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

  *

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

  *

  * Copyright (C) 2003-2009

  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

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

  *

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

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

  * precise terms see the accompanying LICENSE file.

  *

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

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

  * purpose.

  *

  */

 #ifndef LEMON_BFS_H

 #define LEMON_BFS_H

 ///\ingroup search

 ///\file

 ///\brief BFS algorithm.

 #include <lemon/list_graph.h>

 #include <lemon/bits/path_dump.h>

 #include <lemon/core.h>

 #include <lemon/error.h>

 #include <lemon/maps.h>

 #include <lemon/path.h>

 namespace lemon {

   ///Default traits class of Bfs class.

   ///Default traits class of Bfs class.

   ///\tparam GR Digraph type.

   template<class GR>

   struct BfsDefaultTraits

   {

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

     typedef GR Digraph;

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

     ///arcs of the shortest paths.

     ///

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

     ///arcs of the shortest paths.

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

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

     ///Instantiates a \c PredMap.

     ///This function instantiates a \ref PredMap.

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

     ///\ref PredMap.

     static PredMap *createPredMap(const Digraph &g)

     {

       return new PredMap(g);

     }

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

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

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

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

     ///Instantiates a \c ProcessedMap.

     ///This function instantiates a \ref ProcessedMap.

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

     ///we would like to define the \ref ProcessedMap

 #ifdef DOXYGEN

     static ProcessedMap *createProcessedMap(const Digraph &g)

 #else

     static ProcessedMap *createProcessedMap(const Digraph &)

 #endif

     {

       return new ProcessedMap();

     }

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

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

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

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

     ///Instantiates a \c ReachedMap.

     ///This function instantiates a \ref ReachedMap.

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

     ///we would like to define the \ref ReachedMap.

     static ReachedMap *createReachedMap(const Digraph &g)

     {

       return new ReachedMap(g);

     }

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

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

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

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

     ///Instantiates a \c DistMap.

     ///This function instantiates a \ref DistMap.

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

     ///\ref DistMap.

     static DistMap *createDistMap(const Digraph &g)

     {

       return new DistMap(g);

     }

   };

   ///%BFS algorithm class.

   ///\ingroup search

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

   ///

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

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

   ///used easier.

   ///

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

   ///The default type is \ref ListDigraph.

 #ifdef DOXYGEN

   template <typename GR,

             typename TR>

 #else

   template <typename GR=ListDigraph,

             typename TR=BfsDefaultTraits<GR> >

 #endif

   class Bfs {

   public:

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

     typedef typename TR::Digraph Digraph;

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

     ///shortest paths.

     typedef typename TR::PredMap PredMap;

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

     typedef typename TR::DistMap DistMap;

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

     typedef typename TR::ReachedMap ReachedMap;

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

     typedef typename TR::ProcessedMap ProcessedMap;

     ///The type of the paths.

     typedef PredMapPath<Digraph, PredMap> Path;

     ///The \ref BfsDefaultTraits "traits class" of the algorithm.

     typedef TR Traits;

   private:

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::OutArcIt OutArcIt;

     //Pointer to the underlying digraph.

     const Digraph *G;

     //Pointer to the map of predecessor arcs.

     PredMap *_pred;

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

     bool local_pred;

     //Pointer to the map of distances.

     DistMap *_dist;

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

     bool local_dist;

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

     ReachedMap *_reached;

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

     bool local_reached;

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

     ProcessedMap *_processed;

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

     bool local_processed;

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

     int _queue_head,_queue_tail,_queue_next_dist;

     int _curr_dist;

     //Creates the maps if necessary.

     void create_maps()

     {

       if(!_pred) {

         local_pred = true;

         _pred = Traits::createPredMap(*G);

       }

       if(!_dist) {

         local_dist = true;

         _dist = Traits::createDistMap(*G);

       }

       if(!_reached) {

         local_reached = true;

         _reached = Traits::createReachedMap(*G);

       }

       if(!_processed) {

         local_processed = true;

         _processed = Traits::createProcessedMap(*G);

       }

     }

   protected:

     Bfs() {}

   public:

     typedef Bfs Create;

     ///\name Named Template Parameters

     ///@{

     template <class T>

     struct SetPredMapTraits : public Traits {

       typedef T PredMap;

       static PredMap *createPredMap(const Digraph &)

       {

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

         return 0; // ignore warnings

       }

     };

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

     ///\c PredMap type.

     ///

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

     ///\c PredMap type.

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

     template <class T>

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

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

     };

     template <class T>

     struct SetDistMapTraits : public Traits {

       typedef T DistMap;

       static DistMap *createDistMap(const Digraph &)

       {

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

         return 0; // ignore warnings

       }

     };

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

     ///\c DistMap type.

     ///

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

     ///\c DistMap type.

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

     template <class T>

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

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

     };

     template <class T>

     struct SetReachedMapTraits : public Traits {

       typedef T ReachedMap;

       static ReachedMap *createReachedMap(const Digraph &)

       {

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

         return 0; // ignore warnings

       }

     };

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

     ///\c ReachedMap type.

     ///

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

     ///\c ReachedMap type.

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

     template <class T>

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

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

     };

     template <class T>

     struct SetProcessedMapTraits : public Traits {

       typedef T ProcessedMap;

       static ProcessedMap *createProcessedMap(const Digraph &)

       {

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

         return 0; // ignore warnings

       }

     };

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

     ///\c ProcessedMap type.

     ///

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

     ///\c ProcessedMap type.

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

     template <class T>

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

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

     };

     struct SetStandardProcessedMapTraits : public Traits {

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

       static ProcessedMap *createProcessedMap(const Digraph &g)

       {

         return new ProcessedMap(g);

         return 0; // ignore warnings

       }

     };

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

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

     ///

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

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

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

     struct SetStandardProcessedMap :

       public Bfs< Digraph, SetStandardProcessedMapTraits > {

       typedef Bfs< Digraph, SetStandardProcessedMapTraits > Create;

     };

     ///@}

   public:

     ///Constructor.

     ///Constructor.

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

     Bfs(const Digraph &g) :

       G(&g),

       _pred(NULL), local_pred(false),

       _dist(NULL), local_dist(false),

       _reached(NULL), local_reached(false),

       _processed(NULL), local_processed(false)

     { }

     ///Destructor.

     ~Bfs()

     {

       if(local_pred) delete _pred;

       if(local_dist) delete _dist;

       if(local_reached) delete _reached;

       if(local_processed) delete _processed;

     }

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

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

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

     ///or \ref init(), an instance will be allocated automatically.

     ///The destructor deallocates this automatically allocated map,

     ///of course.

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

     Bfs &predMap(PredMap &m)

     {

       if(local_pred) {

         delete _pred;

         local_pred=false;

       }

       _pred = &m;

       return *this;

     }

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

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

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

     ///or \ref init(), an instance will be allocated automatically.

     ///The destructor deallocates this automatically allocated map,

     ///of course.

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

     Bfs &reachedMap(ReachedMap &m)

     {

       if(local_reached) {

         delete _reached;

         local_reached=false;

       }

       _reached = &m;

       return *this;

     }

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

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

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

     ///or \ref init(), an instance will be allocated automatically.

     ///The destructor deallocates this automatically allocated map,

     ///of course.

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

     Bfs &processedMap(ProcessedMap &m)

     {

       if(local_processed) {

         delete _processed;

         local_processed=false;

       }

       _processed = &m;

       return *this;

     }

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

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

     ///the algorithm.

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

     ///or \ref init(), an instance will be allocated automatically.

     ///The destructor deallocates this automatically allocated map,

     ///of course.

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

     Bfs &distMap(DistMap &m)

     {

       if(local_dist) {

         delete _dist;

         local_dist=false;

       }

       _dist = &m;

       return *this;

     }

   public:

     ///\name Execution Control

     ///The simplest way to execute the BFS algorithm is to use one of the

     ///member functions called \ref run(Node) "run()".\n

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

     ///\ref init(), then you can add several source nodes with

     ///\ref addSource(). Finally the actual path computation can be

     ///performed with one of the \ref start() functions.

     ///@{

     ///\brief Initializes the internal data structures.

     ///

     ///Initializes the internal data structures.

     void init()

     {

       create_maps();

       _queue.resize(countNodes(*G));

       _queue_head=_queue_tail=0;

       _curr_dist=1;

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

         _pred->set(u,INVALID);

         _reached->set(u,false);

         _processed->set(u,false);

       }

     }

     ///Adds a new source node.

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

     ///

     void addSource(Node s)

     {

       if(!(*_reached)[s])

         {

           _reached->set(s,true);

           _pred->set(s,INVALID);

           _dist->set(s,0);

           _queue[_queue_head++]=s;

           _queue_next_dist=_queue_head;

         }

     }

     ///Processes the next node.

     ///Processes the next node.

     ///

     ///\return The processed node.

     ///

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

     Node processNextNode()

     {

       if(_queue_tail==_queue_next_dist) {

         _curr_dist++;

         _queue_next_dist=_queue_head;

       }

       Node n=_queue[_queue_tail++];

       _processed->set(n,true);

       Node m;

       for(OutArcIt e(*G,n);e!=INVALID;++e)

         if(!(*_reached)[m=G->target(e)]) {

           _queue[_queue_head++]=m;

           _reached->set(m,true);

           _pred->set(m,e);

           _dist->set(m,_curr_dist);

         }

       return n;

     }

     ///Processes the next node.

     ///Processes the next node and checks if the given target node

     ///is reached. If the target node is reachable from the processed

     ///node, then the \c reach parameter will be set to \c true.

     ///

     ///\param target The target node.

     ///\retval reach Indicates if the target node is reached.

     ///It should be initially \c false.

     ///

     ///\return The processed node.

     ///

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

     Node processNextNode(Node target, bool& reach)

     {

       if(_queue_tail==_queue_next_dist) {

         _curr_dist++;

         _queue_next_dist=_queue_head;

       }

       Node n=_queue[_queue_tail++];

       _processed->set(n,true);

       Node m;

       for(OutArcIt e(*G,n);e!=INVALID;++e)

         if(!(*_reached)[m=G->target(e)]) {

           _queue[_queue_head++]=m;

           _reached->set(m,true);

           _pred->set(m,e);

           _dist->set(m,_curr_dist);

           reach = reach || (target == m);

         }

       return n;

     }

     ///Processes the next node.

     ///Processes the next node and checks if at least one of reached

     ///nodes has \c true value in the \c nm node map. If one node

     ///with \c true value is reachable from the processed node, then the

     ///\c rnode parameter will be set to the first of such nodes.

     ///

     ///\param nm A \c bool (or convertible) node map that indicates the

     ///possible targets.

     ///\retval rnode The reached target node.

     ///It should be initially \c INVALID.

     ///

     ///\return The processed node.

     ///

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

     template<class NM>

     Node processNextNode(const NM& nm, Node& rnode)

     {

       if(_queue_tail==_queue_next_dist) {

         _curr_dist++;

         _queue_next_dist=_queue_head;

       }

       Node n=_queue[_queue_tail++];

       _processed->set(n,true);

       Node m;

       for(OutArcIt e(*G,n);e!=INVALID;++e)

         if(!(*_reached)[m=G->target(e)]) {

           _queue[_queue_head++]=m;

           _reached->set(m,true);

           _pred->set(m,e);

           _dist->set(m,_curr_dist);

           if (nm[m] && rnode == INVALID) rnode = m;

         }

       return n;

     }

     ///The next node to be processed.

     ///Returns the next node to be processed or \c INVALID if the queue

     ///is empty.

     Node nextNode() const

     {

       return _queue_tail<_queue_head?_queue[_queue_tail]:INVALID;

     }

     ///Returns \c false if there are nodes to be processed.

     ///Returns \c false if there are nodes to be processed

     ///in the queue.

     bool emptyQueue() const { return _queue_tail==_queue_head; }

     ///Returns the number of the nodes to be processed.

     ///Returns the number of the nodes to be processed

     ///in the queue.

     int queueSize() const { return _queue_head-_queue_tail; }

     ///Executes the algorithm.

     ///Executes the algorithm.

     ///

     ///This method runs the %BFS algorithm from the root node(s)

     ///in order to compute the shortest path to each node.

     ///

     ///The algorithm computes

     ///- the shortest path tree (forest),

     ///- the distance of each node from the root(s).

     ///

     ///\pre init() must be called and at least one root node should be

     ///added with addSource() before using this function.

     ///

     ///\note <tt>b.start()</tt> is just a shortcut of the following code.

     ///\code

     ///  while ( !b.emptyQueue() ) {

     ///    b.processNextNode();

     ///  }

     ///\endcode

     void start()

     {

       while ( !emptyQueue() ) processNextNode();

     }

     ///Executes the algorithm until the given target node is reached.

     ///Executes the algorithm until the given target node is reached.

     ///

     ///This method runs the %BFS algorithm from the root node(s)

     ///in order to compute the shortest path to \c t.

     ///

     ///The algorithm computes

     ///- the shortest path to \c t,

     ///- the distance of \c t from the root(s).

     ///

     ///\pre init() must be called and at least one root node should be

     ///added with addSource() before using this function.

     ///

     ///\note <tt>b.start(t)</tt> is just a shortcut of the following code.

     ///\code

     ///  bool reach = false;

     ///  while ( !b.emptyQueue() && !reach ) {

     ///    b.processNextNode(t, reach);

     ///  }

     ///\endcode

     void start(Node t)

     {

       bool reach = false;

       while ( !emptyQueue() && !reach ) processNextNode(t, reach);

     }

     ///Executes the algorithm until a condition is met.

     ///Executes the algorithm until a condition is met.

     ///

     ///This method runs the %BFS algorithm from the root node(s) in

     ///order to compute the shortest path to a node \c v with

     /// <tt>nm[v]</tt> true, if such a node can be found.

     ///

     ///\param nm A \c bool (or convertible) node map. The algorithm

     ///will stop when it reaches a node \c v with <tt>nm[v]</tt> true.

     ///

     ///\return The reached node \c v with <tt>nm[v]</tt> true or

     ///\c INVALID if no such node was found.

     ///

     ///\pre init() must be called and at least one root node should be

     ///added with addSource() before using this function.

     ///

     ///\note <tt>b.start(nm)</tt> is just a shortcut of the following code.

     ///\code

     ///  Node rnode = INVALID;

     ///  while ( !b.emptyQueue() && rnode == INVALID ) {

     ///    b.processNextNode(nm, rnode);

     ///  }

     ///  return rnode;

     ///\endcode

     template<class NodeBoolMap>

     Node start(const NodeBoolMap &nm)

     {

       Node rnode = INVALID;

       while ( !emptyQueue() && rnode == INVALID ) {

         processNextNode(nm, rnode);

       }

       return rnode;

     }

     ///Runs the algorithm from the given source node.

     ///This method runs the %BFS algorithm from node \c s

     ///in order to compute the shortest path to each node.

     ///

     ///The algorithm computes

     ///- the shortest path tree,

     ///- the distance of each node from the root.

     ///

     ///\note <tt>b.run(s)</tt> is just a shortcut of the following code.

     ///\code

     ///  b.init();

     ///  b.addSource(s);

     ///  b.start();

     ///\endcode

     void run(Node s) {

       init();

       addSource(s);

       start();

     }

     ///Finds the shortest path between \c s and \c t.

     ///This method runs the %BFS algorithm from node \c s

     ///in order to compute the shortest path to node \c t

     ///(it stops searching when \c t is processed).

     ///

     ///\return \c true if \c t is reachable form \c s.

     ///

     ///\note Apart from the return value, <tt>b.run(s,t)</tt> is just a

     ///shortcut of the following code.

     ///\code

     ///  b.init();

     ///  b.addSource(s);

     ///  b.start(t);

     ///\endcode

     bool run(Node s,Node t) {

       init();

       addSource(s);

       start(t);

       return reached(t);

     }

     ///Runs the algorithm to visit all nodes in the digraph.

     ///This method runs the %BFS algorithm in order to

     ///compute the shortest path to each node.

     ///

     ///The algorithm computes

     ///- the shortest path tree (forest),

     ///- the distance of each node from the root(s).

     ///

     ///\note <tt>b.run(s)</tt> is just a shortcut of the following code.

     ///\code

     ///  b.init();

     ///  for (NodeIt n(gr); n != INVALID; ++n) {

     ///    if (!b.reached(n)) {

     ///      b.addSource(n);

     ///      b.start();

     ///    }

     ///  }

     ///\endcode

     void run() {

       init();

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

         if (!reached(n)) {

           addSource(n);

           start();

         }

       }

     }

     ///@}

     ///\name Query Functions

     ///The results of the BFS algorithm can be obtained using these

     ///functions.\n

     ///Either \ref run(Node) "run()" or \ref start() should be called

     ///before using them.

     ///@{

     ///The shortest path to a node.

     ///Returns the shortest path to a node.

     ///

     ///\warning \c t should be reached from the root(s).

     ///

     ///\pre Either \ref run(Node) "run()" or \ref init()

     ///must be called before using this function.

     Path path(Node t) const { return Path(*G, *_pred, t); }

     ///The distance of a node from the root(s).

     ///Returns the distance of a node from the root(s).

     ///

     ///\warning If node \c v is not reached from the root(s), then

     ///the return value of this function is undefined.

     ///

     ///\pre Either \ref run(Node) "run()" or \ref init()

     ///must be called before using this function.

     int dist(Node v) const { return (*_dist)[v]; }

     ///Returns the 'previous arc' of the shortest path tree for a node.

     ///This function returns the 'previous arc' of the shortest path

     ///tree for the node \c v, i.e. it returns the last arc of a

     ///shortest path from a root to \c v. It is \c INVALID if \c v

     ///is not reached from the root(s) or if \c v is a root.

     ///

     ///The shortest path tree used here is equal to the shortest path

     ///tree used in \ref predNode().

     ///

     ///\pre Either \ref run(Node) "run()" or \ref init()

     ///must be called before using this function.

     Arc predArc(Node v) const { return (*_pred)[v];}

     ///Returns the 'previous node' of the shortest path tree for a node.

     ///This function returns the 'previous node' of the shortest path

     ///tree for the node \c v, i.e. it returns the last but one node

     ///from a shortest path from a root to \c v. It is \c INVALID

     ///if \c v is not reached from the root(s) or if \c v is a root.

     ///

     ///The shortest path tree used here is equal to the shortest path

     ///tree used in \ref predArc().

     ///

     ///\pre Either \ref run(Node) "run()" or \ref init()

     ///must be called before using this function.

     Node predNode(Node v) const { return (*_pred)[v]==INVALID ? INVALID:

                                   G->source((*_pred)[v]); }

     ///\brief Returns a const reference to the node map that stores the

     /// distances of the nodes.

     ///

     ///Returns a const reference to the node map that stores the distances

     ///of the nodes calculated by the algorithm.

     ///

     ///\pre Either \ref run(Node) "run()" or \ref init()

     ///must be called before using this function.

     const DistMap &distMap() const { return *_dist;}

     ///\brief Returns a const reference to the node map that stores the

     ///predecessor arcs.

     ///

     ///Returns a const reference to the node map that stores the predecessor

     ///arcs, which form the shortest path tree.

     ///

     ///\pre Either \ref run(Node) "run()" or \ref init()

     ///must be called before using this function.

     const PredMap &predMap() const { return *_pred;}

     ///Checks if a node is reached from the root(s).

     ///Returns \c true if \c v is reached from the root(s).

     ///

     ///\pre Either \ref run(Node) "run()" or \ref init()

     ///must be called before using this function.

     bool reached(Node v) const { return (*_reached)[v]; }

     ///@}

   };

   ///Default traits class of bfs() function.

   ///Default traits class of bfs() function.

   ///\tparam GR Digraph type.

   template<class GR>

   struct BfsWizardDefaultTraits

   {

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

     typedef GR Digraph;

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

     ///arcs of the shortest paths.

     ///

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

     ///arcs of the shortest paths.

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

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

     ///Instantiates a PredMap.

     ///This function instantiates a PredMap.

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

     ///PredMap.

     static PredMap *createPredMap(const Digraph &g)

     {

       return new PredMap(g);

     }

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

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

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

     ///By default it is a NullMap.

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

     ///Instantiates a ProcessedMap.

     ///This function instantiates a ProcessedMap.

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

     ///we would like to define the ProcessedMap.

 #ifdef DOXYGEN

     static ProcessedMap *createProcessedMap(const Digraph &g)

 #else

     static ProcessedMap *createProcessedMap(const Digraph &)

 #endif

     {

       return new ProcessedMap();

     }

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

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

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

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

     ///Instantiates a ReachedMap.

     ///This function instantiates a ReachedMap.

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

     ///we would like to define the ReachedMap.

     static ReachedMap *createReachedMap(const Digraph &g)

     {

       return new ReachedMap(g);

     }

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

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

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

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

     ///Instantiates a DistMap.

     ///This function instantiates a DistMap.

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

     ///the DistMap

     static DistMap *createDistMap(const Digraph &g)

     {

       return new DistMap(g);

     }

     ///The type of the shortest paths.

     ///The type of the shortest paths.

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

     typedef lemon::Path<Digraph> Path;

   };

   /// Default traits class used by BfsWizard

   /// To make it easier to use Bfs algorithm

   /// we have created a wizard class.

   /// This \ref BfsWizard class needs default traits,

   /// as well as the \ref Bfs class.

   /// The \ref BfsWizardBase is a class to be the default traits of the

   /// \ref BfsWizard class.

   template<class GR>

   class BfsWizardBase : public BfsWizardDefaultTraits<GR>

   {

     typedef BfsWizardDefaultTraits<GR> Base;

   protected:

     //The type of the nodes in the digraph.

     typedef typename Base::Digraph::Node Node;

     //Pointer to the digraph the algorithm runs on.

     void *_g;

     //Pointer to the map of reached nodes.

     void *_reached;

     //Pointer to the map of processed nodes.

     void *_processed;

     //Pointer to the map of predecessors arcs.

     void *_pred;

     //Pointer to the map of distances.

     void *_dist;

     //Pointer to the shortest path to the target node.

     void *_path;

     //Pointer to the distance of the target node.

     int *_di;

     public:

     /// Constructor.

     /// This constructor does not require parameters, therefore it initiates

     /// all of the attributes to \c 0.

     BfsWizardBase() : _g(0), _reached(0), _processed(0), _pred(0),

                       _dist(0), _path(0), _di(0) {}

     /// Constructor.

     /// This constructor requires one parameter,

     /// others are initiated to \c 0.

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

     BfsWizardBase(const GR &g) :

       _g(reinterpret_cast<void*>(const_cast<GR*>(&g))),

       _reached(0), _processed(0), _pred(0), _dist(0),  _path(0), _di(0) {}

   };

   /// Auxiliary class for the function-type interface of BFS algorithm.

   /// This auxiliary class is created to implement the

   /// \ref bfs() "function-type interface" of \ref Bfs algorithm.

   /// It does not have own \ref run(Node) "run()" method, it uses the

   /// functions and features of the plain \ref Bfs.

   ///

   /// This class should only be used through the \ref bfs() function,

   /// which makes it easier to use the algorithm.

   template<class TR>

   class BfsWizard : public TR

   {

     typedef TR Base;

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

     typedef typename TR::Digraph Digraph;

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::OutArcIt OutArcIt;

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

     ///arcs of the shortest paths.

     typedef typename TR::PredMap PredMap;

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

     typedef typename TR::DistMap DistMap;

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

     typedef typename TR::ReachedMap ReachedMap;

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

     typedef typename TR::ProcessedMap ProcessedMap;

     ///The type of the shortest paths

     typedef typename TR::Path Path;

   public:

     /// Constructor.

     BfsWizard() : TR() {}

     /// Constructor that requires parameters.

     /// Constructor that requires parameters.

     /// These parameters will be the default values for the traits class.

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

     BfsWizard(const Digraph &g) :

       TR(g) {}

     ///Copy constructor

     BfsWizard(const TR &b) : TR(b) {}

     ~BfsWizard() {}

     ///Runs BFS algorithm from the given source node.

     ///This method runs BFS algorithm from node \c s

     ///in order to compute the shortest path to each node.

     void run(Node s)

     {

       Bfs<Digraph,TR> alg(*reinterpret_cast<const Digraph*>(Base::_g));

       if (Base::_pred)

         alg.predMap(*reinterpret_cast<PredMap*>(Base::_pred));

       if (Base::_dist)

         alg.distMap(*reinterpret_cast<DistMap*>(Base::_dist));

       if (Base::_reached)

         alg.reachedMap(*reinterpret_cast<ReachedMap*>(Base::_reached));

       if (Base::_processed)

         alg.processedMap(*reinterpret_cast<ProcessedMap*>(Base::_processed));

       if (s!=INVALID)

         alg.run(s);

       else

         alg.run();

     }

     ///Finds the shortest path between \c s and \c t.

     ///This method runs BFS algorithm from node \c s

     ///in order to compute the shortest path to node \c t

     ///(it stops searching when \c t is processed).

     ///

     ///\return \c true if \c t is reachable form \c s.

     bool run(Node s, Node t)

     {

       Bfs<Digraph,TR> alg(*reinterpret_cast<const Digraph*>(Base::_g));

       if (Base::_pred)

         alg.predMap(*reinterpret_cast<PredMap*>(Base::_pred));

       if (Base::_dist)

         alg.distMap(*reinterpret_cast<DistMap*>(Base::_dist));

       if (Base::_reached)

         alg.reachedMap(*reinterpret_cast<ReachedMap*>(Base::_reached));

       if (Base::_processed)

         alg.processedMap(*reinterpret_cast<ProcessedMap*>(Base::_processed));

       alg.run(s,t);

       if (Base::_path)

         *reinterpret_cast<Path*>(Base::_path) = alg.path(t);

       if (Base::_di)

         *Base::_di = alg.dist(t);

       return alg.reached(t);

     }

     ///Runs BFS algorithm to visit all nodes in the digraph.

     ///This method runs BFS algorithm in order to compute

     ///the shortest path to each node.

     void run()

     {

       run(INVALID);

     }

     template<class T>

     struct SetPredMapBase : public Base {

       typedef T PredMap;

       static PredMap *createPredMap(const Digraph &) { return 0; };

       SetPredMapBase(const TR &b) : TR(b) {}

     };

     ///\brief \ref named-func-param "Named parameter"

     ///for setting PredMap object.

     ///

     ///\ref named-func-param "Named parameter"

     ///for setting PredMap object.

     template<class T>

     BfsWizard<SetPredMapBase<T> > predMap(const T &t)

     {

       Base::_pred=reinterpret_cast<void*>(const_cast<T*>(&t));

       return BfsWizard<SetPredMapBase<T> >(*this);

     }

     template<class T>

     struct SetReachedMapBase : public Base {

       typedef T ReachedMap;

       static ReachedMap *createReachedMap(const Digraph &) { return 0; };

       SetReachedMapBase(const TR &b) : TR(b) {}

     };

     ///\brief \ref named-func-param "Named parameter"

     ///for setting ReachedMap object.

     ///

     /// \ref named-func-param "Named parameter"

     ///for setting ReachedMap object.

     template<class T>

     BfsWizard<SetReachedMapBase<T> > reachedMap(const T &t)

     {

       Base::_reached=reinterpret_cast<void*>(const_cast<T*>(&t));

       return BfsWizard<SetReachedMapBase<T> >(*this);

     }

     template<class T>

     struct SetDistMapBase : public Base {

       typedef T DistMap;

       static DistMap *createDistMap(const Digraph &) { return 0; };

       SetDistMapBase(const TR &b) : TR(b) {}

     };

     ///\brief \ref named-func-param "Named parameter"

     ///for setting DistMap object.

     ///

     /// \ref named-func-param "Named parameter"

     ///for setting DistMap object.

     template<class T>

     BfsWizard<SetDistMapBase<T> > distMap(const T &t)

     {

       Base::_dist=reinterpret_cast<void*>(const_cast<T*>(&t));

       return BfsWizard<SetDistMapBase<T> >(*this);

     }

     template<class T>

     struct SetProcessedMapBase : public Base {

       typedef T ProcessedMap;

       static ProcessedMap *createProcessedMap(const Digraph &) { return 0; };

       SetProcessedMapBase(const TR &b) : TR(b) {}

     };

     ///\brief \ref named-func-param "Named parameter"

     ///for setting ProcessedMap object.

     ///

     /// \ref named-func-param "Named parameter"

     ///for setting ProcessedMap object.

     template<class T>

     BfsWizard<SetProcessedMapBase<T> > processedMap(const T &t)

     {

       Base::_processed=reinterpret_cast<void*>(const_cast<T*>(&t));

       return BfsWizard<SetProcessedMapBase<T> >(*this);

     }

     template<class T>

     struct SetPathBase : public Base {

       typedef T Path;

       SetPathBase(const TR &b) : TR(b) {}

     };

     ///\brief \ref named-func-param "Named parameter"

     ///for getting the shortest path to the target node.

     ///

     ///\ref named-func-param "Named parameter"

     ///for getting the shortest path to the target node.

     template<class T>

     BfsWizard<SetPathBase<T> > path(const T &t)

     {

       Base::_path=reinterpret_cast<void*>(const_cast<T*>(&t));

       return BfsWizard<SetPathBase<T> >(*this);

     }

     ///\brief \ref named-func-param "Named parameter"

     ///for getting the distance of the target node.

     ///

     ///\ref named-func-param "Named parameter"

     ///for getting the distance of the target node.

     BfsWizard dist(const int &d)

     {

       Base::_di=const_cast<int*>(&d);

       return *this;

     }

   };

   ///Function-type interface for BFS algorithm.

   /// \ingroup search

   ///Function-type interface for BFS algorithm.

   ///

   ///This function also has several \ref named-func-param "named parameters",

   ///they are declared as the members of class \ref BfsWizard.

   ///The following examples show how to use these parameters.

   ///\code

   ///  // Compute shortest path from node s to each node

   ///  bfs(g).predMap(preds).distMap(dists).run(s);

   ///

   ///  // Compute shortest path from s to t

   ///  bool reached = bfs(g).path(p).dist(d).run(s,t);

   ///\endcode

   ///\warning Don't forget to put the \ref BfsWizard::run(Node) "run()"

   ///to the end of the parameter list.

   ///\sa BfsWizard

   ///\sa Bfs

   template<class GR>

   BfsWizard<BfsWizardBase<GR> >

   bfs(const GR &digraph)

   {

     return BfsWizard<BfsWizardBase<GR> >(digraph);

   }

 #ifdef DOXYGEN

   /// \brief Visitor class for BFS.

   ///

   /// This class defines the interface of the BfsVisit events, and

   /// it could be the base of a real visitor class.

   template <typename GR>

   struct BfsVisitor {

     typedef GR Digraph;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::Node Node;

     /// \brief Called for the source node(s) of the BFS.

     ///

     /// This function is called for the source node(s) of the BFS.

     void start(const Node& node) {}

     /// \brief Called when a node is reached first time.

     ///

     /// This function is called when a node is reached first time.

     void reach(const Node& node) {}

     /// \brief Called when a node is processed.

     ///

     /// This function is called when a node is processed.

     void process(const Node& node) {}

     /// \brief Called when an arc reaches a new node.

     ///

     /// This function is called when the BFS finds an arc whose target node

     /// is not reached yet.

     void discover(const Arc& arc) {}

     /// \brief Called when an arc is examined but its target node is

     /// already discovered.

     ///

     /// This function is called when an arc is examined but its target node is

     /// already discovered.

     void examine(const Arc& arc) {}

   };

 #else

   template <typename GR>

   struct BfsVisitor {

     typedef GR Digraph;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::Node Node;

     void start(const Node&) {}

     void reach(const Node&) {}

     void process(const Node&) {}

     void discover(const Arc&) {}

     void examine(const Arc&) {}

     template <typename _Visitor>

     struct Constraints {

       void constraints() {

         Arc arc;

         Node node;

         visitor.start(node);

         visitor.reach(node);

         visitor.process(node);

         visitor.discover(arc);

         visitor.examine(arc);

       }

       _Visitor& visitor;

       Constraints() {}

     };

   };

 #endif

   /// \brief Default traits class of BfsVisit class.

   ///

   /// Default traits class of BfsVisit class.

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

   template<class GR>

   struct BfsVisitDefaultTraits {

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

     typedef GR Digraph;

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

     ///

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

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

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

     /// \brief Instantiates a ReachedMap.

     ///

     /// This function instantiates a ReachedMap.

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

     /// we would like to define the ReachedMap.

     static ReachedMap *createReachedMap(const Digraph &digraph) {

       return new ReachedMap(digraph);

     }

   };

   /// \ingroup search

   ///

   /// \brief BFS algorithm class with visitor interface.

   ///

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

   /// with visitor interface.

   ///

   /// The BfsVisit class provides an alternative interface to the Bfs

   /// class. It works with callback mechanism, the BfsVisit object calls

   /// the member functions of the \c Visitor class on every BFS event.

   ///

   /// This interface of the BFS algorithm should be used in special cases

   /// when extra actions have to be performed in connection with certain

   /// events of the BFS algorithm. Otherwise consider to use Bfs or bfs()

   /// instead.

   ///

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

   /// The default type is \ref ListDigraph.

   /// The value of GR is not used directly by \ref BfsVisit,

   /// it is only passed to \ref BfsVisitDefaultTraits.

   /// \tparam VS The Visitor type that is used by the algorithm.

   /// \ref BfsVisitor "BfsVisitor<GR>" is an empty visitor, which

   /// does not observe the BFS events. If you want to observe the BFS

   /// events, you should implement your own visitor class.

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

   /// algorithm. The default traits class is

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

   /// See \ref BfsVisitDefaultTraits for the documentation of

   /// a BFS visit traits class.

 #ifdef DOXYGEN

   template <typename GR, typename VS, typename TR>

 #else

   template <typename GR = ListDigraph,

             typename VS = BfsVisitor<GR>,

             typename TR = BfsVisitDefaultTraits<GR> >

 #endif

   class BfsVisit {

   public:

     ///The traits class.

     typedef TR Traits;

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

     typedef typename Traits::Digraph Digraph;

     ///The visitor type used by the algorithm.

     typedef VS Visitor;

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

     typedef typename Traits::ReachedMap ReachedMap;

   private:

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::OutArcIt OutArcIt;

     //Pointer to the underlying digraph.

     const Digraph *_digraph;

     //Pointer to the visitor object.

     Visitor *_visitor;

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

     ReachedMap *_reached;

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

     bool local_reached;

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

     int _list_front, _list_back;

     //Creates the maps if necessary.

     void create_maps() {

       if(!_reached) {

         local_reached = true;

         _reached = Traits::createReachedMap(*_digraph);

       }

     }

   protected:

     BfsVisit() {}

   public:

     typedef BfsVisit Create;

     /// \name Named Template Parameters

     ///@{

     template <class T>

     struct SetReachedMapTraits : public Traits {

       typedef T ReachedMap;

       static ReachedMap *createReachedMap(const Digraph &digraph) {

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

         return 0; // ignore warnings

       }

     };

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

     /// ReachedMap type.

     ///

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

     template <class T>

     struct SetReachedMap : public BfsVisit< Digraph, Visitor,

                                             SetReachedMapTraits<T> > {

       typedef BfsVisit< Digraph, Visitor, SetReachedMapTraits<T> > Create;

     };

     ///@}

   public:

     /// \brief Constructor.

     ///

     /// Constructor.

     ///

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

     /// \param visitor The visitor object of the algorithm.

     BfsVisit(const Digraph& digraph, Visitor& visitor)

       : _digraph(&digraph), _visitor(&visitor),

         _reached(0), local_reached(false) {}

     /// \brief Destructor.

     ~BfsVisit() {

       if(local_reached) delete _reached;

     }

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

     ///

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

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

     /// or \ref init(), an instance will be allocated automatically.

     /// The destructor deallocates this automatically allocated map,

     /// of course.

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

     BfsVisit &reachedMap(ReachedMap &m) {

       if(local_reached) {

         delete _reached;

         local_reached = false;

       }

       _reached = &m;

       return *this;

     }

   public:

     /// \name Execution Control

     /// The simplest way to execute the BFS algorithm is to use one of the

     /// member functions called \ref run(Node) "run()".\n

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

     /// \ref init(), then you can add several source nodes with

     /// \ref addSource(). Finally the actual path computation can be

     /// performed with one of the \ref start() functions.

     /// @{

     /// \brief Initializes the internal data structures.

     ///

     /// Initializes the internal data structures.

     void init() {

       create_maps();

       _list.resize(countNodes(*_digraph));

       _list_front = _list_back = -1;

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

         _reached->set(u, false);

       }

     }

     /// \brief Adds a new source node.

     ///

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

     void addSource(Node s) {

       if(!(*_reached)[s]) {

           _reached->set(s,true);

           _visitor->start(s);

           _visitor->reach(s);

           _list[++_list_back] = s;

         }

     }

     /// \brief Processes the next node.

     ///

     /// Processes the next node.

     ///

     /// \return The processed node.

     ///

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

     Node processNextNode() {

       Node n = _list[++_list_front];

       _visitor->process(n);

       Arc e;

       for (_digraph->firstOut(e, n); e != INVALID; _digraph->nextOut(e)) {

         Node m = _digraph->target(e);

         if (!(*_reached)[m]) {

           _visitor->discover(e);

           _visitor->reach(m);

           _reached->set(m, true);

           _list[++_list_back] = m;

         } else {

           _visitor->examine(e);

         }

       }

       return n;

     }

     /// \brief Processes the next node.

     ///

     /// Processes the next node and checks if the given target node

     /// is reached. If the target node is reachable from the processed

     /// node, then the \c reach parameter will be set to \c true.

     ///

     /// \param target The target node.

     /// \retval reach Indicates if the target node is reached.

     /// It should be initially \c false.

     ///

     /// \return The processed node.

     ///

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

     Node processNextNode(Node target, bool& reach) {

       Node n = _list[++_list_front];

       _visitor->process(n);

       Arc e;

       for (_digraph->firstOut(e, n); e != INVALID; _digraph->nextOut(e)) {

         Node m = _digraph->target(e);

         if (!(*_reached)[m]) {

           _visitor->discover(e);

           _visitor->reach(m);

           _reached->set(m, true);

           _list[++_list_back] = m;

           reach = reach || (target == m);

         } else {

           _visitor->examine(e);

         }

       }

       return n;

     }

     /// \brief Processes the next node.

     ///

     /// Processes the next node and checks if at least one of reached

     /// nodes has \c true value in the \c nm node map. If one node

     /// with \c true value is reachable from the processed node, then the

     /// \c rnode parameter will be set to the first of such nodes.

     ///

     /// \param nm A \c bool (or convertible) node map that indicates the

     /// possible targets.

     /// \retval rnode The reached target node.

     /// It should be initially \c INVALID.

     ///

     /// \return The processed node.

     ///

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

     template <typename NM>

     Node processNextNode(const NM& nm, Node& rnode) {

       Node n = _list[++_list_front];

       _visitor->process(n);

       Arc e;

       for (_digraph->firstOut(e, n); e != INVALID; _digraph->nextOut(e)) {

         Node m = _digraph->target(e);

         if (!(*_reached)[m]) {

           _visitor->discover(e);

           _visitor->reach(m);

           _reached->set(m, true);

           _list[++_list_back] = m;

           if (nm[m] && rnode == INVALID) rnode = m;

         } else {

           _visitor->examine(e);

         }

       }

       return n;

     }

     /// \brief The next node to be processed.

     ///

     /// Returns the next node to be processed or \c INVALID if the queue

     /// is empty.

     Node nextNode() const {

       return _list_front != _list_back ? _list[_list_front + 1] : INVALID;

     }

     /// \brief Returns \c false if there are nodes

     /// to be processed.

     ///

     /// Returns \c false if there are nodes

     /// to be processed in the queue.

     bool emptyQueue() const { return _list_front == _list_back; }

     /// \brief Returns the number of the nodes to be processed.

     ///

     /// Returns the number of the nodes to be processed in the queue.

     int queueSize() const { return _list_back - _list_front; }

     /// \brief Executes the algorithm.

     ///

     /// Executes the algorithm.

     ///

     /// This method runs the %BFS algorithm from the root node(s)

     /// in order to compute the shortest path to each node.

     ///

     /// The algorithm computes

     /// - the shortest path tree (forest),

     /// - the distance of each node from the root(s).

     ///

     /// \pre init() must be called and at least one root node should be added

     /// with addSource() before using this function.

     ///

     /// \note <tt>b.start()</tt> is just a shortcut of the following code.

     /// \code

     ///   while ( !b.emptyQueue() ) {

     ///     b.processNextNode();

     ///   }

     /// \endcode

     void start() {

       while ( !emptyQueue() ) processNextNode();

     }

     /// \brief Executes the algorithm until the given target node is reached.

     ///

     /// Executes the algorithm until the given target node is reached.

     ///

     /// This method runs the %BFS algorithm from the root node(s)

     /// in order to compute the shortest path to \c t.

     ///

     /// The algorithm computes

     /// - the shortest path to \c t,

     /// - the distance of \c t from the root(s).

     ///

     /// \pre init() must be called and at least one root node should be

     /// added with addSource() before using this function.

     ///

     /// \note <tt>b.start(t)</tt> is just a shortcut of the following code.

     /// \code

     ///   bool reach = false;

     ///   while ( !b.emptyQueue() && !reach ) {

     ///     b.processNextNode(t, reach);

     ///   }

     /// \endcode

     void start(Node t) {

       bool reach = false;

       while ( !emptyQueue() && !reach ) processNextNode(t, reach);

     }

     /// \brief Executes the algorithm until a condition is met.

     ///

     /// Executes the algorithm until a condition is met.

     ///

     /// This method runs the %BFS algorithm from the root node(s) in

     /// order to compute the shortest path to a node \c v with

     /// <tt>nm[v]</tt> true, if such a node can be found.

     ///

     /// \param nm must be a bool (or convertible) node map. The

     /// algorithm will stop when it reaches a node \c v with

     /// <tt>nm[v]</tt> true.

     ///

     /// \return The reached node \c v with <tt>nm[v]</tt> true or

     /// \c INVALID if no such node was found.

     ///

     /// \pre init() must be called and at least one root node should be

     /// added with addSource() before using this function.

     ///

     /// \note <tt>b.start(nm)</tt> is just a shortcut of the following code.

     /// \code

     ///   Node rnode = INVALID;

     ///   while ( !b.emptyQueue() && rnode == INVALID ) {

     ///     b.processNextNode(nm, rnode);

     ///   }

     ///   return rnode;

     /// \endcode

     template <typename NM>

     Node start(const NM &nm) {

       Node rnode = INVALID;

       while ( !emptyQueue() && rnode == INVALID ) {

         processNextNode(nm, rnode);

       }

       return rnode;

     }

     /// \brief Runs the algorithm from the given source node.

     ///

     /// This method runs the %BFS algorithm from node \c s

     /// in order to compute the shortest path to each node.

     ///

     /// The algorithm computes

     /// - the shortest path tree,

     /// - the distance of each node from the root.

     ///

     /// \note <tt>b.run(s)</tt> is just a shortcut of the following code.

     ///\code

     ///   b.init();

     ///   b.addSource(s);

     ///   b.start();

     ///\endcode

     void run(Node s) {

       init();

       addSource(s);

       start();

     }

     /// \brief Finds the shortest path between \c s and \c t.

     ///

     /// This method runs the %BFS algorithm from node \c s

     /// in order to compute the shortest path to node \c t

     /// (it stops searching when \c t is processed).

     ///

     /// \return \c true if \c t is reachable form \c s.

     ///

     /// \note Apart from the return value, <tt>b.run(s,t)</tt> is just a

     /// shortcut of the following code.

     ///\code

     ///   b.init();

     ///   b.addSource(s);

     ///   b.start(t);

     ///\endcode

     bool run(Node s,Node t) {

       init();

       addSource(s);

       start(t);

       return reached(t);

     }

     /// \brief Runs the algorithm to visit all nodes in the digraph.

     ///

     /// This method runs the %BFS algorithm in order to

     /// compute the shortest path to each node.

     ///

     /// The algorithm computes

     /// - the shortest path tree (forest),

     /// - the distance of each node from the root(s).

     ///

     /// \note <tt>b.run(s)</tt> is just a shortcut of the following code.

     ///\code

     ///  b.init();

     ///  for (NodeIt n(gr); n != INVALID; ++n) {

     ///    if (!b.reached(n)) {

     ///      b.addSource(n);

     ///      b.start();

     ///    }

     ///  }

     ///\endcode

     void run() {

       init();

       for (NodeIt it(*_digraph); it != INVALID; ++it) {

         if (!reached(it)) {

           addSource(it);

           start();

         }

       }

     }

     ///@}

     /// \name Query Functions

     /// The results of the BFS algorithm can be obtained using these

     /// functions.\n

     /// Either \ref run(Node) "run()" or \ref start() should be called

     /// before using them.

     ///@{

     /// \brief Checks if a node is reached from the root(s).

     ///

     /// Returns \c true if \c v is reached from the root(s).

     ///

     /// \pre Either \ref run(Node) "run()" or \ref init()

     /// must be called before using this function.

     bool reached(Node v) const { return (*_reached)[v]; }

     ///@}

   };

 } //END OF NAMESPACE LEMON

 #endif