lemon/core.h
author Peter Kovacs <kpeter@inf.elte.hu>
Sun, 09 Jan 2011 00:56:52 +0100
changeset 1034 ef200e268af2
parent 883 dc376822c17d
parent 890 bb871cb8ac06
child 919 e0cef67fe565
permissions -rw-r--r--
Unifications and improvements in TSP algorithms (#386)
     1 /* -*- mode: C++; indent-tabs-mode: nil; -*-
     2  *
     3  * This file is a part of LEMON, a generic C++ optimization library.
     4  *
     5  * Copyright (C) 2003-2010
     6  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
     7  * (Egervary Research Group on Combinatorial Optimization, EGRES).
     8  *
     9  * Permission to use, modify and distribute this software is granted
    10  * provided that this copyright notice appears in all copies. For
    11  * precise terms see the accompanying LICENSE file.
    12  *
    13  * This software is provided "AS IS" with no warranty of any kind,
    14  * express or implied, and with no claim as to its suitability for any
    15  * purpose.
    16  *
    17  */
    18 
    19 #ifndef LEMON_CORE_H
    20 #define LEMON_CORE_H
    21 
    22 #include <vector>
    23 #include <algorithm>
    24 
    25 #include <lemon/config.h>
    26 #include <lemon/bits/enable_if.h>
    27 #include <lemon/bits/traits.h>
    28 #include <lemon/assert.h>
    29 
    30 // Disable the following warnings when compiling with MSVC:
    31 // C4250: 'class1' : inherits 'class2::member' via dominance
    32 // C4355: 'this' : used in base member initializer list
    33 // C4503: 'function' : decorated name length exceeded, name was truncated
    34 // C4800: 'type' : forcing value to bool 'true' or 'false' (performance warning)
    35 // C4996: 'function': was declared deprecated
    36 #ifdef _MSC_VER
    37 #pragma warning( disable : 4250 4355 4503 4800 4996 )
    38 #endif
    39 
    40 ///\file
    41 ///\brief LEMON core utilities.
    42 ///
    43 ///This header file contains core utilities for LEMON.
    44 ///It is automatically included by all graph types, therefore it usually
    45 ///do not have to be included directly.
    46 
    47 namespace lemon {
    48 
    49   /// \brief Dummy type to make it easier to create invalid iterators.
    50   ///
    51   /// Dummy type to make it easier to create invalid iterators.
    52   /// See \ref INVALID for the usage.
    53   struct Invalid {
    54   public:
    55     bool operator==(Invalid) { return true;  }
    56     bool operator!=(Invalid) { return false; }
    57     bool operator< (Invalid) { return false; }
    58   };
    59 
    60   /// \brief Invalid iterators.
    61   ///
    62   /// \ref Invalid is a global type that converts to each iterator
    63   /// in such a way that the value of the target iterator will be invalid.
    64 #ifdef LEMON_ONLY_TEMPLATES
    65   const Invalid INVALID = Invalid();
    66 #else
    67   extern const Invalid INVALID;
    68 #endif
    69 
    70   /// \addtogroup gutils
    71   /// @{
    72 
    73   ///Create convenience typedefs for the digraph types and iterators
    74 
    75   ///This \c \#define creates convenient type definitions for the following
    76   ///types of \c Digraph: \c Node,  \c NodeIt, \c Arc, \c ArcIt, \c InArcIt,
    77   ///\c OutArcIt, \c BoolNodeMap, \c IntNodeMap, \c DoubleNodeMap,
    78   ///\c BoolArcMap, \c IntArcMap, \c DoubleArcMap.
    79   ///
    80   ///\note If the graph type is a dependent type, ie. the graph type depend
    81   ///on a template parameter, then use \c TEMPLATE_DIGRAPH_TYPEDEFS()
    82   ///macro.
    83 #define DIGRAPH_TYPEDEFS(Digraph)                                       \
    84   typedef Digraph::Node Node;                                           \
    85   typedef Digraph::NodeIt NodeIt;                                       \
    86   typedef Digraph::Arc Arc;                                             \
    87   typedef Digraph::ArcIt ArcIt;                                         \
    88   typedef Digraph::InArcIt InArcIt;                                     \
    89   typedef Digraph::OutArcIt OutArcIt;                                   \
    90   typedef Digraph::NodeMap<bool> BoolNodeMap;                           \
    91   typedef Digraph::NodeMap<int> IntNodeMap;                             \
    92   typedef Digraph::NodeMap<double> DoubleNodeMap;                       \
    93   typedef Digraph::ArcMap<bool> BoolArcMap;                             \
    94   typedef Digraph::ArcMap<int> IntArcMap;                               \
    95   typedef Digraph::ArcMap<double> DoubleArcMap
    96 
    97   ///Create convenience typedefs for the digraph types and iterators
    98 
    99   ///\see DIGRAPH_TYPEDEFS
   100   ///
   101   ///\note Use this macro, if the graph type is a dependent type,
   102   ///ie. the graph type depend on a template parameter.
   103 #define TEMPLATE_DIGRAPH_TYPEDEFS(Digraph)                              \
   104   typedef typename Digraph::Node Node;                                  \
   105   typedef typename Digraph::NodeIt NodeIt;                              \
   106   typedef typename Digraph::Arc Arc;                                    \
   107   typedef typename Digraph::ArcIt ArcIt;                                \
   108   typedef typename Digraph::InArcIt InArcIt;                            \
   109   typedef typename Digraph::OutArcIt OutArcIt;                          \
   110   typedef typename Digraph::template NodeMap<bool> BoolNodeMap;         \
   111   typedef typename Digraph::template NodeMap<int> IntNodeMap;           \
   112   typedef typename Digraph::template NodeMap<double> DoubleNodeMap;     \
   113   typedef typename Digraph::template ArcMap<bool> BoolArcMap;           \
   114   typedef typename Digraph::template ArcMap<int> IntArcMap;             \
   115   typedef typename Digraph::template ArcMap<double> DoubleArcMap
   116 
   117   ///Create convenience typedefs for the graph types and iterators
   118 
   119   ///This \c \#define creates the same convenient type definitions as defined
   120   ///by \ref DIGRAPH_TYPEDEFS(Graph) and six more, namely it creates
   121   ///\c Edge, \c EdgeIt, \c IncEdgeIt, \c BoolEdgeMap, \c IntEdgeMap,
   122   ///\c DoubleEdgeMap.
   123   ///
   124   ///\note If the graph type is a dependent type, ie. the graph type depend
   125   ///on a template parameter, then use \c TEMPLATE_GRAPH_TYPEDEFS()
   126   ///macro.
   127 #define GRAPH_TYPEDEFS(Graph)                                           \
   128   DIGRAPH_TYPEDEFS(Graph);                                              \
   129   typedef Graph::Edge Edge;                                             \
   130   typedef Graph::EdgeIt EdgeIt;                                         \
   131   typedef Graph::IncEdgeIt IncEdgeIt;                                   \
   132   typedef Graph::EdgeMap<bool> BoolEdgeMap;                             \
   133   typedef Graph::EdgeMap<int> IntEdgeMap;                               \
   134   typedef Graph::EdgeMap<double> DoubleEdgeMap
   135 
   136   ///Create convenience typedefs for the graph types and iterators
   137 
   138   ///\see GRAPH_TYPEDEFS
   139   ///
   140   ///\note Use this macro, if the graph type is a dependent type,
   141   ///ie. the graph type depend on a template parameter.
   142 #define TEMPLATE_GRAPH_TYPEDEFS(Graph)                                  \
   143   TEMPLATE_DIGRAPH_TYPEDEFS(Graph);                                     \
   144   typedef typename Graph::Edge Edge;                                    \
   145   typedef typename Graph::EdgeIt EdgeIt;                                \
   146   typedef typename Graph::IncEdgeIt IncEdgeIt;                          \
   147   typedef typename Graph::template EdgeMap<bool> BoolEdgeMap;           \
   148   typedef typename Graph::template EdgeMap<int> IntEdgeMap;             \
   149   typedef typename Graph::template EdgeMap<double> DoubleEdgeMap
   150 
   151   /// \brief Function to count the items in a graph.
   152   ///
   153   /// This function counts the items (nodes, arcs etc.) in a graph.
   154   /// The complexity of the function is linear because
   155   /// it iterates on all of the items.
   156   template <typename Graph, typename Item>
   157   inline int countItems(const Graph& g) {
   158     typedef typename ItemSetTraits<Graph, Item>::ItemIt ItemIt;
   159     int num = 0;
   160     for (ItemIt it(g); it != INVALID; ++it) {
   161       ++num;
   162     }
   163     return num;
   164   }
   165 
   166   // Node counting:
   167 
   168   namespace _core_bits {
   169 
   170     template <typename Graph, typename Enable = void>
   171     struct CountNodesSelector {
   172       static int count(const Graph &g) {
   173         return countItems<Graph, typename Graph::Node>(g);
   174       }
   175     };
   176 
   177     template <typename Graph>
   178     struct CountNodesSelector<
   179       Graph, typename
   180       enable_if<typename Graph::NodeNumTag, void>::type>
   181     {
   182       static int count(const Graph &g) {
   183         return g.nodeNum();
   184       }
   185     };
   186   }
   187 
   188   /// \brief Function to count the nodes in the graph.
   189   ///
   190   /// This function counts the nodes in the graph.
   191   /// The complexity of the function is <em>O</em>(<em>n</em>), but for some
   192   /// graph structures it is specialized to run in <em>O</em>(1).
   193   ///
   194   /// \note If the graph contains a \c nodeNum() member function and a
   195   /// \c NodeNumTag tag then this function calls directly the member
   196   /// function to query the cardinality of the node set.
   197   template <typename Graph>
   198   inline int countNodes(const Graph& g) {
   199     return _core_bits::CountNodesSelector<Graph>::count(g);
   200   }
   201 
   202   // Arc counting:
   203 
   204   namespace _core_bits {
   205 
   206     template <typename Graph, typename Enable = void>
   207     struct CountArcsSelector {
   208       static int count(const Graph &g) {
   209         return countItems<Graph, typename Graph::Arc>(g);
   210       }
   211     };
   212 
   213     template <typename Graph>
   214     struct CountArcsSelector<
   215       Graph,
   216       typename enable_if<typename Graph::ArcNumTag, void>::type>
   217     {
   218       static int count(const Graph &g) {
   219         return g.arcNum();
   220       }
   221     };
   222   }
   223 
   224   /// \brief Function to count the arcs in the graph.
   225   ///
   226   /// This function counts the arcs in the graph.
   227   /// The complexity of the function is <em>O</em>(<em>m</em>), but for some
   228   /// graph structures it is specialized to run in <em>O</em>(1).
   229   ///
   230   /// \note If the graph contains a \c arcNum() member function and a
   231   /// \c ArcNumTag tag then this function calls directly the member
   232   /// function to query the cardinality of the arc set.
   233   template <typename Graph>
   234   inline int countArcs(const Graph& g) {
   235     return _core_bits::CountArcsSelector<Graph>::count(g);
   236   }
   237 
   238   // Edge counting:
   239 
   240   namespace _core_bits {
   241 
   242     template <typename Graph, typename Enable = void>
   243     struct CountEdgesSelector {
   244       static int count(const Graph &g) {
   245         return countItems<Graph, typename Graph::Edge>(g);
   246       }
   247     };
   248 
   249     template <typename Graph>
   250     struct CountEdgesSelector<
   251       Graph,
   252       typename enable_if<typename Graph::EdgeNumTag, void>::type>
   253     {
   254       static int count(const Graph &g) {
   255         return g.edgeNum();
   256       }
   257     };
   258   }
   259 
   260   /// \brief Function to count the edges in the graph.
   261   ///
   262   /// This function counts the edges in the graph.
   263   /// The complexity of the function is <em>O</em>(<em>m</em>), but for some
   264   /// graph structures it is specialized to run in <em>O</em>(1).
   265   ///
   266   /// \note If the graph contains a \c edgeNum() member function and a
   267   /// \c EdgeNumTag tag then this function calls directly the member
   268   /// function to query the cardinality of the edge set.
   269   template <typename Graph>
   270   inline int countEdges(const Graph& g) {
   271     return _core_bits::CountEdgesSelector<Graph>::count(g);
   272 
   273   }
   274 
   275 
   276   template <typename Graph, typename DegIt>
   277   inline int countNodeDegree(const Graph& _g, const typename Graph::Node& _n) {
   278     int num = 0;
   279     for (DegIt it(_g, _n); it != INVALID; ++it) {
   280       ++num;
   281     }
   282     return num;
   283   }
   284 
   285   /// \brief Function to count the number of the out-arcs from node \c n.
   286   ///
   287   /// This function counts the number of the out-arcs from node \c n
   288   /// in the graph \c g.
   289   template <typename Graph>
   290   inline int countOutArcs(const Graph& g,  const typename Graph::Node& n) {
   291     return countNodeDegree<Graph, typename Graph::OutArcIt>(g, n);
   292   }
   293 
   294   /// \brief Function to count the number of the in-arcs to node \c n.
   295   ///
   296   /// This function counts the number of the in-arcs to node \c n
   297   /// in the graph \c g.
   298   template <typename Graph>
   299   inline int countInArcs(const Graph& g,  const typename Graph::Node& n) {
   300     return countNodeDegree<Graph, typename Graph::InArcIt>(g, n);
   301   }
   302 
   303   /// \brief Function to count the number of the inc-edges to node \c n.
   304   ///
   305   /// This function counts the number of the inc-edges to node \c n
   306   /// in the undirected graph \c g.
   307   template <typename Graph>
   308   inline int countIncEdges(const Graph& g,  const typename Graph::Node& n) {
   309     return countNodeDegree<Graph, typename Graph::IncEdgeIt>(g, n);
   310   }
   311 
   312   namespace _core_bits {
   313 
   314     template <typename Digraph, typename Item, typename RefMap>
   315     class MapCopyBase {
   316     public:
   317       virtual void copy(const Digraph& from, const RefMap& refMap) = 0;
   318 
   319       virtual ~MapCopyBase() {}
   320     };
   321 
   322     template <typename Digraph, typename Item, typename RefMap,
   323               typename FromMap, typename ToMap>
   324     class MapCopy : public MapCopyBase<Digraph, Item, RefMap> {
   325     public:
   326 
   327       MapCopy(const FromMap& map, ToMap& tmap)
   328         : _map(map), _tmap(tmap) {}
   329 
   330       virtual void copy(const Digraph& digraph, const RefMap& refMap) {
   331         typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
   332         for (ItemIt it(digraph); it != INVALID; ++it) {
   333           _tmap.set(refMap[it], _map[it]);
   334         }
   335       }
   336 
   337     private:
   338       const FromMap& _map;
   339       ToMap& _tmap;
   340     };
   341 
   342     template <typename Digraph, typename Item, typename RefMap, typename It>
   343     class ItemCopy : public MapCopyBase<Digraph, Item, RefMap> {
   344     public:
   345 
   346       ItemCopy(const Item& item, It& it) : _item(item), _it(it) {}
   347 
   348       virtual void copy(const Digraph&, const RefMap& refMap) {
   349         _it = refMap[_item];
   350       }
   351 
   352     private:
   353       Item _item;
   354       It& _it;
   355     };
   356 
   357     template <typename Digraph, typename Item, typename RefMap, typename Ref>
   358     class RefCopy : public MapCopyBase<Digraph, Item, RefMap> {
   359     public:
   360 
   361       RefCopy(Ref& map) : _map(map) {}
   362 
   363       virtual void copy(const Digraph& digraph, const RefMap& refMap) {
   364         typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
   365         for (ItemIt it(digraph); it != INVALID; ++it) {
   366           _map.set(it, refMap[it]);
   367         }
   368       }
   369 
   370     private:
   371       Ref& _map;
   372     };
   373 
   374     template <typename Digraph, typename Item, typename RefMap,
   375               typename CrossRef>
   376     class CrossRefCopy : public MapCopyBase<Digraph, Item, RefMap> {
   377     public:
   378 
   379       CrossRefCopy(CrossRef& cmap) : _cmap(cmap) {}
   380 
   381       virtual void copy(const Digraph& digraph, const RefMap& refMap) {
   382         typedef typename ItemSetTraits<Digraph, Item>::ItemIt ItemIt;
   383         for (ItemIt it(digraph); it != INVALID; ++it) {
   384           _cmap.set(refMap[it], it);
   385         }
   386       }
   387 
   388     private:
   389       CrossRef& _cmap;
   390     };
   391 
   392     template <typename Digraph, typename Enable = void>
   393     struct DigraphCopySelector {
   394       template <typename From, typename NodeRefMap, typename ArcRefMap>
   395       static void copy(const From& from, Digraph &to,
   396                        NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {
   397         to.clear();
   398         for (typename From::NodeIt it(from); it != INVALID; ++it) {
   399           nodeRefMap[it] = to.addNode();
   400         }
   401         for (typename From::ArcIt it(from); it != INVALID; ++it) {
   402           arcRefMap[it] = to.addArc(nodeRefMap[from.source(it)],
   403                                     nodeRefMap[from.target(it)]);
   404         }
   405       }
   406     };
   407 
   408     template <typename Digraph>
   409     struct DigraphCopySelector<
   410       Digraph,
   411       typename enable_if<typename Digraph::BuildTag, void>::type>
   412     {
   413       template <typename From, typename NodeRefMap, typename ArcRefMap>
   414       static void copy(const From& from, Digraph &to,
   415                        NodeRefMap& nodeRefMap, ArcRefMap& arcRefMap) {
   416         to.build(from, nodeRefMap, arcRefMap);
   417       }
   418     };
   419 
   420     template <typename Graph, typename Enable = void>
   421     struct GraphCopySelector {
   422       template <typename From, typename NodeRefMap, typename EdgeRefMap>
   423       static void copy(const From& from, Graph &to,
   424                        NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {
   425         to.clear();
   426         for (typename From::NodeIt it(from); it != INVALID; ++it) {
   427           nodeRefMap[it] = to.addNode();
   428         }
   429         for (typename From::EdgeIt it(from); it != INVALID; ++it) {
   430           edgeRefMap[it] = to.addEdge(nodeRefMap[from.u(it)],
   431                                       nodeRefMap[from.v(it)]);
   432         }
   433       }
   434     };
   435 
   436     template <typename Graph>
   437     struct GraphCopySelector<
   438       Graph,
   439       typename enable_if<typename Graph::BuildTag, void>::type>
   440     {
   441       template <typename From, typename NodeRefMap, typename EdgeRefMap>
   442       static void copy(const From& from, Graph &to,
   443                        NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {
   444         to.build(from, nodeRefMap, edgeRefMap);
   445       }
   446     };
   447 
   448   }
   449 
   450   /// Check whether a graph is undirected.
   451   ///
   452   /// This function returns \c true if the given graph is undirected.
   453 #ifdef DOXYGEN
   454   template <typename GR>
   455   bool undirected(const GR& g) { return false; }
   456 #else
   457   template <typename GR>
   458   typename enable_if<UndirectedTagIndicator<GR>, bool>::type
   459   undirected(const GR&) {
   460     return true;
   461   }
   462   template <typename GR>
   463   typename disable_if<UndirectedTagIndicator<GR>, bool>::type
   464   undirected(const GR&) {
   465     return false;
   466   }
   467 #endif
   468 
   469   /// \brief Class to copy a digraph.
   470   ///
   471   /// Class to copy a digraph to another digraph (duplicate a digraph). The
   472   /// simplest way of using it is through the \c digraphCopy() function.
   473   ///
   474   /// This class not only make a copy of a digraph, but it can create
   475   /// references and cross references between the nodes and arcs of
   476   /// the two digraphs, and it can copy maps to use with the newly created
   477   /// digraph.
   478   ///
   479   /// To make a copy from a digraph, first an instance of DigraphCopy
   480   /// should be created, then the data belongs to the digraph should
   481   /// assigned to copy. In the end, the \c run() member should be
   482   /// called.
   483   ///
   484   /// The next code copies a digraph with several data:
   485   ///\code
   486   ///  DigraphCopy<OrigGraph, NewGraph> cg(orig_graph, new_graph);
   487   ///  // Create references for the nodes
   488   ///  OrigGraph::NodeMap<NewGraph::Node> nr(orig_graph);
   489   ///  cg.nodeRef(nr);
   490   ///  // Create cross references (inverse) for the arcs
   491   ///  NewGraph::ArcMap<OrigGraph::Arc> acr(new_graph);
   492   ///  cg.arcCrossRef(acr);
   493   ///  // Copy an arc map
   494   ///  OrigGraph::ArcMap<double> oamap(orig_graph);
   495   ///  NewGraph::ArcMap<double> namap(new_graph);
   496   ///  cg.arcMap(oamap, namap);
   497   ///  // Copy a node
   498   ///  OrigGraph::Node on;
   499   ///  NewGraph::Node nn;
   500   ///  cg.node(on, nn);
   501   ///  // Execute copying
   502   ///  cg.run();
   503   ///\endcode
   504   template <typename From, typename To>
   505   class DigraphCopy {
   506   private:
   507 
   508     typedef typename From::Node Node;
   509     typedef typename From::NodeIt NodeIt;
   510     typedef typename From::Arc Arc;
   511     typedef typename From::ArcIt ArcIt;
   512 
   513     typedef typename To::Node TNode;
   514     typedef typename To::Arc TArc;
   515 
   516     typedef typename From::template NodeMap<TNode> NodeRefMap;
   517     typedef typename From::template ArcMap<TArc> ArcRefMap;
   518 
   519   public:
   520 
   521     /// \brief Constructor of DigraphCopy.
   522     ///
   523     /// Constructor of DigraphCopy for copying the content of the
   524     /// \c from digraph into the \c to digraph.
   525     DigraphCopy(const From& from, To& to)
   526       : _from(from), _to(to) {}
   527 
   528     /// \brief Destructor of DigraphCopy
   529     ///
   530     /// Destructor of DigraphCopy.
   531     ~DigraphCopy() {
   532       for (int i = 0; i < int(_node_maps.size()); ++i) {
   533         delete _node_maps[i];
   534       }
   535       for (int i = 0; i < int(_arc_maps.size()); ++i) {
   536         delete _arc_maps[i];
   537       }
   538 
   539     }
   540 
   541     /// \brief Copy the node references into the given map.
   542     ///
   543     /// This function copies the node references into the given map.
   544     /// The parameter should be a map, whose key type is the Node type of
   545     /// the source digraph, while the value type is the Node type of the
   546     /// destination digraph.
   547     template <typename NodeRef>
   548     DigraphCopy& nodeRef(NodeRef& map) {
   549       _node_maps.push_back(new _core_bits::RefCopy<From, Node,
   550                            NodeRefMap, NodeRef>(map));
   551       return *this;
   552     }
   553 
   554     /// \brief Copy the node cross references into the given map.
   555     ///
   556     /// This function copies the node cross references (reverse references)
   557     /// into the given map. The parameter should be a map, whose key type
   558     /// is the Node type of the destination digraph, while the value type is
   559     /// the Node type of the source digraph.
   560     template <typename NodeCrossRef>
   561     DigraphCopy& nodeCrossRef(NodeCrossRef& map) {
   562       _node_maps.push_back(new _core_bits::CrossRefCopy<From, Node,
   563                            NodeRefMap, NodeCrossRef>(map));
   564       return *this;
   565     }
   566 
   567     /// \brief Make a copy of the given node map.
   568     ///
   569     /// This function makes a copy of the given node map for the newly
   570     /// created digraph.
   571     /// The key type of the new map \c tmap should be the Node type of the
   572     /// destination digraph, and the key type of the original map \c map
   573     /// should be the Node type of the source digraph.
   574     template <typename FromMap, typename ToMap>
   575     DigraphCopy& nodeMap(const FromMap& map, ToMap& tmap) {
   576       _node_maps.push_back(new _core_bits::MapCopy<From, Node,
   577                            NodeRefMap, FromMap, ToMap>(map, tmap));
   578       return *this;
   579     }
   580 
   581     /// \brief Make a copy of the given node.
   582     ///
   583     /// This function makes a copy of the given node.
   584     DigraphCopy& node(const Node& node, TNode& tnode) {
   585       _node_maps.push_back(new _core_bits::ItemCopy<From, Node,
   586                            NodeRefMap, TNode>(node, tnode));
   587       return *this;
   588     }
   589 
   590     /// \brief Copy the arc references into the given map.
   591     ///
   592     /// This function copies the arc references into the given map.
   593     /// The parameter should be a map, whose key type is the Arc type of
   594     /// the source digraph, while the value type is the Arc type of the
   595     /// destination digraph.
   596     template <typename ArcRef>
   597     DigraphCopy& arcRef(ArcRef& map) {
   598       _arc_maps.push_back(new _core_bits::RefCopy<From, Arc,
   599                           ArcRefMap, ArcRef>(map));
   600       return *this;
   601     }
   602 
   603     /// \brief Copy the arc cross references into the given map.
   604     ///
   605     /// This function copies the arc cross references (reverse references)
   606     /// into the given map. The parameter should be a map, whose key type
   607     /// is the Arc type of the destination digraph, while the value type is
   608     /// the Arc type of the source digraph.
   609     template <typename ArcCrossRef>
   610     DigraphCopy& arcCrossRef(ArcCrossRef& map) {
   611       _arc_maps.push_back(new _core_bits::CrossRefCopy<From, Arc,
   612                           ArcRefMap, ArcCrossRef>(map));
   613       return *this;
   614     }
   615 
   616     /// \brief Make a copy of the given arc map.
   617     ///
   618     /// This function makes a copy of the given arc map for the newly
   619     /// created digraph.
   620     /// The key type of the new map \c tmap should be the Arc type of the
   621     /// destination digraph, and the key type of the original map \c map
   622     /// should be the Arc type of the source digraph.
   623     template <typename FromMap, typename ToMap>
   624     DigraphCopy& arcMap(const FromMap& map, ToMap& tmap) {
   625       _arc_maps.push_back(new _core_bits::MapCopy<From, Arc,
   626                           ArcRefMap, FromMap, ToMap>(map, tmap));
   627       return *this;
   628     }
   629 
   630     /// \brief Make a copy of the given arc.
   631     ///
   632     /// This function makes a copy of the given arc.
   633     DigraphCopy& arc(const Arc& arc, TArc& tarc) {
   634       _arc_maps.push_back(new _core_bits::ItemCopy<From, Arc,
   635                           ArcRefMap, TArc>(arc, tarc));
   636       return *this;
   637     }
   638 
   639     /// \brief Execute copying.
   640     ///
   641     /// This function executes the copying of the digraph along with the
   642     /// copying of the assigned data.
   643     void run() {
   644       NodeRefMap nodeRefMap(_from);
   645       ArcRefMap arcRefMap(_from);
   646       _core_bits::DigraphCopySelector<To>::
   647         copy(_from, _to, nodeRefMap, arcRefMap);
   648       for (int i = 0; i < int(_node_maps.size()); ++i) {
   649         _node_maps[i]->copy(_from, nodeRefMap);
   650       }
   651       for (int i = 0; i < int(_arc_maps.size()); ++i) {
   652         _arc_maps[i]->copy(_from, arcRefMap);
   653       }
   654     }
   655 
   656   protected:
   657 
   658     const From& _from;
   659     To& _to;
   660 
   661     std::vector<_core_bits::MapCopyBase<From, Node, NodeRefMap>* >
   662       _node_maps;
   663 
   664     std::vector<_core_bits::MapCopyBase<From, Arc, ArcRefMap>* >
   665       _arc_maps;
   666 
   667   };
   668 
   669   /// \brief Copy a digraph to another digraph.
   670   ///
   671   /// This function copies a digraph to another digraph.
   672   /// The complete usage of it is detailed in the DigraphCopy class, but
   673   /// a short example shows a basic work:
   674   ///\code
   675   /// digraphCopy(src, trg).nodeRef(nr).arcCrossRef(acr).run();
   676   ///\endcode
   677   ///
   678   /// After the copy the \c nr map will contain the mapping from the
   679   /// nodes of the \c from digraph to the nodes of the \c to digraph and
   680   /// \c acr will contain the mapping from the arcs of the \c to digraph
   681   /// to the arcs of the \c from digraph.
   682   ///
   683   /// \see DigraphCopy
   684   template <typename From, typename To>
   685   DigraphCopy<From, To> digraphCopy(const From& from, To& to) {
   686     return DigraphCopy<From, To>(from, to);
   687   }
   688 
   689   /// \brief Class to copy a graph.
   690   ///
   691   /// Class to copy a graph to another graph (duplicate a graph). The
   692   /// simplest way of using it is through the \c graphCopy() function.
   693   ///
   694   /// This class not only make a copy of a graph, but it can create
   695   /// references and cross references between the nodes, edges and arcs of
   696   /// the two graphs, and it can copy maps for using with the newly created
   697   /// graph.
   698   ///
   699   /// To make a copy from a graph, first an instance of GraphCopy
   700   /// should be created, then the data belongs to the graph should
   701   /// assigned to copy. In the end, the \c run() member should be
   702   /// called.
   703   ///
   704   /// The next code copies a graph with several data:
   705   ///\code
   706   ///  GraphCopy<OrigGraph, NewGraph> cg(orig_graph, new_graph);
   707   ///  // Create references for the nodes
   708   ///  OrigGraph::NodeMap<NewGraph::Node> nr(orig_graph);
   709   ///  cg.nodeRef(nr);
   710   ///  // Create cross references (inverse) for the edges
   711   ///  NewGraph::EdgeMap<OrigGraph::Edge> ecr(new_graph);
   712   ///  cg.edgeCrossRef(ecr);
   713   ///  // Copy an edge map
   714   ///  OrigGraph::EdgeMap<double> oemap(orig_graph);
   715   ///  NewGraph::EdgeMap<double> nemap(new_graph);
   716   ///  cg.edgeMap(oemap, nemap);
   717   ///  // Copy a node
   718   ///  OrigGraph::Node on;
   719   ///  NewGraph::Node nn;
   720   ///  cg.node(on, nn);
   721   ///  // Execute copying
   722   ///  cg.run();
   723   ///\endcode
   724   template <typename From, typename To>
   725   class GraphCopy {
   726   private:
   727 
   728     typedef typename From::Node Node;
   729     typedef typename From::NodeIt NodeIt;
   730     typedef typename From::Arc Arc;
   731     typedef typename From::ArcIt ArcIt;
   732     typedef typename From::Edge Edge;
   733     typedef typename From::EdgeIt EdgeIt;
   734 
   735     typedef typename To::Node TNode;
   736     typedef typename To::Arc TArc;
   737     typedef typename To::Edge TEdge;
   738 
   739     typedef typename From::template NodeMap<TNode> NodeRefMap;
   740     typedef typename From::template EdgeMap<TEdge> EdgeRefMap;
   741 
   742     struct ArcRefMap {
   743       ArcRefMap(const From& from, const To& to,
   744                 const EdgeRefMap& edge_ref, const NodeRefMap& node_ref)
   745         : _from(from), _to(to),
   746           _edge_ref(edge_ref), _node_ref(node_ref) {}
   747 
   748       typedef typename From::Arc Key;
   749       typedef typename To::Arc Value;
   750 
   751       Value operator[](const Key& key) const {
   752         bool forward = _from.u(key) != _from.v(key) ?
   753           _node_ref[_from.source(key)] ==
   754           _to.source(_to.direct(_edge_ref[key], true)) :
   755           _from.direction(key);
   756         return _to.direct(_edge_ref[key], forward);
   757       }
   758 
   759       const From& _from;
   760       const To& _to;
   761       const EdgeRefMap& _edge_ref;
   762       const NodeRefMap& _node_ref;
   763     };
   764 
   765   public:
   766 
   767     /// \brief Constructor of GraphCopy.
   768     ///
   769     /// Constructor of GraphCopy for copying the content of the
   770     /// \c from graph into the \c to graph.
   771     GraphCopy(const From& from, To& to)
   772       : _from(from), _to(to) {}
   773 
   774     /// \brief Destructor of GraphCopy
   775     ///
   776     /// Destructor of GraphCopy.
   777     ~GraphCopy() {
   778       for (int i = 0; i < int(_node_maps.size()); ++i) {
   779         delete _node_maps[i];
   780       }
   781       for (int i = 0; i < int(_arc_maps.size()); ++i) {
   782         delete _arc_maps[i];
   783       }
   784       for (int i = 0; i < int(_edge_maps.size()); ++i) {
   785         delete _edge_maps[i];
   786       }
   787     }
   788 
   789     /// \brief Copy the node references into the given map.
   790     ///
   791     /// This function copies the node references into the given map.
   792     /// The parameter should be a map, whose key type is the Node type of
   793     /// the source graph, while the value type is the Node type of the
   794     /// destination graph.
   795     template <typename NodeRef>
   796     GraphCopy& nodeRef(NodeRef& map) {
   797       _node_maps.push_back(new _core_bits::RefCopy<From, Node,
   798                            NodeRefMap, NodeRef>(map));
   799       return *this;
   800     }
   801 
   802     /// \brief Copy the node cross references into the given map.
   803     ///
   804     /// This function copies the node cross references (reverse references)
   805     /// into the given map. The parameter should be a map, whose key type
   806     /// is the Node type of the destination graph, while the value type is
   807     /// the Node type of the source graph.
   808     template <typename NodeCrossRef>
   809     GraphCopy& nodeCrossRef(NodeCrossRef& map) {
   810       _node_maps.push_back(new _core_bits::CrossRefCopy<From, Node,
   811                            NodeRefMap, NodeCrossRef>(map));
   812       return *this;
   813     }
   814 
   815     /// \brief Make a copy of the given node map.
   816     ///
   817     /// This function makes a copy of the given node map for the newly
   818     /// created graph.
   819     /// The key type of the new map \c tmap should be the Node type of the
   820     /// destination graph, and the key type of the original map \c map
   821     /// should be the Node type of the source graph.
   822     template <typename FromMap, typename ToMap>
   823     GraphCopy& nodeMap(const FromMap& map, ToMap& tmap) {
   824       _node_maps.push_back(new _core_bits::MapCopy<From, Node,
   825                            NodeRefMap, FromMap, ToMap>(map, tmap));
   826       return *this;
   827     }
   828 
   829     /// \brief Make a copy of the given node.
   830     ///
   831     /// This function makes a copy of the given node.
   832     GraphCopy& node(const Node& node, TNode& tnode) {
   833       _node_maps.push_back(new _core_bits::ItemCopy<From, Node,
   834                            NodeRefMap, TNode>(node, tnode));
   835       return *this;
   836     }
   837 
   838     /// \brief Copy the arc references into the given map.
   839     ///
   840     /// This function copies the arc references into the given map.
   841     /// The parameter should be a map, whose key type is the Arc type of
   842     /// the source graph, while the value type is the Arc type of the
   843     /// destination graph.
   844     template <typename ArcRef>
   845     GraphCopy& arcRef(ArcRef& map) {
   846       _arc_maps.push_back(new _core_bits::RefCopy<From, Arc,
   847                           ArcRefMap, ArcRef>(map));
   848       return *this;
   849     }
   850 
   851     /// \brief Copy the arc cross references into the given map.
   852     ///
   853     /// This function copies the arc cross references (reverse references)
   854     /// into the given map. The parameter should be a map, whose key type
   855     /// is the Arc type of the destination graph, while the value type is
   856     /// the Arc type of the source graph.
   857     template <typename ArcCrossRef>
   858     GraphCopy& arcCrossRef(ArcCrossRef& map) {
   859       _arc_maps.push_back(new _core_bits::CrossRefCopy<From, Arc,
   860                           ArcRefMap, ArcCrossRef>(map));
   861       return *this;
   862     }
   863 
   864     /// \brief Make a copy of the given arc map.
   865     ///
   866     /// This function makes a copy of the given arc map for the newly
   867     /// created graph.
   868     /// The key type of the new map \c tmap should be the Arc type of the
   869     /// destination graph, and the key type of the original map \c map
   870     /// should be the Arc type of the source graph.
   871     template <typename FromMap, typename ToMap>
   872     GraphCopy& arcMap(const FromMap& map, ToMap& tmap) {
   873       _arc_maps.push_back(new _core_bits::MapCopy<From, Arc,
   874                           ArcRefMap, FromMap, ToMap>(map, tmap));
   875       return *this;
   876     }
   877 
   878     /// \brief Make a copy of the given arc.
   879     ///
   880     /// This function makes a copy of the given arc.
   881     GraphCopy& arc(const Arc& arc, TArc& tarc) {
   882       _arc_maps.push_back(new _core_bits::ItemCopy<From, Arc,
   883                           ArcRefMap, TArc>(arc, tarc));
   884       return *this;
   885     }
   886 
   887     /// \brief Copy the edge references into the given map.
   888     ///
   889     /// This function copies the edge references into the given map.
   890     /// The parameter should be a map, whose key type is the Edge type of
   891     /// the source graph, while the value type is the Edge type of the
   892     /// destination graph.
   893     template <typename EdgeRef>
   894     GraphCopy& edgeRef(EdgeRef& map) {
   895       _edge_maps.push_back(new _core_bits::RefCopy<From, Edge,
   896                            EdgeRefMap, EdgeRef>(map));
   897       return *this;
   898     }
   899 
   900     /// \brief Copy the edge cross references into the given map.
   901     ///
   902     /// This function copies the edge cross references (reverse references)
   903     /// into the given map. The parameter should be a map, whose key type
   904     /// is the Edge type of the destination graph, while the value type is
   905     /// the Edge type of the source graph.
   906     template <typename EdgeCrossRef>
   907     GraphCopy& edgeCrossRef(EdgeCrossRef& map) {
   908       _edge_maps.push_back(new _core_bits::CrossRefCopy<From,
   909                            Edge, EdgeRefMap, EdgeCrossRef>(map));
   910       return *this;
   911     }
   912 
   913     /// \brief Make a copy of the given edge map.
   914     ///
   915     /// This function makes a copy of the given edge map for the newly
   916     /// created graph.
   917     /// The key type of the new map \c tmap should be the Edge type of the
   918     /// destination graph, and the key type of the original map \c map
   919     /// should be the Edge type of the source graph.
   920     template <typename FromMap, typename ToMap>
   921     GraphCopy& edgeMap(const FromMap& map, ToMap& tmap) {
   922       _edge_maps.push_back(new _core_bits::MapCopy<From, Edge,
   923                            EdgeRefMap, FromMap, ToMap>(map, tmap));
   924       return *this;
   925     }
   926 
   927     /// \brief Make a copy of the given edge.
   928     ///
   929     /// This function makes a copy of the given edge.
   930     GraphCopy& edge(const Edge& edge, TEdge& tedge) {
   931       _edge_maps.push_back(new _core_bits::ItemCopy<From, Edge,
   932                            EdgeRefMap, TEdge>(edge, tedge));
   933       return *this;
   934     }
   935 
   936     /// \brief Execute copying.
   937     ///
   938     /// This function executes the copying of the graph along with the
   939     /// copying of the assigned data.
   940     void run() {
   941       NodeRefMap nodeRefMap(_from);
   942       EdgeRefMap edgeRefMap(_from);
   943       ArcRefMap arcRefMap(_from, _to, edgeRefMap, nodeRefMap);
   944       _core_bits::GraphCopySelector<To>::
   945         copy(_from, _to, nodeRefMap, edgeRefMap);
   946       for (int i = 0; i < int(_node_maps.size()); ++i) {
   947         _node_maps[i]->copy(_from, nodeRefMap);
   948       }
   949       for (int i = 0; i < int(_edge_maps.size()); ++i) {
   950         _edge_maps[i]->copy(_from, edgeRefMap);
   951       }
   952       for (int i = 0; i < int(_arc_maps.size()); ++i) {
   953         _arc_maps[i]->copy(_from, arcRefMap);
   954       }
   955     }
   956 
   957   private:
   958 
   959     const From& _from;
   960     To& _to;
   961 
   962     std::vector<_core_bits::MapCopyBase<From, Node, NodeRefMap>* >
   963       _node_maps;
   964 
   965     std::vector<_core_bits::MapCopyBase<From, Arc, ArcRefMap>* >
   966       _arc_maps;
   967 
   968     std::vector<_core_bits::MapCopyBase<From, Edge, EdgeRefMap>* >
   969       _edge_maps;
   970 
   971   };
   972 
   973   /// \brief Copy a graph to another graph.
   974   ///
   975   /// This function copies a graph to another graph.
   976   /// The complete usage of it is detailed in the GraphCopy class,
   977   /// but a short example shows a basic work:
   978   ///\code
   979   /// graphCopy(src, trg).nodeRef(nr).edgeCrossRef(ecr).run();
   980   ///\endcode
   981   ///
   982   /// After the copy the \c nr map will contain the mapping from the
   983   /// nodes of the \c from graph to the nodes of the \c to graph and
   984   /// \c ecr will contain the mapping from the edges of the \c to graph
   985   /// to the edges of the \c from graph.
   986   ///
   987   /// \see GraphCopy
   988   template <typename From, typename To>
   989   GraphCopy<From, To>
   990   graphCopy(const From& from, To& to) {
   991     return GraphCopy<From, To>(from, to);
   992   }
   993 
   994   namespace _core_bits {
   995 
   996     template <typename Graph, typename Enable = void>
   997     struct FindArcSelector {
   998       typedef typename Graph::Node Node;
   999       typedef typename Graph::Arc Arc;
  1000       static Arc find(const Graph &g, Node u, Node v, Arc e) {
  1001         if (e == INVALID) {
  1002           g.firstOut(e, u);
  1003         } else {
  1004           g.nextOut(e);
  1005         }
  1006         while (e != INVALID && g.target(e) != v) {
  1007           g.nextOut(e);
  1008         }
  1009         return e;
  1010       }
  1011     };
  1012 
  1013     template <typename Graph>
  1014     struct FindArcSelector<
  1015       Graph,
  1016       typename enable_if<typename Graph::FindArcTag, void>::type>
  1017     {
  1018       typedef typename Graph::Node Node;
  1019       typedef typename Graph::Arc Arc;
  1020       static Arc find(const Graph &g, Node u, Node v, Arc prev) {
  1021         return g.findArc(u, v, prev);
  1022       }
  1023     };
  1024   }
  1025 
  1026   /// \brief Find an arc between two nodes of a digraph.
  1027   ///
  1028   /// This function finds an arc from node \c u to node \c v in the
  1029   /// digraph \c g.
  1030   ///
  1031   /// If \c prev is \ref INVALID (this is the default value), then
  1032   /// it finds the first arc from \c u to \c v. Otherwise it looks for
  1033   /// the next arc from \c u to \c v after \c prev.
  1034   /// \return The found arc or \ref INVALID if there is no such an arc.
  1035   ///
  1036   /// Thus you can iterate through each arc from \c u to \c v as it follows.
  1037   ///\code
  1038   /// for(Arc e = findArc(g,u,v); e != INVALID; e = findArc(g,u,v,e)) {
  1039   ///   ...
  1040   /// }
  1041   ///\endcode
  1042   ///
  1043   /// \note \ref ConArcIt provides iterator interface for the same
  1044   /// functionality.
  1045   ///
  1046   ///\sa ConArcIt
  1047   ///\sa ArcLookUp, AllArcLookUp, DynArcLookUp
  1048   template <typename Graph>
  1049   inline typename Graph::Arc
  1050   findArc(const Graph &g, typename Graph::Node u, typename Graph::Node v,
  1051           typename Graph::Arc prev = INVALID) {
  1052     return _core_bits::FindArcSelector<Graph>::find(g, u, v, prev);
  1053   }
  1054 
  1055   /// \brief Iterator for iterating on parallel arcs connecting the same nodes.
  1056   ///
  1057   /// Iterator for iterating on parallel arcs connecting the same nodes. It is
  1058   /// a higher level interface for the \ref findArc() function. You can
  1059   /// use it the following way:
  1060   ///\code
  1061   /// for (ConArcIt<Graph> it(g, src, trg); it != INVALID; ++it) {
  1062   ///   ...
  1063   /// }
  1064   ///\endcode
  1065   ///
  1066   ///\sa findArc()
  1067   ///\sa ArcLookUp, AllArcLookUp, DynArcLookUp
  1068   template <typename GR>
  1069   class ConArcIt : public GR::Arc {
  1070     typedef typename GR::Arc Parent;
  1071 
  1072   public:
  1073 
  1074     typedef typename GR::Arc Arc;
  1075     typedef typename GR::Node Node;
  1076 
  1077     /// \brief Constructor.
  1078     ///
  1079     /// Construct a new ConArcIt iterating on the arcs that
  1080     /// connects nodes \c u and \c v.
  1081     ConArcIt(const GR& g, Node u, Node v) : _graph(g) {
  1082       Parent::operator=(findArc(_graph, u, v));
  1083     }
  1084 
  1085     /// \brief Constructor.
  1086     ///
  1087     /// Construct a new ConArcIt that continues the iterating from arc \c a.
  1088     ConArcIt(const GR& g, Arc a) : Parent(a), _graph(g) {}
  1089 
  1090     /// \brief Increment operator.
  1091     ///
  1092     /// It increments the iterator and gives back the next arc.
  1093     ConArcIt& operator++() {
  1094       Parent::operator=(findArc(_graph, _graph.source(*this),
  1095                                 _graph.target(*this), *this));
  1096       return *this;
  1097     }
  1098   private:
  1099     const GR& _graph;
  1100   };
  1101 
  1102   namespace _core_bits {
  1103 
  1104     template <typename Graph, typename Enable = void>
  1105     struct FindEdgeSelector {
  1106       typedef typename Graph::Node Node;
  1107       typedef typename Graph::Edge Edge;
  1108       static Edge find(const Graph &g, Node u, Node v, Edge e) {
  1109         bool b;
  1110         if (u != v) {
  1111           if (e == INVALID) {
  1112             g.firstInc(e, b, u);
  1113           } else {
  1114             b = g.u(e) == u;
  1115             g.nextInc(e, b);
  1116           }
  1117           while (e != INVALID && (b ? g.v(e) : g.u(e)) != v) {
  1118             g.nextInc(e, b);
  1119           }
  1120         } else {
  1121           if (e == INVALID) {
  1122             g.firstInc(e, b, u);
  1123           } else {
  1124             b = true;
  1125             g.nextInc(e, b);
  1126           }
  1127           while (e != INVALID && (!b || g.v(e) != v)) {
  1128             g.nextInc(e, b);
  1129           }
  1130         }
  1131         return e;
  1132       }
  1133     };
  1134 
  1135     template <typename Graph>
  1136     struct FindEdgeSelector<
  1137       Graph,
  1138       typename enable_if<typename Graph::FindEdgeTag, void>::type>
  1139     {
  1140       typedef typename Graph::Node Node;
  1141       typedef typename Graph::Edge Edge;
  1142       static Edge find(const Graph &g, Node u, Node v, Edge prev) {
  1143         return g.findEdge(u, v, prev);
  1144       }
  1145     };
  1146   }
  1147 
  1148   /// \brief Find an edge between two nodes of a graph.
  1149   ///
  1150   /// This function finds an edge from node \c u to node \c v in graph \c g.
  1151   /// If node \c u and node \c v is equal then each loop edge
  1152   /// will be enumerated once.
  1153   ///
  1154   /// If \c prev is \ref INVALID (this is the default value), then
  1155   /// it finds the first edge from \c u to \c v. Otherwise it looks for
  1156   /// the next edge from \c u to \c v after \c prev.
  1157   /// \return The found edge or \ref INVALID if there is no such an edge.
  1158   ///
  1159   /// Thus you can iterate through each edge between \c u and \c v
  1160   /// as it follows.
  1161   ///\code
  1162   /// for(Edge e = findEdge(g,u,v); e != INVALID; e = findEdge(g,u,v,e)) {
  1163   ///   ...
  1164   /// }
  1165   ///\endcode
  1166   ///
  1167   /// \note \ref ConEdgeIt provides iterator interface for the same
  1168   /// functionality.
  1169   ///
  1170   ///\sa ConEdgeIt
  1171   template <typename Graph>
  1172   inline typename Graph::Edge
  1173   findEdge(const Graph &g, typename Graph::Node u, typename Graph::Node v,
  1174             typename Graph::Edge p = INVALID) {
  1175     return _core_bits::FindEdgeSelector<Graph>::find(g, u, v, p);
  1176   }
  1177 
  1178   /// \brief Iterator for iterating on parallel edges connecting the same nodes.
  1179   ///
  1180   /// Iterator for iterating on parallel edges connecting the same nodes.
  1181   /// It is a higher level interface for the findEdge() function. You can
  1182   /// use it the following way:
  1183   ///\code
  1184   /// for (ConEdgeIt<Graph> it(g, u, v); it != INVALID; ++it) {
  1185   ///   ...
  1186   /// }
  1187   ///\endcode
  1188   ///
  1189   ///\sa findEdge()
  1190   template <typename GR>
  1191   class ConEdgeIt : public GR::Edge {
  1192     typedef typename GR::Edge Parent;
  1193 
  1194   public:
  1195 
  1196     typedef typename GR::Edge Edge;
  1197     typedef typename GR::Node Node;
  1198 
  1199     /// \brief Constructor.
  1200     ///
  1201     /// Construct a new ConEdgeIt iterating on the edges that
  1202     /// connects nodes \c u and \c v.
  1203     ConEdgeIt(const GR& g, Node u, Node v) : _graph(g), _u(u), _v(v) {
  1204       Parent::operator=(findEdge(_graph, _u, _v));
  1205     }
  1206 
  1207     /// \brief Constructor.
  1208     ///
  1209     /// Construct a new ConEdgeIt that continues iterating from edge \c e.
  1210     ConEdgeIt(const GR& g, Edge e) : Parent(e), _graph(g) {}
  1211 
  1212     /// \brief Increment operator.
  1213     ///
  1214     /// It increments the iterator and gives back the next edge.
  1215     ConEdgeIt& operator++() {
  1216       Parent::operator=(findEdge(_graph, _u, _v, *this));
  1217       return *this;
  1218     }
  1219   private:
  1220     const GR& _graph;
  1221     Node _u, _v;
  1222   };
  1223 
  1224 
  1225   ///Dynamic arc look-up between given endpoints.
  1226 
  1227   ///Using this class, you can find an arc in a digraph from a given
  1228   ///source to a given target in amortized time <em>O</em>(log<em>d</em>),
  1229   ///where <em>d</em> is the out-degree of the source node.
  1230   ///
  1231   ///It is possible to find \e all parallel arcs between two nodes with
  1232   ///the \c operator() member.
  1233   ///
  1234   ///This is a dynamic data structure. Consider to use \ref ArcLookUp or
  1235   ///\ref AllArcLookUp if your digraph is not changed so frequently.
  1236   ///
  1237   ///This class uses a self-adjusting binary search tree, the Splay tree
  1238   ///of Sleator and Tarjan to guarantee the logarithmic amortized
  1239   ///time bound for arc look-ups. This class also guarantees the
  1240   ///optimal time bound in a constant factor for any distribution of
  1241   ///queries.
  1242   ///
  1243   ///\tparam GR The type of the underlying digraph.
  1244   ///
  1245   ///\sa ArcLookUp
  1246   ///\sa AllArcLookUp
  1247   template <typename GR>
  1248   class DynArcLookUp
  1249     : protected ItemSetTraits<GR, typename GR::Arc>::ItemNotifier::ObserverBase
  1250   {
  1251     typedef typename ItemSetTraits<GR, typename GR::Arc>
  1252     ::ItemNotifier::ObserverBase Parent;
  1253 
  1254     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
  1255 
  1256   public:
  1257 
  1258     /// The Digraph type
  1259     typedef GR Digraph;
  1260 
  1261   protected:
  1262 
  1263     class AutoNodeMap : public ItemSetTraits<GR, Node>::template Map<Arc>::Type
  1264     {
  1265       typedef typename ItemSetTraits<GR, Node>::template Map<Arc>::Type Parent;
  1266 
  1267     public:
  1268 
  1269       AutoNodeMap(const GR& digraph) : Parent(digraph, INVALID) {}
  1270 
  1271       virtual void add(const Node& node) {
  1272         Parent::add(node);
  1273         Parent::set(node, INVALID);
  1274       }
  1275 
  1276       virtual void add(const std::vector<Node>& nodes) {
  1277         Parent::add(nodes);
  1278         for (int i = 0; i < int(nodes.size()); ++i) {
  1279           Parent::set(nodes[i], INVALID);
  1280         }
  1281       }
  1282 
  1283       virtual void build() {
  1284         Parent::build();
  1285         Node it;
  1286         typename Parent::Notifier* nf = Parent::notifier();
  1287         for (nf->first(it); it != INVALID; nf->next(it)) {
  1288           Parent::set(it, INVALID);
  1289         }
  1290       }
  1291     };
  1292 
  1293     class ArcLess {
  1294       const Digraph &g;
  1295     public:
  1296       ArcLess(const Digraph &_g) : g(_g) {}
  1297       bool operator()(Arc a,Arc b) const
  1298       {
  1299         return g.target(a)<g.target(b);
  1300       }
  1301     };
  1302 
  1303   protected:
  1304 
  1305     const Digraph &_g;
  1306     AutoNodeMap _head;
  1307     typename Digraph::template ArcMap<Arc> _parent;
  1308     typename Digraph::template ArcMap<Arc> _left;
  1309     typename Digraph::template ArcMap<Arc> _right;
  1310 
  1311   public:
  1312 
  1313     ///Constructor
  1314 
  1315     ///Constructor.
  1316     ///
  1317     ///It builds up the search database.
  1318     DynArcLookUp(const Digraph &g)
  1319       : _g(g),_head(g),_parent(g),_left(g),_right(g)
  1320     {
  1321       Parent::attach(_g.notifier(typename Digraph::Arc()));
  1322       refresh();
  1323     }
  1324 
  1325   protected:
  1326 
  1327     virtual void add(const Arc& arc) {
  1328       insert(arc);
  1329     }
  1330 
  1331     virtual void add(const std::vector<Arc>& arcs) {
  1332       for (int i = 0; i < int(arcs.size()); ++i) {
  1333         insert(arcs[i]);
  1334       }
  1335     }
  1336 
  1337     virtual void erase(const Arc& arc) {
  1338       remove(arc);
  1339     }
  1340 
  1341     virtual void erase(const std::vector<Arc>& arcs) {
  1342       for (int i = 0; i < int(arcs.size()); ++i) {
  1343         remove(arcs[i]);
  1344       }
  1345     }
  1346 
  1347     virtual void build() {
  1348       refresh();
  1349     }
  1350 
  1351     virtual void clear() {
  1352       for(NodeIt n(_g);n!=INVALID;++n) {
  1353         _head[n] = INVALID;
  1354       }
  1355     }
  1356 
  1357     void insert(Arc arc) {
  1358       Node s = _g.source(arc);
  1359       Node t = _g.target(arc);
  1360       _left[arc] = INVALID;
  1361       _right[arc] = INVALID;
  1362 
  1363       Arc e = _head[s];
  1364       if (e == INVALID) {
  1365         _head[s] = arc;
  1366         _parent[arc] = INVALID;
  1367         return;
  1368       }
  1369       while (true) {
  1370         if (t < _g.target(e)) {
  1371           if (_left[e] == INVALID) {
  1372             _left[e] = arc;
  1373             _parent[arc] = e;
  1374             splay(arc);
  1375             return;
  1376           } else {
  1377             e = _left[e];
  1378           }
  1379         } else {
  1380           if (_right[e] == INVALID) {
  1381             _right[e] = arc;
  1382             _parent[arc] = e;
  1383             splay(arc);
  1384             return;
  1385           } else {
  1386             e = _right[e];
  1387           }
  1388         }
  1389       }
  1390     }
  1391 
  1392     void remove(Arc arc) {
  1393       if (_left[arc] == INVALID) {
  1394         if (_right[arc] != INVALID) {
  1395           _parent[_right[arc]] = _parent[arc];
  1396         }
  1397         if (_parent[arc] != INVALID) {
  1398           if (_left[_parent[arc]] == arc) {
  1399             _left[_parent[arc]] = _right[arc];
  1400           } else {
  1401             _right[_parent[arc]] = _right[arc];
  1402           }
  1403         } else {
  1404           _head[_g.source(arc)] = _right[arc];
  1405         }
  1406       } else if (_right[arc] == INVALID) {
  1407         _parent[_left[arc]] = _parent[arc];
  1408         if (_parent[arc] != INVALID) {
  1409           if (_left[_parent[arc]] == arc) {
  1410             _left[_parent[arc]] = _left[arc];
  1411           } else {
  1412             _right[_parent[arc]] = _left[arc];
  1413           }
  1414         } else {
  1415           _head[_g.source(arc)] = _left[arc];
  1416         }
  1417       } else {
  1418         Arc e = _left[arc];
  1419         if (_right[e] != INVALID) {
  1420           e = _right[e];
  1421           while (_right[e] != INVALID) {
  1422             e = _right[e];
  1423           }
  1424           Arc s = _parent[e];
  1425           _right[_parent[e]] = _left[e];
  1426           if (_left[e] != INVALID) {
  1427             _parent[_left[e]] = _parent[e];
  1428           }
  1429 
  1430           _left[e] = _left[arc];
  1431           _parent[_left[arc]] = e;
  1432           _right[e] = _right[arc];
  1433           _parent[_right[arc]] = e;
  1434 
  1435           _parent[e] = _parent[arc];
  1436           if (_parent[arc] != INVALID) {
  1437             if (_left[_parent[arc]] == arc) {
  1438               _left[_parent[arc]] = e;
  1439             } else {
  1440               _right[_parent[arc]] = e;
  1441             }
  1442           }
  1443           splay(s);
  1444         } else {
  1445           _right[e] = _right[arc];
  1446           _parent[_right[arc]] = e;
  1447           _parent[e] = _parent[arc];
  1448 
  1449           if (_parent[arc] != INVALID) {
  1450             if (_left[_parent[arc]] == arc) {
  1451               _left[_parent[arc]] = e;
  1452             } else {
  1453               _right[_parent[arc]] = e;
  1454             }
  1455           } else {
  1456             _head[_g.source(arc)] = e;
  1457           }
  1458         }
  1459       }
  1460     }
  1461 
  1462     Arc refreshRec(std::vector<Arc> &v,int a,int b)
  1463     {
  1464       int m=(a+b)/2;
  1465       Arc me=v[m];
  1466       if (a < m) {
  1467         Arc left = refreshRec(v,a,m-1);
  1468         _left[me] = left;
  1469         _parent[left] = me;
  1470       } else {
  1471         _left[me] = INVALID;
  1472       }
  1473       if (m < b) {
  1474         Arc right = refreshRec(v,m+1,b);
  1475         _right[me] = right;
  1476         _parent[right] = me;
  1477       } else {
  1478         _right[me] = INVALID;
  1479       }
  1480       return me;
  1481     }
  1482 
  1483     void refresh() {
  1484       for(NodeIt n(_g);n!=INVALID;++n) {
  1485         std::vector<Arc> v;
  1486         for(OutArcIt a(_g,n);a!=INVALID;++a) v.push_back(a);
  1487         if (!v.empty()) {
  1488           std::sort(v.begin(),v.end(),ArcLess(_g));
  1489           Arc head = refreshRec(v,0,v.size()-1);
  1490           _head[n] = head;
  1491           _parent[head] = INVALID;
  1492         }
  1493         else _head[n] = INVALID;
  1494       }
  1495     }
  1496 
  1497     void zig(Arc v) {
  1498       Arc w = _parent[v];
  1499       _parent[v] = _parent[w];
  1500       _parent[w] = v;
  1501       _left[w] = _right[v];
  1502       _right[v] = w;
  1503       if (_parent[v] != INVALID) {
  1504         if (_right[_parent[v]] == w) {
  1505           _right[_parent[v]] = v;
  1506         } else {
  1507           _left[_parent[v]] = v;
  1508         }
  1509       }
  1510       if (_left[w] != INVALID){
  1511         _parent[_left[w]] = w;
  1512       }
  1513     }
  1514 
  1515     void zag(Arc v) {
  1516       Arc w = _parent[v];
  1517       _parent[v] = _parent[w];
  1518       _parent[w] = v;
  1519       _right[w] = _left[v];
  1520       _left[v] = w;
  1521       if (_parent[v] != INVALID){
  1522         if (_left[_parent[v]] == w) {
  1523           _left[_parent[v]] = v;
  1524         } else {
  1525           _right[_parent[v]] = v;
  1526         }
  1527       }
  1528       if (_right[w] != INVALID){
  1529         _parent[_right[w]] = w;
  1530       }
  1531     }
  1532 
  1533     void splay(Arc v) {
  1534       while (_parent[v] != INVALID) {
  1535         if (v == _left[_parent[v]]) {
  1536           if (_parent[_parent[v]] == INVALID) {
  1537             zig(v);
  1538           } else {
  1539             if (_parent[v] == _left[_parent[_parent[v]]]) {
  1540               zig(_parent[v]);
  1541               zig(v);
  1542             } else {
  1543               zig(v);
  1544               zag(v);
  1545             }
  1546           }
  1547         } else {
  1548           if (_parent[_parent[v]] == INVALID) {
  1549             zag(v);
  1550           } else {
  1551             if (_parent[v] == _left[_parent[_parent[v]]]) {
  1552               zag(v);
  1553               zig(v);
  1554             } else {
  1555               zag(_parent[v]);
  1556               zag(v);
  1557             }
  1558           }
  1559         }
  1560       }
  1561       _head[_g.source(v)] = v;
  1562     }
  1563 
  1564 
  1565   public:
  1566 
  1567     ///Find an arc between two nodes.
  1568 
  1569     ///Find an arc between two nodes.
  1570     ///\param s The source node.
  1571     ///\param t The target node.
  1572     ///\param p The previous arc between \c s and \c t. It it is INVALID or
  1573     ///not given, the operator finds the first appropriate arc.
  1574     ///\return An arc from \c s to \c t after \c p or
  1575     ///\ref INVALID if there is no more.
  1576     ///
  1577     ///For example, you can count the number of arcs from \c u to \c v in the
  1578     ///following way.
  1579     ///\code
  1580     ///DynArcLookUp<ListDigraph> ae(g);
  1581     ///...
  1582     ///int n = 0;
  1583     ///for(Arc a = ae(u,v); a != INVALID; a = ae(u,v,a)) n++;
  1584     ///\endcode
  1585     ///
  1586     ///Finding the arcs take at most <em>O</em>(log<em>d</em>)
  1587     ///amortized time, specifically, the time complexity of the lookups
  1588     ///is equal to the optimal search tree implementation for the
  1589     ///current query distribution in a constant factor.
  1590     ///
  1591     ///\note This is a dynamic data structure, therefore the data
  1592     ///structure is updated after each graph alteration. Thus although
  1593     ///this data structure is theoretically faster than \ref ArcLookUp
  1594     ///and \ref AllArcLookUp, it often provides worse performance than
  1595     ///them.
  1596     Arc operator()(Node s, Node t, Arc p = INVALID) const  {
  1597       if (p == INVALID) {
  1598         Arc a = _head[s];
  1599         if (a == INVALID) return INVALID;
  1600         Arc r = INVALID;
  1601         while (true) {
  1602           if (_g.target(a) < t) {
  1603             if (_right[a] == INVALID) {
  1604               const_cast<DynArcLookUp&>(*this).splay(a);
  1605               return r;
  1606             } else {
  1607               a = _right[a];
  1608             }
  1609           } else {
  1610             if (_g.target(a) == t) {
  1611               r = a;
  1612             }
  1613             if (_left[a] == INVALID) {
  1614               const_cast<DynArcLookUp&>(*this).splay(a);
  1615               return r;
  1616             } else {
  1617               a = _left[a];
  1618             }
  1619           }
  1620         }
  1621       } else {
  1622         Arc a = p;
  1623         if (_right[a] != INVALID) {
  1624           a = _right[a];
  1625           while (_left[a] != INVALID) {
  1626             a = _left[a];
  1627           }
  1628           const_cast<DynArcLookUp&>(*this).splay(a);
  1629         } else {
  1630           while (_parent[a] != INVALID && _right[_parent[a]] ==  a) {
  1631             a = _parent[a];
  1632           }
  1633           if (_parent[a] == INVALID) {
  1634             return INVALID;
  1635           } else {
  1636             a = _parent[a];
  1637             const_cast<DynArcLookUp&>(*this).splay(a);
  1638           }
  1639         }
  1640         if (_g.target(a) == t) return a;
  1641         else return INVALID;
  1642       }
  1643     }
  1644 
  1645   };
  1646 
  1647   ///Fast arc look-up between given endpoints.
  1648 
  1649   ///Using this class, you can find an arc in a digraph from a given
  1650   ///source to a given target in time <em>O</em>(log<em>d</em>),
  1651   ///where <em>d</em> is the out-degree of the source node.
  1652   ///
  1653   ///It is not possible to find \e all parallel arcs between two nodes.
  1654   ///Use \ref AllArcLookUp for this purpose.
  1655   ///
  1656   ///\warning This class is static, so you should call refresh() (or at
  1657   ///least refresh(Node)) to refresh this data structure whenever the
  1658   ///digraph changes. This is a time consuming (superlinearly proportional
  1659   ///(<em>O</em>(<em>m</em> log<em>m</em>)) to the number of arcs).
  1660   ///
  1661   ///\tparam GR The type of the underlying digraph.
  1662   ///
  1663   ///\sa DynArcLookUp
  1664   ///\sa AllArcLookUp
  1665   template<class GR>
  1666   class ArcLookUp
  1667   {
  1668     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
  1669 
  1670   public:
  1671 
  1672     /// The Digraph type
  1673     typedef GR Digraph;
  1674 
  1675   protected:
  1676     const Digraph &_g;
  1677     typename Digraph::template NodeMap<Arc> _head;
  1678     typename Digraph::template ArcMap<Arc> _left;
  1679     typename Digraph::template ArcMap<Arc> _right;
  1680 
  1681     class ArcLess {
  1682       const Digraph &g;
  1683     public:
  1684       ArcLess(const Digraph &_g) : g(_g) {}
  1685       bool operator()(Arc a,Arc b) const
  1686       {
  1687         return g.target(a)<g.target(b);
  1688       }
  1689     };
  1690 
  1691   public:
  1692 
  1693     ///Constructor
  1694 
  1695     ///Constructor.
  1696     ///
  1697     ///It builds up the search database, which remains valid until the digraph
  1698     ///changes.
  1699     ArcLookUp(const Digraph &g) :_g(g),_head(g),_left(g),_right(g) {refresh();}
  1700 
  1701   private:
  1702     Arc refreshRec(std::vector<Arc> &v,int a,int b)
  1703     {
  1704       int m=(a+b)/2;
  1705       Arc me=v[m];
  1706       _left[me] = a<m?refreshRec(v,a,m-1):INVALID;
  1707       _right[me] = m<b?refreshRec(v,m+1,b):INVALID;
  1708       return me;
  1709     }
  1710   public:
  1711     ///Refresh the search data structure at a node.
  1712 
  1713     ///Build up the search database of node \c n.
  1714     ///
  1715     ///It runs in time <em>O</em>(<em>d</em> log<em>d</em>), where <em>d</em>
  1716     ///is the number of the outgoing arcs of \c n.
  1717     void refresh(Node n)
  1718     {
  1719       std::vector<Arc> v;
  1720       for(OutArcIt e(_g,n);e!=INVALID;++e) v.push_back(e);
  1721       if(v.size()) {
  1722         std::sort(v.begin(),v.end(),ArcLess(_g));
  1723         _head[n]=refreshRec(v,0,v.size()-1);
  1724       }
  1725       else _head[n]=INVALID;
  1726     }
  1727     ///Refresh the full data structure.
  1728 
  1729     ///Build up the full search database. In fact, it simply calls
  1730     ///\ref refresh(Node) "refresh(n)" for each node \c n.
  1731     ///
  1732     ///It runs in time <em>O</em>(<em>m</em> log<em>D</em>), where <em>m</em> is
  1733     ///the number of the arcs in the digraph and <em>D</em> is the maximum
  1734     ///out-degree of the digraph.
  1735     void refresh()
  1736     {
  1737       for(NodeIt n(_g);n!=INVALID;++n) refresh(n);
  1738     }
  1739 
  1740     ///Find an arc between two nodes.
  1741 
  1742     ///Find an arc between two nodes in time <em>O</em>(log<em>d</em>),
  1743     ///where <em>d</em> is the number of outgoing arcs of \c s.
  1744     ///\param s The source node.
  1745     ///\param t The target node.
  1746     ///\return An arc from \c s to \c t if there exists,
  1747     ///\ref INVALID otherwise.
  1748     ///
  1749     ///\warning If you change the digraph, refresh() must be called before using
  1750     ///this operator. If you change the outgoing arcs of
  1751     ///a single node \c n, then \ref refresh(Node) "refresh(n)" is enough.
  1752     Arc operator()(Node s, Node t) const
  1753     {
  1754       Arc e;
  1755       for(e=_head[s];
  1756           e!=INVALID&&_g.target(e)!=t;
  1757           e = t < _g.target(e)?_left[e]:_right[e]) ;
  1758       return e;
  1759     }
  1760 
  1761   };
  1762 
  1763   ///Fast look-up of all arcs between given endpoints.
  1764 
  1765   ///This class is the same as \ref ArcLookUp, with the addition
  1766   ///that it makes it possible to find all parallel arcs between given
  1767   ///endpoints.
  1768   ///
  1769   ///\warning This class is static, so you should call refresh() (or at
  1770   ///least refresh(Node)) to refresh this data structure whenever the
  1771   ///digraph changes. This is a time consuming (superlinearly proportional
  1772   ///(<em>O</em>(<em>m</em> log<em>m</em>)) to the number of arcs).
  1773   ///
  1774   ///\tparam GR The type of the underlying digraph.
  1775   ///
  1776   ///\sa DynArcLookUp
  1777   ///\sa ArcLookUp
  1778   template<class GR>
  1779   class AllArcLookUp : public ArcLookUp<GR>
  1780   {
  1781     using ArcLookUp<GR>::_g;
  1782     using ArcLookUp<GR>::_right;
  1783     using ArcLookUp<GR>::_left;
  1784     using ArcLookUp<GR>::_head;
  1785 
  1786     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
  1787 
  1788     typename GR::template ArcMap<Arc> _next;
  1789 
  1790     Arc refreshNext(Arc head,Arc next=INVALID)
  1791     {
  1792       if(head==INVALID) return next;
  1793       else {
  1794         next=refreshNext(_right[head],next);
  1795         _next[head]=( next!=INVALID && _g.target(next)==_g.target(head))
  1796           ? next : INVALID;
  1797         return refreshNext(_left[head],head);
  1798       }
  1799     }
  1800 
  1801     void refreshNext()
  1802     {
  1803       for(NodeIt n(_g);n!=INVALID;++n) refreshNext(_head[n]);
  1804     }
  1805 
  1806   public:
  1807 
  1808     /// The Digraph type
  1809     typedef GR Digraph;
  1810 
  1811     ///Constructor
  1812 
  1813     ///Constructor.
  1814     ///
  1815     ///It builds up the search database, which remains valid until the digraph
  1816     ///changes.
  1817     AllArcLookUp(const Digraph &g) : ArcLookUp<GR>(g), _next(g) {refreshNext();}
  1818 
  1819     ///Refresh the data structure at a node.
  1820 
  1821     ///Build up the search database of node \c n.
  1822     ///
  1823     ///It runs in time <em>O</em>(<em>d</em> log<em>d</em>), where <em>d</em> is
  1824     ///the number of the outgoing arcs of \c n.
  1825     void refresh(Node n)
  1826     {
  1827       ArcLookUp<GR>::refresh(n);
  1828       refreshNext(_head[n]);
  1829     }
  1830 
  1831     ///Refresh the full data structure.
  1832 
  1833     ///Build up the full search database. In fact, it simply calls
  1834     ///\ref refresh(Node) "refresh(n)" for each node \c n.
  1835     ///
  1836     ///It runs in time <em>O</em>(<em>m</em> log<em>D</em>), where <em>m</em> is
  1837     ///the number of the arcs in the digraph and <em>D</em> is the maximum
  1838     ///out-degree of the digraph.
  1839     void refresh()
  1840     {
  1841       for(NodeIt n(_g);n!=INVALID;++n) refresh(_head[n]);
  1842     }
  1843 
  1844     ///Find an arc between two nodes.
  1845 
  1846     ///Find an arc between two nodes.
  1847     ///\param s The source node.
  1848     ///\param t The target node.
  1849     ///\param prev The previous arc between \c s and \c t. It it is INVALID or
  1850     ///not given, the operator finds the first appropriate arc.
  1851     ///\return An arc from \c s to \c t after \c prev or
  1852     ///\ref INVALID if there is no more.
  1853     ///
  1854     ///For example, you can count the number of arcs from \c u to \c v in the
  1855     ///following way.
  1856     ///\code
  1857     ///AllArcLookUp<ListDigraph> ae(g);
  1858     ///...
  1859     ///int n = 0;
  1860     ///for(Arc a = ae(u,v); a != INVALID; a=ae(u,v,a)) n++;
  1861     ///\endcode
  1862     ///
  1863     ///Finding the first arc take <em>O</em>(log<em>d</em>) time,
  1864     ///where <em>d</em> is the number of outgoing arcs of \c s. Then the
  1865     ///consecutive arcs are found in constant time.
  1866     ///
  1867     ///\warning If you change the digraph, refresh() must be called before using
  1868     ///this operator. If you change the outgoing arcs of
  1869     ///a single node \c n, then \ref refresh(Node) "refresh(n)" is enough.
  1870     ///
  1871 #ifdef DOXYGEN
  1872     Arc operator()(Node s, Node t, Arc prev=INVALID) const {}
  1873 #else
  1874     using ArcLookUp<GR>::operator() ;
  1875     Arc operator()(Node s, Node t, Arc prev) const
  1876     {
  1877       return prev==INVALID?(*this)(s,t):_next[prev];
  1878     }
  1879 #endif
  1880 
  1881   };
  1882 
  1883   /// @}
  1884 
  1885 } //namespace lemon
  1886 
  1887 #endif