/* -*- mode: C++; indent-tabs-mode: nil; -*-

  *

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

  *

  * Copyright (C) 2003-2009

  * 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