diff -r b67149f0e675 -r a5d8c039f218 lemon/graph_utils.h
--- a/lemon/graph_utils.h	Mon Jul 14 15:40:24 2008 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2760 +0,0 @@
-/* -*- mode: C++; indent-tabs-mode: nil; -*-
- *
- * This file is a part of LEMON, a generic C++ optimization library.
- *
- * Copyright (C) 2003-2008
- * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
- * (Egervary Research Group on Combinatorial Optimization, EGRES).
- *
- * Permission to use, modify and distribute this software is granted
- * provided that this copyright notice appears in all copies. For
- * precise terms see the accompanying LICENSE file.
- *
- * This software is provided "AS IS" with no warranty of any kind,
- * express or implied, and with no claim as to its suitability for any
- * purpose.
- *
- */
-
-#ifndef LEMON_GRAPH_UTILS_H
-#define LEMON_GRAPH_UTILS_H
-
-#include <iterator>
-#include <vector>
-#include <map>
-#include <cmath>
-#include <algorithm>
-
-#include <lemon/bits/invalid.h>
-#include <lemon/bits/utility.h>
-#include <lemon/maps.h>
-#include <lemon/bits/traits.h>
-
-#include <lemon/bits/alteration_notifier.h>
-#include <lemon/bits/default_map.h>
-
-///\ingroup gutils
-///\file
-///\brief Graph utilities.
-
-namespace lemon {
-
-  /// \addtogroup gutils
-  /// @{
-
-  ///Creates convenience typedefs for the digraph types and iterators
-
-  ///This \c \#define creates convenience typedefs for the following types
-  ///of \c Digraph: \c Node,  \c NodeIt, \c Arc, \c ArcIt, \c InArcIt,
-  ///\c OutArcIt, \c BoolNodeMap, \c IntNodeMap, \c DoubleNodeMap,
-  ///\c BoolArcMap, \c IntArcMap, \c DoubleArcMap.
-  ///
-  ///\note If the graph type is a dependent type, ie. the graph type depend
-  ///on a template parameter, then use \c TEMPLATE_DIGRAPH_TYPEDEFS()
-  ///macro.
-#define DIGRAPH_TYPEDEFS(Digraph)                                       \
-  typedef Digraph::Node Node;                                           \
-  typedef Digraph::NodeIt NodeIt;                                       \
-  typedef Digraph::Arc Arc;                                             \
-  typedef Digraph::ArcIt ArcIt;                                         \
-  typedef Digraph::InArcIt InArcIt;                                     \
-  typedef Digraph::OutArcIt OutArcIt;                                   \
-  typedef Digraph::NodeMap<bool> BoolNodeMap;                           \
-  typedef Digraph::NodeMap<int> IntNodeMap;                             \
-  typedef Digraph::NodeMap<double> DoubleNodeMap;                       \
-  typedef Digraph::ArcMap<bool> BoolArcMap;                             \
-  typedef Digraph::ArcMap<int> IntArcMap;                               \
-  typedef Digraph::ArcMap<double> DoubleArcMap
-
-  ///Creates convenience typedefs for the digraph types and iterators
-
-  ///\see DIGRAPH_TYPEDEFS
-  ///
-  ///\note Use this macro, if the graph type is a dependent type,
-  ///ie. the graph type depend on a template parameter.
-#define TEMPLATE_DIGRAPH_TYPEDEFS(Digraph)                              \
-  typedef typename Digraph::Node Node;                                  \
-  typedef typename Digraph::NodeIt NodeIt;                              \
-  typedef typename Digraph::Arc Arc;                                    \
-  typedef typename Digraph::ArcIt ArcIt;                                \
-  typedef typename Digraph::InArcIt InArcIt;                            \
-  typedef typename Digraph::OutArcIt OutArcIt;                          \
-  typedef typename Digraph::template NodeMap<bool> BoolNodeMap;         \
-  typedef typename Digraph::template NodeMap<int> IntNodeMap;           \
-  typedef typename Digraph::template NodeMap<double> DoubleNodeMap;     \
-  typedef typename Digraph::template ArcMap<bool> BoolArcMap;           \
-  typedef typename Digraph::template ArcMap<int> IntArcMap;             \
-  typedef typename Digraph::template ArcMap<double> DoubleArcMap
-
-  ///Creates convenience typedefs for the graph types and iterators
-
-  ///This \c \#define creates the same convenience typedefs as defined
-  ///by \ref DIGRAPH_TYPEDEFS(Graph) and six more, namely it creates
-  ///\c Edge, \c EdgeIt, \c IncEdgeIt, \c BoolEdgeMap, \c IntEdgeMap,
-  ///\c DoubleEdgeMap.
-  ///
-  ///\note If the graph type is a dependent type, ie. the graph type depend
-  ///on a template parameter, then use \c TEMPLATE_DIGRAPH_TYPEDEFS()
-  ///macro.
-#define GRAPH_TYPEDEFS(Graph)                                           \
-  DIGRAPH_TYPEDEFS(Graph);                                              \
-  typedef Graph::Edge Edge;                                             \
-  typedef Graph::EdgeIt EdgeIt;                                         \
-  typedef Graph::IncEdgeIt IncEdgeIt;                                   \
-  typedef Graph::EdgeMap<bool> BoolEdgeMap;                             \
-  typedef Graph::EdgeMap<int> IntEdgeMap;                               \
-  typedef Graph::EdgeMap<double> DoubleEdgeMap
-
-  ///Creates convenience typedefs for the graph types and iterators
-
-  ///\see GRAPH_TYPEDEFS
-  ///
-  ///\note Use this macro, if the graph type is a dependent type,
-  ///ie. the graph type depend on a template parameter.
-#define TEMPLATE_GRAPH_TYPEDEFS(Graph)                                  \
-  TEMPLATE_DIGRAPH_TYPEDEFS(Graph);                                     \
-  typedef typename Graph::Edge Edge;                                    \
-  typedef typename Graph::EdgeIt EdgeIt;                                \
-  typedef typename Graph::IncEdgeIt IncEdgeIt;                          \
-  typedef typename Graph::template EdgeMap<bool> BoolEdgeMap;           \
-  typedef typename Graph::template EdgeMap<int> IntEdgeMap;             \
-  typedef typename Graph::template EdgeMap<double> DoubleEdgeMap
-
-  /// \brief Function to count the items in the graph.
-  ///
-  /// This function counts the items (nodes, arcs etc) in the graph.
-  /// The complexity of the function is O(n) because
-  /// it iterates on all of the items.
-  template <typename Graph, typename Item>
-  inline int countItems(const Graph& g) {
-    typedef typename ItemSetTraits<Graph, Item>::ItemIt ItemIt;
-    int num = 0;
-    for (ItemIt it(g); it != INVALID; ++it) {
-      ++num;
-    }
-    return num;
-  }
-
-  // Node counting:
-
-  namespace _graph_utils_bits {
-
-    template <typename Graph, typename Enable = void>
-    struct CountNodesSelector {
-      static int count(const Graph &g) {
-        return countItems<Graph, typename Graph::Node>(g);
-      }
-    };
-
-    template <typename Graph>
-    struct CountNodesSelector<
-      Graph, typename
-      enable_if<typename Graph::NodeNumTag, void>::type>
-    {
-      static int count(const Graph &g) {
-        return g.nodeNum();
-      }
-    };
-  }
-
-  /// \brief Function to count the nodes in the graph.
-  ///
-  /// This function counts the nodes in the graph.
-  /// The complexity of the function is O(n) but for some
-  /// graph structures it is specialized to run in O(1).
-  ///
-  /// If the graph contains a \e nodeNum() member function and a
-  /// \e NodeNumTag tag then this function calls directly the member
-  /// function to query the cardinality of the node set.
-  template <typename Graph>
-  inline int countNodes(const Graph& g) {
-    return _graph_utils_bits::CountNodesSelector<Graph>::count(g);
-  }
-
-  // Arc counting:
-
-  namespace _graph_utils_bits {
-
-    template <typename Graph, typename Enable = void>
-    struct CountArcsSelector {
-      static int count(const Graph &g) {
-        return countItems<Graph, typename Graph::Arc>(g);
-      }
-    };
-
-    template <typename Graph>
-    struct CountArcsSelector<
-      Graph,
-      typename enable_if<typename Graph::ArcNumTag, void>::type>
-    {
-      static int count(const Graph &g) {
-        return g.arcNum();
-      }
-    };
-  }
-
-  /// \brief Function to count the arcs in the graph.
-  ///
-  /// This function counts the arcs in the graph.
-  /// The complexity of the function is O(e) but for some
-  /// graph structures it is specialized to run in O(1).
-  ///
-  /// If the graph contains a \e arcNum() member function and a
-  /// \e EdgeNumTag tag then this function calls directly the member
-  /// function to query the cardinality of the arc set.
-  template <typename Graph>
-  inline int countArcs(const Graph& g) {
-    return _graph_utils_bits::CountArcsSelector<Graph>::count(g);
-  }
-
-  // Edge counting:
-  namespace _graph_utils_bits {
-
-    template <typename Graph, typename Enable = void>
-    struct CountEdgesSelector {
-      static int count(const Graph &g) {
-        return countItems<Graph, typename Graph::Edge>(g);
-      }
-    };
-
-    template <typename Graph>
-    struct CountEdgesSelector<
-      Graph,
-      typename enable_if<typename Graph::EdgeNumTag, void>::type>
-    {
-      static int count(const Graph &g) {
-        return g.edgeNum();
-      }
-    };
-  }
-
-  /// \brief Function to count the edges in the graph.
-  ///
-  /// This function counts the edges in the graph.
-  /// The complexity of the function is O(m) but for some
-  /// graph structures it is specialized to run in O(1).
-  ///
-  /// If the graph contains a \e edgeNum() member function and a
-  /// \e EdgeNumTag tag then this function calls directly the member
-  /// function to query the cardinality of the edge set.
-  template <typename Graph>
-  inline int countEdges(const Graph& g) {
-    return _graph_utils_bits::CountEdgesSelector<Graph>::count(g);
-
-  }
-
-
-  template <typename Graph, typename DegIt>
-  inline int countNodeDegree(const Graph& _g, const typename Graph::Node& _n) {
-    int num = 0;
-    for (DegIt it(_g, _n); it != INVALID; ++it) {
-      ++num;
-    }
-    return num;
-  }
-
-  /// \brief Function to count the number of the out-arcs from node \c n.
-  ///
-  /// This function counts the number of the out-arcs from node \c n
-  /// in the graph.
-  template <typename Graph>
-  inline int countOutArcs(const Graph& _g,  const typename Graph::Node& _n) {
-    return countNodeDegree<Graph, typename Graph::OutArcIt>(_g, _n);
-  }
-
-  /// \brief Function to count the number of the in-arcs to node \c n.
-  ///
-  /// This function counts the number of the in-arcs to node \c n
-  /// in the graph.
-  template <typename Graph>
-  inline int countInArcs(const Graph& _g,  const typename Graph::Node& _n) {
-    return countNodeDegree<Graph, typename Graph::InArcIt>(_g, _n);
-  }
-
-  /// \brief Function to count the number of the inc-edges to node \c n.
-  ///
-  /// This function counts the number of the inc-edges to node \c n
-  /// in the graph.
-  template <typename Graph>
-  inline int countIncEdges(const Graph& _g,  const typename Graph::Node& _n) {
-    return countNodeDegree<Graph, typename Graph::IncEdgeIt>(_g, _n);
-  }
-
-  namespace _graph_utils_bits {
-
-    template <typename Graph, typename Enable = void>
-    struct FindArcSelector {
-      typedef typename Graph::Node Node;
-      typedef typename Graph::Arc Arc;
-      static Arc find(const Graph &g, Node u, Node v, Arc e) {
-        if (e == INVALID) {
-          g.firstOut(e, u);
-        } else {
-          g.nextOut(e);
-        }
-        while (e != INVALID && g.target(e) != v) {
-          g.nextOut(e);
-        }
-        return e;
-      }
-    };
-
-    template <typename Graph>
-    struct FindArcSelector<
-      Graph,
-      typename enable_if<typename Graph::FindEdgeTag, void>::type>
-    {
-      typedef typename Graph::Node Node;
-      typedef typename Graph::Arc Arc;
-      static Arc find(const Graph &g, Node u, Node v, Arc prev) {
-        return g.findArc(u, v, prev);
-      }
-    };
-  }
-
-  /// \brief Finds an arc between two nodes of a graph.
-  ///
-  /// Finds an arc from node \c u to node \c v in graph \c g.
-  ///
-  /// If \c prev is \ref INVALID (this is the default value), then
-  /// it finds the first arc from \c u to \c v. Otherwise it looks for
-  /// the next arc from \c u to \c v after \c prev.
-  /// \return The found arc or \ref INVALID if there is no such an arc.
-  ///
-  /// Thus you can iterate through each arc from \c u to \c v as it follows.
-  ///\code
-  /// for(Arc e=findArc(g,u,v);e!=INVALID;e=findArc(g,u,v,e)) {
-  ///   ...
-  /// }
-  ///\endcode
-  ///
-  ///\sa ArcLookUp
-  ///\sa AllArcLookUp
-  ///\sa DynArcLookUp
-  ///\sa ConArcIt
-  template <typename Graph>
-  inline typename Graph::Arc
-  findArc(const Graph &g, typename Graph::Node u, typename Graph::Node v,
-           typename Graph::Arc prev = INVALID) {
-    return _graph_utils_bits::FindArcSelector<Graph>::find(g, u, v, prev);
-  }
-
-  /// \brief Iterator for iterating on arcs connected the same nodes.
-  ///
-  /// Iterator for iterating on arcs connected the same nodes. It is
-  /// higher level interface for the findArc() function. You can
-  /// use it the following way:
-  ///\code
-  /// for (ConArcIt<Graph> it(g, src, trg); it != INVALID; ++it) {
-  ///   ...
-  /// }
-  ///\endcode
-  ///
-  ///\sa findArc()
-  ///\sa ArcLookUp
-  ///\sa AllArcLookUp
-  ///\sa DynArcLookUp
-  template <typename _Graph>
-  class ConArcIt : public _Graph::Arc {
-  public:
-
-    typedef _Graph Graph;
-    typedef typename Graph::Arc Parent;
-
-    typedef typename Graph::Arc Arc;
-    typedef typename Graph::Node Node;
-
-    /// \brief Constructor.
-    ///
-    /// Construct a new ConArcIt iterating on the arcs which
-    /// connects the \c u and \c v node.
-    ConArcIt(const Graph& g, Node u, Node v) : _graph(g) {
-      Parent::operator=(findArc(_graph, u, v));
-    }
-
-    /// \brief Constructor.
-    ///
-    /// Construct a new ConArcIt which continues the iterating from
-    /// the \c e arc.
-    ConArcIt(const Graph& g, Arc a) : Parent(a), _graph(g) {}
-
-    /// \brief Increment operator.
-    ///
-    /// It increments the iterator and gives back the next arc.
-    ConArcIt& operator++() {
-      Parent::operator=(findArc(_graph, _graph.source(*this),
-                                _graph.target(*this), *this));
-      return *this;
-    }
-  private:
-    const Graph& _graph;
-  };
-
-  namespace _graph_utils_bits {
-
-    template <typename Graph, typename Enable = void>
-    struct FindEdgeSelector {
-      typedef typename Graph::Node Node;
-      typedef typename Graph::Edge Edge;
-      static Edge find(const Graph &g, Node u, Node v, Edge e) {
-        bool b;
-        if (u != v) {
-          if (e == INVALID) {
-            g.firstInc(e, b, u);
-          } else {
-            b = g.u(e) == u;
-            g.nextInc(e, b);
-          }
-          while (e != INVALID && (b ? g.v(e) : g.u(e)) != v) {
-            g.nextInc(e, b);
-          }
-        } else {
-          if (e == INVALID) {
-            g.firstInc(e, b, u);
-          } else {
-            b = true;
-            g.nextInc(e, b);
-          }
-          while (e != INVALID && (!b || g.v(e) != v)) {
-            g.nextInc(e, b);
-          }
-        }
-        return e;
-      }
-    };
-
-    template <typename Graph>
-    struct FindEdgeSelector<
-      Graph,
-      typename enable_if<typename Graph::FindEdgeTag, void>::type>
-    {
-      typedef typename Graph::Node Node;
-      typedef typename Graph::Edge Edge;
-      static Edge find(const Graph &g, Node u, Node v, Edge prev) {
-        return g.findEdge(u, v, prev);
-      }
-    };
-  }
-
-  /// \brief Finds an edge between two nodes of a graph.
-  ///
-  /// Finds an edge from node \c u to node \c v in graph \c g.
-  /// If the node \c u and node \c v is equal then each loop edge
-  /// will be enumerated once.
-  ///
-  /// If \c prev is \ref INVALID (this is the default value), then
-  /// it finds the first arc from \c u to \c v. Otherwise it looks for
-  /// the next arc from \c u to \c v after \c prev.
-  /// \return The found arc or \ref INVALID if there is no such an arc.
-  ///
-  /// Thus you can iterate through each arc from \c u to \c v as it follows.
-  ///\code
-  /// for(Edge e = findEdge(g,u,v); e != INVALID;
-  ///     e = findEdge(g,u,v,e)) {
-  ///   ...
-  /// }
-  ///\endcode
-  ///
-  ///\sa ConEdgeIt
-
-  template <typename Graph>
-  inline typename Graph::Edge
-  findEdge(const Graph &g, typename Graph::Node u, typename Graph::Node v,
-            typename Graph::Edge p = INVALID) {
-    return _graph_utils_bits::FindEdgeSelector<Graph>::find(g, u, v, p);
-  }
-
-  /// \brief Iterator for iterating on edges connected the same nodes.
-  ///
-  /// Iterator for iterating on edges connected the same nodes. It is
-  /// higher level interface for the findEdge() function. You can
-  /// use it the following way:
-  ///\code
-  /// for (ConEdgeIt<Graph> it(g, src, trg); it != INVALID; ++it) {
-  ///   ...
-  /// }
-  ///\endcode
-  ///
-  ///\sa findEdge()
-  template <typename _Graph>
-  class ConEdgeIt : public _Graph::Edge {
-  public:
-
-    typedef _Graph Graph;
-    typedef typename Graph::Edge Parent;
-
-    typedef typename Graph::Edge Edge;
-    typedef typename Graph::Node Node;
-
-    /// \brief Constructor.
-    ///
-    /// Construct a new ConEdgeIt iterating on the edges which
-    /// connects the \c u and \c v node.
-    ConEdgeIt(const Graph& g, Node u, Node v) : _graph(g) {
-      Parent::operator=(findEdge(_graph, u, v));
-    }
-
-    /// \brief Constructor.
-    ///
-    /// Construct a new ConEdgeIt which continues the iterating from
-    /// the \c e edge.
-    ConEdgeIt(const Graph& g, Edge e) : Parent(e), _graph(g) {}
-
-    /// \brief Increment operator.
-    ///
-    /// It increments the iterator and gives back the next edge.
-    ConEdgeIt& operator++() {
-      Parent::operator=(findEdge(_graph, _graph.u(*this),
-                                 _graph.v(*this), *this));
-      return *this;
-    }
-  private:
-    const Graph& _graph;
-  };
-
-  namespace _graph_utils_bits {
-
-    template <typename Digraph, typename Item, typename RefMap>
-    class MapCopyBase {
-    public:
-      virtual void copy(const Digraph& from, const RefMap& refMap) = 0;
-
-      virtual ~MapCopyBase() {}
-    };
-
-    template <typename Digraph, typename Item, typename RefMap,
-              typename ToMap, typename FromMap>
-    class MapCopy : public MapCopyBase<Digraph, Item, RefMap> {
-    public:
-
-      MapCopy(ToMap& tmap, const FromMap& map)
-        : _tmap(tmap), _map(map) {}
-
-      virtual void copy(const Digraph& digraph, const RefMap& refMap) {
-        typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
-        for (ItemIt it(digraph); it != INVALID; ++it) {
-          _tmap.set(refMap[it], _map[it]);
-        }
-      }
-
-    private:
-      ToMap& _tmap;
-      const FromMap& _map;
-    };
-
-    template <typename Digraph, typename Item, typename RefMap, typename It>
-    class ItemCopy : public MapCopyBase<Digraph, Item, RefMap> {
-    public:
-
-      ItemCopy(It& it, const Item& item) : _it(it), _item(item) {}
-
-      virtual void copy(const Digraph&, const RefMap& refMap) {
-        _it = refMap[_item];
-      }
-
-    private:
-      It& _it;
-      Item _item;
-    };
-
-    template <typename Digraph, typename Item, typename RefMap, typename Ref>
-    class RefCopy : public MapCopyBase<Digraph, Item, RefMap> {
-    public:
-
-      RefCopy(Ref& map) : _map(map) {}
-
-      virtual void copy(const Digraph& digraph, const RefMap& refMap) {
-        typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
-        for (ItemIt it(digraph); it != INVALID; ++it) {
-          _map.set(it, refMap[it]);
-        }
-      }
-
-    private:
-      Ref& _map;
-    };
-
-    template <typename Digraph, typename Item, typename RefMap,
-              typename CrossRef>
-    class CrossRefCopy : public MapCopyBase<Digraph, Item, RefMap> {
-    public:
-
-      CrossRefCopy(CrossRef& cmap) : _cmap(cmap) {}
-
-      virtual void copy(const Digraph& digraph, const RefMap& refMap) {
-        typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
-        for (ItemIt it(digraph); it != INVALID; ++it) {
-          _cmap.set(refMap[it], it);
-        }
-      }
-
-    private:
-      CrossRef& _cmap;
-    };
-
-    template <typename Digraph, typename Enable = void>
-    struct DigraphCopySelector {
-      template <typename From, typename NodeRefMap, typename ArcRefMap>
-      static void copy(Digraph &to, const From& from,
-                       NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {
-        for (typename From::NodeIt it(from); it != INVALID; ++it) {
-          nodeRefMap[it] = to.addNode();
-        }
-        for (typename From::ArcIt it(from); it != INVALID; ++it) {
-          arcRefMap[it] = to.addArc(nodeRefMap[from.source(it)],
-                                    nodeRefMap[from.target(it)]);
-        }
-      }
-    };
-
-    template <typename Digraph>
-    struct DigraphCopySelector<
-      Digraph,
-      typename enable_if<typename Digraph::BuildTag, void>::type>
-    {
-      template <typename From, typename NodeRefMap, typename ArcRefMap>
-      static void copy(Digraph &to, const From& from,
-                       NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {
-        to.build(from, nodeRefMap, arcRefMap);
-      }
-    };
-
-    template <typename Graph, typename Enable = void>
-    struct GraphCopySelector {
-      template <typename From, typename NodeRefMap, typename EdgeRefMap>
-      static void copy(Graph &to, const From& from,
-                       NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {
-        for (typename From::NodeIt it(from); it != INVALID; ++it) {
-          nodeRefMap[it] = to.addNode();
-        }
-        for (typename From::EdgeIt it(from); it != INVALID; ++it) {
-          edgeRefMap[it] = to.addEdge(nodeRefMap[from.u(it)],
-                                      nodeRefMap[from.v(it)]);
-        }
-      }
-    };
-
-    template <typename Graph>
-    struct GraphCopySelector<
-      Graph,
-      typename enable_if<typename Graph::BuildTag, void>::type>
-    {
-      template <typename From, typename NodeRefMap, typename EdgeRefMap>
-      static void copy(Graph &to, const From& from,
-                       NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {
-        to.build(from, nodeRefMap, edgeRefMap);
-      }
-    };
-
-  }
-
-  /// \brief Class to copy a digraph.
-  ///
-  /// Class to copy a digraph to another digraph (duplicate a digraph). The
-  /// simplest way of using it is through the \c copyDigraph() function.
-  ///
-  /// This class not just make a copy of a graph, but it can create
-  /// references and cross references between the nodes and arcs of
-  /// the two graphs, it can copy maps for use with the newly created
-  /// graph and copy nodes and arcs.
-  ///
-  /// To make a copy from a graph, first an instance of DigraphCopy
-  /// should be created, then the data belongs to the graph should
-  /// assigned to copy. In the end, the \c run() member should be
-  /// called.
-  ///
-  /// The next code copies a graph with several data:
-  ///\code
-  ///  DigraphCopy<NewGraph, OrigGraph> dc(new_graph, orig_graph);
-  ///  // create a reference for the nodes
-  ///  OrigGraph::NodeMap<NewGraph::Node> nr(orig_graph);
-  ///  dc.nodeRef(nr);
-  ///  // create a cross reference (inverse) for the arcs
-  ///  NewGraph::ArcMap<OrigGraph::Arc> acr(new_graph);
-  ///  dc.arcCrossRef(acr);
-  ///  // copy an arc map
-  ///  OrigGraph::ArcMap<double> oamap(orig_graph);
-  ///  NewGraph::ArcMap<double> namap(new_graph);
-  ///  dc.arcMap(namap, oamap);
-  ///  // copy a node
-  ///  OrigGraph::Node on;
-  ///  NewGraph::Node nn;
-  ///  dc.node(nn, on);
-  ///  // Executions of copy
-  ///  dc.run();
-  ///\endcode
-  template <typename To, typename From>
-  class DigraphCopy {
-  private:
-
-    typedef typename From::Node Node;
-    typedef typename From::NodeIt NodeIt;
-    typedef typename From::Arc Arc;
-    typedef typename From::ArcIt ArcIt;
-
-    typedef typename To::Node TNode;
-    typedef typename To::Arc TArc;
-
-    typedef typename From::template NodeMap<TNode> NodeRefMap;
-    typedef typename From::template ArcMap<TArc> ArcRefMap;
-
-
-  public:
-
-
-    /// \brief Constructor for the DigraphCopy.
-    ///
-    /// It copies the content of the \c _from digraph into the
-    /// \c _to digraph.
-    DigraphCopy(To& to, const From& from)
-      : _from(from), _to(to) {}
-
-    /// \brief Destructor of the DigraphCopy
-    ///
-    /// Destructor of the DigraphCopy
-    ~DigraphCopy() {
-      for (int i = 0; i < int(_node_maps.size()); ++i) {
-        delete _node_maps[i];
-      }
-      for (int i = 0; i < int(_arc_maps.size()); ++i) {
-        delete _arc_maps[i];
-      }
-
-    }
-
-    /// \brief Copies the node references into the given map.
-    ///
-    /// Copies the node references into the given map. The parameter
-    /// should be a map, which key type is the Node type of the source
-    /// graph, while the value type is the Node type of the
-    /// destination graph.
-    template <typename NodeRef>
-    DigraphCopy& nodeRef(NodeRef& map) {
-      _node_maps.push_back(new _graph_utils_bits::RefCopy<From, Node,
-                           NodeRefMap, NodeRef>(map));
-      return *this;
-    }
-
-    /// \brief Copies the node cross references into the given map.
-    ///
-    ///  Copies the node cross references (reverse references) into
-    ///  the given map. The parameter should be a map, which key type
-    ///  is the Node type of the destination graph, while the value type is
-    ///  the Node type of the source graph.
-    template <typename NodeCrossRef>
-    DigraphCopy& nodeCrossRef(NodeCrossRef& map) {
-      _node_maps.push_back(new _graph_utils_bits::CrossRefCopy<From, Node,
-                           NodeRefMap, NodeCrossRef>(map));
-      return *this;
-    }
-
-    /// \brief Make copy of the given map.
-    ///
-    /// Makes copy of the given map for the newly created digraph.
-    /// The new map's key type is the destination graph's node type,
-    /// and the copied map's key type is the source graph's node type.
-    template <typename ToMap, typename FromMap>
-    DigraphCopy& nodeMap(ToMap& tmap, const FromMap& map) {
-      _node_maps.push_back(new _graph_utils_bits::MapCopy<From, Node,
-                           NodeRefMap, ToMap, FromMap>(tmap, map));
-      return *this;
-    }
-
-    /// \brief Make a copy of the given node.
-    ///
-    /// Make a copy of the given node.
-    DigraphCopy& node(TNode& tnode, const Node& snode) {
-      _node_maps.push_back(new _graph_utils_bits::ItemCopy<From, Node,
-                           NodeRefMap, TNode>(tnode, snode));
-      return *this;
-    }
-
-    /// \brief Copies the arc references into the given map.
-    ///
-    /// Copies the arc references into the given map.
-    template <typename ArcRef>
-    DigraphCopy& arcRef(ArcRef& map) {
-      _arc_maps.push_back(new _graph_utils_bits::RefCopy<From, Arc,
-                          ArcRefMap, ArcRef>(map));
-      return *this;
-    }
-
-    /// \brief Copies the arc cross references into the given map.
-    ///
-    ///  Copies the arc cross references (reverse references) into
-    ///  the given map.
-    template <typename ArcCrossRef>
-    DigraphCopy& arcCrossRef(ArcCrossRef& map) {
-      _arc_maps.push_back(new _graph_utils_bits::CrossRefCopy<From, Arc,
-                          ArcRefMap, ArcCrossRef>(map));
-      return *this;
-    }
-
-    /// \brief Make copy of the given map.
-    ///
-    /// Makes copy of the given map for the newly created digraph.
-    /// The new map's key type is the to digraph's arc type,
-    /// and the copied map's key type is the from digraph's arc
-    /// type.
-    template <typename ToMap, typename FromMap>
-    DigraphCopy& arcMap(ToMap& tmap, const FromMap& map) {
-      _arc_maps.push_back(new _graph_utils_bits::MapCopy<From, Arc,
-                          ArcRefMap, ToMap, FromMap>(tmap, map));
-      return *this;
-    }
-
-    /// \brief Make a copy of the given arc.
-    ///
-    /// Make a copy of the given arc.
-    DigraphCopy& arc(TArc& tarc, const Arc& sarc) {
-      _arc_maps.push_back(new _graph_utils_bits::ItemCopy<From, Arc,
-                          ArcRefMap, TArc>(tarc, sarc));
-      return *this;
-    }
-
-    /// \brief Executes the copies.
-    ///
-    /// Executes the copies.
-    void run() {
-      NodeRefMap nodeRefMap(_from);
-      ArcRefMap arcRefMap(_from);
-      _graph_utils_bits::DigraphCopySelector<To>::
-        copy(_to, _from, nodeRefMap, arcRefMap);
-      for (int i = 0; i < int(_node_maps.size()); ++i) {
-        _node_maps[i]->copy(_from, nodeRefMap);
-      }
-      for (int i = 0; i < int(_arc_maps.size()); ++i) {
-        _arc_maps[i]->copy(_from, arcRefMap);
-      }
-    }
-
-  protected:
-
-
-    const From& _from;
-    To& _to;
-
-    std::vector<_graph_utils_bits::MapCopyBase<From, Node, NodeRefMap>* >
-    _node_maps;
-
-    std::vector<_graph_utils_bits::MapCopyBase<From, Arc, ArcRefMap>* >
-    _arc_maps;
-
-  };
-
-  /// \brief Copy a digraph to another digraph.
-  ///
-  /// Copy a digraph to another digraph. The complete usage of the
-  /// function is detailed in the DigraphCopy class, but a short
-  /// example shows a basic work:
-  ///\code
-  /// copyDigraph(trg, src).nodeRef(nr).arcCrossRef(ecr).run();
-  ///\endcode
-  ///
-  /// After the copy the \c nr map will contain the mapping from the
-  /// nodes of the \c from digraph to the nodes of the \c to digraph and
-  /// \c ecr will contain the mapping from the arcs of the \c to digraph
-  /// to the arcs of the \c from digraph.
-  ///
-  /// \see DigraphCopy
-  template <typename To, typename From>
-  DigraphCopy<To, From> copyDigraph(To& to, const From& from) {
-    return DigraphCopy<To, From>(to, from);
-  }
-
-  /// \brief Class to copy a graph.
-  ///
-  /// Class to copy a graph to another graph (duplicate a graph). The
-  /// simplest way of using it is through the \c copyGraph() function.
-  ///
-  /// This class not just make a copy of a graph, but it can create
-  /// references and cross references between the nodes, edges and arcs of
-  /// the two graphs, it can copy maps for use with the newly created
-  /// graph and copy nodes, edges and arcs.
-  ///
-  /// To make a copy from a graph, first an instance of GraphCopy
-  /// should be created, then the data belongs to the graph should
-  /// assigned to copy. In the end, the \c run() member should be
-  /// called.
-  ///
-  /// The next code copies a graph with several data:
-  ///\code
-  ///  GraphCopy<NewGraph, OrigGraph> dc(new_graph, orig_graph);
-  ///  // create a reference for the nodes
-  ///  OrigGraph::NodeMap<NewGraph::Node> nr(orig_graph);
-  ///  dc.nodeRef(nr);
-  ///  // create a cross reference (inverse) for the edges
-  ///  NewGraph::EdgeMap<OrigGraph::Arc> ecr(new_graph);
-  ///  dc.edgeCrossRef(ecr);
-  ///  // copy an arc map
-  ///  OrigGraph::ArcMap<double> oamap(orig_graph);
-  ///  NewGraph::ArcMap<double> namap(new_graph);
-  ///  dc.arcMap(namap, oamap);
-  ///  // copy a node
-  ///  OrigGraph::Node on;
-  ///  NewGraph::Node nn;
-  ///  dc.node(nn, on);
-  ///  // Executions of copy
-  ///  dc.run();
-  ///\endcode
-  template <typename To, typename From>
-  class GraphCopy {
-  private:
-
-    typedef typename From::Node Node;
-    typedef typename From::NodeIt NodeIt;
-    typedef typename From::Arc Arc;
-    typedef typename From::ArcIt ArcIt;
-    typedef typename From::Edge Edge;
-    typedef typename From::EdgeIt EdgeIt;
-
-    typedef typename To::Node TNode;
-    typedef typename To::Arc TArc;
-    typedef typename To::Edge TEdge;
-
-    typedef typename From::template NodeMap<TNode> NodeRefMap;
-    typedef typename From::template EdgeMap<TEdge> EdgeRefMap;
-
-    struct ArcRefMap {
-      ArcRefMap(const To& to, const From& from,
-                const EdgeRefMap& edge_ref, const NodeRefMap& node_ref)
-        : _to(to), _from(from),
-          _edge_ref(edge_ref), _node_ref(node_ref) {}
-
-      typedef typename From::Arc Key;
-      typedef typename To::Arc Value;
-
-      Value operator[](const Key& key) const {
-        bool forward = _from.u(key) != _from.v(key) ?
-          _node_ref[_from.source(key)] ==
-          _to.source(_to.direct(_edge_ref[key], true)) :
-          _from.direction(key);
-        return _to.direct(_edge_ref[key], forward);
-      }
-
-      const To& _to;
-      const From& _from;
-      const EdgeRefMap& _edge_ref;
-      const NodeRefMap& _node_ref;
-    };
-
-
-  public:
-
-
-    /// \brief Constructor for the GraphCopy.
-    ///
-    /// It copies the content of the \c _from graph into the
-    /// \c _to graph.
-    GraphCopy(To& to, const From& from)
-      : _from(from), _to(to) {}
-
-    /// \brief Destructor of the GraphCopy
-    ///
-    /// Destructor of the GraphCopy
-    ~GraphCopy() {
-      for (int i = 0; i < int(_node_maps.size()); ++i) {
-        delete _node_maps[i];
-      }
-      for (int i = 0; i < int(_arc_maps.size()); ++i) {
-        delete _arc_maps[i];
-      }
-      for (int i = 0; i < int(_edge_maps.size()); ++i) {
-        delete _edge_maps[i];
-      }
-
-    }
-
-    /// \brief Copies the node references into the given map.
-    ///
-    /// Copies the node references into the given map.
-    template <typename NodeRef>
-    GraphCopy& nodeRef(NodeRef& map) {
-      _node_maps.push_back(new _graph_utils_bits::RefCopy<From, Node,
-                           NodeRefMap, NodeRef>(map));
-      return *this;
-    }
-
-    /// \brief Copies the node cross references into the given map.
-    ///
-    ///  Copies the node cross references (reverse references) into
-    ///  the given map.
-    template <typename NodeCrossRef>
-    GraphCopy& nodeCrossRef(NodeCrossRef& map) {
-      _node_maps.push_back(new _graph_utils_bits::CrossRefCopy<From, Node,
-                           NodeRefMap, NodeCrossRef>(map));
-      return *this;
-    }
-
-    /// \brief Make copy of the given map.
-    ///
-    /// Makes copy of the given map for the newly created graph.
-    /// The new map's key type is the to graph's node type,
-    /// and the copied map's key type is the from graph's node
-    /// type.
-    template <typename ToMap, typename FromMap>
-    GraphCopy& nodeMap(ToMap& tmap, const FromMap& map) {
-      _node_maps.push_back(new _graph_utils_bits::MapCopy<From, Node,
-                           NodeRefMap, ToMap, FromMap>(tmap, map));
-      return *this;
-    }
-
-    /// \brief Make a copy of the given node.
-    ///
-    /// Make a copy of the given node.
-    GraphCopy& node(TNode& tnode, const Node& snode) {
-      _node_maps.push_back(new _graph_utils_bits::ItemCopy<From, Node,
-                           NodeRefMap, TNode>(tnode, snode));
-      return *this;
-    }
-
-    /// \brief Copies the arc references into the given map.
-    ///
-    /// Copies the arc references into the given map.
-    template <typename ArcRef>
-    GraphCopy& arcRef(ArcRef& map) {
-      _arc_maps.push_back(new _graph_utils_bits::RefCopy<From, Arc,
-                          ArcRefMap, ArcRef>(map));
-      return *this;
-    }
-
-    /// \brief Copies the arc cross references into the given map.
-    ///
-    ///  Copies the arc cross references (reverse references) into
-    ///  the given map.
-    template <typename ArcCrossRef>
-    GraphCopy& arcCrossRef(ArcCrossRef& map) {
-      _arc_maps.push_back(new _graph_utils_bits::CrossRefCopy<From, Arc,
-                          ArcRefMap, ArcCrossRef>(map));
-      return *this;
-    }
-
-    /// \brief Make copy of the given map.
-    ///
-    /// Makes copy of the given map for the newly created graph.
-    /// The new map's key type is the to graph's arc type,
-    /// and the copied map's key type is the from graph's arc
-    /// type.
-    template <typename ToMap, typename FromMap>
-    GraphCopy& arcMap(ToMap& tmap, const FromMap& map) {
-      _arc_maps.push_back(new _graph_utils_bits::MapCopy<From, Arc,
-                          ArcRefMap, ToMap, FromMap>(tmap, map));
-      return *this;
-    }
-
-    /// \brief Make a copy of the given arc.
-    ///
-    /// Make a copy of the given arc.
-    GraphCopy& arc(TArc& tarc, const Arc& sarc) {
-      _arc_maps.push_back(new _graph_utils_bits::ItemCopy<From, Arc,
-                          ArcRefMap, TArc>(tarc, sarc));
-      return *this;
-    }
-
-    /// \brief Copies the edge references into the given map.
-    ///
-    /// Copies the edge references into the given map.
-    template <typename EdgeRef>
-    GraphCopy& edgeRef(EdgeRef& map) {
-      _edge_maps.push_back(new _graph_utils_bits::RefCopy<From, Edge,
-                           EdgeRefMap, EdgeRef>(map));
-      return *this;
-    }
-
-    /// \brief Copies the edge cross references into the given map.
-    ///
-    /// Copies the edge cross references (reverse
-    /// references) into the given map.
-    template <typename EdgeCrossRef>
-    GraphCopy& edgeCrossRef(EdgeCrossRef& map) {
-      _edge_maps.push_back(new _graph_utils_bits::CrossRefCopy<From,
-                           Edge, EdgeRefMap, EdgeCrossRef>(map));
-      return *this;
-    }
-
-    /// \brief Make copy of the given map.
-    ///
-    /// Makes copy of the given map for the newly created graph.
-    /// The new map's key type is the to graph's edge type,
-    /// and the copied map's key type is the from graph's edge
-    /// type.
-    template <typename ToMap, typename FromMap>
-    GraphCopy& edgeMap(ToMap& tmap, const FromMap& map) {
-      _edge_maps.push_back(new _graph_utils_bits::MapCopy<From, Edge,
-                           EdgeRefMap, ToMap, FromMap>(tmap, map));
-      return *this;
-    }
-
-    /// \brief Make a copy of the given edge.
-    ///
-    /// Make a copy of the given edge.
-    GraphCopy& edge(TEdge& tedge, const Edge& sedge) {
-      _edge_maps.push_back(new _graph_utils_bits::ItemCopy<From, Edge,
-                           EdgeRefMap, TEdge>(tedge, sedge));
-      return *this;
-    }
-
-    /// \brief Executes the copies.
-    ///
-    /// Executes the copies.
-    void run() {
-      NodeRefMap nodeRefMap(_from);
-      EdgeRefMap edgeRefMap(_from);
-      ArcRefMap arcRefMap(_to, _from, edgeRefMap, nodeRefMap);
-      _graph_utils_bits::GraphCopySelector<To>::
-        copy(_to, _from, nodeRefMap, edgeRefMap);
-      for (int i = 0; i < int(_node_maps.size()); ++i) {
-        _node_maps[i]->copy(_from, nodeRefMap);
-      }
-      for (int i = 0; i < int(_edge_maps.size()); ++i) {
-        _edge_maps[i]->copy(_from, edgeRefMap);
-      }
-      for (int i = 0; i < int(_arc_maps.size()); ++i) {
-        _arc_maps[i]->copy(_from, arcRefMap);
-      }
-    }
-
-  private:
-
-    const From& _from;
-    To& _to;
-
-    std::vector<_graph_utils_bits::MapCopyBase<From, Node, NodeRefMap>* >
-    _node_maps;
-
-    std::vector<_graph_utils_bits::MapCopyBase<From, Arc, ArcRefMap>* >
-    _arc_maps;
-
-    std::vector<_graph_utils_bits::MapCopyBase<From, Edge, EdgeRefMap>* >
-    _edge_maps;
-
-  };
-
-  /// \brief Copy a graph to another graph.
-  ///
-  /// Copy a graph to another graph. The complete usage of the
-  /// function is detailed in the GraphCopy class, but a short
-  /// example shows a basic work:
-  ///\code
-  /// copyGraph(trg, src).nodeRef(nr).arcCrossRef(ecr).run();
-  ///\endcode
-  ///
-  /// After the copy the \c nr map will contain the mapping from the
-  /// nodes of the \c from graph to the nodes of the \c to graph and
-  /// \c ecr will contain the mapping from the arcs of the \c to graph
-  /// to the arcs of the \c from graph.
-  ///
-  /// \see GraphCopy
-  template <typename To, typename From>
-  GraphCopy<To, From>
-  copyGraph(To& to, const From& from) {
-    return GraphCopy<To, From>(to, from);
-  }
-
-  /// @}
-
-  /// \addtogroup graph_maps
-  /// @{
-
-  /// Provides an immutable and unique id for each item in the graph.
-
-  /// The IdMap class provides a unique and immutable id for each item of the
-  /// same type (e.g. node) in the graph. This id is <ul><li>\b unique:
-  /// different items (nodes) get different ids <li>\b immutable: the id of an
-  /// item (node) does not change (even if you delete other nodes).  </ul>
-  /// Through this map you get access (i.e. can read) the inner id values of
-  /// the items stored in the graph. This map can be inverted with its member
-  /// class \c InverseMap or with the \c operator() member.
-  ///
-  template <typename _Graph, typename _Item>
-  class IdMap {
-  public:
-    typedef _Graph Graph;
-    typedef int Value;
-    typedef _Item Item;
-    typedef _Item Key;
-
-    /// \brief Constructor.
-    ///
-    /// Constructor of the map.
-    explicit IdMap(const Graph& graph) : _graph(&graph) {}
-
-    /// \brief Gives back the \e id of the item.
-    ///
-    /// Gives back the immutable and unique \e id of the item.
-    int operator[](const Item& item) const { return _graph->id(item);}
-
-    /// \brief Gives back the item by its id.
-    ///
-    /// Gives back the item by its id.
-    Item operator()(int id) { return _graph->fromId(id, Item()); }
-
-  private:
-    const Graph* _graph;
-
-  public:
-
-    /// \brief The class represents the inverse of its owner (IdMap).
-    ///
-    /// The class represents the inverse of its owner (IdMap).
-    /// \see inverse()
-    class InverseMap {
-    public:
-
-      /// \brief Constructor.
-      ///
-      /// Constructor for creating an id-to-item map.
-      explicit InverseMap(const Graph& graph) : _graph(&graph) {}
-
-      /// \brief Constructor.
-      ///
-      /// Constructor for creating an id-to-item map.
-      explicit InverseMap(const IdMap& map) : _graph(map._graph) {}
-
-      /// \brief Gives back the given item from its id.
-      ///
-      /// Gives back the given item from its id.
-      ///
-      Item operator[](int id) const { return _graph->fromId(id, Item());}
-
-    private:
-      const Graph* _graph;
-    };
-
-    /// \brief Gives back the inverse of the map.
-    ///
-    /// Gives back the inverse of the IdMap.
-    InverseMap inverse() const { return InverseMap(*_graph);}
-
-  };
-
-
-  /// \brief General invertable graph-map type.
-
-  /// This type provides simple invertable graph-maps.
-  /// The InvertableMap wraps an arbitrary ReadWriteMap
-  /// and if a key is set to a new value then store it
-  /// in the inverse map.
-  ///
-  /// The values of the map can be accessed
-  /// with stl compatible forward iterator.
-  ///
-  /// \tparam _Graph The graph type.
-  /// \tparam _Item The item type of the graph.
-  /// \tparam _Value The value type of the map.
-  ///
-  /// \see IterableValueMap
-  template <typename _Graph, typename _Item, typename _Value>
-  class InvertableMap : protected DefaultMap<_Graph, _Item, _Value> {
-  private:
-
-    typedef DefaultMap<_Graph, _Item, _Value> Map;
-    typedef _Graph Graph;
-
-    typedef std::map<_Value, _Item> Container;
-    Container _inv_map;
-
-  public:
-
-    /// The key type of InvertableMap (Node, Arc, Edge).
-    typedef typename Map::Key Key;
-    /// The value type of the InvertableMap.
-    typedef typename Map::Value Value;
-
-
-
-    /// \brief Constructor.
-    ///
-    /// Construct a new InvertableMap for the graph.
-    ///
-    explicit InvertableMap(const Graph& graph) : Map(graph) {}
-
-    /// \brief Forward iterator for values.
-    ///
-    /// This iterator is an stl compatible forward
-    /// iterator on the values of the map. The values can
-    /// be accessed in the [beginValue, endValue) range.
-    ///
-    class ValueIterator
-      : public std::iterator<std::forward_iterator_tag, Value> {
-      friend class InvertableMap;
-    private:
-      ValueIterator(typename Container::const_iterator _it)
-        : it(_it) {}
-    public:
-
-      ValueIterator() {}
-
-      ValueIterator& operator++() { ++it; return *this; }
-      ValueIterator operator++(int) {
-        ValueIterator tmp(*this);
-        operator++();
-        return tmp;
-      }
-
-      const Value& operator*() const { return it->first; }
-      const Value* operator->() const { return &(it->first); }
-
-      bool operator==(ValueIterator jt) const { return it == jt.it; }
-      bool operator!=(ValueIterator jt) const { return it != jt.it; }
-
-    private:
-      typename Container::const_iterator it;
-    };
-
-    /// \brief Returns an iterator to the first value.
-    ///
-    /// Returns an stl compatible iterator to the
-    /// first value of the map. The values of the
-    /// map can be accessed in the [beginValue, endValue)
-    /// range.
-    ValueIterator beginValue() const {
-      return ValueIterator(_inv_map.begin());
-    }
-
-    /// \brief Returns an iterator after the last value.
-    ///
-    /// Returns an stl compatible iterator after the
-    /// last value of the map. The values of the
-    /// map can be accessed in the [beginValue, endValue)
-    /// range.
-    ValueIterator endValue() const {
-      return ValueIterator(_inv_map.end());
-    }
-
-    /// \brief The setter function of the map.
-    ///
-    /// Sets the mapped value.
-    void set(const Key& key, const Value& val) {
-      Value oldval = Map::operator[](key);
-      typename Container::iterator it = _inv_map.find(oldval);
-      if (it != _inv_map.end() && it->second == key) {
-        _inv_map.erase(it);
-      }
-      _inv_map.insert(make_pair(val, key));
-      Map::set(key, val);
-    }
-
-    /// \brief The getter function of the map.
-    ///
-    /// It gives back the value associated with the key.
-    typename MapTraits<Map>::ConstReturnValue
-    operator[](const Key& key) const {
-      return Map::operator[](key);
-    }
-
-    /// \brief Gives back the item by its value.
-    ///
-    /// Gives back the item by its value.
-    Key operator()(const Value& key) const {
-      typename Container::const_iterator it = _inv_map.find(key);
-      return it != _inv_map.end() ? it->second : INVALID;
-    }
-
-  protected:
-
-    /// \brief Erase the key from the map.
-    ///
-    /// Erase the key to the map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void erase(const Key& key) {
-      Value val = Map::operator[](key);
-      typename Container::iterator it = _inv_map.find(val);
-      if (it != _inv_map.end() && it->second == key) {
-        _inv_map.erase(it);
-      }
-      Map::erase(key);
-    }
-
-    /// \brief Erase more keys from the map.
-    ///
-    /// Erase more keys from the map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void erase(const std::vector<Key>& keys) {
-      for (int i = 0; i < int(keys.size()); ++i) {
-        Value val = Map::operator[](keys[i]);
-        typename Container::iterator it = _inv_map.find(val);
-        if (it != _inv_map.end() && it->second == keys[i]) {
-          _inv_map.erase(it);
-        }
-      }
-      Map::erase(keys);
-    }
-
-    /// \brief Clear the keys from the map and inverse map.
-    ///
-    /// Clear the keys from the map and inverse map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void clear() {
-      _inv_map.clear();
-      Map::clear();
-    }
-
-  public:
-
-    /// \brief The inverse map type.
-    ///
-    /// The inverse of this map. The subscript operator of the map
-    /// gives back always the item what was last assigned to the value.
-    class InverseMap {
-    public:
-      /// \brief Constructor of the InverseMap.
-      ///
-      /// Constructor of the InverseMap.
-      explicit InverseMap(const InvertableMap& inverted)
-        : _inverted(inverted) {}
-
-      /// The value type of the InverseMap.
-      typedef typename InvertableMap::Key Value;
-      /// The key type of the InverseMap.
-      typedef typename InvertableMap::Value Key;
-
-      /// \brief Subscript operator.
-      ///
-      /// Subscript operator. It gives back always the item
-      /// what was last assigned to the value.
-      Value operator[](const Key& key) const {
-        return _inverted(key);
-      }
-
-    private:
-      const InvertableMap& _inverted;
-    };
-
-    /// \brief It gives back the just readable inverse map.
-    ///
-    /// It gives back the just readable inverse map.
-    InverseMap inverse() const {
-      return InverseMap(*this);
-    }
-
-
-
-  };
-
-  /// \brief Provides a mutable, continuous and unique descriptor for each
-  /// item in the graph.
-  ///
-  /// The DescriptorMap class provides a unique and continuous (but mutable)
-  /// descriptor (id) for each item of the same type (e.g. node) in the
-  /// graph. This id is <ul><li>\b unique: different items (nodes) get
-  /// different ids <li>\b continuous: the range of the ids is the set of
-  /// integers between 0 and \c n-1, where \c n is the number of the items of
-  /// this type (e.g. nodes) (so the id of a node can change if you delete an
-  /// other node, i.e. this id is mutable).  </ul> This map can be inverted
-  /// with its member class \c InverseMap, or with the \c operator() member.
-  ///
-  /// \tparam _Graph The graph class the \c DescriptorMap belongs to.
-  /// \tparam _Item The Item is the Key of the Map. It may be Node, Arc or
-  /// Edge.
-  template <typename _Graph, typename _Item>
-  class DescriptorMap : protected DefaultMap<_Graph, _Item, int> {
-
-    typedef _Item Item;
-    typedef DefaultMap<_Graph, _Item, int> Map;
-
-  public:
-    /// The graph class of DescriptorMap.
-    typedef _Graph Graph;
-
-    /// The key type of DescriptorMap (Node, Arc, Edge).
-    typedef typename Map::Key Key;
-    /// The value type of DescriptorMap.
-    typedef typename Map::Value Value;
-
-    /// \brief Constructor.
-    ///
-    /// Constructor for descriptor map.
-    explicit DescriptorMap(const Graph& _graph) : Map(_graph) {
-      Item it;
-      const typename Map::Notifier* nf = Map::notifier();
-      for (nf->first(it); it != INVALID; nf->next(it)) {
-        Map::set(it, _inv_map.size());
-        _inv_map.push_back(it);
-      }
-    }
-
-  protected:
-
-    /// \brief Add a new key to the map.
-    ///
-    /// Add a new key to the map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void add(const Item& item) {
-      Map::add(item);
-      Map::set(item, _inv_map.size());
-      _inv_map.push_back(item);
-    }
-
-    /// \brief Add more new keys to the map.
-    ///
-    /// Add more new keys to the map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void add(const std::vector<Item>& items) {
-      Map::add(items);
-      for (int i = 0; i < int(items.size()); ++i) {
-        Map::set(items[i], _inv_map.size());
-        _inv_map.push_back(items[i]);
-      }
-    }
-
-    /// \brief Erase the key from the map.
-    ///
-    /// Erase the key from the map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void erase(const Item& item) {
-      Map::set(_inv_map.back(), Map::operator[](item));
-      _inv_map[Map::operator[](item)] = _inv_map.back();
-      _inv_map.pop_back();
-      Map::erase(item);
-    }
-
-    /// \brief Erase more keys from the map.
-    ///
-    /// Erase more keys from the map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void erase(const std::vector<Item>& items) {
-      for (int i = 0; i < int(items.size()); ++i) {
-        Map::set(_inv_map.back(), Map::operator[](items[i]));
-        _inv_map[Map::operator[](items[i])] = _inv_map.back();
-        _inv_map.pop_back();
-      }
-      Map::erase(items);
-    }
-
-    /// \brief Build the unique map.
-    ///
-    /// Build the unique map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void build() {
-      Map::build();
-      Item it;
-      const typename Map::Notifier* nf = Map::notifier();
-      for (nf->first(it); it != INVALID; nf->next(it)) {
-        Map::set(it, _inv_map.size());
-        _inv_map.push_back(it);
-      }
-    }
-
-    /// \brief Clear the keys from the map.
-    ///
-    /// Clear the keys from the map. It is called by the
-    /// \c AlterationNotifier.
-    virtual void clear() {
-      _inv_map.clear();
-      Map::clear();
-    }
-
-  public:
-
-    /// \brief Returns the maximal value plus one.
-    ///
-    /// Returns the maximal value plus one in the map.
-    unsigned int size() const {
-      return _inv_map.size();
-    }
-
-    /// \brief Swaps the position of the two items in the map.
-    ///
-    /// Swaps the position of the two items in the map.
-    void swap(const Item& p, const Item& q) {
-      int pi = Map::operator[](p);
-      int qi = Map::operator[](q);
-      Map::set(p, qi);
-      _inv_map[qi] = p;
-      Map::set(q, pi);
-      _inv_map[pi] = q;
-    }
-
-    /// \brief Gives back the \e descriptor of the item.
-    ///
-    /// Gives back the mutable and unique \e descriptor of the map.
-    int operator[](const Item& item) const {
-      return Map::operator[](item);
-    }
-
-    /// \brief Gives back the item by its descriptor.
-    ///
-    /// Gives back th item by its descriptor.
-    Item operator()(int id) const {
-      return _inv_map[id];
-    }
-
-  private:
-
-    typedef std::vector<Item> Container;
-    Container _inv_map;
-
-  public:
-    /// \brief The inverse map type of DescriptorMap.
-    ///
-    /// The inverse map type of DescriptorMap.
-    class InverseMap {
-    public:
-      /// \brief Constructor of the InverseMap.
-      ///
-      /// Constructor of the InverseMap.
-      explicit InverseMap(const DescriptorMap& inverted)
-        : _inverted(inverted) {}
-
-
-      /// The value type of the InverseMap.
-      typedef typename DescriptorMap::Key Value;
-      /// The key type of the InverseMap.
-      typedef typename DescriptorMap::Value Key;
-
-      /// \brief Subscript operator.
-      ///
-      /// Subscript operator. It gives back the item
-      /// that the descriptor belongs to currently.
-      Value operator[](const Key& key) const {
-        return _inverted(key);
-      }
-
-      /// \brief Size of the map.
-      ///
-      /// Returns the size of the map.
-      unsigned int size() const {
-        return _inverted.size();
-      }
-
-    private:
-      const DescriptorMap& _inverted;
-    };
-
-    /// \brief Gives back the inverse of the map.
-    ///
-    /// Gives back the inverse of the map.
-    const InverseMap inverse() const {
-      return InverseMap(*this);
-    }
-  };
-
-  /// \brief Returns the source of the given arc.
-  ///
-  /// The SourceMap gives back the source Node of the given arc.
-  /// \see TargetMap
-  template <typename Digraph>
-  class SourceMap {
-  public:
-
-    typedef typename Digraph::Node Value;
-    typedef typename Digraph::Arc Key;
-
-    /// \brief Constructor
-    ///
-    /// Constructor
-    /// \param _digraph The digraph that the map belongs to.
-    explicit SourceMap(const Digraph& digraph) : _digraph(digraph) {}
-
-    /// \brief The subscript operator.
-    ///
-    /// The subscript operator.
-    /// \param arc The arc
-    /// \return The source of the arc
-    Value operator[](const Key& arc) const {
-      return _digraph.source(arc);
-    }
-
-  private:
-    const Digraph& _digraph;
-  };
-
-  /// \brief Returns a \ref SourceMap class.
-  ///
-  /// This function just returns an \ref SourceMap class.
-  /// \relates SourceMap
-  template <typename Digraph>
-  inline SourceMap<Digraph> sourceMap(const Digraph& digraph) {
-    return SourceMap<Digraph>(digraph);
-  }
-
-  /// \brief Returns the target of the given arc.
-  ///
-  /// The TargetMap gives back the target Node of the given arc.
-  /// \see SourceMap
-  template <typename Digraph>
-  class TargetMap {
-  public:
-
-    typedef typename Digraph::Node Value;
-    typedef typename Digraph::Arc Key;
-
-    /// \brief Constructor
-    ///
-    /// Constructor
-    /// \param _digraph The digraph that the map belongs to.
-    explicit TargetMap(const Digraph& digraph) : _digraph(digraph) {}
-
-    /// \brief The subscript operator.
-    ///
-    /// The subscript operator.
-    /// \param e The arc
-    /// \return The target of the arc
-    Value operator[](const Key& e) const {
-      return _digraph.target(e);
-    }
-
-  private:
-    const Digraph& _digraph;
-  };
-
-  /// \brief Returns a \ref TargetMap class.
-  ///
-  /// This function just returns a \ref TargetMap class.
-  /// \relates TargetMap
-  template <typename Digraph>
-  inline TargetMap<Digraph> targetMap(const Digraph& digraph) {
-    return TargetMap<Digraph>(digraph);
-  }
-
-  /// \brief Returns the "forward" directed arc view of an edge.
-  ///
-  /// Returns the "forward" directed arc view of an edge.
-  /// \see BackwardMap
-  template <typename Graph>
-  class ForwardMap {
-  public:
-
-    typedef typename Graph::Arc Value;
-    typedef typename Graph::Edge Key;
-
-    /// \brief Constructor
-    ///
-    /// Constructor
-    /// \param _graph The graph that the map belongs to.
-    explicit ForwardMap(const Graph& graph) : _graph(graph) {}
-
-    /// \brief The subscript operator.
-    ///
-    /// The subscript operator.
-    /// \param key An edge
-    /// \return The "forward" directed arc view of edge
-    Value operator[](const Key& key) const {
-      return _graph.direct(key, true);
-    }
-
-  private:
-    const Graph& _graph;
-  };
-
-  /// \brief Returns a \ref ForwardMap class.
-  ///
-  /// This function just returns an \ref ForwardMap class.
-  /// \relates ForwardMap
-  template <typename Graph>
-  inline ForwardMap<Graph> forwardMap(const Graph& graph) {
-    return ForwardMap<Graph>(graph);
-  }
-
-  /// \brief Returns the "backward" directed arc view of an edge.
-  ///
-  /// Returns the "backward" directed arc view of an edge.
-  /// \see ForwardMap
-  template <typename Graph>
-  class BackwardMap {
-  public:
-
-    typedef typename Graph::Arc Value;
-    typedef typename Graph::Edge Key;
-
-    /// \brief Constructor
-    ///
-    /// Constructor
-    /// \param _graph The graph that the map belongs to.
-    explicit BackwardMap(const Graph& graph) : _graph(graph) {}
-
-    /// \brief The subscript operator.
-    ///
-    /// The subscript operator.
-    /// \param key An edge
-    /// \return The "backward" directed arc view of edge
-    Value operator[](const Key& key) const {
-      return _graph.direct(key, false);
-    }
-
-  private:
-    const Graph& _graph;
-  };
-
-  /// \brief Returns a \ref BackwardMap class
-
-  /// This function just returns a \ref BackwardMap class.
-  /// \relates BackwardMap
-  template <typename Graph>
-  inline BackwardMap<Graph> backwardMap(const Graph& graph) {
-    return BackwardMap<Graph>(graph);
-  }
-
-  /// \brief Potential difference map
-  ///
-  /// If there is an potential map on the nodes then we
-  /// can get an arc map as we get the substraction of the
-  /// values of the target and source.
-  template <typename Digraph, typename NodeMap>
-  class PotentialDifferenceMap {
-  public:
-    typedef typename Digraph::Arc Key;
-    typedef typename NodeMap::Value Value;
-
-    /// \brief Constructor
-    ///
-    /// Contructor of the map
-    explicit PotentialDifferenceMap(const Digraph& digraph,
-                                    const NodeMap& potential)
-      : _digraph(digraph), _potential(potential) {}
-
-    /// \brief Const subscription operator
-    ///
-    /// Const subscription operator
-    Value operator[](const Key& arc) const {
-      return _potential[_digraph.target(arc)] -
-        _potential[_digraph.source(arc)];
-    }
-
-  private:
-    const Digraph& _digraph;
-    const NodeMap& _potential;
-  };
-
-  /// \brief Returns a PotentialDifferenceMap.
-  ///
-  /// This function just returns a PotentialDifferenceMap.
-  /// \relates PotentialDifferenceMap
-  template <typename Digraph, typename NodeMap>
-  PotentialDifferenceMap<Digraph, NodeMap>
-  potentialDifferenceMap(const Digraph& digraph, const NodeMap& potential) {
-    return PotentialDifferenceMap<Digraph, NodeMap>(digraph, potential);
-  }
-
-  /// \brief Map of the node in-degrees.
-  ///
-  /// This map returns the in-degree of a node. Once it is constructed,
-  /// the degrees are stored in a standard NodeMap, so each query is done
-  /// in constant time. On the other hand, the values are updated automatically
-  /// whenever the digraph changes.
-  ///
-  /// \warning Besides addNode() and addArc(), a digraph structure may provide
-  /// alternative ways to modify the digraph. The correct behavior of InDegMap
-  /// is not guarantied if these additional features are used. For example
-  /// the functions \ref ListDigraph::changeSource() "changeSource()",
-  /// \ref ListDigraph::changeTarget() "changeTarget()" and
-  /// \ref ListDigraph::reverseArc() "reverseArc()"
-  /// of \ref ListDigraph will \e not update the degree values correctly.
-  ///
-  /// \sa OutDegMap
-
-  template <typename _Digraph>
-  class InDegMap
-    : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>
-      ::ItemNotifier::ObserverBase {
-
-  public:
-
-    typedef _Digraph Digraph;
-    typedef int Value;
-    typedef typename Digraph::Node Key;
-
-    typedef typename ItemSetTraits<Digraph, typename Digraph::Arc>
-    ::ItemNotifier::ObserverBase Parent;
-
-  private:
-
-    class AutoNodeMap : public DefaultMap<Digraph, Key, int> {
-    public:
-
-      typedef DefaultMap<Digraph, Key, int> Parent;
-
-      AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}
-
-      virtual void add(const Key& key) {
-        Parent::add(key);
-        Parent::set(key, 0);
-      }
-
-      virtual void add(const std::vector<Key>& keys) {
-        Parent::add(keys);
-        for (int i = 0; i < int(keys.size()); ++i) {
-          Parent::set(keys[i], 0);
-        }
-      }
-
-      virtual void build() {
-        Parent::build();
-        Key it;
-        typename Parent::Notifier* nf = Parent::notifier();
-        for (nf->first(it); it != INVALID; nf->next(it)) {
-          Parent::set(it, 0);
-        }
-      }
-    };
-
-  public:
-
-    /// \brief Constructor.
-    ///
-    /// Constructor for creating in-degree map.
-    explicit InDegMap(const Digraph& digraph)
-      : _digraph(digraph), _deg(digraph) {
-      Parent::attach(_digraph.notifier(typename Digraph::Arc()));
-
-      for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {
-        _deg[it] = countInArcs(_digraph, it);
-      }
-    }
-
-    /// Gives back the in-degree of a Node.
-    int operator[](const Key& key) const {
-      return _deg[key];
-    }
-
-  protected:
-
-    typedef typename Digraph::Arc Arc;
-
-    virtual void add(const Arc& arc) {
-      ++_deg[_digraph.target(arc)];
-    }
-
-    virtual void add(const std::vector<Arc>& arcs) {
-      for (int i = 0; i < int(arcs.size()); ++i) {
-        ++_deg[_digraph.target(arcs[i])];
-      }
-    }
-
-    virtual void erase(const Arc& arc) {
-      --_deg[_digraph.target(arc)];
-    }
-
-    virtual void erase(const std::vector<Arc>& arcs) {
-      for (int i = 0; i < int(arcs.size()); ++i) {
-        --_deg[_digraph.target(arcs[i])];
-      }
-    }
-
-    virtual void build() {
-      for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {
-        _deg[it] = countInArcs(_digraph, it);
-      }
-    }
-
-    virtual void clear() {
-      for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {
-        _deg[it] = 0;
-      }
-    }
-  private:
-
-    const Digraph& _digraph;
-    AutoNodeMap _deg;
-  };
-
-  /// \brief Map of the node out-degrees.
-  ///
-  /// This map returns the out-degree of a node. Once it is constructed,
-  /// the degrees are stored in a standard NodeMap, so each query is done
-  /// in constant time. On the other hand, the values are updated automatically
-  /// whenever the digraph changes.
-  ///
-  /// \warning Besides addNode() and addArc(), a digraph structure may provide
-  /// alternative ways to modify the digraph. The correct behavior of OutDegMap
-  /// is not guarantied if these additional features are used. For example
-  /// the functions \ref ListDigraph::changeSource() "changeSource()",
-  /// \ref ListDigraph::changeTarget() "changeTarget()" and
-  /// \ref ListDigraph::reverseArc() "reverseArc()"
-  /// of \ref ListDigraph will \e not update the degree values correctly.
-  ///
-  /// \sa InDegMap
-
-  template <typename _Digraph>
-  class OutDegMap
-    : protected ItemSetTraits<_Digraph, typename _Digraph::Arc>
-      ::ItemNotifier::ObserverBase {
-
-  public:
-
-    typedef _Digraph Digraph;
-    typedef int Value;
-    typedef typename Digraph::Node Key;
-
-    typedef typename ItemSetTraits<Digraph, typename Digraph::Arc>
-    ::ItemNotifier::ObserverBase Parent;
-
-  private:
-
-    class AutoNodeMap : public DefaultMap<Digraph, Key, int> {
-    public:
-
-      typedef DefaultMap<Digraph, Key, int> Parent;
-
-      AutoNodeMap(const Digraph& digraph) : Parent(digraph, 0) {}
-
-      virtual void add(const Key& key) {
-        Parent::add(key);
-        Parent::set(key, 0);
-      }
-      virtual void add(const std::vector<Key>& keys) {
-        Parent::add(keys);
-        for (int i = 0; i < int(keys.size()); ++i) {
-          Parent::set(keys[i], 0);
-        }
-      }
-      virtual void build() {
-        Parent::build();
-        Key it;
-        typename Parent::Notifier* nf = Parent::notifier();
-        for (nf->first(it); it != INVALID; nf->next(it)) {
-          Parent::set(it, 0);
-        }
-      }
-    };
-
-  public:
-
-    /// \brief Constructor.
-    ///
-    /// Constructor for creating out-degree map.
-    explicit OutDegMap(const Digraph& digraph)
-      : _digraph(digraph), _deg(digraph) {
-      Parent::attach(_digraph.notifier(typename Digraph::Arc()));
-
-      for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {
-        _deg[it] = countOutArcs(_digraph, it);
-      }
-    }
-
-    /// Gives back the out-degree of a Node.
-    int operator[](const Key& key) const {
-      return _deg[key];
-    }
-
-  protected:
-
-    typedef typename Digraph::Arc Arc;
-
-    virtual void add(const Arc& arc) {
-      ++_deg[_digraph.source(arc)];
-    }
-
-    virtual void add(const std::vector<Arc>& arcs) {
-      for (int i = 0; i < int(arcs.size()); ++i) {
-        ++_deg[_digraph.source(arcs[i])];
-      }
-    }
-
-    virtual void erase(const Arc& arc) {
-      --_deg[_digraph.source(arc)];
-    }
-
-    virtual void erase(const std::vector<Arc>& arcs) {
-      for (int i = 0; i < int(arcs.size()); ++i) {
-        --_deg[_digraph.source(arcs[i])];
-      }
-    }
-
-    virtual void build() {
-      for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {
-        _deg[it] = countOutArcs(_digraph, it);
-      }
-    }
-
-    virtual void clear() {
-      for(typename Digraph::NodeIt it(_digraph); it != INVALID; ++it) {
-        _deg[it] = 0;
-      }
-    }
-  private:
-
-    const Digraph& _digraph;
-    AutoNodeMap _deg;
-  };
-
-
-  ///Dynamic arc look up between given endpoints.
-
-  ///\ingroup gutils
-  ///Using this class, you can find an arc in a digraph from a given
-  ///source to a given target in amortized time <em>O(log d)</em>,
-  ///where <em>d</em> is the out-degree of the source node.
-  ///
-  ///It is possible to find \e all parallel arcs between two nodes with
-  ///the \c findFirst() and \c findNext() members.
-  ///
-  ///See the \ref ArcLookUp and \ref AllArcLookUp classes if your
-  ///digraph is not changed so frequently.
-  ///
-  ///This class uses a self-adjusting binary search tree, Sleator's
-  ///and Tarjan's Splay tree for guarantee the logarithmic amortized
-  ///time bound for arc lookups. This class also guarantees the
-  ///optimal time bound in a constant factor for any distribution of
-  ///queries.
-  ///
-  ///\tparam G The type of the underlying digraph.
-  ///
-  ///\sa ArcLookUp
-  ///\sa AllArcLookUp
-  template<class G>
-  class DynArcLookUp
-    : protected ItemSetTraits<G, typename G::Arc>::ItemNotifier::ObserverBase
-  {
-  public:
-    typedef typename ItemSetTraits<G, typename G::Arc>
-    ::ItemNotifier::ObserverBase Parent;
-
-    TEMPLATE_DIGRAPH_TYPEDEFS(G);
-    typedef G Digraph;
-
-  protected:
-
-    class AutoNodeMap : public DefaultMap<G, Node, Arc> {
-    public:
-
-      typedef DefaultMap<G, Node, Arc> Parent;
-
-      AutoNodeMap(const G& digraph) : Parent(digraph, INVALID) {}
-
-      virtual void add(const Node& node) {
-        Parent::add(node);
-        Parent::set(node, INVALID);
-      }
-
-      virtual void add(const std::vector<Node>& nodes) {
-        Parent::add(nodes);
-        for (int i = 0; i < int(nodes.size()); ++i) {
-          Parent::set(nodes[i], INVALID);
-        }
-      }
-
-      virtual void build() {
-        Parent::build();
-        Node it;
-        typename Parent::Notifier* nf = Parent::notifier();
-        for (nf->first(it); it != INVALID; nf->next(it)) {
-          Parent::set(it, INVALID);
-        }
-      }
-    };
-
-    const Digraph &_g;
-    AutoNodeMap _head;
-    typename Digraph::template ArcMap<Arc> _parent;
-    typename Digraph::template ArcMap<Arc> _left;
-    typename Digraph::template ArcMap<Arc> _right;
-
-    class ArcLess {
-      const Digraph &g;
-    public:
-      ArcLess(const Digraph &_g) : g(_g) {}
-      bool operator()(Arc a,Arc b) const
-      {
-        return g.target(a)<g.target(b);
-      }
-    };
-
-  public:
-
-    ///Constructor
-
-    ///Constructor.
-    ///
-    ///It builds up the search database.
-    DynArcLookUp(const Digraph &g)
-      : _g(g),_head(g),_parent(g),_left(g),_right(g)
-    {
-      Parent::attach(_g.notifier(typename Digraph::Arc()));
-      refresh();
-    }
-
-  protected:
-
-    virtual void add(const Arc& arc) {
-      insert(arc);
-    }
-
-    virtual void add(const std::vector<Arc>& arcs) {
-      for (int i = 0; i < int(arcs.size()); ++i) {
-        insert(arcs[i]);
-      }
-    }
-
-    virtual void erase(const Arc& arc) {
-      remove(arc);
-    }
-
-    virtual void erase(const std::vector<Arc>& arcs) {
-      for (int i = 0; i < int(arcs.size()); ++i) {
-        remove(arcs[i]);
-      }
-    }
-
-    virtual void build() {
-      refresh();
-    }
-
-    virtual void clear() {
-      for(NodeIt n(_g);n!=INVALID;++n) {
-        _head.set(n, INVALID);
-      }
-    }
-
-    void insert(Arc arc) {
-      Node s = _g.source(arc);
-      Node t = _g.target(arc);
-      _left.set(arc, INVALID);
-      _right.set(arc, INVALID);
-
-      Arc e = _head[s];
-      if (e == INVALID) {
-        _head.set(s, arc);
-        _parent.set(arc, INVALID);
-        return;
-      }
-      while (true) {
-        if (t < _g.target(e)) {
-          if (_left[e] == INVALID) {
-            _left.set(e, arc);
-            _parent.set(arc, e);
-            splay(arc);
-            return;
-          } else {
-            e = _left[e];
-          }
-        } else {
-          if (_right[e] == INVALID) {
-            _right.set(e, arc);
-            _parent.set(arc, e);
-            splay(arc);
-            return;
-          } else {
-            e = _right[e];
-          }
-        }
-      }
-    }
-
-    void remove(Arc arc) {
-      if (_left[arc] == INVALID) {
-        if (_right[arc] != INVALID) {
-          _parent.set(_right[arc], _parent[arc]);
-        }
-        if (_parent[arc] != INVALID) {
-          if (_left[_parent[arc]] == arc) {
-            _left.set(_parent[arc], _right[arc]);
-          } else {
-            _right.set(_parent[arc], _right[arc]);
-          }
-        } else {
-          _head.set(_g.source(arc), _right[arc]);
-        }
-      } else if (_right[arc] == INVALID) {
-        _parent.set(_left[arc], _parent[arc]);
-        if (_parent[arc] != INVALID) {
-          if (_left[_parent[arc]] == arc) {
-            _left.set(_parent[arc], _left[arc]);
-          } else {
-            _right.set(_parent[arc], _left[arc]);
-          }
-        } else {
-          _head.set(_g.source(arc), _left[arc]);
-        }
-      } else {
-        Arc e = _left[arc];
-        if (_right[e] != INVALID) {
-          e = _right[e];
-          while (_right[e] != INVALID) {
-            e = _right[e];
-          }
-          Arc s = _parent[e];
-          _right.set(_parent[e], _left[e]);
-          if (_left[e] != INVALID) {
-            _parent.set(_left[e], _parent[e]);
-          }
-
-          _left.set(e, _left[arc]);
-          _parent.set(_left[arc], e);
-          _right.set(e, _right[arc]);
-          _parent.set(_right[arc], e);
-
-          _parent.set(e, _parent[arc]);
-          if (_parent[arc] != INVALID) {
-            if (_left[_parent[arc]] == arc) {
-              _left.set(_parent[arc], e);
-            } else {
-              _right.set(_parent[arc], e);
-            }
-          }
-          splay(s);
-        } else {
-          _right.set(e, _right[arc]);
-          _parent.set(_right[arc], e);
-
-          if (_parent[arc] != INVALID) {
-            if (_left[_parent[arc]] == arc) {
-              _left.set(_parent[arc], e);
-            } else {
-              _right.set(_parent[arc], e);
-            }
-          } else {
-            _head.set(_g.source(arc), e);
-          }
-        }
-      }
-    }
-
-    Arc refreshRec(std::vector<Arc> &v,int a,int b)
-    {
-      int m=(a+b)/2;
-      Arc me=v[m];
-      if (a < m) {
-        Arc left = refreshRec(v,a,m-1);
-        _left.set(me, left);
-        _parent.set(left, me);
-      } else {
-        _left.set(me, INVALID);
-      }
-      if (m < b) {
-        Arc right = refreshRec(v,m+1,b);
-        _right.set(me, right);
-        _parent.set(right, me);
-      } else {
-        _right.set(me, INVALID);
-      }
-      return me;
-    }
-
-    void refresh() {
-      for(NodeIt n(_g);n!=INVALID;++n) {
-        std::vector<Arc> v;
-        for(OutArcIt e(_g,n);e!=INVALID;++e) v.push_back(e);
-        if(v.size()) {
-          std::sort(v.begin(),v.end(),ArcLess(_g));
-          Arc head = refreshRec(v,0,v.size()-1);
-          _head.set(n, head);
-          _parent.set(head, INVALID);
-        }
-        else _head.set(n, INVALID);
-      }
-    }
-
-    void zig(Arc v) {
-      Arc w = _parent[v];
-      _parent.set(v, _parent[w]);
-      _parent.set(w, v);
-      _left.set(w, _right[v]);
-      _right.set(v, w);
-      if (_parent[v] != INVALID) {
-        if (_right[_parent[v]] == w) {
-          _right.set(_parent[v], v);
-        } else {
-          _left.set(_parent[v], v);
-        }
-      }
-      if (_left[w] != INVALID){
-        _parent.set(_left[w], w);
-      }
-    }
-
-    void zag(Arc v) {
-      Arc w = _parent[v];
-      _parent.set(v, _parent[w]);
-      _parent.set(w, v);
-      _right.set(w, _left[v]);
-      _left.set(v, w);
-      if (_parent[v] != INVALID){
-        if (_left[_parent[v]] == w) {
-          _left.set(_parent[v], v);
-        } else {
-          _right.set(_parent[v], v);
-        }
-      }
-      if (_right[w] != INVALID){
-        _parent.set(_right[w], w);
-      }
-    }
-
-    void splay(Arc v) {
-      while (_parent[v] != INVALID) {
-        if (v == _left[_parent[v]]) {
-          if (_parent[_parent[v]] == INVALID) {
-            zig(v);
-          } else {
-            if (_parent[v] == _left[_parent[_parent[v]]]) {
-              zig(_parent[v]);
-              zig(v);
-            } else {
-              zig(v);
-              zag(v);
-            }
-          }
-        } else {
-          if (_parent[_parent[v]] == INVALID) {
-            zag(v);
-          } else {
-            if (_parent[v] == _left[_parent[_parent[v]]]) {
-              zag(v);
-              zig(v);
-            } else {
-              zag(_parent[v]);
-              zag(v);
-            }
-          }
-        }
-      }
-      _head[_g.source(v)] = v;
-    }
-
-
-  public:
-
-    ///Find an arc between two nodes.
-
-    ///Find an arc between two nodes in time <em>O(</em>log<em>d)</em>, where
-    /// <em>d</em> is the number of outgoing arcs of \c s.
-    ///\param s The source node
-    ///\param t The target node
-    ///\return An arc from \c s to \c t if there exists,
-    ///\ref INVALID otherwise.
-    Arc operator()(Node s, Node t) const
-    {
-      Arc a = _head[s];
-      while (true) {
-        if (_g.target(a) == t) {
-          const_cast<DynArcLookUp&>(*this).splay(a);
-          return a;
-        } else if (t < _g.target(a)) {
-          if (_left[a] == INVALID) {
-            const_cast<DynArcLookUp&>(*this).splay(a);
-            return INVALID;
-          } else {
-            a = _left[a];
-          }
-        } else  {
-          if (_right[a] == INVALID) {
-            const_cast<DynArcLookUp&>(*this).splay(a);
-            return INVALID;
-          } else {
-            a = _right[a];
-          }
-        }
-      }
-    }
-
-    ///Find the first arc between two nodes.
-
-    ///Find the first arc between two nodes in time
-    /// <em>O(</em>log<em>d)</em>, where <em>d</em> is the number of
-    /// outgoing arcs of \c s.
-    ///\param s The source node
-    ///\param t The target node
-    ///\return An arc from \c s to \c t if there exists, \ref INVALID
-    /// otherwise.
-    Arc findFirst(Node s, Node t) const
-    {
-      Arc a = _head[s];
-      Arc r = INVALID;
-      while (true) {
-        if (_g.target(a) < t) {
-          if (_right[a] == INVALID) {
-            const_cast<DynArcLookUp&>(*this).splay(a);
-            return r;
-          } else {
-            a = _right[a];
-          }
-        } else {
-          if (_g.target(a) == t) {
-            r = a;
-          }
-          if (_left[a] == INVALID) {
-            const_cast<DynArcLookUp&>(*this).splay(a);
-            return r;
-          } else {
-            a = _left[a];
-          }
-        }
-      }
-    }
-
-    ///Find the next arc between two nodes.
-
-    ///Find the next arc between two nodes in time
-    /// <em>O(</em>log<em>d)</em>, where <em>d</em> is the number of
-    /// outgoing arcs of \c s.
-    ///\param s The source node
-    ///\param t The target node
-    ///\return An arc from \c s to \c t if there exists, \ref INVALID
-    /// otherwise.
-
-    ///\note If \c e is not the result of the previous \c findFirst()
-    ///operation then the amorized time bound can not be guaranteed.
-#ifdef DOXYGEN
-    Arc findNext(Node s, Node t, Arc a) const
-#else
-    Arc findNext(Node, Node t, Arc a) const
-#endif
-    {
-      if (_right[a] != INVALID) {
-        a = _right[a];
-        while (_left[a] != INVALID) {
-          a = _left[a];
-        }
-        const_cast<DynArcLookUp&>(*this).splay(a);
-      } else {
-        while (_parent[a] != INVALID && _right[_parent[a]] ==  a) {
-          a = _parent[a];
-        }
-        if (_parent[a] == INVALID) {
-          return INVALID;
-        } else {
-          a = _parent[a];
-          const_cast<DynArcLookUp&>(*this).splay(a);
-        }
-      }
-      if (_g.target(a) == t) return a;
-      else return INVALID;
-    }
-
-  };
-
-  ///Fast arc look up between given endpoints.
-
-  ///\ingroup gutils
-  ///Using this class, you can find an arc in a digraph from a given
-  ///source to a given target in time <em>O(log d)</em>,
-  ///where <em>d</em> is the out-degree of the source node.
-  ///
-  ///It is not possible to find \e all parallel arcs between two nodes.
-  ///Use \ref AllArcLookUp for this purpose.
-  ///
-  ///\warning This class is static, so you should refresh() (or at least
-  ///refresh(Node)) this data structure
-  ///whenever the digraph changes. This is a time consuming (superlinearly
-  ///proportional (<em>O(m</em>log<em>m)</em>) to the number of arcs).
-  ///
-  ///\tparam G The type of the underlying digraph.
-  ///
-  ///\sa DynArcLookUp
-  ///\sa AllArcLookUp
-  template<class G>
-  class ArcLookUp
-  {
-  public:
-    TEMPLATE_DIGRAPH_TYPEDEFS(G);
-    typedef G Digraph;
-
-  protected:
-    const Digraph &_g;
-    typename Digraph::template NodeMap<Arc> _head;
-    typename Digraph::template ArcMap<Arc> _left;
-    typename Digraph::template ArcMap<Arc> _right;
-
-    class ArcLess {
-      const Digraph &g;
-    public:
-      ArcLess(const Digraph &_g) : g(_g) {}
-      bool operator()(Arc a,Arc b) const
-      {
-        return g.target(a)<g.target(b);
-      }
-    };
-
-  public:
-
-    ///Constructor
-
-    ///Constructor.
-    ///
-    ///It builds up the search database, which remains valid until the digraph
-    ///changes.
-    ArcLookUp(const Digraph &g) :_g(g),_head(g),_left(g),_right(g) {refresh();}
-
-  private:
-    Arc refreshRec(std::vector<Arc> &v,int a,int b)
-    {
-      int m=(a+b)/2;
-      Arc me=v[m];
-      _left[me] = a<m?refreshRec(v,a,m-1):INVALID;
-      _right[me] = m<b?refreshRec(v,m+1,b):INVALID;
-      return me;
-    }
-  public:
-    ///Refresh the data structure at a node.
-
-    ///Build up the search database of node \c n.
-    ///
-    ///It runs in time <em>O(d</em>log<em>d)</em>, where <em>d</em> is
-    ///the number of the outgoing arcs of \c n.
-    void refresh(Node n)
-    {
-      std::vector<Arc> v;
-      for(OutArcIt e(_g,n);e!=INVALID;++e) v.push_back(e);
-      if(v.size()) {
-        std::sort(v.begin(),v.end(),ArcLess(_g));
-        _head[n]=refreshRec(v,0,v.size()-1);
-      }
-      else _head[n]=INVALID;
-    }
-    ///Refresh the full data structure.
-
-    ///Build up the full search database. In fact, it simply calls
-    ///\ref refresh(Node) "refresh(n)" for each node \c n.
-    ///
-    ///It runs in time <em>O(m</em>log<em>D)</em>, where <em>m</em> is
-    ///the number of the arcs of \c n and <em>D</em> is the maximum
-    ///out-degree of the digraph.
-
-    void refresh()
-    {
-      for(NodeIt n(_g);n!=INVALID;++n) refresh(n);
-    }
-
-    ///Find an arc between two nodes.
-
-    ///Find an arc between two nodes in time <em>O(</em>log<em>d)</em>, where
-    /// <em>d</em> is the number of outgoing arcs of \c s.
-    ///\param s The source node
-    ///\param t The target node
-    ///\return An arc from \c s to \c t if there exists,
-    ///\ref INVALID otherwise.
-    ///
-    ///\warning If you change the digraph, refresh() must be called before using
-    ///this operator. If you change the outgoing arcs of
-    ///a single node \c n, then
-    ///\ref refresh(Node) "refresh(n)" is enough.
-    ///
-    Arc operator()(Node s, Node t) const
-    {
-      Arc e;
-      for(e=_head[s];
-          e!=INVALID&&_g.target(e)!=t;
-          e = t < _g.target(e)?_left[e]:_right[e]) ;
-      return e;
-    }
-
-  };
-
-  ///Fast look up of all arcs between given endpoints.
-
-  ///\ingroup gutils
-  ///This class is the same as \ref ArcLookUp, with the addition
-  ///that it makes it possible to find all arcs between given endpoints.
-  ///
-  ///\warning This class is static, so you should refresh() (or at least
-  ///refresh(Node)) this data structure
-  ///whenever the digraph changes. This is a time consuming (superlinearly
-  ///proportional (<em>O(m</em>log<em>m)</em>) to the number of arcs).
-  ///
-  ///\tparam G The type of the underlying digraph.
-  ///
-  ///\sa DynArcLookUp
-  ///\sa ArcLookUp
-  template<class G>
-  class AllArcLookUp : public ArcLookUp<G>
-  {
-    using ArcLookUp<G>::_g;
-    using ArcLookUp<G>::_right;
-    using ArcLookUp<G>::_left;
-    using ArcLookUp<G>::_head;
-
-    TEMPLATE_DIGRAPH_TYPEDEFS(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