/* -*- 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_DIJKSTRA_H

#define LEMON_DIJKSTRA_H

///\ingroup shortest_path

///\file

///\brief Dijkstra algorithm.

#include <limits>

#include <lemon/list_graph.h>

#include <lemon/bin_heap.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 {

  /// \brief Default operation traits for the Dijkstra algorithm class.

  ///

  /// This operation traits class defines all computational operations and

  /// constants which are used in the Dijkstra algorithm.

  template <typename V>

  struct DijkstraDefaultOperationTraits {

    /// \e

    typedef V Value;

    /// \brief Gives back the zero value of the type.

    static Value zero() {

      return static_cast<Value>(0);

    }

    /// \brief Gives back the sum of the given two elements.

    static Value plus(const Value& left, const Value& right) {

      return left + right;

    }

    /// \brief Gives back true only if the first value is less than the second.

    static bool less(const Value& left, const Value& right) {

      return left < right;

    }

  };

  ///Default traits class of Dijkstra class.

  ///Default traits class of Dijkstra class.

  ///\tparam GR The type of the digraph.

  ///\tparam LEN The type of the length map.

  template<typename GR, typename LEN>

  struct DijkstraDefaultTraits

  {

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

    typedef GR Digraph;

    ///The type of the map that stores the arc lengths.

    ///The type of the map that stores the arc lengths.

    ///It must conform to the \ref concepts::ReadMap "ReadMap" concept.

    typedef LEN LengthMap;

    ///The type of the arc lengths.

    typedef typename LEN::Value Value;

    /// Operation traits for %Dijkstra algorithm.

    /// This class defines the operations that are used in the algorithm.

    /// \see DijkstraDefaultOperationTraits

    typedef DijkstraDefaultOperationTraits<Value> OperationTraits;

    /// The cross reference type used by the heap.

    /// The cross reference type used by the heap.

    /// Usually it is \c Digraph::NodeMap<int>.

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

    ///Instantiates a \c HeapCrossRef.

    ///This function instantiates a \ref HeapCrossRef.

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

    /// \ref HeapCrossRef.

    static HeapCrossRef *createHeapCrossRef(const Digraph &g)

    {

      return new HeapCrossRef(g);

    }

    ///The heap type used by the %Dijkstra algorithm.

    ///The heap type used by the Dijkstra algorithm.

    ///

    ///\sa BinHeap

    ///\sa Dijkstra

    typedef BinHeap<typename LEN::Value, HeapCrossRef, std::less<Value> > Heap;

    ///Instantiates a \c Heap.

    ///This function instantiates a \ref Heap.

    static Heap *createHeap(HeapCrossRef& r)

    {

      return new Heap(r);

    }

    ///\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 conform to 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 conform to the \ref concepts::WriteMap "WriteMap" concept.

    ///By default it is a NullMap.

    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 stores the distances of the nodes.

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

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

    typedef typename Digraph::template NodeMap<typename LEN::Value> 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);

    }

  };

  ///%Dijkstra algorithm class.

  /// \ingroup shortest_path

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

  ///

  ///The %Dijkstra algorithm solves the single-source shortest path problem

  ///when all arc lengths are non-negative. If there are negative lengths,

  ///the BellmanFord algorithm should be used instead.

  ///

  ///The arc lengths are passed to the algorithm using a

  ///\ref concepts::ReadMap "ReadMap",

  ///so it is easy to change it to any kind of length.

  ///The type of the length is determined by the

  ///\ref concepts::ReadMap::Value "Value" of the length map.

  ///It is also possible to change the underlying priority heap.

  ///

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

  ///%Dijkstra 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.

  ///\tparam LEN A \ref concepts::ReadMap "readable" arc map that specifies

  ///the lengths of the arcs.

  ///It is read once for each arc, so the map may involve in

  ///relatively time consuming process to compute the arc lengths if

  ///it is necessary. The default map type is \ref

  ///concepts::Digraph::ArcMap "GR::ArcMap<int>".

#ifdef DOXYGEN

  template <typename GR, typename LEN, typename TR>

#else

  template <typename GR=ListDigraph,

            typename LEN=typename GR::template ArcMap<int>,

            typename TR=DijkstraDefaultTraits<GR,LEN> >

#endif

  class Dijkstra {

  public:

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

    typedef typename TR::Digraph Digraph;

    ///The type of the arc lengths.

    typedef typename TR::LengthMap::Value Value;

    ///The type of the map that stores the arc lengths.

    typedef typename TR::LengthMap LengthMap;

    ///\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 processed.

    typedef typename TR::ProcessedMap ProcessedMap;

    ///The type of the paths.

    typedef PredMapPath<Digraph, PredMap> Path;

    ///The cross reference type used for the current heap.

    typedef typename TR::HeapCrossRef HeapCrossRef;

    ///The heap type used by the algorithm.

    typedef typename TR::Heap Heap;

    ///\brief The \ref DijkstraDefaultOperationTraits "operation traits class"

    ///of the algorithm.

    typedef typename TR::OperationTraits OperationTraits;

    ///The \ref DijkstraDefaultTraits "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 length map.

    const LengthMap *_length;

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

    ProcessedMap *_processed;

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

    bool local_processed;

    //Pointer to the heap cross references.

    HeapCrossRef *_heap_cross_ref;

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

    bool local_heap_cross_ref;

    //Pointer to the heap.

    Heap *_heap;

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

    bool local_heap;

    //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(!_processed) {

        local_processed = true;

        _processed = Traits::createProcessedMap(*G);

      }

      if (!_heap_cross_ref) {

        local_heap_cross_ref = true;

        _heap_cross_ref = Traits::createHeapCrossRef(*G);

      }

      if (!_heap) {

        local_heap = true;

        _heap = Traits::createHeap(*_heap_cross_ref);

      }

    }

  public:

    typedef Dijkstra 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 conform to the \ref concepts::WriteMap "WriteMap" concept.

    template <class T>

    struct SetPredMap

      : public Dijkstra< Digraph, LengthMap, SetPredMapTraits<T> > {

      typedef Dijkstra< Digraph, LengthMap, 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 conform to the \ref concepts::WriteMap "WriteMap" concept.

    template <class T>

    struct SetDistMap

      : public Dijkstra< Digraph, LengthMap, SetDistMapTraits<T> > {

      typedef Dijkstra< Digraph, LengthMap, SetDistMapTraits<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 conform to the \ref concepts::WriteMap "WriteMap" concept.

    template <class T>

    struct SetProcessedMap

      : public Dijkstra< Digraph, LengthMap, SetProcessedMapTraits<T> > {

      typedef Dijkstra< Digraph, LengthMap, SetProcessedMapTraits<T> > Create;

    };

    struct SetStandardProcessedMapTraits : public Traits {

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

      static ProcessedMap *createProcessedMap(const Digraph &g)

      {

        return new ProcessedMap(g);

      }

    };

    ///\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 Dijkstra< Digraph, LengthMap, SetStandardProcessedMapTraits > {

      typedef Dijkstra< Digraph, LengthMap, SetStandardProcessedMapTraits >

      Create;

    };

    template <class H, class CR>

    struct SetHeapTraits : public Traits {

      typedef CR HeapCrossRef;

      typedef H Heap;

      static HeapCrossRef *createHeapCrossRef(const Digraph &) {

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

        return 0; // ignore warnings

      }

      static Heap *createHeap(HeapCrossRef &)

      {

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

        return 0; // ignore warnings

      }

    };

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

    ///heap and cross reference types

    ///

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

    ///reference types. If this named parameter is used, then external

    ///heap and cross reference objects must be passed to the algorithm

    ///using the \ref heap() function before calling \ref run(Node) "run()"

    ///or \ref init().

    ///\sa SetStandardHeap

    template <class H, class CR = typename Digraph::template NodeMap<int> >

    struct SetHeap

      : public Dijkstra< Digraph, LengthMap, SetHeapTraits<H, CR> > {

      typedef Dijkstra< Digraph, LengthMap, SetHeapTraits<H, CR> > Create;

    };

    template <class H, class CR>

    struct SetStandardHeapTraits : public Traits {

      typedef CR HeapCrossRef;

      typedef H Heap;

      static HeapCrossRef *createHeapCrossRef(const Digraph &G) {

        return new HeapCrossRef(G);

      }

      static Heap *createHeap(HeapCrossRef &R)

      {

        return new Heap(R);

      }

    };

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

    ///heap and cross reference types with automatic allocation

    ///

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

    ///reference types with automatic allocation.

    ///They should have standard constructor interfaces to be able to

    ///automatically created by the algorithm (i.e. the digraph should be

    ///passed to the constructor of the cross reference and the cross

    ///reference should be passed to the constructor of the heap).

    ///However external heap and cross reference objects could also be

    ///passed to the algorithm using the \ref heap() function before

    ///calling \ref run(Node) "run()" or \ref init().

    ///\sa SetHeap

    template <class H, class CR = typename Digraph::template NodeMap<int> >

    struct SetStandardHeap

      : public Dijkstra< Digraph, LengthMap, SetStandardHeapTraits<H, CR> > {

      typedef Dijkstra< Digraph, LengthMap, SetStandardHeapTraits<H, CR> >

      Create;

    };

    template <class T>

    struct SetOperationTraitsTraits : public Traits {

      typedef T OperationTraits;

    };

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

    ///\c OperationTraits type

    ///

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

    ///\c OperationTraits type.

    /// For more information see \ref DijkstraDefaultOperationTraits.

    template <class T>

    struct SetOperationTraits

      : public Dijkstra<Digraph, LengthMap, SetOperationTraitsTraits<T> > {

      typedef Dijkstra<Digraph, LengthMap, SetOperationTraitsTraits<T> >

      Create;

    };

    ///@}

  protected:

    Dijkstra() {}

  public:

    ///Constructor.

    ///Constructor.

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

    ///\param length The length map used by the algorithm.

    Dijkstra(const Digraph& g, const LengthMap& length) :

      G(&g), _length(&length),

      _pred(NULL), local_pred(false),

      _dist(NULL), local_dist(false),

      _processed(NULL), local_processed(false),

      _heap_cross_ref(NULL), local_heap_cross_ref(false),

      _heap(NULL), local_heap(false)

    { }

    ///Destructor.

    ~Dijkstra()

    {

      if(local_pred) delete _pred;

      if(local_dist) delete _dist;

      if(local_processed) delete _processed;

      if(local_heap_cross_ref) delete _heap_cross_ref;

      if(local_heap) delete _heap;

    }

    ///Sets the length map.

    ///Sets the length map.

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

    Dijkstra &lengthMap(const LengthMap &m)

    {

      _length = &m;

      return *this;

    }

    ///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>

    Dijkstra &predMap(PredMap &m)

    {

      if(local_pred) {

        delete _pred;

        local_pred=false;

      }

      _pred = &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>

    Dijkstra &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>

    Dijkstra &distMap(DistMap &m)

    {

      if(local_dist) {

        delete _dist;

        local_dist=false;

      }

      _dist = &m;

      return *this;

    }

    ///Sets the heap and the cross reference used by algorithm.

    ///Sets the heap and the cross reference used by algorithm.

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

    ///or \ref init(), heap and cross reference instances will be

    ///allocated automatically.

    ///The destructor deallocates these automatically allocated objects,

    ///of course.

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

    Dijkstra &heap(Heap& hp, HeapCrossRef &cr)

    {

      if(local_heap_cross_ref) {

        delete _heap_cross_ref;

        local_heap_cross_ref=false;

      }

      _heap_cross_ref = &cr;

      if(local_heap) {

        delete _heap;

        local_heap=false;

      }

      _heap = &hp;

      return *this;

    }

  private:

    void finalizeNodeData(Node v,Value dst)

    {

      _processed->set(v,true);

      _dist->set(v, dst);

    }

  public:

    ///\name Execution Control

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

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

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

    ///\ref init() first, 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();

      _heap->clear();

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

        _pred->set(u,INVALID);

        _processed->set(u,false);

        _heap_cross_ref->set(u,Heap::PRE_HEAP);

      }

    }

    ///Adds a new source node.

    ///Adds a new source node to the priority heap.

    ///The optional second parameter is the initial distance of the node.

    ///

    ///The function checks if the node has already been added to the heap and

    ///it is pushed to the heap only if either it was not in the heap

    ///or the shortest path found till then is shorter than \c dst.

    void addSource(Node s,Value dst=OperationTraits::zero())

    {

      if(_heap->state(s) != Heap::IN_HEAP) {

        _heap->push(s,dst);

      } else if(OperationTraits::less((*_heap)[s], dst)) {

        _heap->set(s,dst);

        _pred->set(s,INVALID);

      }

    }

    ///Processes the next node in the priority heap

    ///Processes the next node in the priority heap.

    ///

    ///\return The processed node.

    ///

    ///\warning The priority heap must not be empty.

    Node processNextNode()

    {

      Node v=_heap->top();

      Value oldvalue=_heap->prio();

      _heap->pop();

      finalizeNodeData(v,oldvalue);

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

        Node w=G->target(e);

        switch(_heap->state(w)) {

        case Heap::PRE_HEAP:

          _heap->push(w,OperationTraits::plus(oldvalue, (*_length)[e]));

          _pred->set(w,e);

          break;

        case Heap::IN_HEAP:

          {

            Value newvalue = OperationTraits::plus(oldvalue, (*_length)[e]);

            if ( OperationTraits::less(newvalue, (*_heap)[w]) ) {

              _heap->decrease(w, newvalue);

              _pred->set(w,e);

            }

          }

          break;

        case Heap::POST_HEAP:

          break;

        }

      }

      return v;

    }

    ///The next node to be processed.

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

    ///priority heap is empty.

    Node nextNode() const

    {

      return !_heap->empty()?_heap->top():INVALID;

    }

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

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

    ///in the priority heap.

    bool emptyQueue() const { return _heap->empty(); }

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

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

    ///in the priority heap.

    int queueSize() const { return _heap->size(); }

    ///Executes the algorithm.

    ///Executes the algorithm.

    ///

    ///This method runs the %Dijkstra 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>d.start()</tt> is just a shortcut of the following code.

    ///\code

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

    ///    d.processNextNode();

    ///  }

    ///\endcode

    void start()

    {

      while ( !emptyQueue() ) processNextNode();

    }

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

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

    ///

    ///This method runs the %Dijkstra 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.

    void start(Node t)

    {

      while ( !_heap->empty() && _heap->top()!=t ) processNextNode();

      if ( !_heap->empty() ) {

        finalizeNodeData(_heap->top(),_heap->prio());

        _heap->pop();

      }

    }

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

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

    ///

    ///This method runs the %Dijkstra 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.

    template<class NodeBoolMap>

    Node start(const NodeBoolMap &nm)

    {

      while ( !_heap->empty() && !nm[_heap->top()] ) processNextNode();

      if ( _heap->empty() ) return INVALID;

      finalizeNodeData(_heap->top(),_heap->prio());

      return _heap->top();

    }

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

    ///This method runs the %Dijkstra 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>d.run(s)</tt> is just a shortcut of the following code.

    ///\code

    ///  d.init();

    ///  d.addSource(s);

    ///  d.start();

    ///\endcode

    void run(Node s) {

      init();

      addSource(s);

      start();

    }

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

    ///This method runs the %Dijkstra 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>d.run(s,t)</tt> is just a

    ///shortcut of the following code.

    ///\code

    ///  d.init();

    ///  d.addSource(s);

    ///  d.start(t);

    ///\endcode

    bool run(Node s,Node t) {

      init();

      addSource(s);

      start(t);

      return (*_heap_cross_ref)[t] == Heap::POST_HEAP;

    }

    ///@}

    ///\name Query Functions

    ///The results of the %Dijkstra algorithm can be obtained using these

    ///functions.\n

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

    ///before using them.

    ///@{

    ///The shortest path to the given node.

    ///Returns the shortest path to the given node from the root(s).

    ///

    ///\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 the given node from the root(s).

    ///Returns the distance of the given 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.

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

    ///\brief Returns the 'previous arc' of the shortest path tree for

    ///the given 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() and \ref predMap().

    ///

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

    ///must be called before using this function.

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

    ///\brief Returns the 'previous node' of the shortest path tree for

    ///the given 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

    ///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 predArc() and \ref predMap().

    ///

    ///\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 (forest).

    ///

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

    ///must be called before using this function.

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

    ///Checks if the given 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 (*_heap_cross_ref)[v] !=

                                        Heap::PRE_HEAP; }

    ///Checks if a node is processed.

    ///Returns \c true if \c v is processed, i.e. the shortest

    ///path to \c v has already found.

    ///

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

    ///must be called before using this function.

    bool processed(Node v) const { return (*_heap_cross_ref)[v] ==

                                          Heap::POST_HEAP; }

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

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

    ///It may be decreased in the following processes.

    ///

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

    ///must be called before using this function and

    ///node \c v must be reached but not necessarily processed.

    Value currentDist(Node v) const {

      return processed(v) ? (*_dist)[v] : (*_heap)[v];

    }

    ///@}

  };

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

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

  ///\tparam GR The type of the digraph.

  ///\tparam LEN The type of the length map.

  template<class GR, class LEN>

  struct DijkstraWizardDefaultTraits

  {

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

    typedef GR Digraph;

    ///The type of the map that stores the arc lengths.

    ///The type of the map that stores the arc lengths.

    ///It must conform to the \ref concepts::ReadMap "ReadMap" concept.

    typedef LEN LengthMap;

    ///The type of the arc lengths.

    typedef typename LEN::Value Value;

    /// Operation traits for Dijkstra algorithm.

    /// This class defines the operations that are used in the algorithm.

    /// \see DijkstraDefaultOperationTraits

    typedef DijkstraDefaultOperationTraits<Value> OperationTraits;

    /// The cross reference type used by the heap.

    /// The cross reference type used by the heap.

    /// Usually it is \c Digraph::NodeMap<int>.

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

    ///Instantiates a \ref HeapCrossRef.

    ///This function instantiates a \ref HeapCrossRef.

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

    /// HeapCrossRef.

    static HeapCrossRef *createHeapCrossRef(const Digraph &g)

    {

      return new HeapCrossRef(g);

    }

    ///The heap type used by the Dijkstra algorithm.

    ///The heap type used by the Dijkstra algorithm.

    ///

    ///\sa BinHeap

    ///\sa Dijkstra

    typedef BinHeap<Value, typename Digraph::template NodeMap<int>,

                    std::less<Value> > Heap;

    ///Instantiates a \ref Heap.

    ///This function instantiates a \ref Heap.

    /// \param r is the HeapCrossRef which is used.

    static Heap *createHeap(HeapCrossRef& r)

    {

      return new Heap(r);

    }

    ///\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 conform to 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);