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

 *

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

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

#ifndef LEMON_BFS_H

#define LEMON_BFS_H

///\ingroup search

///\file

///\brief BFS algorithm.

#include <lemon/list_graph.h>

#include <lemon/bits/path_dump.h>

#include <lemon/core.h>

#include <lemon/error.h>

#include <lemon/maps.h>

#include <lemon/path.h>

namespace lemon {

  ///Default traits class of Bfs class.

  ///Default traits class of Bfs class.

  ///\tparam GR Digraph type.

  template<class GR>

  struct BfsDefaultTraits

  {

    ///The type of the digraph the algorithm runs on.

    typedef GR Digraph;

    ///\brief The type of the map that stores the predecessor

    ///arcs of the shortest paths.

    ///

    ///The type of the map that stores the predecessor

    ///arcs of the shortest paths.

    ///It must meet the \ref concepts::WriteMap "WriteMap" concept.

    typedef typename Digraph::template NodeMap<typename Digraph::Arc> PredMap;

    ///Instantiates a PredMap.

    ///\param g is the digraph, to which we would like to define the

    ///PredMap.

    static PredMap *createPredMap(const Digraph &g)

    {

      return new PredMap(g);

    }

    ///The type of the map that indicates which nodes are processed.

    ///The type of the map that indicates which nodes are processed.

    ///It must meet the \ref concepts::WriteMap "WriteMap" concept.

    typedef NullMap<typename Digraph::Node,bool> ProcessedMap;

    ///Instantiates a ProcessedMap.

    ///This function instantiates a ProcessedMap.

    ///\param g is the digraph, to which

    ///we would like to define the ProcessedMap

#ifdef DOXYGEN

    static ProcessedMap *createProcessedMap(const Digraph &g)

#else

    static ProcessedMap *createProcessedMap(const Digraph &)

#endif

    {

      return new ProcessedMap();

    }

    ///The type of the map that indicates which nodes are reached.

    typedef typename Digraph::template NodeMap<bool> ReachedMap;

    ///Instantiates a ReachedMap.

    ///This function instantiates a ReachedMap.

    ///\param g is the digraph, to which

    ///we would like to define the ReachedMap.

    static ReachedMap *createReachedMap(const Digraph &g)

    {

      return new ReachedMap(g);

    }

    ///The type of the map that stores the distances of the nodes.

    ///The type of the map that stores the distances of the nodes.

    ///It must meet the \ref concepts::WriteMap "WriteMap" concept.

    typedef typename Digraph::template NodeMap<int> DistMap;

    ///Instantiates a DistMap.

    ///This function instantiates a DistMap.

    ///\param g is the digraph, to which we would like to define the

    ///DistMap.

    static DistMap *createDistMap(const Digraph &g)

    {

      return new DistMap(g);

    }

  };

  ///%BFS algorithm class.

  ///\ingroup search

  ///This class provides an efficient implementation of the %BFS algorithm.

  ///

  ///There is also a \ref bfs() "function-type interface" for the BFS

  ///algorithm, which is convenient in the simplier cases and it can be

  ///used easier.

  ///

  ///\tparam GR The type of the digraph the algorithm runs on.

  ///The default value is \ref ListDigraph. The value of GR is not used

  ///directly by \ref Bfs, it is only passed to \ref BfsDefaultTraits.

  ///\tparam TR Traits class to set various data types used by the algorithm.

  ///The default traits class is

  ///\ref BfsDefaultTraits "BfsDefaultTraits<GR>".

  ///See \ref BfsDefaultTraits for the documentation of

  ///a Bfs traits class.

#ifdef DOXYGEN

  template <typename GR,

            typename TR>

#else

  template <typename GR=ListDigraph,

            typename TR=BfsDefaultTraits<GR> >

#endif

  class Bfs {

  public:

    ///The type of the digraph the algorithm runs on.

    typedef typename TR::Digraph Digraph;

    ///\brief The type of the map that stores the predecessor arcs of the

    ///shortest paths.

    typedef typename TR::PredMap PredMap;

    ///The type of the map that stores the distances of the nodes.

    typedef typename TR::DistMap DistMap;

    ///The type of the map that indicates which nodes are reached.

    typedef typename TR::ReachedMap ReachedMap;

    ///The type of the map that indicates which nodes are processed.

    typedef typename TR::ProcessedMap ProcessedMap;

    ///The type of the paths.

    typedef PredMapPath<Digraph, PredMap> Path;

    ///The traits class.

    typedef TR Traits;

  private:

    typedef typename Digraph::Node Node;

    typedef typename Digraph::NodeIt NodeIt;

    typedef typename Digraph::Arc Arc;

    typedef typename Digraph::OutArcIt OutArcIt;

    //Pointer to the underlying digraph.

    const Digraph *G;

    //Pointer to the map of predecessor arcs.

    PredMap *_pred;

    //Indicates if _pred is locally allocated (true) or not.

    bool local_pred;

    //Pointer to the map of distances.

    DistMap *_dist;

    //Indicates if _dist is locally allocated (true) or not.

    bool local_dist;

    //Pointer to the map of reached status of the nodes.

    ReachedMap *_reached;

    //Indicates if _reached is locally allocated (true) or not.

    bool local_reached;

    //Pointer to the map of processed status of the nodes.

    ProcessedMap *_processed;

    //Indicates if _processed is locally allocated (true) or not.

    bool local_processed;

    std::vector<typename Digraph::Node> _queue;

    int _queue_head,_queue_tail,_queue_next_dist;

    int _curr_dist;

    //Creates the maps if necessary.

    void create_maps()

    {

      if(!_pred) {

        local_pred = true;

        _pred = Traits::createPredMap(*G);

      }

      if(!_dist) {

        local_dist = true;

        _dist = Traits::createDistMap(*G);

      }

      if(!_reached) {

        local_reached = true;

        _reached = Traits::createReachedMap(*G);

      }

      if(!_processed) {

        local_processed = true;

        _processed = Traits::createProcessedMap(*G);

      }

    }

  protected:

    Bfs() {}

  public:

    typedef Bfs Create;

    ///\name Named template parameters

    ///@{

    template <class T>

    struct SetPredMapTraits : public Traits {

      typedef T PredMap;

      static PredMap *createPredMap(const Digraph &)

      {

        LEMON_ASSERT(false, "PredMap is not initialized");

        return 0; // ignore warnings

      }

    };

    ///\brief \ref named-templ-param "Named parameter" for setting

    ///PredMap type.

    ///

    ///\ref named-templ-param "Named parameter" for setting

    ///PredMap type.

    template <class T>

    struct SetPredMap : public Bfs< Digraph, SetPredMapTraits<T> > {

      typedef Bfs< Digraph, SetPredMapTraits<T> > Create;

    };

    template <class T>

    struct SetDistMapTraits : public Traits {

      typedef T DistMap;

      static DistMap *createDistMap(const Digraph &)

      {

        LEMON_ASSERT(false, "DistMap is not initialized");

        return 0; // ignore warnings

      }

    };

    ///\brief \ref named-templ-param "Named parameter" for setting

    ///DistMap type.

    ///

    ///\ref named-templ-param "Named parameter" for setting

    ///DistMap type.

    template <class T>

    struct SetDistMap : public Bfs< Digraph, SetDistMapTraits<T> > {

      typedef Bfs< Digraph, SetDistMapTraits<T> > Create;

    };

    template <class T>

    struct SetReachedMapTraits : public Traits {

      typedef T ReachedMap;

      static ReachedMap *createReachedMap(const Digraph &)

      {

        LEMON_ASSERT(false, "ReachedMap is not initialized");

        return 0; // ignore warnings

      }

    };

    ///\brief \ref named-templ-param "Named parameter" for setting

    ///ReachedMap type.

    ///

    ///\ref named-templ-param "Named parameter" for setting

    ///ReachedMap type.

    template <class T>

    struct SetReachedMap : public Bfs< Digraph, SetReachedMapTraits<T> > {

      typedef Bfs< Digraph, SetReachedMapTraits<T> > Create;

    };

    template <class T>

    struct SetProcessedMapTraits : public Traits {

      typedef T ProcessedMap;

      static ProcessedMap *createProcessedMap(const Digraph &)

      {

        LEMON_ASSERT(false, "ProcessedMap is not initialized");

        return 0; // ignore warnings

      }

    };

    ///\brief \ref named-templ-param "Named parameter" for setting

    ///ProcessedMap type.

    ///

    ///\ref named-templ-param "Named parameter" for setting

    ///ProcessedMap type.

    template <class T>

    struct SetProcessedMap : public Bfs< Digraph, SetProcessedMapTraits<T> > {

      typedef Bfs< Digraph, SetProcessedMapTraits<T> > Create;

    };

    struct SetStandardProcessedMapTraits : public Traits {

      typedef typename Digraph::template NodeMap<bool> ProcessedMap;

      static ProcessedMap *createProcessedMap(const Digraph &g)

      {

        return new ProcessedMap(g);

        return 0; // ignore warnings

      }

    };

    ///\brief \ref named-templ-param "Named parameter" for setting

    ///ProcessedMap type to be <tt>Digraph::NodeMap<bool></tt>.

    ///

    ///\ref named-templ-param "Named parameter" for setting

    ///ProcessedMap type to be <tt>Digraph::NodeMap<bool></tt>.

    ///If you don't set it explicitly, it will be automatically allocated.

    struct SetStandardProcessedMap :

      public Bfs< Digraph, SetStandardProcessedMapTraits > {

      typedef Bfs< Digraph, SetStandardProcessedMapTraits > Create;

    };

    ///@}

  public:

    ///Constructor.

    ///Constructor.

    ///\param g The digraph the algorithm runs on.

    Bfs(const Digraph &g) :

      G(&g),

      _pred(NULL), local_pred(false),

      _dist(NULL), local_dist(false),

      _reached(NULL), local_reached(false),

      _processed(NULL), local_processed(false)

    { }

    ///Destructor.

    ~Bfs()

    {

      if(local_pred) delete _pred;

      if(local_dist) delete _dist;

      if(local_reached) delete _reached;

      if(local_processed) delete _processed;

    }

    ///Sets the map that stores the predecessor arcs.

    ///Sets the map that stores the predecessor arcs.

    ///If you don't use this function before calling \ref run(),

    ///it will allocate one. The destructor deallocates this

    ///automatically allocated map, of course.

    ///\return <tt> (*this) </tt>

    Bfs &predMap(PredMap &m)

    {

      if(local_pred) {

        delete _pred;

        local_pred=false;

      }

      _pred = &m;

      return *this;

    }

    ///Sets the map that indicates which nodes are reached.

    ///Sets the map that indicates which nodes are reached.

    ///If you don't use this function before calling \ref run(),

    ///it will allocate one. The destructor deallocates this

    ///automatically allocated map, of course.

    ///\return <tt> (*this) </tt>

    Bfs &reachedMap(ReachedMap &m)

    {

      if(local_reached) {

        delete _reached;

        local_reached=false;

      }

      _reached = &m;

      return *this;

    }

    ///Sets the map that indicates which nodes are processed.

    ///Sets the map that indicates which nodes are processed.

    ///If you don't use this function before calling \ref run(),

    ///it will allocate one. The destructor deallocates this

    ///automatically allocated map, of course.

    ///\return <tt> (*this) </tt>

    Bfs &processedMap(ProcessedMap &m)

    {

      if(local_processed) {

        delete _processed;

        local_processed=false;

      }

      _processed = &m;

      return *this;

    }

    ///Sets the map that stores the distances of the nodes.

    ///Sets the map that stores the distances of the nodes calculated by

    ///the algorithm.

    ///If you don't use this function before calling \ref run(),

    ///it will allocate one. The destructor deallocates this

    ///automatically allocated map, of course.

    ///\return <tt> (*this) </tt>

    Bfs &distMap(DistMap &m)

    {

      if(local_dist) {

        delete _dist;

        local_dist=false;

      }

      _dist = &m;

      return *this;

    }

  public:

    ///\name Execution control

    ///The simplest way to execute the algorithm is to use

    ///one of the member functions called \ref lemon::Bfs::run() "run()".

    ///\n

    ///If you need more control on the execution, first you must call

    ///\ref lemon::Bfs::init() "init()", then you can add several source

    ///nodes with \ref lemon::Bfs::addSource() "addSource()".

    ///Finally \ref lemon::Bfs::start() "start()" will perform the

    ///actual path computation.

    ///@{

    ///Initializes the internal data structures.

    ///Initializes the internal data structures.

    ///

    void init()

    {

      create_maps();

      _queue.resize(countNodes(*G));

      _queue_head=_queue_tail=0;

      _curr_dist=1;

      for ( NodeIt u(*G) ; u!=INVALID ; ++u ) {

        _pred->set(u,INVALID);

        _reached->set(u,false);

        _processed->set(u,false);

      }

    }

    ///Adds a new source node.

    ///Adds a new source node to the set of nodes to be processed.

    ///

    void addSource(Node s)

    {

      if(!(*_reached)[s])

        {

          _reached->set(s,true);

          _pred->set(s,INVALID);

          _dist->set(s,0);

          _queue[_queue_head++]=s;

          _queue_next_dist=_queue_head;

        }

    }

    ///Processes the next node.

    ///Processes the next node.

    ///

    ///\return The processed node.

    ///

    ///\pre The queue must not be empty.

    Node processNextNode()

    {

      if(_queue_tail==_queue_next_dist) {

        _curr_dist++;

        _queue_next_dist=_queue_head;

      }

    ///Contract two nodes.

    ///This function contracts two nodes.

    ///Node \p b will be removed but instead of deleting

    ///incident arcs, they will be joined to \p a.

    ///The last parameter \p r controls whether to remove loops. \c true

    ///means that loops will be removed.

    ///

    ///\note The <tt>ArcIt</tt>s referencing a moved arc remain

    ///valid. However <tt>InArcIt</tt>s and <tt>OutArcIt</tt>s

    ///may be invalidated.

    ///

    ///\warning This functionality cannot be used together with the Snapshot

    ///feature.

    void contract(Node a, Node b, bool r = true)

    {

      for(OutArcIt e(*this,b);e!=INVALID;) {

        OutArcIt f=e;

        ++f;

        if(r && target(e)==a) erase(e);

        else changeSource(e,a);

        e=f;

      }

      for(InArcIt e(*this,b);e!=INVALID;) {

        InArcIt f=e;

        ++f;

        if(r && source(e)==a) erase(e);

        else changeTarget(e,a);

        e=f;

      }

      erase(b);

    }

    ///Split a node.

    ///This function splits a node. First a new node is added to the digraph,

    ///then the source of each outgoing arc of \c n is moved to this new node.

    ///If \c connect is \c true (this is the default value), then a new arc

    ///from \c n to the newly created node is also added.

    ///\return The newly created node.

    ///

    ///\note The <tt>ArcIt</tt>s referencing a moved arc remain

    ///valid. However <tt>InArcIt</tt>s and <tt>OutArcIt</tt>s may

    ///be invalidated.

    ///

    ///\warning This functionality cannot be used in conjunction with the

    ///Snapshot feature.

    Node split(Node n, bool connect = true) {

      Node b = addNode();

      for(OutArcIt e(*this,n);e!=INVALID;) {

        OutArcIt f=e;

        ++f;

        changeSource(e,b);

        e=f;

      }

      if (connect) addArc(n,b);

      return b;

    }

    ///Split an arc.

    ///This function splits an arc. First a new node \c b is added to

    ///the digraph, then the original arc is re-targeted to \c

    ///b. Finally an arc from \c b to the original target is added.

    ///

    ///\return The newly created node.

    ///

    ///\warning This functionality cannot be used together with the

    ///Snapshot feature.

    Node split(Arc e) {

      Node b = addNode();

      addArc(b,target(e));

      changeTarget(e,b);

      return b;

    }

    /// \brief Class to make a snapshot of the digraph and restore

    /// it later.

    ///

    /// Class to make a snapshot of the digraph and restore it later.

    ///

    /// The newly added nodes and arcs can be removed using the

    /// restore() function.

    ///

    /// \warning Arc and node deletions and other modifications (e.g.

    /// contracting, splitting, reversing arcs or nodes) cannot be

    /// restored. These events invalidate the snapshot.

    class Snapshot {

    protected:

      typedef Parent::NodeNotifier NodeNotifier;

      class NodeObserverProxy : public NodeNotifier::ObserverBase {

      public:

        NodeObserverProxy(Snapshot& _snapshot)

          : snapshot(_snapshot) {}

        using NodeNotifier::ObserverBase::attach;

        using NodeNotifier::ObserverBase::detach;

        using NodeNotifier::ObserverBase::attached;

      protected:

        virtual void add(const Node& node) {

          snapshot.addNode(node);

        }

        virtual void add(const std::vector<Node>& nodes) {

          for (int i = nodes.size() - 1; i >= 0; ++i) {

            snapshot.addNode(nodes[i]);

          }

        }

        virtual void erase(const Node& node) {

          snapshot.eraseNode(node);

        }

        virtual void erase(const std::vector<Node>& nodes) {

          for (int i = 0; i < int(nodes.size()); ++i) {

            snapshot.eraseNode(nodes[i]);

          }

        }

        virtual void build() {

          Node node;

          std::vector<Node> nodes;

          for (notifier()->first(node); node != INVALID;

               notifier()->next(node)) {

            nodes.push_back(node);

          }

          for (int i = nodes.size() - 1; i >= 0; --i) {

            snapshot.addNode(nodes[i]);

          }

        }

        virtual void clear() {

          Node node;

          for (notifier()->first(node); node != INVALID;

               notifier()->next(node)) {

            snapshot.eraseNode(node);

          }

        }

        Snapshot& snapshot;

      };

      class ArcObserverProxy : public ArcNotifier::ObserverBase {

      public:

        ArcObserverProxy(Snapshot& _snapshot)

          : snapshot(_snapshot) {}

        using ArcNotifier::ObserverBase::attach;

        using ArcNotifier::ObserverBase::detach;

        using ArcNotifier::ObserverBase::attached;

      protected:

        virtual void add(const Arc& arc) {

          snapshot.addArc(arc);

        }

        virtual void add(const std::vector<Arc>& arcs) {

          for (int i = arcs.size() - 1; i >= 0; ++i) {

            snapshot.addArc(arcs[i]);

          }

        }

        virtual void erase(const Arc& arc) {

          snapshot.eraseArc(arc);

        }

        virtual void erase(const std::vector<Arc>& arcs) {

          for (int i = 0; i < int(arcs.size()); ++i) {

            snapshot.eraseArc(arcs[i]);

          }

        }

        virtual void build() {

          Arc arc;

          std::vector<Arc> arcs;

          for (notifier()->first(arc); arc != INVALID;

               notifier()->next(arc)) {

            arcs.push_back(arc);

          }

          for (int i = arcs.size() - 1; i >= 0; --i) {

            snapshot.addArc(arcs[i]);

          }

        }

        virtual void clear() {

          Arc arc;

          for (notifier()->first(arc); arc != INVALID;

               notifier()->next(arc)) {

            snapshot.eraseArc(arc);

          }

        }

        Snapshot& snapshot;

      };

      ListDigraph *digraph;

      NodeObserverProxy node_observer_proxy;

      ArcObserverProxy arc_observer_proxy;

      std::list<Node> added_nodes;

      std::list<Arc> added_arcs;

      void addNode(const Node& node) {

        added_nodes.push_front(node);

      }

      void eraseNode(const Node& node) {

        std::list<Node>::iterator it =

          std::find(added_nodes.begin(), added_nodes.end(), node);

        if (it == added_nodes.end()) {

          clear();

          arc_observer_proxy.detach();

          throw NodeNotifier::ImmediateDetach();

        } else {

          added_nodes.erase(it);

        }

      }

      void addArc(const Arc& arc) {

        added_arcs.push_front(arc);

      }

      void eraseArc(const Arc& arc) {

        std::list<Arc>::iterator it =

          std::find(added_arcs.begin(), added_arcs.end(), arc);

        if (it == added_arcs.end()) {

          clear();

          node_observer_proxy.detach();

          throw ArcNotifier::ImmediateDetach();

        } else {

          added_arcs.erase(it);

        }

      }

      void attach(ListDigraph &_digraph) {

        digraph = &_digraph;

        node_observer_proxy.attach(digraph->notifier(Node()));

        arc_observer_proxy.attach(digraph->notifier(Arc()));

      }

      void detach() {

        node_observer_proxy.detach();

        arc_observer_proxy.detach();

      }

      bool attached() const {

        return node_observer_proxy.attached();

      }

      void clear() {

        added_nodes.clear();

        added_arcs.clear();

      }

    public:

      /// \brief Default constructor.

      ///

      /// Default constructor.

      /// To actually make a snapshot you must call save().

      Snapshot()

        : digraph(0), node_observer_proxy(*this),

          arc_observer_proxy(*this) {}

      /// \brief Constructor that immediately makes a snapshot.

      ///

      /// This constructor immediately makes a snapshot of the digraph.

      /// \param _digraph The digraph we make a snapshot of.

      Snapshot(ListDigraph &_digraph)

        : node_observer_proxy(*this),

          arc_observer_proxy(*this) {

        attach(_digraph);

      }

      /// \brief Make a snapshot.

      ///

      /// Make a snapshot of the digraph.

      ///

      /// This function can be called more than once. In case of a repeated

      /// call, the previous snapshot gets lost.

      /// \param _digraph The digraph we make the snapshot of.

      void save(ListDigraph &_digraph) {

        if (attached()) {

          detach();

          clear();

        }

        attach(_digraph);

      }

      /// \brief Undo the changes until the last snapshot.

      //

      /// Undo the changes until the last snapshot created by save().

      void restore() {

        detach();

        for(std::list<Arc>::iterator it = added_arcs.begin();

            it != added_arcs.end(); ++it) {

          digraph->erase(*it);

        }

        for(std::list<Node>::iterator it = added_nodes.begin();

            it != added_nodes.end(); ++it) {

          digraph->erase(*it);

        }

        clear();

      }

      /// \brief Gives back true when the snapshot is valid.

      ///

      /// Gives back true when the snapshot is valid.

      bool valid() const {

        return attached();

      }

    };

  };

  ///@}

  class ListGraphBase {

  protected:

    struct NodeT {

      int first_out;

      int prev, next;

    };

    struct ArcT {

      int target;

      int prev_out, next_out;

    };

    std::vector<NodeT> nodes;

    int first_node;

    int first_free_node;

    std::vector<ArcT> arcs;

    int first_free_arc;

  public:

    typedef ListGraphBase Digraph;

    class Node;

    class Arc;

    class Edge;

    class Node {

      friend class ListGraphBase;

    protected:

      int id;

      explicit Node(int pid) { id = pid;}

    public:

      Node() {}

      Node (Invalid) { id = -1; }

      bool operator==(const Node& node) const {return id == node.id;}

      bool operator!=(const Node& node) const {return id != node.id;}

      bool operator<(const Node& node) const {return id < node.id;}

    };

    class Edge {

      friend class ListGraphBase;

    protected:

      int id;

      explicit Edge(int pid) { id = pid;}

    public:

      Edge() {}

      Edge (Invalid) { id = -1; }

      bool operator==(const Edge& edge) const {return id == edge.id;}

      bool operator!=(const Edge& edge) const {return id != edge.id;}

      bool operator<(const Edge& edge) const {return id < edge.id;}

    };

    class Arc {

      friend class ListGraphBase;

    protected:

      int id;

      explicit Arc(int pid) { id = pid;}

    public:

      }

      Arc() {}

      Arc (Invalid) { id = -1; }

      bool operator==(const Arc& arc) const {return id == arc.id;}

      bool operator!=(const Arc& arc) const {return id != arc.id;}

      bool operator<(const Arc& arc) const {return id < arc.id;}

    };

    ListGraphBase()

      : nodes(), first_node(-1),

        first_free_node(-1), arcs(), first_free_arc(-1) {}

    int maxNodeId() const { return nodes.size()-1; }

    int maxEdgeId() const { return arcs.size() / 2 - 1; }

    int maxArcId() const { return arcs.size()-1; }

    Node source(Arc e) const { return Node(arcs[e.id ^ 1].target); }

    Node target(Arc e) const { return Node(arcs[e.id].target); }

    Node u(Edge e) const { return Node(arcs[2 * e.id].target); }

    Node v(Edge e) const { return Node(arcs[2 * e.id + 1].target); }

    static bool direction(Arc e) {

      return (e.id & 1) == 1;

    }

    static Arc direct(Edge e, bool d) {

      return Arc(e.id * 2 + (d ? 1 : 0));

    }

    void first(Node& node) const {

      node.id = first_node;

    }

    void next(Node& node) const {

      node.id = nodes[node.id].next;

    }

    void first(Arc& e) const {

      int n = first_node;

      while (n != -1 && nodes[n].first_out == -1) {

        n = nodes[n].next;

      }

      e.id = (n == -1) ? -1 : nodes[n].first_out;

    }

    void next(Arc& e) const {

      if (arcs[e.id].next_out != -1) {

        e.id = arcs[e.id].next_out;

      } else {

        int n = nodes[arcs[e.id ^ 1].target].next;

        while(n != -1 && nodes[n].first_out == -1) {

          n = nodes[n].next;

        }

        e.id = (n == -1) ? -1 : nodes[n].first_out;

      }

    }

    void first(Edge& e) const {

      int n = first_node;

      while (n != -1) {

        e.id = nodes[n].first_out;

        while ((e.id & 1) != 1) {

          e.id = arcs[e.id].next_out;

        }

        if (e.id != -1) {

          e.id /= 2;

          return;

        }

        n = nodes[n].next;

      }

      e.id = -1;

    }

    void next(Edge& e) const {

      int n = arcs[e.id * 2].target;

      e.id = arcs[(e.id * 2) | 1].next_out;

      while ((e.id & 1) != 1) {

        e.id = arcs[e.id].next_out;

      }

      if (e.id != -1) {

        e.id /= 2;

        return;

      }

      n = nodes[n].next;

      while (n != -1) {

        e.id = nodes[n].first_out;

        while ((e.id & 1) != 1) {

          e.id = arcs[e.id].next_out;

        }

        if (e.id != -1) {

          e.id /= 2;

          return;

        }

        n = nodes[n].next;

      }

      e.id = -1;

    }

    void firstOut(Arc &e, const Node& v) const {

      e.id = nodes[v.id].first_out;

    }

    void nextOut(Arc &e) const {

      e.id = arcs[e.id].next_out;

    }

    void firstIn(Arc &e, const Node& v) const {

      e.id = ((nodes[v.id].first_out) ^ 1);

      if (e.id == -2) e.id = -1;

    }

    void nextIn(Arc &e) const {

      e.id = ((arcs[e.id ^ 1].next_out) ^ 1);

      if (e.id == -2) e.id = -1;

    }

    void firstInc(Edge &e, bool& d, const Node& v) const {

      int a = nodes[v.id].first_out;

      if (a != -1 ) {

        e.id = a / 2;

        d = ((a & 1) == 1);

      } else {

        e.id = -1;

        d = true;

      }

    }

    void nextInc(Edge &e, bool& d) const {

      int a = (arcs[(e.id * 2) | (d ? 1 : 0)].next_out);

      if (a != -1 ) {

        e.id = a / 2;

        d = ((a & 1) == 1);

      } else {

        e.id = -1;

        d = true;

      }

    }

    static int id(Node v) { return v.id; }

    static int id(Arc e) { return e.id; }

    static int id(Edge e) { return e.id; }

    static Node nodeFromId(int id) { return Node(id);}

    static Arc arcFromId(int id) { return Arc(id);}

    static Edge edgeFromId(int id) { return Edge(id);}

    bool valid(Node n) const {

      return n.id >= 0 && n.id < static_cast<int>(nodes.size()) &&

        nodes[n.id].prev != -2;

    }

    bool valid(Arc a) const {

      return a.id >= 0 && a.id < static_cast<int>(arcs.size()) &&

        arcs[a.id].prev_out != -2;

    }

    bool valid(Edge e) const {

      return e.id >= 0 && 2 * e.id < static_cast<int>(arcs.size()) &&

        arcs[2 * e.id].prev_out != -2;

    }

    Node addNode() {

      int n;

      if(first_free_node==-1) {

        n = nodes.size();

        nodes.push_back(NodeT());

      } else {

        n = first_free_node;

        first_free_node = nodes[n].next;

      }

      nodes[n].next = first_node;

      if (first_node != -1) nodes[first_node].prev = n;

      first_node = n;

      nodes[n].prev = -1;

      nodes[n].first_out = -1;

      return Node(n);

    }

    Edge addEdge(Node u, Node v) {

      int n;

      if (first_free_arc == -1) {

        n = arcs.size();

        arcs.push_back(ArcT());

        arcs.push_back(ArcT());

      } else {

        n = first_free_arc;

        first_free_arc = arcs[n].next_out;

      }

      arcs[n].target = u.id;

      arcs[n | 1].target = v.id;

      arcs[n].next_out = nodes[v.id].first_out;

      if (nodes[v.id].first_out != -1) {

        arcs[nodes[v.id].first_out].prev_out = n;

      }

      arcs[n].prev_out = -1;

      nodes[v.id].first_out = n;

      arcs[n | 1].next_out = nodes[u.id].first_out;

      if (nodes[u.id].first_out != -1) {

        arcs[nodes[u.id].first_out].prev_out = (n | 1);

      }

      arcs[n | 1].prev_out = -1;

      nodes[u.id].first_out = (n | 1);

      return Edge(n / 2);

    }

    void erase(const Node& node) {

      int n = node.id;

      if(nodes[n].next != -1) {

        nodes[nodes[n].next].prev = nodes[n].prev;

      }

      if(nodes[n].prev != -1) {

        nodes[nodes[n].prev].next = nodes[n].next;

      } else {

        first_node = nodes[n].next;

      }

      nodes[n].next = first_free_node;

      first_free_node = n;

      nodes[n].prev = -2;

    }

    void erase(const Edge& edge) {

      int n = edge.id * 2;

      if (arcs[n].next_out != -1) {

        arcs[arcs[n].next_out].prev_out = arcs[n].prev_out;

      }

      if (arcs[n].prev_out != -1) {

        arcs[arcs[n].prev_out].next_out = arcs[n].next_out;

      } else {

        nodes[arcs[n | 1].target].first_out = arcs[n].next_out;

      }

      if (arcs[n | 1].next_out != -1) {

        arcs[arcs[n | 1].next_out].prev_out = arcs[n | 1].prev_out;

      }

      if (arcs[n | 1].prev_out != -1) {

        arcs[arcs[n | 1].prev_out].next_out = arcs[n | 1].next_out;