/* -*- 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;

         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;

       } else {

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

       }

       arcs[n].next_out = first_free_arc;

       first_free_arc = n;

       arcs[n].prev_out = -2;

       arcs[n | 1].prev_out = -2;

     }

     void clear() {

       arcs.clear();

       nodes.clear();

       first_node = first_free_node = first_free_arc = -1;

     }

   protected:

     void changeV(Edge e, Node n) {

       if(arcs[2 * e.id].next_out != -1) {

         arcs[arcs[2 * e.id].next_out].prev_out = arcs[2 * e.id].prev_out;

       }

       if(arcs[2 * e.id].prev_out != -1) {

         arcs[arcs[2 * e.id].prev_out].next_out =

           arcs[2 * e.id].next_out;

       } else {

         nodes[arcs[(2 * e.id) | 1].target].first_out =

           arcs[2 * e.id].next_out;

       }

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

         arcs[nodes[n.id].first_out].prev_out = 2 * e.id;

       }

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

       arcs[2 * e.id].prev_out = -1;

       arcs[2 * e.id].next_out = nodes[n.id].first_out;

       nodes[n.id].first_out = 2 * e.id;

     }

     void changeU(Edge e, Node n) {

       if(arcs[(2 * e.id) | 1].next_out != -1) {

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

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

       }

       if(arcs[(2 * e.id) | 1].prev_out != -1) {

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

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

       } else {

         nodes[arcs[2 * e.id].target].first_out =

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

       }

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

         arcs[nodes[n.id].first_out].prev_out = ((2 * e.id) | 1);

       }

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

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

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

       nodes[n.id].first_out = ((2 * e.id) | 1);

     }

   };

   typedef GraphExtender<ListGraphBase> ExtendedListGraphBase;

   /// \addtogroup graphs

   /// @{

   ///A general undirected graph structure.

   ///\ref ListGraph is a versatile and fast undirected graph

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

   ///\c std::vector structures.

   ///

   ///This type fully conforms to the \ref concepts::Graph "Graph 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, edges and arcs.

   ///

   ///\sa concepts::Graph

   ///\sa ListDigraph

   class ListGraph : public ExtendedListGraphBase {

     typedef ExtendedListGraphBase Parent;

   private:

     /// Graphs are \e not copy constructible. Use GraphCopy instead.

     ListGraph(const ListGraph &) :ExtendedListGraphBase()  {};

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

     /// Use GraphCopy instead.

     void operator=(const ListGraph &) {}

   public:

     /// Constructor

     /// Constructor.

     ///

     ListGraph() {}

     typedef Parent::OutArcIt IncEdgeIt;

     /// \brief Add a new node to the graph.

     ///

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

     /// \return The new node.

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

     /// \brief Add a new edge to the graph.

     ///

     /// This function adds a new edge to the graph between nodes

     /// \c u and \c v with inherent orientation from node \c u to

     /// node \c v.

     /// \return The new edge.

     Edge addEdge(Node u, Node v) {

       return Parent::addEdge(u, v);

     }

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

     ///

     /// This function erases the given node along with its incident arcs

     /// from the graph.

     ///

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

     /// edges are invalidated, of course.

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

     ///\brief Erase an edge from the graph.

     ///

     /// This function erases the given edge from the graph.

     ///

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

     /// of course.

     void erase(Edge e) { Parent::erase(e); }

     /// Node validity check

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

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

     ///

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

     /// added to the graph.

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

     /// Edge validity check

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

     /// i.e. it is a real edge of the graph.

     ///

     /// \warning A removed edge could become valid again if new edges are

     /// added to the graph.

     bool valid(Edge e) const { return Parent::valid(e); }

     /// Arc validity check

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

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

     ///

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

     /// added to the graph.

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

     /// \brief Change the first node of an edge.

     ///

     /// This function changes the first node of the given edge \c e to \c n.

     ///

     ///\note \c EdgeIt and \c ArcIt iterators referencing the

     ///changed edge are invalidated and all other iterators whose

     ///base node is the changed node are also invalidated.

     ///

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

     ///Snapshot feature.

     void changeU(Edge e, Node n) {

       Parent::changeU(e,n);

     }

     /// \brief Change the second node of an edge.

     ///

     /// This function changes the second node of the given edge \c e to \c n.

     ///

     ///\note \c EdgeIt iterators referencing the changed edge remain

     ///valid, but \c ArcIt iterators referencing the changed edge and

     ///all other iterators whose base node is the changed node are also

     ///invalidated.

     ///

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

     ///Snapshot feature.

     void changeV(Edge e, Node n) {

       Parent::changeV(e,n);

     }

     /// \brief Contract two nodes.

     ///

     /// This function contracts the given two nodes.

     /// Node \c b is removed, but instead of deleting

     /// its incident edges, they are joined to node \c a.

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

     /// then the newly created loops are removed.

     ///

     /// \note The moved edges are joined to node \c a using changeU()

     /// or changeV(), thus all edge and arc iterators whose base node is

     /// \c b are invalidated.

     /// Moreover all iterators referencing node \c b 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 a, Node b, bool r = true) {

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

         IncEdgeIt f = e; ++f;

         if (r && runningNode(e) == a) {

           erase(e);

         } else if (u(e) == b) {

           changeU(e, a);

         } else {

           changeV(e, a);

         }

         e = f;

       }

       erase(b);

     }

     ///Clear the graph.

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

     ///

     ///\note All iterators of the graph 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 graph you want to build will

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

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

     /// to build the graph.

     /// \sa reserveEdge()

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

     /// Reserve memory for edges.

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

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

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

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

     /// to build the graph.

     /// \sa reserveNode()

     void reserveEdge(int m) { arcs.reserve(2 * m); };

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

     /// it later.

     ///

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

     ///

     /// The newly added nodes and edges 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 edges again using

     /// another Snapshot instance.

     ///

     /// \warning Node and edge deletions and other modifications

     /// (e.g. changing the end-nodes of edges or contracting nodes)

     /// cannot be restored. These events invalidate the snapshot.

     /// However, the edges and nodes that were added to the graph 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 EdgeObserverProxy : public EdgeNotifier::ObserverBase {

       public:

         EdgeObserverProxy(Snapshot& _snapshot)

           : snapshot(_snapshot) {}

         using EdgeNotifier::ObserverBase::attach;

         using EdgeNotifier::ObserverBase::detach;

         using EdgeNotifier::ObserverBase::attached;

       protected:

         virtual void add(const Edge& edge) {

           snapshot.addEdge(edge);

         }

         virtual void add(const std::vector<Edge>& edges) {

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

             snapshot.addEdge(edges[i]);

           }

         }

         virtual void erase(const Edge& edge) {

           snapshot.eraseEdge(edge);

         }

         virtual void erase(const std::vector<Edge>& edges) {

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

             snapshot.eraseEdge(edges[i]);

           }

         }

         virtual void build() {

           Edge edge;

           std::vector<Edge> edges;

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

                notifier()->next(edge)) {

             edges.push_back(edge);

           }

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

             snapshot.addEdge(edges[i]);

           }

         }

         virtual void clear() {

           Edge edge;

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

                notifier()->next(edge)) {

             snapshot.eraseEdge(edge);

           }

         }

         Snapshot& snapshot;

       };

       ListGraph *graph;

       NodeObserverProxy node_observer_proxy;

       EdgeObserverProxy edge_observer_proxy;

       std::list<Node> added_nodes;

       std::list<Edge> added_edges;

       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();

           edge_observer_proxy.detach();

           throw NodeNotifier::ImmediateDetach();

         } else {

           added_nodes.erase(it);

         }

       }

       void addEdge(const Edge& edge) {

         added_edges.push_front(edge);

       }

       void eraseEdge(const Edge& edge) {

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

           std::find(added_edges.begin(), added_edges.end(), edge);

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

           clear();

           node_observer_proxy.detach();

           throw EdgeNotifier::ImmediateDetach();

         } else {

           added_edges.erase(it);

         }

       }

       void attach(ListGraph &_graph) {

         graph = &_graph;

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

         edge_observer_proxy.attach(graph->notifier(Edge()));

       }

       void detach() {

         node_observer_proxy.detach();

         edge_observer_proxy.detach();

       }

       bool attached() const {

         return node_observer_proxy.attached();

       }

       void clear() {

         added_nodes.clear();

         added_edges.clear();

       }

     public:

       /// \brief Default constructor.

       ///

       /// Default constructor.

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

       Snapshot()

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

           edge_observer_proxy(*this) {}

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

       ///

       /// This constructor immediately makes a snapshot of the given graph.

       Snapshot(ListGraph &gr)

         : node_observer_proxy(*this),

           edge_observer_proxy(*this) {

         attach(gr);

       }

       /// \brief Make a snapshot.

       ///

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

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

       /// call, the previous snapshot gets lost.

       void save(ListGraph &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(ListGraph&).

       ///

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

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

       void restore() {

         detach();

         for(std::list<Edge>::iterator it = added_edges.begin();

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

           graph->erase(*it);

         }

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

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

           graph->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 ListBpGraphBase {

   protected:

     struct NodeT {

       int first_out;

       int prev, next;

       int partition_prev, partition_next;

       int partition_index;

       bool red;

     };

     struct ArcT {

       int target;

       int prev_out, next_out;

     };

     std::vector<NodeT> nodes;

     int first_node, first_red, first_blue;

     int max_red, max_blue;

     int first_free_red, first_free_blue;

     std::vector<ArcT> arcs;

     int first_free_arc;

   public:

     typedef ListBpGraphBase BpGraph;

     class Node {

       friend class ListBpGraphBase;

     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 RedNode : public Node {

       friend class ListBpGraphBase;

     protected:

       explicit RedNode(int pid) : Node(pid) {}

     public:

       RedNode() {}

       RedNode(const RedNode& node) : Node(node) {}

       RedNode(Invalid) : Node(INVALID){}

     };

     class BlueNode : public Node {

       friend class ListBpGraphBase;

     protected:

       explicit BlueNode(int pid) : Node(pid) {}

     public:

       BlueNode() {}

       BlueNode(const BlueNode& node) : Node(node) {}

       BlueNode(Invalid) : Node(INVALID){}

     };

     class Edge {

       friend class ListBpGraphBase;

     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 ListBpGraphBase;

     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;}

     };

     ListBpGraphBase()

       : nodes(), first_node(-1),

         first_red(-1), first_blue(-1),

         max_red(-1), max_blue(-1),

         first_free_red(-1), first_free_blue(-1),

         arcs(), first_free_arc(-1) {}

     bool red(Node n) const { return nodes[n.id].red; }

     bool blue(Node n) const { return !nodes[n.id].red; }

     static RedNode asRedNodeUnsafe(Node n) { return RedNode(n.id); }

     static BlueNode asBlueNodeUnsafe(Node n) { return BlueNode(n.id); }

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

     int maxRedId() const { return max_red; }

     int maxBlueId() const { return max_blue; }

     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); }

     RedNode redNode(Edge e) const {

       return RedNode(arcs[2 * e.id].target);

     }

     BlueNode blueNode(Edge e) const {

       return BlueNode(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(RedNode& node) const {

       node.id = first_red;

     }

     void next(RedNode& node) const {

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

     }

     void first(BlueNode& node) const {

       node.id = first_blue;

     }

     void next(BlueNode& node) const {

       node.id = nodes[node.id].partition_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; }

     int id(RedNode v) const { return nodes[v.id].partition_index; }

     int id(BlueNode v) const { return nodes[v.id].partition_index; }

     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;

     }

     RedNode addRedNode() {

       int n;

       if(first_free_red==-1) {

         n = nodes.size();

         nodes.push_back(NodeT());

         nodes[n].partition_index = ++max_red;

         nodes[n].red = true;

       } else {

         n = first_free_red;

         first_free_red = 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].partition_next = first_red;

       if (first_red != -1) nodes[first_red].partition_prev = n;

       first_red = n;

       nodes[n].partition_prev = -1;

       nodes[n].first_out = -1;

       return RedNode(n);

     }

     BlueNode addBlueNode() {

       int n;

       if(first_free_blue==-1) {

         n = nodes.size();

         nodes.push_back(NodeT());

         nodes[n].partition_index = ++max_blue;

         nodes[n].red = false;

       } else {

         n = first_free_blue;

         first_free_blue = 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].partition_next = first_blue;

       if (first_blue != -1) nodes[first_blue].partition_prev = n;

       first_blue = n;

       nodes[n].partition_prev = -1;

       nodes[n].first_out = -1;

       return BlueNode(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;

       }

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

         nodes[nodes[n].partition_next].partition_prev = nodes[n].partition_prev;

       }

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

         nodes[nodes[n].partition_prev].partition_next = nodes[n].partition_next;

       } else {

         if (nodes[n].red) {

           first_red = nodes[n].partition_next;

         } else {

           first_blue = nodes[n].partition_next;

         }

       }

       if (nodes[n].red) {

         nodes[n].next = first_free_red;

         first_free_red = n;

       } else {

         nodes[n].next = first_free_blue;

         first_free_blue = 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;

       } else {

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

       }

       arcs[n].next_out = first_free_arc;

       first_free_arc = n;

       arcs[n].prev_out = -2;

       arcs[n | 1].prev_out = -2;

     }

     void clear() {

       arcs.clear();

       nodes.clear();

       first_node = first_free_arc = first_red = first_blue =

         max_red = max_blue = first_free_red = first_free_blue = -1;

     }

   protected:

     void changeRed(Edge e, RedNode n) {

       if(arcs[(2 * e.id) | 1].next_out != -1) {

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

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

       }

       if(arcs[(2 * e.id) | 1].prev_out != -1) {

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

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

       } else {

         nodes[arcs[2 * e.id].target].first_out =

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

       }

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

         arcs[nodes[n.id].first_out].prev_out = ((2 * e.id) | 1);

       }

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

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

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

       nodes[n.id].first_out = ((2 * e.id) | 1);

     }

     void changeBlue(Edge e, BlueNode n) {

        if(arcs[2 * e.id].next_out != -1) {

         arcs[arcs[2 * e.id].next_out].prev_out = arcs[2 * e.id].prev_out;

       }

       if(arcs[2 * e.id].prev_out != -1) {

         arcs[arcs[2 * e.id].prev_out].next_out =

           arcs[2 * e.id].next_out;

       } else {

         nodes[arcs[(2 * e.id) | 1].target].first_out =

           arcs[2 * e.id].next_out;

       }

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

         arcs[nodes[n.id].first_out].prev_out = 2 * e.id;

       }

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

       arcs[2 * e.id].prev_out = -1;

       arcs[2 * e.id].next_out = nodes[n.id].first_out;

       nodes[n.id].first_out = 2 * e.id;

     }

   };

   typedef BpGraphExtender<ListBpGraphBase> ExtendedListBpGraphBase;

   /// \addtogroup graphs

   /// @{

   ///A general undirected graph structure.

   ///\ref ListBpGraph is a versatile and fast undirected graph

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

   ///\c std::vector structures.

   ///

   ///This type fully conforms to the \ref concepts::BpGraph "BpGraph 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, edges and arcs.

   ///

   ///\sa concepts::BpGraph

   ///\sa ListDigraph

   class ListBpGraph : public ExtendedListBpGraphBase {

     typedef ExtendedListBpGraphBase Parent;

   private:

     /// BpGraphs are \e not copy constructible. Use BpGraphCopy instead.

     ListBpGraph(const ListBpGraph &) :ExtendedListBpGraphBase()  {};

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

     /// Use BpGraphCopy instead.

     void operator=(const ListBpGraph &) {}

   public:

     /// Constructor

     /// Constructor.

     ///

     ListBpGraph() {}

     typedef Parent::OutArcIt IncEdgeIt;

     /// \brief Add a new red node to the graph.

     ///

     /// This function adds a red new node to the graph.

     /// \return The new node.

     RedNode addRedNode() { return Parent::addRedNode(); }

     /// \brief Add a new blue node to the graph.

     ///

     /// This function adds a blue new node to the graph.

     /// \return The new node.

     BlueNode addBlueNode() { return Parent::addBlueNode(); }

     /// \brief Add a new edge to the graph.

     ///

     /// This function adds a new edge to the graph between nodes

     /// \c u and \c v with inherent orientation from node \c u to

     /// node \c v.

     /// \return The new edge.

     Edge addEdge(RedNode u, BlueNode v) {

       return Parent::addEdge(u, v);

     }

     Edge addEdge(BlueNode v, RedNode u) {

       return Parent::addEdge(u, v);

     }

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

     ///

     /// This function erases the given node along with its incident arcs

     /// from the graph.

     ///

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

     /// edges are invalidated, of course.

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

     ///\brief Erase an edge from the graph.

     ///

     /// This function erases the given edge from the graph.

     ///

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

     /// of course.

     void erase(Edge e) { Parent::erase(e); }

     /// Node validity check

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

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

     ///

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

     /// added to the graph.

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

     /// Edge validity check

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

     /// i.e. it is a real edge of the graph.

     ///

     /// \warning A removed edge could become valid again if new edges are

     /// added to the graph.

     bool valid(Edge e) const { return Parent::valid(e); }

     /// Arc validity check

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

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

     ///

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

     /// added to the graph.

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

     /// \brief Change the red node of an edge.

     ///

     /// This function changes the red node of the given edge \c e to \c n.

     ///

     ///\note \c EdgeIt and \c ArcIt iterators referencing the

     ///changed edge are invalidated and all other iterators whose

     ///base node is the changed node are also invalidated.

     ///

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

     ///Snapshot feature.

     void changeRed(Edge e, RedNode n) {

       Parent::changeRed(e, n);

     }

     /// \brief Change the blue node of an edge.

     ///

     /// This function changes the blue node of the given edge \c e to \c n.

     ///

     ///\note \c EdgeIt iterators referencing the changed edge remain

     ///valid, but \c ArcIt iterators referencing the changed edge and

     ///all other iterators whose base node is the changed node are also

     ///invalidated.

     ///

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

     ///Snapshot feature.

     void changeBlue(Edge e, BlueNode n) {

       Parent::changeBlue(e, n);

     }

     ///Clear the graph.

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

     ///

     ///\note All iterators of the graph 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 graph you want to build will

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

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

     /// to build the graph.

     /// \sa reserveEdge()

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

     /// Reserve memory for edges.

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

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

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

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

     /// to build the graph.

     /// \sa reserveNode()

     void reserveEdge(int m) { arcs.reserve(2 * m); };

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

     /// it later.

     ///

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

     ///

     /// The newly added nodes and edges 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 edges again using

     /// another Snapshot instance.

     ///

     /// \warning Node and edge deletions and other modifications

     /// (e.g. changing the end-nodes of edges or contracting nodes)

     /// cannot be restored. These events invalidate the snapshot.

     /// However, the edges and nodes that were added to the graph 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 EdgeObserverProxy : public EdgeNotifier::ObserverBase {

       public:

         EdgeObserverProxy(Snapshot& _snapshot)

           : snapshot(_snapshot) {}

         using EdgeNotifier::ObserverBase::attach;

         using EdgeNotifier::ObserverBase::detach;

         using EdgeNotifier::ObserverBase::attached;

       protected:

         virtual void add(const Edge& edge) {

           snapshot.addEdge(edge);

         }

         virtual void add(const std::vector<Edge>& edges) {

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

             snapshot.addEdge(edges[i]);

           }

         }

         virtual void erase(const Edge& edge) {

           snapshot.eraseEdge(edge);

         }

         virtual void erase(const std::vector<Edge>& edges) {

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

             snapshot.eraseEdge(edges[i]);

           }

         }

         virtual void build() {

           Edge edge;

           std::vector<Edge> edges;

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

                notifier()->next(edge)) {

             edges.push_back(edge);

           }

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

             snapshot.addEdge(edges[i]);

           }

         }

         virtual void clear() {

           Edge edge;

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

                notifier()->next(edge)) {

             snapshot.eraseEdge(edge);

           }

         }

         Snapshot& snapshot;

       };

       ListBpGraph *graph;

       NodeObserverProxy node_observer_proxy;

       EdgeObserverProxy edge_observer_proxy;

       std::list<Node> added_nodes;

       std::list<Edge> added_edges;

       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();

           edge_observer_proxy.detach();

           throw NodeNotifier::ImmediateDetach();

         } else {

           added_nodes.erase(it);

         }

       }

       void addEdge(const Edge& edge) {

         added_edges.push_front(edge);

       }

       void eraseEdge(const Edge& edge) {

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

           std::find(added_edges.begin(), added_edges.end(), edge);

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

           clear();

           node_observer_proxy.detach();

           throw EdgeNotifier::ImmediateDetach();

         } else {

           added_edges.erase(it);

         }

       }

       void attach(ListBpGraph &_graph) {

         graph = &_graph;

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

         edge_observer_proxy.attach(graph->notifier(Edge()));

       }

       void detach() {

         node_observer_proxy.detach();

         edge_observer_proxy.detach();

       }

       bool attached() const {

         return node_observer_proxy.attached();

       }

       void clear() {

         added_nodes.clear();

         added_edges.clear();

       }

     public:

       /// \brief Default constructor.

       ///

       /// Default constructor.

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

       Snapshot()

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

           edge_observer_proxy(*this) {}

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

       ///

       /// This constructor immediately makes a snapshot of the given graph.

       Snapshot(ListBpGraph &gr)

         : node_observer_proxy(*this),

           edge_observer_proxy(*this) {

         attach(gr);

       }

       /// \brief Make a snapshot.

       ///

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

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

       /// call, the previous snapshot gets lost.

       void save(ListBpGraph &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(ListBpGraph&).

       ///

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

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

       void restore() {

         detach();

         for(std::list<Edge>::iterator it = added_edges.begin();

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

           graph->erase(*it);

         }

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

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

           graph->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();

       }

     };

   };

   /// @}

 } //namespace lemon

 #endif