RhodeCode

%%%%% Defining LEMON %%%%%

@misc{lemon,

  key =          {LEMON},

  title =        {{LEMON} -- {L}ibrary for {E}fficient {M}odeling and

                  {O}ptimization in {N}etworks},

  howpublished = {\url{http://lemon.cs.elte.hu/}},

  year =         2009

}

@misc{egres,

  key =          {EGRES},

  title =        {{EGRES} -- {E}gerv{\'a}ry {R}esearch {G}roup on

                  {C}ombinatorial {O}ptimization},

  url =          {http://www.cs.elte.hu/egres/}

}

@misc{coinor,

  key =          {COIN-OR},

  title =        {{COIN-OR} -- {C}omputational {I}nfrastructure for

                  {O}perations {R}esearch},

  url =          {http://www.coin-or.org/}

}

%%%%% Other libraries %%%%%%

@misc{boost,

  key =          {Boost},

  title =        {{B}oost {C++} {L}ibraries},

  url =          {http://www.boost.org/}

}

@book{bglbook,

  author =       {Jeremy G. Siek and Lee-Quan Lee and Andrew

                  Lumsdaine},

  title =        {The Boost Graph Library: User Guide and Reference

                  Manual},

  publisher =    {Addison-Wesley},

  year =         2002

}

@misc{leda,

  key =          {LEDA},

  title =        {{LEDA} -- {L}ibrary of {E}fficient {D}ata {T}ypes and

                  {A}lgorithms},

  url =          {http://www.algorithmic-solutions.com/}

}

@book{ledabook,

  author =       {Kurt Mehlhorn and Stefan N{\"a}her},

  title =        {{LEDA}: {A} platform for combinatorial and geometric

                  computing},

  isbn =         {0-521-56329-1},

  publisher =    {Cambridge University Press},

  address =      {New York, NY, USA},

  year =         1999

}

%%%%% Tools that LEMON depends on %%%%%

@misc{cmake,

  key =          {CMake},

  title =        {{CMake} -- {C}ross {P}latform {M}ake},

  url =          {http://www.cmake.org/}

}

@misc{doxygen,

  key =          {Doxygen},

  title =        {{Doxygen} -- {S}ource code documentation generator

                  tool},

  url =          {http://www.doxygen.org/}

}

%%%%% LP/MIP libraries %%%%%

@misc{glpk,

  key =          {GLPK},

  title =        {{GLPK} -- {GNU} {L}inear {P}rogramming {K}it},

  url =          {http://www.gnu.org/software/glpk/}

}

@misc{clp,

  key =          {Clp},

  title =        {{Clp} -- {Coin-Or} {L}inear {P}rogramming},

  url =          {http://projects.coin-or.org/Clp/}

}

@misc{cbc,

  key =          {Cbc},

  title =        {{Cbc} -- {Coin-Or} {B}ranch and {C}ut},

  url =          {http://projects.coin-or.org/Cbc/}

}

@misc{cplex,

  key =          {CPLEX},

  title =        {{ILOG} {CPLEX}},

  url =          {http://www.ilog.com/}

}

@misc{soplex,

  key =          {SoPlex},

  title =        {{SoPlex} -- {T}he {S}equential {O}bject-{O}riented

                  {S}implex},

  url =          {http://soplex.zib.de/}

}

%%%%% General books %%%%%

@book{amo93networkflows,

  author =       {Ravindra K. Ahuja and Thomas L. Magnanti and James

                  B. Orlin},

  title =        {Network Flows: Theory, Algorithms, and Applications},

  publisher =    {Prentice-Hall, Inc.},

  year =         1993,

  month =        feb,

  isbn =         {978-0136175490}

}

@book{schrijver03combinatorial,

  author =       {Alexander Schrijver},

  title =        {Combinatorial Optimization: Polyhedra and Efficiency},

  publisher =    {Springer-Verlag},

  year =         2003,

  isbn =         {978-3540443896}

}

@book{clrs01algorithms,

  author =       {Thomas H. Cormen and Charles E. Leiserson and Ronald

                  L. Rivest and Clifford Stein},

  title =        {Introduction to Algorithms},

  publisher =    {The MIT Press},

  year =         2001,

  edition =      {2nd}

}

@book{stroustrup00cpp,

  author =       {Bjarne Stroustrup},

  title =        {The C++ Programming Language},

  edition =      {3rd},

  publisher =    {Addison-Wesley Professional},

  isbn =         0201700735,

  month =        {February},

  year =         2000

}

%%%%% Maximum flow algorithms %%%%%

@article{edmondskarp72theoretical,

  author =       {Jack Edmonds and Richard M. Karp},

  title =        {Theoretical improvements in algorithmic efficiency

                  for network flow problems},

  journal =      {Journal of the ACM},

  year =         1972,

  volume =       19,

  number =       2,

  pages =        {248-264}

}

@article{goldberg88newapproach,

  author =       {Andrew V. Goldberg and Robert E. Tarjan},

  title =        {A new approach to the maximum flow problem},

  journal =      {Journal of the ACM},

  year =         1988,

  volume =       35,

  number =       4,

  pages =        {921-940}

}

@article{dinic70algorithm,

  author =       {E. A. Dinic},

  title =        {Algorithm for solution of a problem of maximum flow

                  in a network with power estimation},

  journal =      {Soviet Math. Doklady},

  year =         1970,

  volume =       11,

  pages =        {1277-1280}

}

@article{goldberg08partial,

  author =       {Andrew V. Goldberg},

  title =        {The Partial Augment-Relabel Algorithm for the

                  Maximum Flow Problem},

  journal =      {16th Annual European Symposium on Algorithms},

  year =         2008,

  pages =        {466-477}

}

@article{sleator83dynamic,

  author =       {Daniel D. Sleator and Robert E. Tarjan},

  title =        {A data structure for dynamic trees},

  journal =      {Journal of Computer and System Sciences},

  year =         1983,

  volume =       26,

  number =       3,

  pages =        {362-391}

}

%%%%% Minimum mean cycle algorithms %%%%%

@article{karp78characterization,

  author =       {Richard M. Karp},

  title =        {A characterization of the minimum cycle mean in a

                  digraph},

  journal =      {Discrete Math.},

  year =         1978,

  volume =       23,

  pages =        {309-311}

}

@article{dasdan98minmeancycle,

  author =       {Ali Dasdan and Rajesh K. Gupta},

  title =        {Faster Maximum and Minimum Mean Cycle Alogrithms for

                  System Performance Analysis},

  journal =      {IEEE Transactions on Computer-Aided Design of

                  Integrated Circuits and Systems},

  year =         1998,

  volume =       17,

  number =       10,

  pages =        {889-899}

}

%%%%% Minimum cost flow algorithms %%%%%

@article{klein67primal,

  author =       {Morton Klein},

  title =        {A primal method for minimal cost flows with

                  applications to the assignment and transportation

                  problems},

  journal =      {Management Science},

  year =         1967,

  volume =       14,

  pages =        {205-220}

}

@article{goldberg89cyclecanceling,

  author =       {Andrew V. Goldberg and Robert E. Tarjan},

  title =        {Finding minimum-cost circulations by canceling

                  negative cycles},

  journal =      {Journal of the ACM},

  year =         1989,

  volume =       36,

  number =       4,

  pages =        {873-886}

}

@article{goldberg90approximation,

  author =       {Andrew V. Goldberg and Robert E. Tarjan},

  title =        {Finding Minimum-Cost Circulations by Successive

                  Approximation},

  journal =      {Mathematics of Operations Research},

  year =         1990,

  volume =       15,

  number =       3,

  pages =        {430-466}

}

@article{goldberg97efficient,

  author =       {Andrew V. Goldberg},

  title =        {An Efficient Implementation of a Scaling

                  Minimum-Cost Flow Algorithm},

  journal =      {Journal of Algorithms},

  year =         1997,

  volume =       22,

  number =       1,

  pages =        {1-29}

}

@article{bunnagel98efficient,

  author =       {Ursula B{\"u}nnagel and Bernhard Korte and Jens

                  Vygen},

  title =        {Efficient implementation of the {G}oldberg-{T}arjan

                  minimum-cost flow algorithm},

  journal =      {Optimization Methods and Software},

  year =         1998,

  volume =       10,

  pages =        {157-174}

}

@book{dantzig63linearprog,

  author =       {George B. Dantzig},

  title =        {Linear Programming and Extensions},

  publisher =    {Princeton University Press},

  year =         1963

}

@mastersthesis{kellyoneill91netsimplex,

  author =       {Damian J. Kelly and Garrett M. O'Neill},

  title =        {The Minimum Cost Flow Problem and The Network

                  Simplex Method},

  school =       {University College},

  address =      {Dublin, Ireland},

  year =         1991,

  month =        sep,

}

/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2008

 * 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/path.h>

#include <limits>

namespace lemon {

  /// \brief Default OperationTraits 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.

  template <

    typename V, 

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

  struct BellmanFordDefaultOperationTraits {

    /// \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 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 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

    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>".

#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);

      }

      _mask = new MaskMap(*_gr, false);

    }

  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);

	}

      }

    }

    /// \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]; }