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

 *

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

 *

 * Copyright (C) 2003-2010

 * 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_BELLMAN_FORD_H

#define LEMON_BELLMAN_FORD_H

/// \ingroup shortest_path

/// \file

/// \brief Bellman-Ford 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/tolerance.h>

#include <lemon/path.h>

#include <limits>

namespace lemon {

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

  ///

  /// This operation traits class defines all computational operations

  /// and constants that are used in the Bellman-Ford algorithm.

  /// The default implementation is based on the \c numeric_limits class.

  /// If the numeric type does not have infinity value, then the maximum

  /// value is used as extremal infinity value.

  ///

  /// \see BellmanFordToleranceOperationTraits

  template <

    typename V,

    bool has_inf = std::numeric_limits<V>::has_infinity>

  struct BellmanFordDefaultOperationTraits {

    /// \brief Value type for the algorithm.

    typedef V Value;

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

    static Value zero() {

      return static_cast<Value>(0);

    }

    /// \brief Gives back the positive infinity value of the type.

    static Value infinity() {

      return std::numeric_limits<Value>::infinity();

    }

    /// \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 \c true only if the first value is less than

    /// the second.

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

      return left < right;

    }

  };

  template <typename V>

  struct BellmanFordDefaultOperationTraits<V, false> {

    typedef V Value;

    static Value zero() {

      return static_cast<Value>(0);

    }

    static Value infinity() {

      return std::numeric_limits<Value>::max();

    }

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

      if (left == infinity() || right == infinity()) return infinity();

      return left + right;

    }

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

      return left < right;

    }

  };

  /// \brief Operation traits for the BellmanFord algorithm class

  /// using tolerance.

  ///

  /// This operation traits class defines all computational operations

  /// and constants that are used in the Bellman-Ford algorithm.

  /// The only difference between this implementation and

  /// \ref BellmanFordDefaultOperationTraits is that this class uses

  /// the \ref Tolerance "tolerance technique" in its \ref less()

  /// function.

  ///

  /// \tparam V The value type.

  /// \tparam eps The epsilon value for the \ref less() function.

  /// By default, it is the epsilon value used by \ref Tolerance

  /// "Tolerance<V>".

  ///

  /// \see BellmanFordDefaultOperationTraits

#ifdef DOXYGEN

  template <typename V, V eps>

#else

  template <

    typename V,

    V eps = Tolerance<V>::def_epsilon>

#endif

  struct BellmanFordToleranceOperationTraits {

    /// \brief Value type for the algorithm.

    typedef V Value;

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

    static Value zero() {

      return static_cast<Value>(0);

    }

    /// \brief Gives back the positive infinity value of the type.

    static Value infinity() {

      return std::numeric_limits<Value>::infinity();

    }

    /// \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 \c true only if the first value is less than

    /// the second.

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

      return left + eps < right;

    }

  };

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

  ///

  /// Default traits class of BellmanFord class.

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

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

  template<typename GR, typename LEN>

  struct BellmanFordDefaultTraits {

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

    typedef GR Digraph;

    /// \brief 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;

    /// \brief Operation traits for Bellman-Ford algorithm.

    ///

    /// It defines the used operations and the infinity value for the

    /// given \c Value type.

    /// \see BellmanFordDefaultOperationTraits,

    /// BellmanFordToleranceOperationTraits

    typedef BellmanFordDefaultOperationTraits<Value> OperationTraits;

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

    /// shortest paths.

    ///

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

    /// arcs of the shortest paths.

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

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

    /// \brief 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 GR& g) {

      return new PredMap(g);

    }

    /// \brief 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 GR::template NodeMap<typename LEN::Value> DistMap;

    /// \brief 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 GR& g) {

      return new DistMap(g);

    }

  };

  /// \brief %BellmanFord algorithm class.

  ///

  /// \ingroup shortest_path

  /// This class provides an efficient implementation of the Bellman-Ford

  /// algorithm. The maximum time complexity of the algorithm is

  /// <tt>O(ne)</tt>.

  ///

  /// The Bellman-Ford algorithm solves the single-source shortest path

  /// problem when the arcs can have negative lengths, but the digraph

  /// should not contain directed cycles with negative total length.

  /// If all arc costs are non-negative, consider to use the Dijkstra

  /// algorithm instead, since it is more efficient.

  ///

  /// 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 values is determined by the

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

  ///

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

  /// Bellman-Ford 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. The default map type is

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

  /// \tparam TR The traits class that defines various types used by the

  /// algorithm. By default, it is \ref BellmanFordDefaultTraits

  /// "BellmanFordDefaultTraits<GR, LEN>".

  /// In most cases, this parameter should not be set directly,

  /// consider to use the named template parameters instead.

#ifdef DOXYGEN

  template <typename GR, typename LEN, typename TR>

#else

  template <typename GR=ListDigraph,

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

            typename TR=BellmanFordDefaultTraits<GR,LEN> >

#endif

  class BellmanFord {

  public:

    ///The type of the underlying digraph.

    typedef typename TR::Digraph Digraph;

    /// \brief The type of the arc lengths.

    typedef typename TR::LengthMap::Value Value;

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

    typedef typename TR::LengthMap LengthMap;

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

    /// 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;

    /// The type of the paths.

    typedef PredMapPath<Digraph, PredMap> Path;

    ///\brief The \ref BellmanFordDefaultOperationTraits

    /// "operation traits class" of the algorithm.

    typedef typename TR::OperationTraits OperationTraits;

    ///The \ref BellmanFordDefaultTraits "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 *_gr;

    // 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;

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

    MaskMap *_mask;

    std::vector<Node> _process;

    // Creates the maps if necessary.

    void create_maps() {

      if(!_pred) {

        _local_pred = true;

        _pred = Traits::createPredMap(*_gr);

      }

      if(!_dist) {

        _local_dist = true;

        _dist = Traits::createDistMap(*_gr);

      }

      if(!_mask) {

        _mask = new MaskMap(*_gr);

      }

    }

  public :

    typedef BellmanFord 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 BellmanFord< Digraph, LengthMap, SetPredMapTraits<T> > {

      typedef BellmanFord< 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 BellmanFord< Digraph, LengthMap, SetDistMapTraits<T> > {

      typedef BellmanFord< Digraph, LengthMap, SetDistMapTraits<T> > 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 BellmanFordDefaultOperationTraits.

    template <class T>

    struct SetOperationTraits

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

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

      Create;

    };

    ///@}

  protected:

    BellmanFord() {}

  public:

    /// \brief Constructor.

    ///

    /// Constructor.

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

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

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

      _gr(&g), _length(&length),

      _pred(0), _local_pred(false),

      _dist(0), _local_dist(false), _mask(0) {}

    ///Destructor.

    ~BellmanFord() {

      if(_local_pred) delete _pred;

      if(_local_dist) delete _dist;

      if(_mask) delete _mask;

    }

    /// \brief Sets the length map.

    ///

    /// Sets the length map.

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

    BellmanFord &lengthMap(const LengthMap &map) {

      _length = ↦

      return *this;

    }

    /// \brief 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()

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

    /// The destructor deallocates this automatically allocated map,

    /// of course.

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

    BellmanFord &predMap(PredMap &map) {

      if(_local_pred) {

        delete _pred;

        _local_pred=false;

      }

      _pred = ↦

      return *this;

    }

    /// \brief 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()

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

    /// The destructor deallocates this automatically allocated map,

    /// of course.

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

    BellmanFord &distMap(DistMap &map) {

      if(_local_dist) {

        delete _dist;

        _local_dist=false;

      }

      _dist = ↦

      return *this;

    }

    /// \name Execution Control

    /// The simplest way to execute the Bellman-Ford algorithm is to use

    /// one of the member functions called \ref 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 \ref start(), \ref checkedStart() or

    /// \ref limitedStart().

    ///@{

    /// \brief Initializes the internal data structures.

    ///

    /// Initializes the internal data structures. The optional parameter

    /// is the initial distance of each node.

    void init(const Value value = OperationTraits::infinity()) {

      create_maps();

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

        _pred->set(it, INVALID);

        _dist->set(it, value);

      }

      _process.clear();

      if (OperationTraits::less(value, OperationTraits::infinity())) {

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

          _process.push_back(it);

          _mask->set(it, true);

        }

      } else {

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

          _mask->set(it, false);

        }

      }

    }

    /// \brief Adds a new source node.

    ///

    /// This function adds a new source node. The optional second parameter

    /// is the initial distance of the node.

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

      _dist->set(source, dst);

      if (!(*_mask)[source]) {

        _process.push_back(source);

        _mask->set(source, true);

      }

    }

    /// \brief Executes one round from the Bellman-Ford algorithm.

    ///

    /// If the algoritm calculated the distances in the previous round

    /// exactly for the paths of at most \c k arcs, then this function

    /// will calculate the distances exactly for the paths of at most

    /// <tt>k+1</tt> arcs. Performing \c k iterations using this function

    /// calculates the shortest path distances exactly for the paths

    /// consisting of at most \c k arcs.

    ///

    /// \warning The paths with limited arc number cannot be retrieved

    /// easily with \ref path() or \ref predArc() functions. If you also

    /// need the shortest paths and not only the distances, you should

    /// store the \ref predMap() "predecessor map" after each iteration

    /// and build the path manually.

    ///

    /// \return \c true when the algorithm have not found more shorter

    /// paths.

    ///

    /// \see ActiveIt

    bool processNextRound() {

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

        _mask->set(_process[i], false);

      }

      std::vector<Node> nextProcess;

      std::vector<Value> values(_process.size());

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

        values[i] = (*_dist)[_process[i]];

      }

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

        for (OutArcIt it(*_gr, _process[i]); it != INVALID; ++it) {

          Node target = _gr->target(it);

          Value relaxed = OperationTraits::plus(values[i], (*_length)[it]);

          if (OperationTraits::less(relaxed, (*_dist)[target])) {

            _pred->set(target, it);

            _dist->set(target, relaxed);

            if (!(*_mask)[target]) {

              _mask->set(target, true);

              nextProcess.push_back(target);

            }

          }

        }

      }

      _process.swap(nextProcess);

      return _process.empty();

    }

    /// \brief Executes one weak round from the Bellman-Ford algorithm.

    ///

    /// If the algorithm calculated the distances in the previous round

    /// at least for the paths of at most \c k arcs, then this function

    /// will calculate the distances at least for the paths of at most

    /// <tt>k+1</tt> arcs.

    /// This function does not make it possible to calculate the shortest

    /// path distances exactly for paths consisting of at most \c k arcs,

    /// this is why it is called weak round.

    ///

    /// \return \c true when the algorithm have not found more shorter

    /// paths.

    ///

    /// \see ActiveIt

    bool processNextWeakRound() {

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

        _mask->set(_process[i], false);

      }

      std::vector<Node> nextProcess;

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

        for (OutArcIt it(*_gr, _process[i]); it != INVALID; ++it) {

          Node target = _gr->target(it);

          Value relaxed =

            OperationTraits::plus((*_dist)[_process[i]], (*_length)[it]);

          if (OperationTraits::less(relaxed, (*_dist)[target])) {

            _pred->set(target, it);

            _dist->set(target, relaxed);

            if (!(*_mask)[target]) {

              _mask->set(target, true);

              nextProcess.push_back(target);

            }

          }

        }

      }

      _process.swap(nextProcess);

      return _process.empty();

    }

    /// \brief Executes the algorithm.

    ///

    /// Executes the algorithm.

    ///

    /// This method runs the Bellman-Ford 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.

    void start() {

      int num = countNodes(*_gr) - 1;

      for (int i = 0; i < num; ++i) {

        if (processNextWeakRound()) break;

      }

    }

    /// \brief Executes the algorithm and checks the negative cycles.

    ///

    /// Executes the algorithm and checks the negative cycles.

    ///

    /// This method runs the Bellman-Ford algorithm from the root node(s)

    /// in order to compute the shortest path to each node and also checks

    /// if the digraph contains cycles with negative total length.

    ///

    /// The algorithm computes

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

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

    ///

    /// \return \c false if there is a negative cycle in the digraph.

    ///

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

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

    bool checkedStart() {

      int num = countNodes(*_gr);

      for (int i = 0; i < num; ++i) {

        if (processNextWeakRound()) return true;

      }

      return _process.empty();

    }

    /// \brief Executes the algorithm with arc number limit.

    ///

    /// Executes the algorithm with arc number limit.

    ///

    /// This method runs the Bellman-Ford algorithm from the root node(s)

    /// in order to compute the shortest path distance for each node

    /// using only the paths consisting of at most \c num arcs.

    ///

    /// The algorithm computes

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

    /// - the predecessor arc for each node.

    ///

    /// \warning The paths with limited arc number cannot be retrieved

    /// easily with \ref path() or \ref predArc() functions. If you also

    /// need the shortest paths and not only the distances, you should

    /// store the \ref predMap() "predecessor map" after each iteration

    /// and build the path manually.

    ///

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

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

    void limitedStart(int num) {

      for (int i = 0; i < num; ++i) {

        if (processNextRound()) break;

      }

    }

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

    ///

    /// This method runs the Bellman-Ford algorithm from the given root

    /// node \c 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).

    ///

    /// \note bf.run(s) is just a shortcut of the following code.

    /// \code

    ///   bf.init();

    ///   bf.addSource(s);

    ///   bf.start();

    /// \endcode

    void run(Node s) {

      init();

      addSource(s);

      start();

    }

    /// \brief Runs the algorithm from the given root node with arc

    /// number limit.

    ///

    /// This method runs the Bellman-Ford algorithm from the given root

    /// node \c s in order to compute the shortest path distance for each

    /// node using only the paths consisting of at most \c num arcs.

    ///

    /// The algorithm computes

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

    /// - the predecessor arc for each node.

    ///

    /// \warning The paths with limited arc number cannot be retrieved

    /// easily with \ref path() or \ref predArc() functions. If you also

    /// need the shortest paths and not only the distances, you should

    /// store the \ref predMap() "predecessor map" after each iteration

    /// and build the path manually.

    ///

    /// \note bf.run(s, num) is just a shortcut of the following code.

    /// \code

    ///   bf.init();

    ///   bf.addSource(s);

    ///   bf.limitedStart(num);

    /// \endcode

    void run(Node s, int num) {

      init();

      addSource(s);

      limitedStart(num);

    }

    ///@}

    /// \brief LEMON iterator for getting the active nodes.

    ///

    /// This class provides a common style LEMON iterator that traverses

    /// the active nodes of the Bellman-Ford algorithm after the last

    /// phase. These nodes should be checked in the next phase to

    /// find augmenting arcs outgoing from them.

    class ActiveIt {

    public:

      /// \brief Constructor.

      ///

      /// Constructor for getting the active nodes of the given BellmanFord

      /// instance.

      ActiveIt(const BellmanFord& algorithm) : _algorithm(&algorithm)

      {

        _index = _algorithm->_process.size() - 1;

      }

      /// \brief Invalid constructor.

      ///

      /// Invalid constructor.

      ActiveIt(Invalid) : _algorithm(0), _index(-1) {}

      /// \brief Conversion to \c Node.

      ///

      /// Conversion to \c Node.

      operator Node() const {

        return _index >= 0 ? _algorithm->_process[_index] : INVALID;

      }

      /// \brief Increment operator.

      ///

      /// Increment operator.

      ActiveIt& operator++() {

        --_index;

        return *this;

      }

      bool operator==(const ActiveIt& it) const {

        return static_cast<Node>(*this) == static_cast<Node>(it);

      }

      bool operator!=(const ActiveIt& it) const {

        return static_cast<Node>(*this) != static_cast<Node>(it);

      }

      bool operator<(const ActiveIt& it) const {

        return static_cast<Node>(*this) < static_cast<Node>(it);

      }

    private:

      const BellmanFord* _algorithm;

      int _index;

    };

    /// \name Query Functions

    /// The result of the Bellman-Ford algorithm can be obtained using these

    /// functions.\n

    /// Either \ref run() or \ref init() should be called before using them.

    ///@{

    /// \brief The shortest path to the given node.

    ///

    /// Gives back 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() or \ref init() must be called before

    /// using this function.

    Path path(Node t) const

    {

      return Path(*_gr, *_pred, t);

    }

    /// \brief 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() 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 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() 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 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() or \ref init() must be called before

    /// using this function.

    Node predNode(Node v) const {

      return (*_pred)[v] == INVALID ? INVALID : _gr->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() 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() or \ref init() must be called before

    /// using this function.

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

    /// \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() or \ref init() must be called before

    /// using this function.

    bool reached(Node v) const {

      return (*_dist)[v] != OperationTraits::infinity();

    }

    /// \brief Gives back a negative cycle.

    ///

    /// This function gives back a directed cycle with negative total

    /// length if the algorithm has already found one.

    /// Otherwise it gives back an empty path.

    lemon::Path<Digraph> negativeCycle() const {

      typename Digraph::template NodeMap<int> state(*_gr, -1);

      lemon::Path<Digraph> cycle;

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

        if (state[_process[i]] != -1) continue;

        for (Node v = _process[i]; (*_pred)[v] != INVALID;

             v = _gr->source((*_pred)[v])) {

          if (state[v] == i) {

            cycle.addFront((*_pred)[v]);

            for (Node u = _gr->source((*_pred)[v]); u != v;

                 u = _gr->source((*_pred)[u])) {

              cycle.addFront((*_pred)[u]);

            }

            return cycle;

          }

          else if (state[v] >= 0) {

            break;

          }

          state[v] = i;

        }

      }

      return cycle;

    }

    ///@}

  };

  /// \brief Default traits class of bellmanFord() function.

  ///

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

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

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

  template <typename GR, typename LEN>

  struct BellmanFordWizardDefaultTraits {

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

    typedef GR Digraph;

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

    ///

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

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

    typedef LEN LengthMap;

    /// The type of the arc lengths.

    typedef typename LEN::Value Value;

    /// \brief Operation traits for Bellman-Ford algorithm.

    ///

    /// It defines the used operations and the infinity value for the

    /// given \c Value type.

    /// \see BellmanFordDefaultOperationTraits,

    /// BellmanFordToleranceOperationTraits

    typedef BellmanFordDefaultOperationTraits<Value> OperationTraits;

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

    /// arcs of the shortest paths.

    ///

    /// The type of the map that stores the last arcs of the shortest paths.

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

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

    /// \brief 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 GR &g) {

      return new PredMap(g);

    }

    /// \brief 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 GR::template NodeMap<Value> DistMap;

    /// \brief 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 GR &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;

  };

  /// \brief Default traits class used by BellmanFordWizard.

  ///

  /// Default traits class used by BellmanFordWizard.

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

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

  template <typename GR, typename LEN>

  class BellmanFordWizardBase

    : public BellmanFordWizardDefaultTraits<GR, LEN> {

    typedef BellmanFordWizardDefaultTraits<GR, LEN> Base;

  protected:

    // Type of the nodes in the digraph.

    typedef typename Base::Digraph::Node Node;

    // Pointer to the underlying digraph.

    void *_graph;

    // Pointer to the length map

    void *_length;

    // 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.

    void *_di;

    public:

    /// Constructor.

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

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

    BellmanFordWizardBase() :

      _graph(0), _length(0), _pred(0), _dist(0), _path(0), _di(0) {}

    /// Constructor.

    /// This constructor requires two parameters,

    /// others are initiated to \c 0.

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

    /// \param len The length map.

    BellmanFordWizardBase(const GR& gr,

                          const LEN& len) :

      _graph(reinterpret_cast<void*>(const_cast<GR*>(&gr))),

      _length(reinterpret_cast<void*>(const_cast<LEN*>(&len))),

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

  };

  /// \brief Auxiliary class for the function-type interface of the

  /// \ref BellmanFord "Bellman-Ford" algorithm.

  ///

  /// This auxiliary class is created to implement the

  /// \ref bellmanFord() "function-type interface" of the

  /// \ref BellmanFord "Bellman-Ford" algorithm.

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

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

  ///

  /// This class should only be used through the \ref bellmanFord()

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

  ///

  /// \tparam TR The traits class that defines various types used by the

  /// algorithm.