text: /* -*- mode: C++; indent-tabs-mode: nil; -*-
   *
   * This file is a part of LEMON, a generic C++ optimization library.
   *
   * Copyright (C) 2003-2010
   * 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_CORE_H
  #define LEMON_CORE_H
  
  #include <vector>
  #include <algorithm>
  
  #include <lemon/config.h>
  #include <lemon/bits/enable_if.h>
  #include <lemon/bits/traits.h>
  #include <lemon/assert.h>
  
  // Disable the following warnings when compiling with MSVC:
  // C4250: 'class1' : inherits 'class2::member' via dominance
  // C4355: 'this' : used in base member initializer list
  // C4503: 'function' : decorated name length exceeded, name was truncated
  // C4800: 'type' : forcing value to bool 'true' or 'false' (performance warning)
  // C4996: 'function': was declared deprecated
  #ifdef _MSC_VER
  #pragma warning( disable : 4250 4355 4503 4800 4996 )
  #endif
  
  #ifdef __GNUC__
  #define GCC_VERSION (__GNUC__ * 10000                   \
                       + __GNUC_MINOR__ * 100             \
                       + __GNUC_PATCHLEVEL__)
  #endif
  
  #if GCC_VERSION >= 40800
  // Needed by the [DI]GRAPH_TYPEDEFS marcos for gcc 4.8
  #pragma GCC diagnostic ignored "-Wunused-local-typedefs"
  #endif
  
  ///\file
  ///\brief LEMON core utilities.
  ///
  ///This header file contains core utilities for LEMON.
  ///It is automatically included by all graph types, therefore it usually
  ///do not have to be included directly.
  
  namespace lemon {
  
    /// \brief Dummy type to make it easier to create invalid iterators.
    ///
    /// Dummy type to make it easier to create invalid iterators.
    /// See \ref INVALID for the usage.
    struct Invalid {
    public:
      bool operator==(Invalid) { return true;  }
      bool operator!=(Invalid) { return false; }
      bool operator< (Invalid) { return false; }
    };
  
    /// \brief Invalid iterators.
    ///
    /// \ref Invalid is a global type that converts to each iterator
    /// in such a way that the value of the target iterator will be invalid.
  #ifdef LEMON_ONLY_TEMPLATES
    const Invalid INVALID = Invalid();
  #else
    extern const Invalid INVALID;
  #endif
  
    /// \addtogroup gutils
    /// @{
  
    ///Create convenience typedefs for the digraph types and iterators
  
    ///This \c \#define creates convenient type definitions 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
  
    ///Create 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
  
    ///Create convenience typedefs for the graph types and iterators
  
    ///This \c \#define creates the same convenient type definitions 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_GRAPH_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
  
    ///Create 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 a graph.
    ///
    /// This function counts the items (nodes, arcs etc.) in a graph.
    /// The complexity of the function is linear 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 _core_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 <em>O</em>(<em>n</em>), but for some
    /// graph structures it is specialized to run in <em>O</em>(1).
    ///
    /// \note If the graph contains a \c nodeNum() member function and a
    /// \c 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 _core_bits::CountNodesSelector<Graph>::count(g);
    }
  
    // Arc counting:
  
    namespace _core_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 <em>O</em>(<em>m</em>), but for some
    /// graph structures it is specialized to run in <em>O</em>(1).
    ///
    /// \note If the graph contains a \c arcNum() member function and a
    /// \c ArcNumTag 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 _core_bits::CountArcsSelector<Graph>::count(g);
    }
  
    // Edge counting:
  
    namespace _core_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 <em>O</em>(<em>m</em>), but for some
    /// graph structures it is specialized to run in <em>O</em>(1).
    ///
    /// \note If the graph contains a \c edgeNum() member function and a
    /// \c 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 _core_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 \c g.
    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 \c g.
    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 undirected graph \c g.
    template <typename Graph>
    inline int countIncEdges(const Graph& g,  const typename Graph::Node& n) {
      return countNodeDegree<Graph, typename Graph::IncEdgeIt>(g, n);
    }
  
    namespace _core_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 FromMap, typename ToMap>
      class MapCopy : public MapCopyBase<Digraph, Item, RefMap> {
      public:
  
        MapCopy(const FromMap& map, ToMap& tmap)
          : _map(map), _tmap(tmap) {}
  
        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:
        const FromMap& _map;
        ToMap& _tmap;
      };
  
      template <typename Digraph, typename Item, typename RefMap, typename It>
      class ItemCopy : public MapCopyBase<Digraph, Item, RefMap> {
      public:
  
        ItemCopy(const Item& item, It& it) : _item(item), _it(it) {}
  
        virtual void copy(const Digraph&, const RefMap& refMap) {
          _it = refMap[_item];
        }
  
      private:
        Item _item;
        It& _it;
      };
  
      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(const From& from, Digraph &to,
                         NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {
          to.clear();
          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(const From& from, Digraph &to,
                         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(const From& from, Graph &to,
                         NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {
          to.clear();
          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(const From& from, Graph &to,
                         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 digraphCopy() function.
    ///
    /// This class not only make a copy of a digraph, but it can create
    /// references and cross references between the nodes and arcs of
    /// the two digraphs, and it can copy maps to use with the newly created
    /// digraph.
    ///
    /// To make a copy from a digraph, first an instance of DigraphCopy
    /// should be created, then the data belongs to the digraph should
    /// assigned to copy. In the end, the \c run() member should be
    /// called.
    ///
    /// The next code copies a digraph with several data:
    ///\code
    ///  DigraphCopy<OrigGraph, NewGraph> cg(orig_graph, new_graph);
    ///  // Create references for the nodes
    ///  OrigGraph::NodeMap<NewGraph::Node> nr(orig_graph);
    ///  cg.nodeRef(nr);
    ///  // Create cross references (inverse) for the arcs
    ///  NewGraph::ArcMap<OrigGraph::Arc> acr(new_graph);
    ///  cg.arcCrossRef(acr);
    ///  // Copy an arc map
    ///  OrigGraph::ArcMap<double> oamap(orig_graph);
    ///  NewGraph::ArcMap<double> namap(new_graph);
    ///  cg.arcMap(oamap, namap);
    ///  // Copy a node
    ///  OrigGraph::Node on;
    ///  NewGraph::Node nn;
    ///  cg.node(on, nn);
    ///  // Execute copying
    ///  cg.run();
    ///\endcode
    template <typename From, typename To>
    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 of DigraphCopy.
      ///
      /// Constructor of DigraphCopy for copying the content of the
      /// \c from digraph into the \c to digraph.
      DigraphCopy(const From& from, To& to)
        : _from(from), _to(to) {}
  
      /// \brief Destructor of DigraphCopy
      ///
      /// Destructor of 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 Copy the node references into the given map.
      ///
      /// This function copies the node references into the given map.
      /// The parameter should be a map, whose key type is the Node type of
      /// the source digraph, while the value type is the Node type of the
      /// destination digraph.
      template <typename NodeRef>
      DigraphCopy& nodeRef(NodeRef& map) {
        _node_maps.push_back(new _core_bits::RefCopy<From, Node,
                             NodeRefMap, NodeRef>(map));
        return *this;
      }
  
      /// \brief Copy the node cross references into the given map.
      ///
      /// This function copies the node cross references (reverse references)
      /// into the given map. The parameter should be a map, whose key type
      /// is the Node type of the destination digraph, while the value type is
      /// the Node type of the source digraph.
      template <typename NodeCrossRef>
      DigraphCopy& nodeCrossRef(NodeCrossRef& map) {
        _node_maps.push_back(new _core_bits::CrossRefCopy<From, Node,
                             NodeRefMap, NodeCrossRef>(map));
        return *this;
      }
  
      /// \brief Make a copy of the given node map.
      ///
      /// This function makes a copy of the given node map for the newly
      /// created digraph.
      /// The key type of the new map \c tmap should be the Node type of the
      /// destination digraph, and the key type of the original map \c map
      /// should be the Node type of the source digraph.
      template <typename FromMap, typename ToMap>
      DigraphCopy& nodeMap(const FromMap& map, ToMap& tmap) {
        _node_maps.push_back(new _core_bits::MapCopy<From, Node,
                             NodeRefMap, FromMap, ToMap>(map, tmap));
        return *this;
      }
  
      /// \brief Make a copy of the given node.
      ///
      /// This function makes a copy of the given node.
      DigraphCopy& node(const Node& node, TNode& tnode) {
        _node_maps.push_back(new _core_bits::ItemCopy<From, Node,
                             NodeRefMap, TNode>(node, tnode));
        return *this;
      }
  
      /// \brief Copy the arc references into the given map.
      ///
      /// This function copies the arc references into the given map.
      /// The parameter should be a map, whose key type is the Arc type of
      /// the source digraph, while the value type is the Arc type of the
      /// destination digraph.
      template <typename ArcRef>
      DigraphCopy& arcRef(ArcRef& map) {
        _arc_maps.push_back(new _core_bits::RefCopy<From, Arc,
                            ArcRefMap, ArcRef>(map));
        return *this;
      }
  
      /// \brief Copy the arc cross references into the given map.
      ///
      /// This function copies the arc cross references (reverse references)
      /// into the given map. The parameter should be a map, whose key type
      /// is the Arc type of the destination digraph, while the value type is
      /// the Arc type of the source digraph.
      template <typename ArcCrossRef>
      DigraphCopy& arcCrossRef(ArcCrossRef& map) {
        _arc_maps.push_back(new _core_bits::CrossRefCopy<From, Arc,
                            ArcRefMap, ArcCrossRef>(map));
        return *this;
      }
  
      /// \brief Make a copy of the given arc map.
      ///
      /// This function makes a copy of the given arc map for the newly
      /// created digraph.
      /// The key type of the new map \c tmap should be the Arc type of the
      /// destination digraph, and the key type of the original map \c map
      /// should be the Arc type of the source digraph.
      template <typename FromMap, typename ToMap>
      DigraphCopy& arcMap(const FromMap& map, ToMap& tmap) {
        _arc_maps.push_back(new _core_bits::MapCopy<From, Arc,
                            ArcRefMap, FromMap, ToMap>(map, tmap));
        return *this;
      }
  
      /// \brief Make a copy of the given arc.
      ///
      /// This function makes a copy of the given arc.
      DigraphCopy& arc(const Arc& arc, TArc& tarc) {
        _arc_maps.push_back(new _core_bits::ItemCopy<From, Arc,
                            ArcRefMap, TArc>(arc, tarc));
        return *this;
      }
  
      /// \brief Execute copying.
      ///
      /// This function executes the copying of the digraph along with the
      /// copying of the assigned data.
      void run() {
        NodeRefMap nodeRefMap(_from);
        ArcRefMap arcRefMap(_from);
        _core_bits::DigraphCopySelector<To>::
          copy(_from, _to, 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<_core_bits::MapCopyBase<From, Node, NodeRefMap>* >
        _node_maps;
  
      std::vector<_core_bits::MapCopyBase<From, Arc, ArcRefMap>* >
        _arc_maps;
  
    };
  
    /// \brief Copy a digraph to another digraph.
    ///
    /// This function copies a digraph to another digraph.
    /// The complete usage of it is detailed in the DigraphCopy class, but
    /// a short example shows a basic work:
    ///\code
    /// digraphCopy(src, trg).nodeRef(nr).arcCrossRef(acr).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 acr will contain the mapping from the arcs of the \c to digraph
    /// to the arcs of the \c from digraph.
    ///
    /// \see DigraphCopy
    template <typename From, typename To>
    DigraphCopy<From, To> digraphCopy(const From& from, To& to) {
      return DigraphCopy<From, To>(from, to);
    }
  
    /// \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 graphCopy() function.
    ///
    /// This class not only make a copy of a graph, but it can create
    /// references and cross references between the nodes, edges and arcs of
    /// the two graphs, and it can copy maps for using with the newly created
    /// graph.
    ///
    /// 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<OrigGraph, NewGraph> cg(orig_graph, new_graph);
    ///  // Create references for the nodes
    ///  OrigGraph::NodeMap<NewGraph::Node> nr(orig_graph);
    ///  cg.nodeRef(nr);
    ///  // Create cross references (inverse) for the edges
    ///  NewGraph::EdgeMap<OrigGraph::Edge> ecr(new_graph);
    ///  cg.edgeCrossRef(ecr);
    ///  // Copy an edge map
    ///  OrigGraph::EdgeMap<double> oemap(orig_graph);
    ///  NewGraph::EdgeMap<double> nemap(new_graph);
    ///  cg.edgeMap(oemap, nemap);
    ///  // Copy a node
    ///  OrigGraph::Node on;
    ///  NewGraph::Node nn;
    ///  cg.node(on, nn);
    ///  // Execute copying
    ///  cg.run();
    ///\endcode
    template <typename From, typename To>
    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 From& from, const To& to,
                  const EdgeRefMap& edge_ref, const NodeRefMap& node_ref)
          : _from(from), _to(to),
            _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 From& _from;
        const To& _to;
        const EdgeRefMap& _edge_ref;
        const NodeRefMap& _node_ref;
      };
  
    public:
  
      /// \brief Constructor of GraphCopy.
      ///
      /// Constructor of GraphCopy for copying the content of the
      /// \c from graph into the \c to graph.
      GraphCopy(const From& from, To& to)
        : _from(from), _to(to) {}
  
      /// \brief Destructor of GraphCopy
      ///
      /// Destructor of 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 Copy the node references into the given map.
      ///
      /// This function copies the node references into the given map.
      /// The parameter should be a map, whose 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>
      GraphCopy& nodeRef(NodeRef& map) {
        _node_maps.push_back(new _core_bits::RefCopy<From, Node,
                             NodeRefMap, NodeRef>(map));
        return *this;
      }
  
      /// \brief Copy the node cross references into the given map.
      ///
      /// This function copies the node cross references (reverse references)
      /// into the given map. The parameter should be a map, whose 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>
      GraphCopy& nodeCrossRef(NodeCrossRef& map) {
        _node_maps.push_back(new _core_bits::CrossRefCopy<From, Node,
                             NodeRefMap, NodeCrossRef>(map));
        return *this;
      }
  
      /// \brief Make a copy of the given node map.
      ///
      /// This function makes a copy of the given node map for the newly
      /// created graph.
      /// The key type of the new map \c tmap should be the Node type of the
      /// destination graph, and the key type of the original map \c map
      /// should be the Node type of the source graph.
      template <typename FromMap, typename ToMap>
      GraphCopy& nodeMap(const FromMap& map, ToMap& tmap) {
        _node_maps.push_back(new _core_bits::MapCopy<From, Node,
                             NodeRefMap, FromMap, ToMap>(map, tmap));
        return *this;
      }
  
      /// \brief Make a copy of the given node.
      ///
      /// This function makes a copy of the given node.
      GraphCopy& node(const Node& node, TNode& tnode) {
        _node_maps.push_back(new _core_bits::ItemCopy<From, Node,
                             NodeRefMap, TNode>(node, tnode));
        return *this;
      }
  
      /// \brief Copy the arc references into the given map.
      ///
      /// This function copies the arc references into the given map.
      /// The parameter should be a map, whose key type is the Arc type of
      /// the source graph, while the value type is the Arc type of the
      /// destination graph.
      template <typename ArcRef>
      GraphCopy& arcRef(ArcRef& map) {
        _arc_maps.push_back(new _core_bits::RefCopy<From, Arc,
                            ArcRefMap, ArcRef>(map));
        return *this;
      }
  
      /// \brief Copy the arc cross references into the given map.
      ///
      /// This function copies the arc cross references (reverse references)
      /// into the given map. The parameter should be a map, whose key type
      /// is the Arc type of the destination graph, while the value type is
      /// the Arc type of the source graph.
      template <typename ArcCrossRef>
      GraphCopy& arcCrossRef(ArcCrossRef& map) {
        _arc_maps.push_back(new _core_bits::CrossRefCopy<From, Arc,
                            ArcRefMap, ArcCrossRef>(map));
        return *this;
      }
  
      /// \brief Make a copy of the given arc map.
      ///
      /// This function makes a copy of the given arc map for the newly
      /// created graph.
      /// The key type of the new map \c tmap should be the Arc type of the
      /// destination graph, and the key type of the original map \c map
      /// should be the Arc type of the source graph.
      template <typename FromMap, typename ToMap>
      GraphCopy& arcMap(const FromMap& map, ToMap& tmap) {
        _arc_maps.push_back(new _core_bits::MapCopy<From, Arc,
                            ArcRefMap, FromMap, ToMap>(map, tmap));
        return *this;
      }
  
      /// \brief Make a copy of the given arc.
      ///
      /// This function makes a copy of the given arc.
      GraphCopy& arc(const Arc& arc, TArc& tarc) {
        _arc_maps.push_back(new _core_bits::ItemCopy<From, Arc,
                            ArcRefMap, TArc>(arc, tarc));
        return *this;
      }
  
      /// \brief Copy the edge references into the given map.
      ///
      /// This function copies the edge references into the given map.
      /// The parameter should be a map, whose key type is the Edge type of
      /// the source graph, while the value type is the Edge type of the
      /// destination graph.
      template <typename EdgeRef>
      GraphCopy& edgeRef(EdgeRef& map) {
        _edge_maps.push_back(new _core_bits::RefCopy<From, Edge,
                             EdgeRefMap, EdgeRef>(map));
        return *this;
      }
  
      /// \brief Copy the edge cross references into the given map.
      ///
      /// This function copies the edge cross references (reverse references)
      /// into the given map. The parameter should be a map, whose key type
      /// is the Edge type of the destination graph, while the value type is
      /// the Edge type of the source graph.
      template <typename EdgeCrossRef>
      GraphCopy& edgeCrossRef(EdgeCrossRef& map) {
        _edge_maps.push_back(new _core_bits::CrossRefCopy<From,
                             Edge, EdgeRefMap, EdgeCrossRef>(map));
        return *this;
      }
  
      /// \brief Make a copy of the given edge map.
      ///
      /// This function makes a copy of the given edge map for the newly
      /// created graph.
      /// The key type of the new map \c tmap should be the Edge type of the
      /// destination graph, and the key type of the original map \c map
      /// should be the Edge type of the source graph.
      template <typename FromMap, typename ToMap>
      GraphCopy& edgeMap(const FromMap& map, ToMap& tmap) {
        _edge_maps.push_back(new _core_bits::MapCopy<From, Edge,
                             EdgeRefMap, FromMap, ToMap>(map, tmap));
        return *this;
      }
  
      /// \brief Make a copy of the given edge.
      ///
      /// This function makes a copy of the given edge.
      GraphCopy& edge(const Edge& edge, TEdge& tedge) {
        _edge_maps.push_back(new _core_bits::ItemCopy<From, Edge,
                             EdgeRefMap, TEdge>(edge, tedge));
        return *this;
      }
  
      /// \brief Execute copying.
      ///
      /// This function executes the copying of the graph along with the
      /// copying of the assigned data.
      void run() {
        NodeRefMap nodeRefMap(_from);
        EdgeRefMap edgeRefMap(_from);
        ArcRefMap arcRefMap(_from, _to, edgeRefMap, nodeRefMap);
        _core_bits::GraphCopySelector<To>::
          copy(_from, _to, 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<_core_bits::MapCopyBase<From, Node, NodeRefMap>* >
        _node_maps;
  
      std::vector<_core_bits::MapCopyBase<From, Arc, ArcRefMap>* >
        _arc_maps;
  
      std::vector<_core_bits::MapCopyBase<From, Edge, EdgeRefMap>* >
        _edge_maps;
  
    };
  
    /// \brief Copy a graph to another graph.
    ///
    /// This function copies a graph to another graph.
    /// The complete usage of it is detailed in the GraphCopy class,
    /// but a short example shows a basic work:
    ///\code
    /// graphCopy(src, trg).nodeRef(nr).edgeCrossRef(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 edges of the \c to graph
    /// to the edges of the \c from graph.
    ///
    /// \see GraphCopy
    template <typename From, typename To>
    GraphCopy<From, To>
    graphCopy(const From& from, To& to) {
      return GraphCopy<From, To>(from, to);
    }
  
    namespace _core_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::FindArcTag, 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 Find an arc between two nodes of a digraph.
    ///
    /// This function finds an arc from node \c u to node \c v in the
    /// 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
    ///
    /// \note \ref ConArcIt provides iterator interface for the same
    /// functionality.
    ///
    ///\sa ConArcIt
    ///\sa ArcLookUp, AllArcLookUp, DynArcLookUp
    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 _core_bits::FindArcSelector<Graph>::find(g, u, v, prev);
    }
  
    /// \brief Iterator for iterating on parallel arcs connecting the same nodes.
    ///
    /// Iterator for iterating on parallel arcs connecting the same nodes. It is
    /// a higher level interface for the \ref 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, AllArcLookUp, DynArcLookUp
    template <typename GR>
    class ConArcIt : public GR::Arc {
      typedef typename GR::Arc Parent;
  
    public:
  
      typedef typename GR::Arc Arc;
      typedef typename GR::Node Node;
  
      /// \brief Constructor.
      ///
      /// Construct a new ConArcIt iterating on the arcs that
      /// connects nodes \c u and \c v.
      ConArcIt(const GR& g, Node u, Node v) : _graph(g) {
        Parent::operator=(findArc(_graph, u, v));
      }
  
      /// \brief Constructor.
      ///
      /// Construct a new ConArcIt that continues the iterating from arc \c a.
      ConArcIt(const GR& 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 GR& _graph;
    };
  
    namespace _core_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 Find an edge between two nodes of a graph.
    ///
    /// This function finds an edge from node \c u to node \c v in graph \c g.
    /// If 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 edge from \c u to \c v. Otherwise it looks for
    /// the next edge from \c u to \c v after \c prev.
    /// \return The found edge or \ref INVALID if there is no such an edge.
    ///
    /// Thus you can iterate through each edge between \c u and \c v
    /// as it follows.
    ///\code
    /// for(Edge e = findEdge(g,u,v); e != INVALID; e = findEdge(g,u,v,e)) {
    ///   ...
    /// }
    ///\endcode
    ///
    /// \note \ref ConEdgeIt provides iterator interface for the same
    /// functionality.
    ///
    ///\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 _core_bits::FindEdgeSelector<Graph>::find(g, u, v, p);
    }
  
    /// \brief Iterator for iterating on parallel edges connecting the same nodes.
    ///
    /// Iterator for iterating on parallel edges connecting the same nodes.
    /// It is a higher level interface for the findEdge() function. You can
    /// use it the following way:
    ///\code
    /// for (ConEdgeIt<Graph> it(g, u, v); it != INVALID; ++it) {
    ///   ...
    /// }
    ///\endcode
    ///
    ///\sa findEdge()
    template <typename GR>
    class ConEdgeIt : public GR::Edge {
      typedef typename GR::Edge Parent;
  
    public:
  
      typedef typename GR::Edge Edge;
      typedef typename GR::Node Node;
  
      /// \brief Constructor.
      ///
      /// Construct a new ConEdgeIt iterating on the edges that
      /// connects nodes \c u and \c v.
      ConEdgeIt(const GR& g, Node u, Node v) : _graph(g), _u(u), _v(v) {
        Parent::operator=(findEdge(_graph, _u, _v));
      }
  
      /// \brief Constructor.
      ///
      /// Construct a new ConEdgeIt that continues iterating from edge \c e.
      ConEdgeIt(const GR& 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, _u, _v, *this));
        return *this;
      }
    private:
      const GR& _graph;
      Node _u, _v;
    };
  
  
    ///Dynamic arc look-up between given endpoints.
  
    ///Using this class, you can find an arc in a digraph from a given
    ///source to a given target in amortized time <em>O</em>(log<em>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 operator() member.
    ///
    ///This is a dynamic data structure. Consider to use \ref ArcLookUp or
    ///\ref AllArcLookUp if your digraph is not changed so frequently.
    ///
    ///This class uses a self-adjusting binary search tree, the Splay tree
    ///of Sleator and Tarjan to guarantee the logarithmic amortized
    ///time bound for arc look-ups. This class also guarantees the
    ///optimal time bound in a constant factor for any distribution of
    ///queries.
    ///
    ///\tparam GR The type of the underlying digraph.
    ///
    ///\sa ArcLookUp
    ///\sa AllArcLookUp
    template <typename GR>
    class DynArcLookUp
      : protected ItemSetTraits<GR, typename GR::Arc>::ItemNotifier::ObserverBase
    {
      typedef typename ItemSetTraits<GR, typename GR::Arc>
      ::ItemNotifier::ObserverBase Parent;
  
      TEMPLATE_DIGRAPH_TYPEDEFS(GR);
  
    public:
  
      /// The Digraph type
      typedef GR Digraph;
  
    protected:
  
      class AutoNodeMap : public ItemSetTraits<GR, Node>::template Map<Arc>::Type
      {
        typedef typename ItemSetTraits<GR, Node>::template Map<Arc>::Type Parent;
  
      public:
  
        AutoNodeMap(const GR& 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);
          }
        }
      };
  
      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);
        }
      };
  
    protected:
  
      const Digraph &_g;
      AutoNodeMap _head;
      typename Digraph::template ArcMap<Arc> _parent;
      typename Digraph::template ArcMap<Arc> _left;
      typename Digraph::template ArcMap<Arc> _right;
  
    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[n] = INVALID;
        }
      }
  
      void insert(Arc arc) {
        Node s = _g.source(arc);
        Node t = _g.target(arc);
        _left[arc] = INVALID;
        _right[arc] = INVALID;
  
        Arc e = _head[s];
        if (e == INVALID) {
          _head[s] = arc;
          _parent[arc] = INVALID;
          return;
        }
        while (true) {
          if (t < _g.target(e)) {
            if (_left[e] == INVALID) {
              _left[e] = arc;
              _parent[arc] = e;
              splay(arc);
              return;
            } else {
              e = _left[e];
            }
          } else {
            if (_right[e] == INVALID) {
              _right[e] = arc;
              _parent[arc] = e;
              splay(arc);
              return;
            } else {
              e = _right[e];
            }
          }
        }
      }
  
      void remove(Arc arc) {
        if (_left[arc] == INVALID) {
          if (_right[arc] != INVALID) {
            _parent[_right[arc]] = _parent[arc];
          }
          if (_parent[arc] != INVALID) {
            if (_left[_parent[arc]] == arc) {
              _left[_parent[arc]] = _right[arc];
            } else {
              _right[_parent[arc]] = _right[arc];
            }
          } else {
            _head[_g.source(arc)] = _right[arc];
          }
        } else if (_right[arc] == INVALID) {
          _parent[_left[arc]] = _parent[arc];
          if (_parent[arc] != INVALID) {
            if (_left[_parent[arc]] == arc) {
              _left[_parent[arc]] = _left[arc];
            } else {
              _right[_parent[arc]] = _left[arc];
            }
          } else {
            _head[_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[_parent[e]] = _left[e];
            if (_left[e] != INVALID) {
              _parent[_left[e]] = _parent[e];
            }
  
            _left[e] = _left[arc];
            _parent[_left[arc]] = e;
            _right[e] = _right[arc];
            _parent[_right[arc]] = e;
  
            _parent[e] = _parent[arc];
            if (_parent[arc] != INVALID) {
              if (_left[_parent[arc]] == arc) {
                _left[_parent[arc]] = e;
              } else {
                _right[_parent[arc]] = e;
              }
            }
            splay(s);
          } else {
            _right[e] = _right[arc];
            _parent[_right[arc]] = e;
            _parent[e] = _parent[arc];
  
            if (_parent[arc] != INVALID) {
              if (_left[_parent[arc]] == arc) {
                _left[_parent[arc]] = e;
              } else {
                _right[_parent[arc]] = e;
              }
            } else {
              _head[_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[me] = left;
          _parent[left] = me;
        } else {
          _left[me] = INVALID;
        }
        if (m < b) {
          Arc right = refreshRec(v,m+1,b);
          _right[me] = right;
          _parent[right] = me;
        } else {
          _right[me] = INVALID;
        }
        return me;
      }
  
      void refresh() {
        for(NodeIt n(_g);n!=INVALID;++n) {
          std::vector<Arc> v;
          for(OutArcIt a(_g,n);a!=INVALID;++a) v.push_back(a);
          if (!v.empty()) {
            std::sort(v.begin(),v.end(),ArcLess(_g));
            Arc head = refreshRec(v,0,v.size()-1);
            _head[n] = head;
            _parent[head] = INVALID;
          }
          else _head[n] = INVALID;
        }
      }
  
      void zig(Arc v) {
        Arc w = _parent[v];
        _parent[v] = _parent[w];
        _parent[w] = v;
        _left[w] = _right[v];
        _right[v] = w;
        if (_parent[v] != INVALID) {
          if (_right[_parent[v]] == w) {
            _right[_parent[v]] = v;
          } else {
            _left[_parent[v]] = v;
          }
        }
        if (_left[w] != INVALID){
          _parent[_left[w]] = w;
        }
      }
  
      void zag(Arc v) {
        Arc w = _parent[v];
        _parent[v] = _parent[w];
        _parent[w] = v;
        _right[w] = _left[v];
        _left[v] = w;
        if (_parent[v] != INVALID){
          if (_left[_parent[v]] == w) {
            _left[_parent[v]] = v;
          } else {
            _right[_parent[v]] = v;
          }
        }
        if (_right[w] != INVALID){
          _parent[_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.
      ///\param s The source node.
      ///\param t The target node.
      ///\param p 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 p 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
      ///DynArcLookUp<ListDigraph> ae(g);
      ///...
      ///int n = 0;
      ///for(Arc a = ae(u,v); a != INVALID; a = ae(u,v,a)) n++;
      ///\endcode
      ///
      ///Finding the arcs take at most <em>O</em>(log<em>d</em>)
      ///amortized time, specifically, the time complexity of the lookups
      ///is equal to the optimal search tree implementation for the
      ///current query distribution in a constant factor.
      ///
      ///\note This is a dynamic data structure, therefore the data
      ///structure is updated after each graph alteration. Thus although
      ///this data structure is theoretically faster than \ref ArcLookUp
      ///and \ref AllArcLookUp, it often provides worse performance than
      ///them.
      Arc operator()(Node s, Node t, Arc p = INVALID) const  {
        if (p == INVALID) {
          Arc a = _head[s];
          if (a == INVALID) return INVALID;
          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];
              }
            }
          }
        } else {
          Arc a = p;
          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.
  
    ///Using this class, you can find an arc in a digraph from a given
    ///source to a given target in time <em>O</em>(log<em>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 call refresh() (or at
    ///least refresh(Node)) to refresh this data structure whenever the
    ///digraph changes. This is a time consuming (superlinearly proportional
    ///(<em>O</em>(<em>m</em> log<em>m</em>)) to the number of arcs).
    ///
    ///\tparam GR The type of the underlying digraph.
    ///
    ///\sa DynArcLookUp
    ///\sa AllArcLookUp
    template<class GR>
    class ArcLookUp
    {
      TEMPLATE_DIGRAPH_TYPEDEFS(GR);
  
    public:
  
      /// The Digraph type
      typedef GR 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 search data structure at a node.
  
      ///Build up the search database of node \c n.
      ///
      ///It runs in time <em>O</em>(<em>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</em>(<em>m</em> log<em>D</em>), where <em>m</em> is
      ///the number of the arcs in the digraph 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.
  
    ///This class is the same as \ref ArcLookUp, with the addition
    ///that it makes it possible to find all parallel arcs between given
    ///endpoints.
    ///
    ///\warning This class is static, so you should call refresh() (or at
    ///least refresh(Node)) to refresh this data structure whenever the
    ///digraph changes. This is a time consuming (superlinearly proportional
    ///(<em>O</em>(<em>m</em> log<em>m</em>)) to the number of arcs).
    ///
    ///\tparam GR The type of the underlying digraph.
    ///
    ///\sa DynArcLookUp
    ///\sa ArcLookUp
    template<class GR>
    class AllArcLookUp : public ArcLookUp<GR>
    {
      using ArcLookUp<GR>::_g;
      using ArcLookUp<GR>::_right;
      using ArcLookUp<GR>::_left;
      using ArcLookUp<GR>::_head;
  
      TEMPLATE_DIGRAPH_TYPEDEFS(GR);
  
      typename GR::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!=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:
  
      /// The Digraph type
      typedef GR Digraph;
  
      ///Constructor
  
      ///Constructor.
      ///
      ///It builds up the search database, which remains valid until the digraph
      ///changes.
      AllArcLookUp(const Digraph &g) : ArcLookUp<GR>(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</em>(<em>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<GR>::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</em>(<em>m</em> log<em>D</em>), where <em>m</em> is
      ///the number of the arcs in the digraph 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 a = ae(u,v); a != INVALID; a=ae(u,v,a)) 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.
      ///
      Arc operator()(Node s, Node t, Arc prev=INVALID) const
      {
        if(prev==INVALID)
          {
            Arc f=INVALID;
            Arc e;
            for(e=_head[s];
                e!=INVALID&&_g.target(e)!=t;
                e = t < _g.target(e)?_left[e]:_right[e]) ;
            while(e!=INVALID)
              if(_g.target(e)==t)
                {
                  f = e;
                  e = _left[e];
                }
              else e = _right[e];
            return f;
          }
        else return _next[prev];
      }
  
    };
  
    /// @}
  
  } //namespace lemon
  
  #endif
  