/* -*- 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.

   template<class TR>

   class BellmanFordWizard : public TR {

     typedef TR Base;

     typedef typename TR::Digraph Digraph;

     typedef typename Digraph::Node Node;

     typedef typename Digraph::NodeIt NodeIt;

     typedef typename Digraph::Arc Arc;

     typedef typename Digraph::OutArcIt ArcIt;

     typedef typename TR::LengthMap LengthMap;

     typedef typename LengthMap::Value Value;

     typedef typename TR::PredMap PredMap;

     typedef typename TR::DistMap DistMap;

     typedef typename TR::Path Path;

   public:

     /// Constructor.

     BellmanFordWizard() : TR() {}

     /// \brief Constructor that requires parameters.

     ///

     /// Constructor that requires parameters.

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

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

     /// \param len The length map.

     BellmanFordWizard(const Digraph& gr, const LengthMap& len)

       : TR(gr, len) {}

     /// \brief Copy constructor

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

     ~BellmanFordWizard() {}

     /// \brief Runs the Bellman-Ford algorithm from the given source node.

     ///

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

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

     void run(Node s) {

       BellmanFord<Digraph,LengthMap,TR>

         bf(*reinterpret_cast<const Digraph*>(Base::_graph),

            *reinterpret_cast<const LengthMap*>(Base::_length));

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

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

       bf.run(s);

     }

     /// \brief Runs the Bellman-Ford algorithm to find the shortest path

     /// between \c s and \c t.

     ///

     /// This method runs the Bellman-Ford algorithm from node \c s

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

     /// Actually, it computes the shortest path to each node, but using

     /// this function you can retrieve the distance and the shortest path

     /// for a single target node easier.

     ///

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

     bool run(Node s, Node t) {

       BellmanFord<Digraph,LengthMap,TR>

         bf(*reinterpret_cast<const Digraph*>(Base::_graph),

            *reinterpret_cast<const LengthMap*>(Base::_length));

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

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

       bf.run(s);

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

       if (Base::_di) *reinterpret_cast<Value*>(Base::_di) = bf.dist(t);

       return bf.reached(t);

     }

     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-templ-param "Named parameter" for setting

     /// the predecessor map.

     ///

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

     /// the map that stores the predecessor arcs of the nodes.

     template<class T>

     BellmanFordWizard<SetPredMapBase<T> > predMap(const T &t) {

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

       return BellmanFordWizard<SetPredMapBase<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-templ-param "Named parameter" for setting

     /// the distance map.

     ///

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

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

     /// by the algorithm.

     template<class T>

     BellmanFordWizard<SetDistMapBase<T> > distMap(const T &t) {

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

       return BellmanFordWizard<SetDistMapBase<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>

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

     {

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

       return BellmanFordWizard<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.

     BellmanFordWizard dist(const Value &d)

     {

       Base::_di=reinterpret_cast<void*>(const_cast<Value*>(&d));

       return *this;

     }

   };

   /// \brief Function type interface for the \ref BellmanFord "Bellman-Ford"

   /// algorithm.

   ///

   /// \ingroup shortest_path

   /// Function type interface for the \ref BellmanFord "Bellman-Ford"

   /// algorithm.

   ///

   /// This function also has several \ref named-templ-func-param

   /// "named parameters", they are declared as the members of class

   /// \ref BellmanFordWizard.

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

   /// \code

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

   ///   bellmanFord(g,length).predMap(preds).distMap(dists).run(s);

   ///

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

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

   /// \endcode

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

   /// to the end of the parameter list.

   /// \sa BellmanFordWizard

   /// \sa BellmanFord

   template<typename GR, typename LEN>

   BellmanFordWizard<BellmanFordWizardBase<GR,LEN> >

   bellmanFord(const GR& digraph,

               const LEN& length)

   {

     return BellmanFordWizard<BellmanFordWizardBase<GR,LEN> >(digraph, length);

   }

 } //END OF NAMESPACE LEMON

 #endif