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

 *

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

 *

 * Copyright (C) 2003-2013

 * 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_LIST_GRAPH_H

#define LEMON_LIST_GRAPH_H

///\ingroup graphs

///\file

///\brief ListDigraph and ListGraph classes.

#include <lemon/core.h>

#include <lemon/error.h>

#include <lemon/bits/graph_extender.h>

#include <vector>

#include <list>

namespace lemon {

  class ListDigraph;

  class ListDigraphBase {

  protected:

    struct NodeT {

      int first_in, first_out;

      int prev, next;

    };

    struct ArcT {

      int target, source;

      int prev_in, prev_out;

      int next_in, next_out;

    };

    std::vector<NodeT> _nodes;

    int first_node;

    int first_free_node;

    std::vector<ArcT> _arcs;

    int first_free_arc;

  public:

    typedef ListDigraphBase Digraph;

    class Node {

      friend class ListDigraphBase;

      friend class ListDigraph;

    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 Arc {

      friend class ListDigraphBase;

      friend class ListDigraph;

    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;}

    };

    ListDigraphBase()

      : _nodes(), first_node(-1),

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

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

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

    Node source(Arc e) const { return Node(_arcs[e.id].source); }

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

    void first(Node& node) const {

      node.id = first_node;

    }

    void next(Node& node) const {

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

    }

    void first(Arc& arc) const {

      int n;

      for(n = first_node;

          n != -1 && _nodes[n].first_out == -1;

          n = _nodes[n].next) {}

      arc.id = (n == -1) ? -1 : _nodes[n].first_out;

    }

    void next(Arc& arc) const {

      if (_arcs[arc.id].next_out != -1) {

        arc.id = _arcs[arc.id].next_out;

      } else {

        int n;

        for(n = _nodes[_arcs[arc.id].source].next;

            n != -1 && _nodes[n].first_out == -1;

            n = _nodes[n].next) {}

        arc.id = (n == -1) ? -1 : _nodes[n].first_out;

      }

    }

    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_in;

    }

    void nextIn(Arc &e) const {

      e.id=_arcs[e.id].next_in;

    }

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

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

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

    static Arc arcFromId(int id) { return Arc(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_in != -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_in = _nodes[n].first_out = -1;

      return Node(n);

    }

    Arc addArc(Node u, Node v) {

      int n;

      if (first_free_arc == -1) {

        n = _arcs.size();

        _arcs.push_back(ArcT());

      } else {

        n = first_free_arc;

        first_free_arc = _arcs[n].next_in;

      }

      _arcs[n].source = u.id;

      _arcs[n].target = v.id;

      _arcs[n].next_out = _nodes[u.id].first_out;

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

        _arcs[_nodes[u.id].first_out].prev_out = n;

      }

      _arcs[n].next_in = _nodes[v.id].first_in;

      if(_nodes[v.id].first_in != -1) {

        _arcs[_nodes[v.id].first_in].prev_in = n;

      }

      _arcs[n].prev_in = _arcs[n].prev_out = -1;

      _nodes[u.id].first_out = _nodes[v.id].first_in = n;

      return Arc(n);

    }

    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 Arc& arc) {

      int n = arc.id;

      if(_arcs[n].next_in!=-1) {

        _arcs[_arcs[n].next_in].prev_in = _arcs[n].prev_in;

      }

      if(_arcs[n].prev_in!=-1) {

        _arcs[_arcs[n].prev_in].next_in = _arcs[n].next_in;

      } else {

        _nodes[_arcs[n].target].first_in = _arcs[n].next_in;

      }

      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].source].first_out = _arcs[n].next_out;

      }

      _arcs[n].next_in = first_free_arc;

      first_free_arc = n;

      _arcs[n].prev_in = -2;

    }

    void clear() {

      _arcs.clear();

      _nodes.clear();

      first_node = first_free_node = first_free_arc = -1;

    }

  protected:

    void changeTarget(Arc e, Node n)

    {

      if(_arcs[e.id].next_in != -1)

        _arcs[_arcs[e.id].next_in].prev_in = _arcs[e.id].prev_in;

      if(_arcs[e.id].prev_in != -1)

        _arcs[_arcs[e.id].prev_in].next_in = _arcs[e.id].next_in;

      else _nodes[_arcs[e.id].target].first_in = _arcs[e.id].next_in;

      if (_nodes[n.id].first_in != -1) {

        _arcs[_nodes[n.id].first_in].prev_in = e.id;

      }

      _arcs[e.id].target = n.id;

      _arcs[e.id].prev_in = -1;

      _arcs[e.id].next_in = _nodes[n.id].first_in;

      _nodes[n.id].first_in = e.id;

    }

    void changeSource(Arc e, Node n)

    {

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

        _arcs[_arcs[e.id].next_out].prev_out = _arcs[e.id].prev_out;

      if(_arcs[e.id].prev_out != -1)

        _arcs[_arcs[e.id].prev_out].next_out = _arcs[e.id].next_out;

      else _nodes[_arcs[e.id].source].first_out = _arcs[e.id].next_out;

      if (_nodes[n.id].first_out != -1) {

        _arcs[_nodes[n.id].first_out].prev_out = e.id;

      }

      _arcs[e.id].source = n.id;

      _arcs[e.id].prev_out = -1;

      _arcs[e.id].next_out = _nodes[n.id].first_out;

      _nodes[n.id].first_out = e.id;

    }

  };

  typedef DigraphExtender<ListDigraphBase> ExtendedListDigraphBase;

  /// \addtogroup graphs

  /// @{

  ///A general directed graph structure.

  ///\ref ListDigraph is a versatile and fast directed graph

  ///implementation based on linked lists that are stored in

  ///\c std::vector structures.

  ///

  ///This type fully conforms to the \ref concepts::Digraph "Digraph concept"

  ///and it also provides several useful additional functionalities.

  ///Most of its member functions and nested classes are documented

  ///only in the concept class.

  ///

  ///This class provides only linear time counting for nodes and arcs.

  ///

  ///\sa concepts::Digraph

  ///\sa ListGraph

  class ListDigraph : public ExtendedListDigraphBase {

    typedef ExtendedListDigraphBase Parent;

  private:

    /// Digraphs are \e not copy constructible. Use DigraphCopy instead.

    ListDigraph(const ListDigraph &) :ExtendedListDigraphBase() {};

    /// \brief Assignment of a digraph to another one is \e not allowed.

    /// Use DigraphCopy instead.

    void operator=(const ListDigraph &) {}

  public:

    /// Constructor

    /// Constructor.

    ///

    ListDigraph() {}

    ///Add a new node to the digraph.

    ///This function adds a new node to the digraph.

    ///\return The new node.

    Node addNode() { return Parent::addNode(); }

    ///Add a new arc to the digraph.

    ///This function adds a new arc to the digraph with source node \c s

    ///and target node \c t.

    ///\return The new arc.

    Arc addArc(Node s, Node t) {

      return Parent::addArc(s, t);

    }

    ///\brief Erase a node from the digraph.

    ///

    ///This function erases the given node along with its outgoing and

    ///incoming arcs from the digraph.

    ///

    ///\note All iterators referencing the removed node or the connected

    ///arcs are invalidated, of course.

    void erase(Node n) { Parent::erase(n); }

    ///\brief Erase an arc from the digraph.

    ///

    ///This function erases the given arc from the digraph.

    ///

    ///\note All iterators referencing the removed arc are invalidated,

    ///of course.

    void erase(Arc a) { Parent::erase(a); }

    /// Node validity check

    /// This function gives back \c true if the given node is valid,

    /// i.e. it is a real node of the digraph.

    ///

    /// \warning A removed node could become valid again if new nodes are

    /// added to the digraph.

    bool valid(Node n) const { return Parent::valid(n); }

    /// Arc validity check

    /// This function gives back \c true if the given arc is valid,

    /// i.e. it is a real arc of the digraph.

    ///

    /// \warning A removed arc could become valid again if new arcs are

    /// added to the digraph.

    bool valid(Arc a) const { return Parent::valid(a); }

    /// Change the target node of an arc

    /// This function changes the target node of the given arc \c a to \c n.

    ///

    ///\note \c ArcIt and \c OutArcIt iterators referencing the changed

    ///arc remain valid, but \c InArcIt iterators are invalidated.

    ///

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

    ///feature.

    void changeTarget(Arc a, Node n) {

      Parent::changeTarget(a,n);

    }

    /// Change the source node of an arc

    /// This function changes the source node of the given arc \c a to \c n.

    ///

    ///\note \c InArcIt iterators referencing the changed arc remain

    ///valid, but \c ArcIt and \c OutArcIt iterators are invalidated.

    ///

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

    ///feature.

    void changeSource(Arc a, Node n) {

      Parent::changeSource(a,n);

    }

    /// Reverse the direction of an arc.

    /// This function reverses the direction of the given arc.

    ///\note \c ArcIt, \c OutArcIt and \c InArcIt iterators referencing

    ///the changed arc are invalidated.

    ///

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

    ///feature.

    void reverseArc(Arc a) {

      Node t=target(a);

      changeTarget(a,source(a));

      changeSource(a,t);

    }

    ///Contract two nodes.

    ///This function contracts the given two nodes.

    ///Node \c v is removed, but instead of deleting its

    ///incident arcs, they are joined to node \c u.

    ///If the last parameter \c r is \c true (this is the default value),

    ///then the newly created loops are removed.

    ///

    ///\note The moved arcs are joined to node \c u using changeSource()

    ///or changeTarget(), thus \c ArcIt and \c OutArcIt iterators are

    ///invalidated for the outgoing arcs of node \c v and \c InArcIt

    ///iterators are invalidated for the incoming arcs of \c v.

    ///Moreover all iterators referencing node \c v or the removed

    ///loops are also invalidated. Other iterators remain valid.

    ///

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

    ///feature.

    void contract(Node u, Node v, bool r = true)

    {

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

        OutArcIt f=e;

        ++f;

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

        else changeSource(e,u);

        e=f;

      }

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

        InArcIt f=e;

        ++f;

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

        else changeTarget(e,u);

        e=f;

      }

      erase(v);

    }

    ///Split a node.

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

    ///to the digraph, then the source of each outgoing arc of node \c n

    ///is moved to this new node.

    ///If the second parameter \c connect is \c true (this is the default

    ///value), then a new arc from node \c n to the newly created node

    ///is also added.

    ///\return The newly created node.

    ///

    ///\note All iterators remain valid.

    ///

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

    ///Snapshot feature.

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

      Node b = addNode();

      _nodes[b.id].first_out=_nodes[n.id].first_out;

      _nodes[n.id].first_out=-1;

      for(int i=_nodes[b.id].first_out; i!=-1; i=_arcs[i].next_out) {

        _arcs[i].source=b.id;

      }

      if (connect) addArc(n,b);

      return b;

    }

    ///Split an arc.

    ///This function splits the given arc. First, a new node \c v is

    ///added to the digraph, then the target node of the original arc

    ///is set to \c v. Finally, an arc from \c v to the original target

    ///is added.

    ///\return The newly created node.

    ///

    ///\note \c InArcIt iterators referencing the original arc are

    ///invalidated. Other iterators remain valid.

    ///

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

    ///Snapshot feature.

    Node split(Arc a) {

      Node v = addNode();

      addArc(v,target(a));

      changeTarget(a,v);

      return v;

    }

    ///Clear the digraph.

    ///This function erases all nodes and arcs from the digraph.

    ///

    ///\note All iterators of the digraph are invalidated, of course.

    void clear() {

      Parent::clear();

    }

    /// Reserve memory for nodes.

    /// Using this function, it is possible to avoid superfluous memory

    /// allocation: if you know that the digraph you want to build will

    /// be large (e.g. it will contain millions of nodes and/or arcs),

    /// then it is worth reserving space for this amount before starting

    /// to build the digraph.

    /// \sa reserveArc()

    void reserveNode(int n) { _nodes.reserve(n); };

    /// Reserve memory for arcs.

    /// Using this function, it is possible to avoid superfluous memory

    /// allocation: if you know that the digraph you want to build will

    /// be large (e.g. it will contain millions of nodes and/or arcs),

    /// then it is worth reserving space for this amount before starting

    /// to build the digraph.

    /// \sa reserveNode()

    void reserveArc(int m) { _arcs.reserve(m); };

    /// \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.

    ///

    /// \note After a state is restored, you cannot restore a later state,

    /// i.e. you cannot add the removed nodes and arcs again using

    /// another Snapshot instance.

    ///

    /// \warning Node and arc deletions and other modifications (e.g.

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

    /// restored. These events invalidate the snapshot.

    /// However, the arcs and nodes that were added to the digraph after

    /// making the current snapshot can be removed without invalidating it.

    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.

      /// You have to call save() to actually make a snapshot.

      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 given digraph.

      Snapshot(ListDigraph &gr)

        : node_observer_proxy(*this),

          arc_observer_proxy(*this) {

        attach(gr);

      }

      /// \brief Make a snapshot.

      ///

      /// This function makes a snapshot of the given digraph.

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

      /// call, the previous snapshot gets lost.

      void save(ListDigraph &gr) {

        if (attached()) {

          detach();

          clear();

        }

        attach(gr);

      }

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

      ///

      /// This function undos the changes until the last snapshot

      /// created by save() or Snapshot(ListDigraph&).

      ///

      /// \warning This method invalidates the snapshot, i.e. repeated

      /// restoring is not supported unless you call save() again.

      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 Returns \c true if the snapshot is valid.

      ///

      /// This function returns \c true if 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 Graph;

    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:

      operator Edge() const {

        return id != -1 ? edgeFromId(id / 2) : INVALID;

      }

      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;