& map) {
+      edgeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Edge, 
+                               EdgeRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Make a copy of the given edge.
+    ///
+    /// Make a copy of the given edge.
+    GraphCopy& edge(TEdge& tedge, const Edge& sedge) {
+      edgeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Edge, 
+                               EdgeRefMap, TEdge>(tedge, sedge));
+      return *this;
+    }
+
+    /// \brief Executes the copies.
+    ///
+    /// Executes the copies.
+    void run() {
+      NodeRefMap nodeRefMap(from);
+      EdgeRefMap edgeRefMap(from);
+      ArcRefMap arcRefMap(to, from, edgeRefMap, nodeRefMap);
+      _digraph_utils_bits::GraphCopySelector<To>::
+        copy(to, from, nodeRefMap, edgeRefMap);
+      for (int i = 0; i < int(nodeMapCopies.size()); ++i) {
+        nodeMapCopies[i]->copy(from, nodeRefMap);
+      }
+      for (int i = 0; i < int(edgeMapCopies.size()); ++i) {
+        edgeMapCopies[i]->copy(from, edgeRefMap);
+      }
+      for (int i = 0; i < int(arcMapCopies.size()); ++i) {
+        arcMapCopies[i]->copy(from, arcRefMap);
+      }
+    }
+
+  private:
+    
+    const From& from;
+    To& to;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Node, NodeRefMap>* > 
+    nodeMapCopies;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Arc, ArcRefMap>* > 
+    arcMapCopies;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Edge, EdgeRefMap>* > 
+    edgeMapCopies;
+
+  };
+
+  /// \brief Copy an graph to another digraph.
+  ///
+  /// Copy an graph to another digraph.
+  /// The usage of the function:
+  /// 
+  ///\code
+  /// copyGraph(trg, src).nodeRef(nr).arcCrossRef(ecr).run();
+  ///\endcode
+  /// 
+  /// After the copy the \c nr map will contain the mapping from the
+  /// nodes of the \c from digraph to the nodes of the \c to digraph and
+  /// \c ecr will contain the mapping from the arcs of the \c to digraph
+  /// to the arcs of the \c from digraph.
+  ///
+  /// \see GraphCopy 
+  template <typename To, typename From>
+  GraphCopy<To, From> 
+  copyGraph(To& to, const From& from) {
+    return GraphCopy<To, From>(to, from);
+  }
+
+  /// \brief Class to copy a bipartite digraph.
+  ///
+  /// Class to copy a bipartite digraph to another digraph
+  /// (duplicate a digraph).  The simplest way of using it is through
+  /// the \c copyBpGraph() function.
+  template <typename To, typename From>
+  class BpGraphCopy {
+  private:
+
+    typedef typename From::Node Node;
+    typedef typename From::Red Red;
+    typedef typename From::Blue Blue;
+    typedef typename From::NodeIt NodeIt;
+    typedef typename From::Arc Arc;
+    typedef typename From::ArcIt ArcIt;
+    typedef typename From::Edge Edge;
+    typedef typename From::EdgeIt EdgeIt;
+
+    typedef typename To::Node TNode;
+    typedef typename To::Arc TArc;
+    typedef typename To::Edge TEdge;
+
+    typedef typename From::template RedMap<TNode> RedRefMap;
+    typedef typename From::template BlueMap<TNode> BlueRefMap;
+    typedef typename From::template EdgeMap<TEdge> EdgeRefMap;
+
+    struct NodeRefMap {
+      NodeRefMap(const From& _from, const RedRefMap& _red_ref,
+                 const BlueRefMap& _blue_ref)
+        : from(_from), red_ref(_red_ref), blue_ref(_blue_ref) {}
+
+      typedef typename From::Node Key;
+      typedef typename To::Node Value;
+
+      Value operator[](const Key& key) const {
+	return from.red(key) ? red_ref[key] : blue_ref[key]; 
+      }
+      
+      const From& from;
+      const RedRefMap& red_ref;
+      const BlueRefMap& blue_ref;
+    };
+
+    struct ArcRefMap {
+      ArcRefMap(const To& _to, const From& _from,
+                 const EdgeRefMap& _edge_ref, const NodeRefMap& _node_ref) 
+        : to(_to), from(_from), 
+          edge_ref(_edge_ref), node_ref(_node_ref) {}
+
+      typedef typename From::Arc Key;
+      typedef typename To::Arc Value;
+
+      Value operator[](const Key& key) const {
+        bool forward = 
+          (from.direction(key) == 
+           (node_ref[from.source(static_cast<const Edge&>(key))] == 
+            to.source(edge_ref[static_cast<const Edge&>(key)])));
+	return to.direct(edge_ref[key], forward); 
+      }
+      
+      const To& to;
+      const From& from;
+      const EdgeRefMap& edge_ref;
+      const NodeRefMap& node_ref;
+    };
+    
+  public: 
+
+
+    /// \brief Constructor for the DigraphCopy.
+    ///
+    /// It copies the content of the \c _from digraph into the
+    /// \c _to digraph.
+    BpGraphCopy(To& _to, const From& _from) 
+      : from(_from), to(_to) {}
+
+    /// \brief Destructor of the DigraphCopy
+    ///
+    /// Destructor of the DigraphCopy
+    ~BpGraphCopy() {
+      for (int i = 0; i < int(redMapCopies.size()); ++i) {
+        delete redMapCopies[i];
+      }
+      for (int i = 0; i < int(blueMapCopies.size()); ++i) {
+        delete blueMapCopies[i];
+      }
+      for (int i = 0; i < int(nodeMapCopies.size()); ++i) {
+        delete nodeMapCopies[i];
+      }
+      for (int i = 0; i < int(arcMapCopies.size()); ++i) {
+        delete arcMapCopies[i];
+      }
+      for (int i = 0; i < int(edgeMapCopies.size()); ++i) {
+        delete edgeMapCopies[i];
+      }
+
+    }
+
+    /// \brief Copies the A-node references into the given map.
+    ///
+    /// Copies the A-node references into the given map.
+    template <typename RedRef>
+    BpGraphCopy& redRef(RedRef& map) {
+      redMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Red, 
+                               RedRefMap, RedRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the A-node cross references into the given map.
+    ///
+    /// Copies the A-node cross references (reverse references) into
+    /// the given map.
+    template <typename RedCrossRef>
+    BpGraphCopy& redCrossRef(RedCrossRef& map) {
+      redMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, 
+                               Red, RedRefMap, RedCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given A-node map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's node type,
+    /// and the copied map's key type is the from digraph's node
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    BpGraphCopy& redMap(ToMap& tmap, const FromMap& map) {
+      redMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Red, 
+                               RedRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Copies the B-node references into the given map.
+    ///
+    /// Copies the B-node references into the given map.
+    template <typename BlueRef>
+    BpGraphCopy& blueRef(BlueRef& map) {
+      blueMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Blue, 
+                               BlueRefMap, BlueRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the B-node cross references into the given map.
+    ///
+    ///  Copies the B-node cross references (reverse references) into
+    ///  the given map.
+    template <typename BlueCrossRef>
+    BpGraphCopy& blueCrossRef(BlueCrossRef& map) {
+      blueMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, 
+                              Blue, BlueRefMap, BlueCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given B-node map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's node type,
+    /// and the copied map's key type is the from digraph's node
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    BpGraphCopy& blueMap(ToMap& tmap, const FromMap& map) {
+      blueMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Blue, 
+                               BlueRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+    /// \brief Copies the node references into the given map.
+    ///
+    /// Copies the node references into the given map.
+    template <typename NodeRef>
+    BpGraphCopy& nodeRef(NodeRef& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Node, 
+                              NodeRefMap, NodeRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the node cross references into the given map.
+    ///
+    ///  Copies the node cross references (reverse references) into
+    ///  the given map.
+    template <typename NodeCrossRef>
+    BpGraphCopy& nodeCrossRef(NodeCrossRef& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Node,
+                              NodeRefMap, NodeCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's node type,
+    /// and the copied map's key type is the from digraph's node
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    BpGraphCopy& nodeMap(ToMap& tmap, const FromMap& map) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Node, 
+                              NodeRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Make a copy of the given node.
+    ///
+    /// Make a copy of the given node.
+    BpGraphCopy& node(TNode& tnode, const Node& snode) {
+      nodeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Node, 
+                              NodeRefMap, TNode>(tnode, snode));
+      return *this;
+    }
+
+    /// \brief Copies the arc references into the given map.
+    ///
+    /// Copies the arc references into the given map.
+    template <typename ArcRef>
+    BpGraphCopy& arcRef(ArcRef& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Arc, 
+                              ArcRefMap, ArcRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the arc cross references into the given map.
+    ///
+    ///  Copies the arc cross references (reverse references) into
+    ///  the given map.
+    template <typename ArcCrossRef>
+    BpGraphCopy& arcCrossRef(ArcCrossRef& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, Arc,
+                              ArcRefMap, ArcCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's arc type,
+    /// and the copied map's key type is the from digraph's arc
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    BpGraphCopy& arcMap(ToMap& tmap, const FromMap& map) {
+      arcMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Arc, 
+                              ArcRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Make a copy of the given arc.
+    ///
+    /// Make a copy of the given arc.
+    BpGraphCopy& arc(TArc& tarc, const Arc& sarc) {
+      arcMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Arc, 
+                              ArcRefMap, TArc>(tarc, sarc));
+      return *this;
+    }
+
+    /// \brief Copies the edge references into the given map.
+    ///
+    /// Copies the edge references into the given map.
+    template <typename EdgeRef>
+    BpGraphCopy& edgeRef(EdgeRef& map) {
+      edgeMapCopies.push_back(new _digraph_utils_bits::RefCopy<From, Edge, 
+                               EdgeRefMap, EdgeRef>(map));
+      return *this;
+    }
+
+    /// \brief Copies the edge cross references into the given map.
+    ///
+    /// Copies the edge cross references (reverse
+    /// references) into the given map.
+    template <typename EdgeCrossRef>
+    BpGraphCopy& edgeCrossRef(EdgeCrossRef& map) {
+      edgeMapCopies.push_back(new _digraph_utils_bits::CrossRefCopy<From, 
+                               Edge, EdgeRefMap, EdgeCrossRef>(map));
+      return *this;
+    }
+
+    /// \brief Make copy of the given map.
+    ///
+    /// Makes copy of the given map for the newly created digraph. 
+    /// The new map's key type is the to digraph's edge type,
+    /// and the copied map's key type is the from digraph's edge
+    /// type.  
+    template <typename ToMap, typename FromMap>
+    BpGraphCopy& edgeMap(ToMap& tmap, const FromMap& map) {
+      edgeMapCopies.push_back(new _digraph_utils_bits::MapCopy<From, Edge, 
+                               EdgeRefMap, ToMap, FromMap>(tmap, map));
+      return *this;
+    }
+
+    /// \brief Make a copy of the given edge.
+    ///
+    /// Make a copy of the given edge.
+    BpGraphCopy& edge(TEdge& tedge, const Edge& sedge) {
+      edgeMapCopies.push_back(new _digraph_utils_bits::ItemCopy<From, Edge, 
+                               EdgeRefMap, TEdge>(tedge, sedge));
+      return *this;
+    }
+
+    /// \brief Executes the copies.
+    ///
+    /// Executes the copies.
+    void run() {
+      RedRefMap redRefMap(from);
+      BlueRefMap blueRefMap(from);
+      NodeRefMap nodeRefMap(from, redRefMap, blueRefMap);
+      EdgeRefMap edgeRefMap(from);
+      ArcRefMap arcRefMap(to, from, edgeRefMap, nodeRefMap);
+      _digraph_utils_bits::BpGraphCopySelector<To>::
+        copy(to, from, redRefMap, blueRefMap, edgeRefMap);
+      for (int i = 0; i < int(redMapCopies.size()); ++i) {
+        redMapCopies[i]->copy(from, redRefMap);
+      }
+      for (int i = 0; i < int(blueMapCopies.size()); ++i) {
+        blueMapCopies[i]->copy(from, blueRefMap);
+      }
+      for (int i = 0; i < int(nodeMapCopies.size()); ++i) {
+        nodeMapCopies[i]->copy(from, nodeRefMap);
+      }
+      for (int i = 0; i < int(edgeMapCopies.size()); ++i) {
+        edgeMapCopies[i]->copy(from, edgeRefMap);
+      }
+      for (int i = 0; i < int(arcMapCopies.size()); ++i) {
+        arcMapCopies[i]->copy(from, arcRefMap);
+      }
+    }
+
+  private:
+    
+    const From& from;
+    To& to;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Red, RedRefMap>* > 
+    redMapCopies;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Blue, BlueRefMap>* > 
+    blueMapCopies;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Node, NodeRefMap>* > 
+    nodeMapCopies;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Arc, ArcRefMap>* > 
+    arcMapCopies;
+
+    std::vector<_digraph_utils_bits::MapCopyBase<From, Edge, EdgeRefMap>* > 
+    edgeMapCopies;
+
+  };
+
+  /// \brief Copy a bipartite digraph to another digraph.
+  ///
+  /// Copy a bipartite digraph to another digraph.
+  /// The usage of the function:
+  /// 
+  ///\code
+  /// copyBpGraph(trg, src).redRef(anr).arcCrossRef(ecr).run();
+  ///\endcode
+  /// 
+  /// After the copy the \c nr map will contain the mapping from the
+  /// nodes of the \c from digraph to the nodes of the \c to digraph and
+  /// \c ecr will contain the mapping from the arcs of the \c to digraph
+  /// to the arcs of the \c from digraph.
+  ///
+  /// \see BpGraphCopy
+  template <typename To, typename From>
+  BpGraphCopy<To, From> 
+  copyBpGraph(To& to, const From& from) {
+    return BpGraphCopy<To, From>(to, from);
+  }
+
+
+  /// @}
+
+  /// \addtogroup digraph_maps
+  /// @{
+
+  /// Provides an immutable and unique id for each item in the digraph.
+
+  /// The IdMap class provides a unique and immutable id for each item of the
+  /// same type (e.g. node) in the digraph. This id is <ul><li>\b unique:
+  /// different items (nodes) get different ids <li>\b immutable: the id of an
+  /// item (node) does not change (even if you delete other nodes).  </ul>
+  /// Through this map you get access (i.e. can read) the inner id values of
+  /// the items stored in the digraph. This map can be inverted with its member
+  /// class \c InverseMap.
+  ///
+  template <typename _Digraph, typename _Item>
+  class IdMap {
+  public:
+    typedef _Digraph Digraph;
+    typedef int Value;
+    typedef _Item Item;
+    typedef _Item Key;
+
+    /// \brief Constructor.
+    ///
+    /// Constructor of the map.
+    explicit IdMap(const Digraph& _digraph) : digraph(&_digraph) {}
+
+    /// \brief Gives back the \e id of the item.
+    ///
+    /// Gives back the immutable and unique \e id of the item.
+    int operator[](const Item& item) const { return digraph->id(item);}
+
+    /// \brief Gives back the item by its id.
+    ///
+    /// Gives back the item by its id.
+    Item operator()(int id) { return digraph->fromId(id, Item()); }
+
+  private:
+    const Digraph* digraph;
+
+  public:
+
+    /// \brief The class represents the inverse of its owner (IdMap).
+    ///
+    /// The class represents the inverse of its owner (IdMap).
+    /// \see inverse()
+    class InverseMap {
+    public:
+
+      /// \brief Constructor.
+      ///
+      /// Constructor for creating an id-to-item map.
+      explicit InverseMap(const Digraph& _digraph) : digraph(&_digraph) {}
+
+      /// \brief Constructor.
+      ///
+      /// Constructor for creating an id-to-item map.
+      explicit InverseMap(const IdMap& idMap) : digraph(idMap.digraph) {}
+
+      /// \brief Gives back the given item from its id.
+      ///
+      /// Gives back the given item from its id.
+      /// 
+      Item operator[](int id) const { return digraph->fromId(id, Item());}
+
+    private:
+      const Digraph* digraph;
+    };
+
+    /// \brief Gives back the inverse of the map.
+    ///
+    /// Gives back the inverse of the IdMap.
+    InverseMap inverse() const { return InverseMap(*digraph);} 
+
+  };
+
+  
+  /// \brief General invertable digraph-map type.
+
+  /// This type provides simple invertable digraph-maps. 
+  /// The InvertableMap wraps an arbitrary ReadWriteMap 
+  /// and if a key is set to a new value then store it
+  /// in the inverse map.
+  ///
+  /// The values of the map can be accessed
+  /// with stl compatible forward iterator.
+  ///
+  /// \param _Digraph The digraph type.
+  /// \param _Item The item type of the digraph.
+  /// \param _Value The value type of the map.
+  ///
+  /// \see IterableValueMap
+  template <typename _Digraph, typename _Item, typename _Value>
+  class InvertableMap : protected DefaultMap<_Digraph, _Item, _Value> {
+  private:
+    
+    typedef DefaultMap<_Digraph, _Item, _Value> Map;
+    typedef _Digraph Digraph;
+
+    typedef std::map<_Value, _Item> Container;
+    Container invMap;    
+
+  public:
+ 
+    /// The key type of InvertableMap (Node, Arc, Edge).
+    typedef typename Map::Key Key;
+    /// The value type of the InvertableMap.
+    typedef typename Map::Value Value;
+
+
+
+    /// \brief Constructor.
+    ///
+    /// Construct a new InvertableMap for the digraph.
+    ///
+    explicit InvertableMap(const Digraph& digraph) : Map(digraph) {} 
+
+    /// \brief Forward iterator for values.
+    ///
+    /// This iterator is an stl compatible forward
+    /// iterator on the values of the map. The values can
+    /// be accessed in the [beginValue, endValue) range.
+    ///
+    class ValueIterator 
+      : public std::iterator<std::forward_iterator_tag, Value> {
+      friend class InvertableMap;
+    private:
+      ValueIterator(typename Container::const_iterator _it) 
+        : it(_it) {}
+    public:
+      
+      ValueIterator() {}
+
+      ValueIterator& operator++() { ++it; return *this; }
+      ValueIterator operator++(int) { 
+        ValueIterator tmp(*this); 
+        operator++();
+        return tmp; 
+      }
+
+      const Value& operator*() const { return it->first; }
+      const Value* operator->() const { return &(it->first); }
+
+      bool operator==(ValueIterator jt) const { return it == jt.it; }
+      bool operator!=(ValueIterator jt) const { return it != jt.it; }
+      
+    private:
+      typename Container::const_iterator it;
+    };
+
+    /// \brief Returns an iterator to the first value.
+    ///
+    /// Returns an stl compatible iterator to the 
+    /// first value of the map. The values of the
+    /// map can be accessed in the [beginValue, endValue)
+    /// range.
+    ValueIterator beginValue() const {
+      return ValueIterator(invMap.begin());
+    }
+
+    /// \brief Returns an iterator after the last value.
+    ///
+    /// Returns an stl compatible iterator after the 
+    /// last value of the map. The values of the
+    /// map can be accessed in the [beginValue, endValue)
+    /// range.
+    ValueIterator endValue() const {
+      return ValueIterator(invMap.end());
+    }
+    
+    /// \brief The setter function of the map.
+    ///
+    /// Sets the mapped value.
+    void set(const Key& key, const Value& val) {
+      Value oldval = Map::operator[](key);
+      typename Container::iterator it = invMap.find(oldval);
+      if (it != invMap.end() && it->second == key) {
+	invMap.erase(it);
+      }      
+      invMap.insert(make_pair(val, key));
+      Map::set(key, val);
+    }
+
+    /// \brief The getter function of the map.
+    ///
+    /// It gives back the value associated with the key.
+    typename MapTraits<Map>::ConstReturnValue 
+    operator[](const Key& key) const {
+      return Map::operator[](key);
+    }
+
+    /// \brief Gives back the item by its value.
+    ///
+    /// Gives back the item by its value.
+    Key operator()(const Value& key) const {
+      typename Container::const_iterator it = invMap.find(key);
+      return it != invMap.end() ? it->second : INVALID;
+    }
+
+  protected:
+
+    /// \brief Erase the key from the map.
+    ///
+    /// Erase the key to the map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void erase(const Key& key) {
+      Value val = Map::operator[](key);
+      typename Container::iterator it = invMap.find(val);
+      if (it != invMap.end() && it->second == key) {
+	invMap.erase(it);
+      }
+      Map::erase(key);
+    }
+
+    /// \brief Erase more keys from the map.
+    ///
+    /// Erase more keys from the map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void erase(const std::vector<Key>& keys) {
+      for (int i = 0; i < int(keys.size()); ++i) {
+	Value val = Map::operator[](keys[i]);
+	typename Container::iterator it = invMap.find(val);
+	if (it != invMap.end() && it->second == keys[i]) {
+	  invMap.erase(it);
+	}
+      }
+      Map::erase(keys);
+    }
+
+    /// \brief Clear the keys from the map and inverse map.
+    ///
+    /// Clear the keys from the map and inverse map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void clear() {
+      invMap.clear();
+      Map::clear();
+    }
+
+  public:
+
+    /// \brief The inverse map type.
+    ///
+    /// The inverse of this map. The subscript operator of the map
+    /// gives back always the item what was last assigned to the value. 
+    class InverseMap {
+    public:
+      /// \brief Constructor of the InverseMap.
+      ///
+      /// Constructor of the InverseMap.
+      explicit InverseMap(const InvertableMap& _inverted) 
+        : inverted(_inverted) {}
+
+      /// The value type of the InverseMap.
+      typedef typename InvertableMap::Key Value;
+      /// The key type of the InverseMap.
+      typedef typename InvertableMap::Value Key; 
+
+      /// \brief Subscript operator. 
+      ///
+      /// Subscript operator. It gives back always the item 
+      /// what was last assigned to the value.
+      Value operator[](const Key& key) const {
+	return inverted(key);
+      }
+      
+    private:
+      const InvertableMap& inverted;
+    };
+
+    /// \brief It gives back the just readable inverse map.
+    ///
+    /// It gives back the just readable inverse map.
+    InverseMap inverse() const {
+      return InverseMap(*this);
+    } 
+
+
+    
+  };
+
+  /// \brief Provides a mutable, continuous and unique descriptor for each 
+  /// item in the digraph.
+  ///
+  /// The DescriptorMap class provides a unique and continuous (but mutable)
+  /// descriptor (id) for each item of the same type (e.g. node) in the
+  /// digraph. This id is <ul><li>\b unique: different items (nodes) get
+  /// different ids <li>\b continuous: the range of the ids is the set of
+  /// integers between 0 and \c n-1, where \c n is the number of the items of
+  /// this type (e.g. nodes) (so the id of a node can change if you delete an
+  /// other node, i.e. this id is mutable).  </ul> This map can be inverted
+  /// with its member class \c InverseMap.
+  ///
+  /// \param _Digraph The digraph class the \c DescriptorMap belongs to.
+  /// \param _Item The Item is the Key of the Map. It may be Node, Arc or 
+  /// Edge.
+  template <typename _Digraph, typename _Item>
+  class DescriptorMap : protected DefaultMap<_Digraph, _Item, int> {
+
+    typedef _Item Item;
+    typedef DefaultMap<_Digraph, _Item, int> Map;
+
+  public:
+    /// The digraph class of DescriptorMap.
+    typedef _Digraph Digraph;
+
+    /// The key type of DescriptorMap (Node, Arc, Edge).
+    typedef typename Map::Key Key;
+    /// The value type of DescriptorMap.
+    typedef typename Map::Value Value;
+
+    /// \brief Constructor.
+    ///
+    /// Constructor for descriptor map.
+    explicit DescriptorMap(const Digraph& _digraph) : Map(_digraph) {
+      Item it;
+      const typename Map::Notifier* nf = Map::notifier(); 
+      for (nf->first(it); it != INVALID; nf->next(it)) {
+	Map::set(it, invMap.size());
+	invMap.push_back(it);	
+      }      
+    }
+
+  protected:
+
+    /// \brief Add a new key to the map.
+    ///
+    /// Add a new key to the map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void add(const Item& item) {
+      Map::add(item);
+      Map::set(item, invMap.size());
+      invMap.push_back(item);
+    }
+
+    /// \brief Add more new keys to the map.
+    ///
+    /// Add more new keys to the map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void add(const std::vector<Item>& items) {
+      Map::add(items);
+      for (int i = 0; i < int(items.size()); ++i) {
+	Map::set(items[i], invMap.size());
+	invMap.push_back(items[i]);
+      }
+    }
+
+    /// \brief Erase the key from the map.
+    ///
+    /// Erase the key from the map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void erase(const Item& item) {
+      Map::set(invMap.back(), Map::operator[](item));
+      invMap[Map::operator[](item)] = invMap.back();
+      invMap.pop_back();
+      Map::erase(item);
+    }
+
+    /// \brief Erase more keys from the map.
+    ///
+    /// Erase more keys from the map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void erase(const std::vector<Item>& items) {
+      for (int i = 0; i < int(items.size()); ++i) {
+	Map::set(invMap.back(), Map::operator[](items[i]));
+	invMap[Map::operator[](items[i])] = invMap.back();
+	invMap.pop_back();
+      }
+      Map::erase(items);
+    }
+
+    /// \brief Build the unique map.
+    ///
+    /// Build the unique map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void build() {
+      Map::build();
+      Item it;
+      const typename Map::Notifier* nf = Map::notifier(); 
+      for (nf->first(it); it != INVALID; nf->next(it)) {
+	Map::set(it, invMap.size());
+	invMap.push_back(it);	
+      }      
+    }
+    
+    /// \brief Clear the keys from the map.
+    ///
+    /// Clear the keys from the map. It is called by the
+    /// \c AlterationNotifier.
+    virtual void clear() {
+      invMap.clear();
+      Map::clear();
+    }
+
+  public:
+
+    /// \brief Returns the maximal value plus one.
+    ///
+    /// Returns the maximal value plus one in the map.
+    unsigned int size() const {
+      return invMap.size();
+    }
+
+    /// \brief Swaps the position of the two items in the map.
+    ///
+    /// Swaps the position of the two items in the map.
+    void swap(const Item& p, const Item& q) {
+      int pi = Map::operator[](p);
+      int qi = Map::operator[](q);
+      Map::set(p, qi);
+      invMap[qi] = p;
+      Map::set(q, pi);
+      invMap[pi] = q;
+    }
+
+    /// \brief Gives back the \e descriptor of the item.
+    ///
+    /// Gives back the mutable and unique \e descriptor of the map.
+    int operator[](const Item& item) const {
+      return Map::operator[](item);
+    }
+
+    /// \brief Gives back the item by its descriptor.
+    ///
+    /// Gives back th item by its descriptor.
+    Item operator()(int id) const {
+      return invMap[id];
+    }
+    
+  private:
+
+    typedef std::vector<Item> Container;
+    Container invMap;
+
+  public:
+    /// \brief The inverse map type of DescriptorMap.
+    ///
+    /// The inverse map type of DescriptorMap.
+    class InverseMap {
+    public:
+      /// \brief Constructor of the InverseMap.
+      ///
+      /// Constructor of the InverseMap.
+      explicit InverseMap(const DescriptorMap& _inverted) 
+	: inverted(_inverted) {}
+
+
+      /// The value type of the InverseMap.
+      typedef typename DescriptorMap::Key Value;
+      /// The key type of the InverseMap.
+      typedef typename DescriptorMap::Value Key; 
+
+      /// \brief Subscript operator. 
+      ///
+      /// Subscript operator. It gives back the item 
+      /// that the descriptor belongs to currently.
+      Value operator[](const Key& key) const {
+	return inverted(key);
+      }
+
+      /// \brief Size of the map.
+      ///
+      /// Returns the size of the map.
+      unsigned int size() const {
+	return inverted.size();
+      }
+      
+    private:
+      const DescriptorMap& inverted;
+    };
+
+    /// \brief Gives back the inverse of the map.
+    ///
+    /// Gives back the inverse of the map.
+    const InverseMap inverse() const {
+      return InverseMap(*this);
+    }
+  };
+
+  /// \brief Returns the source of the given arc.
+  ///
+  /// The SourceMap gives back the source Node of the given arc. 
+  /// \see TargetMap
+  /// \author Balazs Dezso
+  template <typename Digraph>
+  class SourceMap {
+  public:
+
+    typedef typename Digraph::Node Value;
+    typedef typename Digraph::Arc Key;
+
+    /// \brief Constructor
+    ///
+    /// Constructor
+    /// \param _digraph The digraph that the map belongs to.
+    explicit SourceMap(const Digraph& _digraph) : digraph(_digraph) {}
+
+    /// \brief The subscript operator.
+    ///
+    /// The subscript operator.
+    /// \param arc The arc 
+    /// \return The source of the arc 
+    Value operator[](const Key& arc) const {
+      return digraph.source(arc);
+    }
+
+  private:
+    const Digraph& digraph;
+  };
+
+  /// \brief Returns a \ref SourceMap class.
+  ///
+  /// This function just returns an \ref SourceMap class.
+  /// \relates SourceMap
+  template <typename Digraph>
+  inline SourceMap<Digraph> sourceMap(const Digraph& digraph) {
+    return SourceMap<Digraph>(digraph);
+  } 
+
+  /// \brief Returns the target of the given arc.
+  ///
+  /// The TargetMap gives back the target Node of the given arc. 
+  /// \see SourceMap
+  /// \author Balazs Dezso
+  template <typename Digraph>
+  class TargetMap {
+  public:
+
+    typedef typename Digraph::Node Value;
+    typedef typename Digraph::Arc Key;
+
+    /// \brief Constructor
+    ///
+    /// Constructor
+    /// \param _digraph The digraph that the map belongs to.
+    explicit TargetMap(const Digraph& _digraph) : digraph(_digraph) {}
+
+    /// \brief The subscript operator.
+    ///
+    /// The subscript operator.
+    /// \param e The arc 
+    /// \return The target of the arc 
+    Value operator[](const Key& e) const {
+      return digraph.target(e);
+    }
+
+  private:
+    const Digraph& digraph;
+  };
+
+  /// \brief Returns a \ref TargetMap class.
+  ///
+  /// This function just returns a \ref TargetMap class.
+  /// \relates TargetMap
+  template <typename Digraph>
+  inline TargetMap<Digraph> targetMap(const Digraph& digraph) {
+    return TargetMap<Digraph>(digraph);
+  }
+
+  /// \brief Returns the "forward" directed arc view of an edge.
+  ///
+  /// Returns the "forward" directed arc view of an edge.
+  /// \see BackwardMap
+  /// \author Balazs Dezso
+  template <typename Digraph>
+  class ForwardMap {
+  public:
+
+    typedef typename Digraph::Arc Value;
+    typedef typename Digraph::Edge Key;
+
+    /// \brief Constructor
+    ///
+    /// Constructor
+    /// \param _digraph The digraph that the map belongs to.
+    explicit ForwardMap(const Digraph& _digraph) : digraph(_digraph) {}
+
+    /// \brief The subscript operator.
+    ///
+    /// The subscript operator.
+    /// \param key An edge 
+    /// \return The "forward" directed arc view of edge 
+    Value operator[](const Key& key) const {
+      return digraph.direct(key, true);
+    }
+
+  private:
+    const Digraph& digraph;
+  };
+
+  /// \brief Returns a \ref ForwardMap class.
+  ///
+  /// This function just returns an \ref ForwardMap class.
+  /// \relates ForwardMap
+  template <typename Digraph>
+  inline ForwardMap<Digraph> forwardMap(const Digraph& digraph) {
+    return ForwardMap<Digraph>(digraph);
+  }
+
+  /// \brief Returns the "backward" directed arc view of an edge.
+  ///
+  /// Returns the "backward" directed arc view of an edge.
+  /// \see ForwardMap
+  /// \author Balazs Dezso
+  template <typename Digraph>
+  class BackwardMap {
+  public:
+
+    typedef typename Digraph::Arc Value;
+    typedef typename Digraph::Edge Key;
+
+    /// \brief Constructor
+    ///
+    /// Constructor
+    /// \param _digraph The digraph that the map belongs to.
+    explicit BackwardMap(const Digraph& _digraph) : digraph(_digraph) {}
+
+    /// \brief The subscript operator.
+    ///
+    /// The subscript operator.
+    /// \param key An edge 
+    /// \return The "backward" directed arc view of edge 
+    Value operator[](const Key& key) const {
+      return digraph.direct(key, false);
+    }
+
+  private:
+    const Digraph& digraph;
+  };
+
+  /// \brief Returns a \ref BackwardMap class
+
+  /// This function just returns a \ref BackwardMap class.
+  /// \relates BackwardMap
+  template <typename Digraph>
+  inline BackwardMap<Digraph> backwardMap(const Digraph& digraph) {
+    return BackwardMap<Digraph>(digraph);
+  }
+
+  /// \brief Potential difference map
+  ///
+  /// If there is an potential map on the nodes then we
+  /// can get an arc map as we get the substraction of the
+  /// values of the target and source.
+  template <typename Digraph, typename NodeMap>
+  class PotentialDifferenceMap {
+  public:
+    typedef typename Digraph::Arc Key;
+    typedef typename NodeMap::Value Value;
+
+    /// \brief Constructor
+    ///
+    /// Contructor of the map
+    explicit PotentialDifferenceMap(const Digraph& _digraph, 
+                                    const NodeMap& _potential) 
+      : digraph(_digraph), potential(_potential) {}
+
+    /// \brief Const subscription operator
+    ///
+    /// Const subscription operator
+    Value operator[](const Key& arc) const {
+      return potential[digraph.target(arc)] - potential[digraph.source(arc)];
+    }
+
+  private:
+    const Digraph& digraph;
+    const NodeMap& potential;
+  };
+
+  /// \brief Returns a PotentialDifferenceMap.
+  ///
+  /// This function just returns a PotentialDifferenceMap.
+  /// \relates PotentialDifferenceMap
+  template <typename Digraph, typename NodeMap>
+  PotentialDifferenceMap<Digraph, NodeMap> 
+  potentialDifferenceMap(const Digraph& digraph, const NodeMap& potential) {
+    return PotentialDifferenceMap<Digraph, NodeMap>(digraph, potential);
+  }
+
+  /// \brief Map of the node in-degrees.
+  ///
+  /// This map returns the in-degree of a node. Once it is constructed,
+  /// the degrees are stored in a standard NodeMap, so each query is done
+  /// in constant time. On the other hand, the values are updated automatically
+  /// whenever the digraph changes.
+  ///
+  /// \warning Besides addNode() and addArc(), a digraph structure may provide
+  /// alternative ways to modify the digraph. The correct behavior of InDegMap
+  /// is not guarantied if these additional features are used. For example
+  /// the functions \ref ListDigraph::changeSource() "changeSource()",
+  /// \ref ListDigraph::changeTarget() "changeTarget()" and
+  /// \ref ListDigraph::reverseArc() "reverseArc()"
+  /// of \ref ListDigraph will \e not update the degree values correctly.
+  ///
+  /// \sa OutDegMap
+
+  template <typename _Digraph>
+  class InDegMap  
+    : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>
+      ::ItemNotifier::ObserverBase {
+
+  public:
+    
+    typedef _Digraph Digraph;
+    typedef int Value;
+    typedef typename Digraph::Node Key;
+
+    typedef typename ItemSetTraits<_Digraph, typename _Digraph::Arc>
+    ::ItemNotifier::ObserverBase Parent;
+
+  private:
+
+    class AutoNodeMap : public DefaultMap<_Digraph, Key, int> {
+    public:
+
+      typedef DefaultMap<_Digraph, Key, int> Parent;
+      typedef typename Parent::Digraph Digraph;
+
+      AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}
+      
+      virtual void add(const Key& key) {
+	Parent::add(key);
+	Parent::set(key, 0);
+      }
+
+      virtual void add(const std::vector<Key>& keys) {
+	Parent::add(keys);
+	for (int i = 0; i < int(keys.size()); ++i) {
+	  Parent::set(keys[i], 0);
+	}
+      }
+
+      virtual void build() {
+	Parent::build();
+	Key it;
+	typename Parent::Notifier* nf = Parent::notifier();
+	for (nf->first(it); it != INVALID; nf->next(it)) {
+	  Parent::set(it, 0);
+	}
+      }
+    };
+
+  public:
+
+    /// \brief Constructor.
+    ///
+    /// Constructor for creating in-degree map.
+    explicit InDegMap(const Digraph& _digraph) : digraph(_digraph), deg(_digraph) {
+      Parent::attach(digraph.notifier(typename _Digraph::Arc()));
+      
+      for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {
+	deg[it] = countInArcs(digraph, it);
+      }
+    }
+    
+    /// Gives back the in-degree of a Node.
+    int operator[](const Key& key) const {
+      return deg[key];
+    }
+
+  protected:
+    
+    typedef typename Digraph::Arc Arc;
+
+    virtual void add(const Arc& arc) {
+      ++deg[digraph.target(arc)];
+    }
+
+    virtual void add(const std::vector<Arc>& arcs) {
+      for (int i = 0; i < int(arcs.size()); ++i) {
+        ++deg[digraph.target(arcs[i])];
+      }
+    }
+
+    virtual void erase(const Arc& arc) {
+      --deg[digraph.target(arc)];
+    }
+
+    virtual void erase(const std::vector<Arc>& arcs) {
+      for (int i = 0; i < int(arcs.size()); ++i) {
+        --deg[digraph.target(arcs[i])];
+      }
+    }
+
+    virtual void build() {
+      for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {
+	deg[it] = countInArcs(digraph, it);
+      }      
+    }
+
+    virtual void clear() {
+      for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {
+	deg[it] = 0;
+      }
+    }
+  private:
+    
+    const _Digraph& digraph;
+    AutoNodeMap deg;
+  };
+
+  /// \brief Map of the node out-degrees.
+  ///
+  /// This map returns the out-degree of a node. Once it is constructed,
+  /// the degrees are stored in a standard NodeMap, so each query is done
+  /// in constant time. On the other hand, the values are updated automatically
+  /// whenever the digraph changes.
+  ///
+  /// \warning Besides addNode() and addArc(), a digraph structure may provide
+  /// alternative ways to modify the digraph. The correct behavior of OutDegMap
+  /// is not guarantied if these additional features are used. For example
+  /// the functions \ref ListDigraph::changeSource() "changeSource()",
+  /// \ref ListDigraph::changeTarget() "changeTarget()" and
+  /// \ref ListDigraph::reverseArc() "reverseArc()"
+  /// of \ref ListDigraph will \e not update the degree values correctly.
+  ///
+  /// \sa InDegMap
+
+  template <typename _Digraph>
+  class OutDegMap  
+    : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>
+      ::ItemNotifier::ObserverBase {
+
+  public:
+
+    typedef typename ItemSetTraits<_Digraph, typename _Digraph::Arc>
+    ::ItemNotifier::ObserverBase Parent;
+    
+    typedef _Digraph Digraph;
+    typedef int Value;
+    typedef typename Digraph::Node Key;
+
+  private:
+
+    class AutoNodeMap : public DefaultMap<_Digraph, Key, int> {
+    public:
+
+      typedef DefaultMap<_Digraph, Key, int> Parent;
+      typedef typename Parent::Digraph Digraph;
+
+      AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}
+      
+      virtual void add(const Key& key) {
+	Parent::add(key);
+	Parent::set(key, 0);
+      }
+      virtual void add(const std::vector<Key>& keys) {
+	Parent::add(keys);
+	for (int i = 0; i < int(keys.size()); ++i) {
+	  Parent::set(keys[i], 0);
+	}
+      }
+      virtual void build() {
+	Parent::build();
+	Key it;
+	typename Parent::Notifier* nf = Parent::notifier();
+	for (nf->first(it); it != INVALID; nf->next(it)) {
+	  Parent::set(it, 0);
+	}
+      }
+    };
+
+  public:
+
+    /// \brief Constructor.
+    ///
+    /// Constructor for creating out-degree map.
+    explicit OutDegMap(const Digraph& _digraph) : digraph(_digraph), deg(_digraph) {
+      Parent::attach(digraph.notifier(typename _Digraph::Arc()));
+      
+      for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {
+	deg[it] = countOutArcs(digraph, it);
+      }
+    }
+
+    /// Gives back the out-degree of a Node.
+    int operator[](const Key& key) const {
+      return deg[key];
+    }
+
+  protected:
+    
+    typedef typename Digraph::Arc Arc;
+
+    virtual void add(const Arc& arc) {
+      ++deg[digraph.source(arc)];
+    }
+
+    virtual void add(const std::vector<Arc>& arcs) {
+      for (int i = 0; i < int(arcs.size()); ++i) {
+        ++deg[digraph.source(arcs[i])];
+      }
+    }
+
+    virtual void erase(const Arc& arc) {
+      --deg[digraph.source(arc)];
+    }
+
+    virtual void erase(const std::vector<Arc>& arcs) {
+      for (int i = 0; i < int(arcs.size()); ++i) {
+        --deg[digraph.source(arcs[i])];
+      }
+    }
+
+    virtual void build() {
+      for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {
+	deg[it] = countOutArcs(digraph, it);
+      }      
+    }
+
+    virtual void clear() {
+      for(typename _Digraph::NodeIt it(digraph); it != INVALID; ++it) {
+	deg[it] = 0;
+      }
+    }
+  private:
+    
+    const _Digraph& digraph;
+    AutoNodeMap deg;
+  };
+
+
+  ///Dynamic arc look up between given endpoints.
+  
+  ///\ingroup gutils
+  ///Using this class, you can find an arc in a digraph from a given
+  ///source to a given target in amortized time <em>O(log d)</em>,
+  ///where <em>d</em> is the out-degree of the source node.
+  ///
+  ///It is possible to find \e all parallel arcs between two nodes with
+  ///the \c findFirst() and \c findNext() members.
+  ///
+  ///See the \ref ArcLookUp and \ref AllArcLookUp classes if your
+  ///digraph do not changed so frequently.
+  ///
+  ///This class uses a self-adjusting binary search tree, Sleator's
+  ///and Tarjan's Splay tree for guarantee the logarithmic amortized
+  ///time bound for arc lookups. This class also guarantees the
+  ///optimal time bound in a constant factor for any distribution of
+  ///queries.
+  ///
+  ///\param G The type of the underlying digraph.  
+  ///
+  ///\sa ArcLookUp  
+  ///\sa AllArcLookUp  
+  template<class G>
+  class DynArcLookUp 
+    : protected ItemSetTraits<G, typename G::Arc>::ItemNotifier::ObserverBase
+  {
+  public:
+    typedef typename ItemSetTraits<G, typename G::Arc>
+    ::ItemNotifier::ObserverBase Parent;
+
+    GRAPH_TYPEDEFS(typename G);
+    typedef G Digraph;
+
+  protected:
+
+    class AutoNodeMap : public DefaultMap<G, Node, Arc> {
+    public:
+
+      typedef DefaultMap<G, Node, Arc> Parent;
+
+      AutoNodeMap(const G& digraph) : Parent(digraph, INVALID) {}
+      
+      virtual void add(const Node& node) {
+	Parent::add(node);
+	Parent::set(node, INVALID);
+      }
+
+      virtual void add(const std::vector<Node>& nodes) {
+	Parent::add(nodes);
+	for (int i = 0; i < int(nodes.size()); ++i) {
+	  Parent::set(nodes[i], INVALID);
+	}
+      }
+
+      virtual void build() {
+	Parent::build();
+	Node it;
+	typename Parent::Notifier* nf = Parent::notifier();
+	for (nf->first(it); it != INVALID; nf->next(it)) {
+	  Parent::set(it, INVALID);
+	}
+      }
+    };
+
+    const Digraph &_g;
+    AutoNodeMap _head;
+    typename Digraph::template ArcMap<Arc> _parent;
+    typename Digraph::template ArcMap<Arc> _left;
+    typename Digraph::template ArcMap<Arc> _right;
+    
+    class ArcLess {
+      const Digraph &g;
+    public:
+      ArcLess(const Digraph &_g) : g(_g) {}
+      bool operator()(Arc a,Arc b) const 
+      {
+	return g.target(a)<g.target(b);
+      }
+    };
+    
+  public:
+    
+    ///Constructor
+
+    ///Constructor.
+    ///
+    ///It builds up the search database.
+    DynArcLookUp(const Digraph &g) 
+      : _g(g),_head(g),_parent(g),_left(g),_right(g) 
+    { 
+      Parent::attach(_g.notifier(typename Digraph::Arc()));
+      refresh(); 
+    }
+    
+  protected:
+
+    virtual void add(const Arc& arc) {
+      insert(arc);
+    }
+
+    virtual void add(const std::vector<Arc>& arcs) {
+      for (int i = 0; i < int(arcs.size()); ++i) {
+	insert(arcs[i]);
+      }
+    }
+
+    virtual void erase(const Arc& arc) {
+      remove(arc);
+    }
+
+    virtual void erase(const std::vector<Arc>& arcs) {
+      for (int i = 0; i < int(arcs.size()); ++i) {
+	remove(arcs[i]);
+      }     
+    }
+
+    virtual void build() {
+      refresh();
+    }
+
+    virtual void clear() {
+      for(NodeIt n(_g);n!=INVALID;++n) {
+	_head.set(n, INVALID);
+      }
+    }
+
+    void insert(Arc arc) {
+      Node s = _g.source(arc);
+      Node t = _g.target(arc);
+      _left.set(arc, INVALID);
+      _right.set(arc, INVALID);
+      
+      Arc e = _head[s];
+      if (e == INVALID) {
+	_head.set(s, arc);
+	_parent.set(arc, INVALID);
+	return;
+      }
+      while (true) {
+	if (t < _g.target(e)) {
+	  if (_left[e] == INVALID) {
+	    _left.set(e, arc);
+	    _parent.set(arc, e);
+	    splay(arc);
+	    return;
+	  } else {
+	    e = _left[e];
+	  }
+	} else {
+	  if (_right[e] == INVALID) {
+	    _right.set(e, arc);
+	    _parent.set(arc, e);
+	    splay(arc);
+	    return;
+	  } else {
+	    e = _right[e];
+	  }
+	}
+      }
+    }
+
+    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 e = _head[s];
+      while (true) {
+	if (_g.target(e) == t) {
+	  const_cast<DynArcLookUp&>(*this).splay(e);
+	  return e;
+	} else if (t < _g.target(e)) {
+	  if (_left[e] == INVALID) {
+	    const_cast<DynArcLookUp&>(*this).splay(e);
+	    return INVALID;
+	  } else {
+	    e = _left[e];
+	  }
+	} else  {
+	  if (_right[e] == INVALID) {
+	    const_cast<DynArcLookUp&>(*this).splay(e);
+	    return INVALID;
+	  } else {
+	    e = _right[e];
+	  }
+	}
+      }
+    }
+
+    ///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 e = _head[s];
+      Arc r = INVALID;
+      while (true) {
+	if (_g.target(e) < t) {
+	  if (_right[e] == INVALID) {
+	    const_cast<DynArcLookUp&>(*this).splay(e);
+	    return r;
+	  } else {
+	    e = _right[e];
+	  }
+	} else {
+	  if (_g.target(e) == t) {
+	    r = e;
+	  }
+	  if (_left[e] == INVALID) {
+	    const_cast<DynArcLookUp&>(*this).splay(e);
+	    return r;
+	  } else {
+	    e = _left[e];
+	  }
+	}
+      }
+    }
+
+    ///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 e) const
+#else
+    Arc findNext(Node, Node t, Arc e) const
+#endif
+    {
+      if (_right[e] != INVALID) {
+	e = _right[e];
+	while (_left[e] != INVALID) {
+	  e = _left[e];
+	}
+	const_cast<DynArcLookUp&>(*this).splay(e);
+      } else {
+	while (_parent[e] != INVALID && _right[_parent[e]] ==  e) {
+	  e = _parent[e];
+	}
+	if (_parent[e] == INVALID) {
+	  return INVALID;
+	} else {
+	  e = _parent[e];
+	  const_cast<DynArcLookUp&>(*this).splay(e);
+	}
+      }
+      if (_g.target(e) == t) return e;
+      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).
+  ///
+  ///\param G The type of the underlying digraph.
+  ///
+  ///\sa DynArcLookUp
+  ///\sa AllArcLookUp  
+  template<class G>
+  class ArcLookUp 
+  {
+  public:
+    GRAPH_TYPEDEFS(typename 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).
+  ///
+  ///\param G The type of the underlying digraph.
+  ///
+  ///\sa DynArcLookUp
+  ///\sa ArcLookUp  
+  template<class G>
+  class AllArcLookUp : public ArcLookUp<G>
+  {
+    using ArcLookUp<G>::_g;
+    using ArcLookUp<G>::_right;
+    using ArcLookUp<G>::_left;
+    using ArcLookUp<G>::_head;
+
+    GRAPH_TYPEDEFS(typename G);
+    typedef G Digraph;
+    
+    typename Digraph::template ArcMap<Arc> _next;
+    
+    Arc refreshNext(Arc head,Arc 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 digraph
+    ///changes.
+    AllArcLookUp(const Digraph &g) : ArcLookUp<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 arcs of \c n.
+    
+    void refresh(Node n) 
+    {
+      ArcLookUp<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 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(_head[n]);
+    }
+    
+    ///Find an arc between two nodes.
+    
+    ///Find an arc between two nodes.
+    ///\param s The source node
+    ///\param t The target node
+    ///\param prev The previous arc between \c s and \c t. It it is INVALID or
+    ///not given, the operator finds the first appropriate arc.
+    ///\return An arc from \c s to \c t after \c prev or
+    ///\ref INVALID if there is no more.
+    ///
+    ///For example, you can count the number of arcs from \c u to \c v in the
+    ///following way.
+    ///\code
+    ///AllArcLookUp<ListDigraph> ae(g);
+    ///...
+    ///int n=0;
+    ///for(Arc e=ae(u,v);e!=INVALID;e=ae(u,v,e)) n++;
+    ///\endcode
+    ///
+    ///Finding the first arc take <em>O(</em>log<em>d)</em> time, where
+    /// <em>d</em> is the number of outgoing arcs of \c s. Then, the
+    ///consecutive arcs are found in constant time.
+    ///
+    ///\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.
+    ///
+#ifdef DOXYGEN
+    Arc operator()(Node s, Node t, Arc prev=INVALID) const {}
+#else
+    using ArcLookUp<G>::operator() ;
+    Arc operator()(Node s, Node t, Arc prev) const
+    {
+      return prev==INVALID?(*this)(s,t):_next[prev];
+    }
+#endif
+      
+  };
+
+  /// @}
+
+} //END OF NAMESPACE LEMON
+
+#endif