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_HARTMANN_ORLIN_H

#define LEMON_HARTMANN_ORLIN_H

/// \ingroup min_mean_cycle

///

/// \file

/// \brief Hartmann-Orlin's algorithm for finding a minimum mean cycle.

#include <vector>

#include <limits>

#include <lemon/core.h>

#include <lemon/path.h>

#include <lemon/tolerance.h>

#include <lemon/connectivity.h>

namespace lemon {

  /// \brief Default traits class of HartmannOrlin algorithm.

  ///

  /// Default traits class of HartmannOrlin algorithm.

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

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

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

#ifdef DOXYGEN

  template <typename GR, typename LEN>

#else

  template <typename GR, typename LEN,

    bool integer = std::numeric_limits<typename LEN::Value>::is_integer>

#endif

  struct HartmannOrlinDefaultTraits

  {

    /// The type of the digraph

    typedef GR Digraph;

    /// The type of the length map

    typedef LEN LengthMap;

    /// The type of the arc lengths

    typedef typename LengthMap::Value Value;

    /// \brief The large value type used for internal computations

    ///

    /// The large value type used for internal computations.

    /// It is \c long \c long if the \c Value type is integer,

    /// otherwise it is \c double.

    /// \c Value must be convertible to \c LargeValue.

    typedef double LargeValue;

    /// The tolerance type used for internal computations

    typedef lemon::Tolerance<LargeValue> Tolerance;

    /// \brief The path type of the found cycles

    ///

    /// The path type of the found cycles.

    /// It must conform to the \ref lemon::concepts::Path "Path" concept

    /// and it must have an \c addFront() function.

    typedef lemon::Path<Digraph> Path;

  };

  // Default traits class for integer value types

  template <typename GR, typename LEN>

  struct HartmannOrlinDefaultTraits<GR, LEN, true>

  {

    typedef GR Digraph;

    typedef LEN LengthMap;

    typedef typename LengthMap::Value Value;

#ifdef LEMON_HAVE_LONG_LONG

    typedef long long LargeValue;

#else

    typedef long LargeValue;

#endif

    typedef lemon::Tolerance<LargeValue> Tolerance;

    typedef lemon::Path<Digraph> Path;

  };

  /// \addtogroup min_mean_cycle

  /// @{

  /// \brief Implementation of the Hartmann-Orlin algorithm for finding

  /// a minimum mean cycle.

  ///

  /// This class implements the Hartmann-Orlin algorithm for finding

  /// a directed cycle of minimum mean length (cost) in a digraph

  /// \ref amo93networkflows, \ref dasdan98minmeancycle.

  /// It is an improved version of \ref Karp "Karp"'s original algorithm,

  /// it applies an efficient early termination scheme.

  /// It runs in time O(ne) and uses space O(n<sup>2</sup>+e).

  ///

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

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

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

             typename TR = HartmannOrlinDefaultTraits<GR, LEN> >

#endif

  class HartmannOrlin

  {

  public:

    /// The type of the digraph

    typedef typename TR::Digraph Digraph;

    /// The type of the length map

    typedef typename TR::LengthMap LengthMap;

    /// The type of the arc lengths

    typedef typename TR::Value Value;

    /// \brief The large value type

    ///

    /// The large value type used for internal computations.

    /// Using the \ref HartmannOrlinDefaultTraits "default traits class",

    /// it is \c long \c long if the \c Value type is integer,

    /// otherwise it is \c double.

    typedef typename TR::LargeValue LargeValue;

    /// The tolerance type

    typedef typename TR::Tolerance Tolerance;

    /// \brief The path type of the found cycles

    ///

    /// The path type of the found cycles.

    /// Using the \ref HartmannOrlinDefaultTraits "default traits class",

    /// it is \ref lemon::Path "Path<Digraph>".

    typedef typename TR::Path Path;

    /// The \ref HartmannOrlinDefaultTraits "traits class" of the algorithm

    typedef TR Traits;

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    // Data sturcture for path data

    struct PathData

    {

      LargeValue dist;

      Arc pred;

      PathData(LargeValue d, Arc p = INVALID) :

        dist(d), pred(p) {}

    };

    typedef typename Digraph::template NodeMap<std::vector<PathData> >

      PathDataNodeMap;

  private:

    // The digraph the algorithm runs on

    const Digraph &_gr;

    // The length of the arcs

    const LengthMap &_length;

    // Data for storing the strongly connected components

    int _comp_num;

    typename Digraph::template NodeMap<int> _comp;

    std::vector<std::vector<Node> > _comp_nodes;

    std::vector<Node>* _nodes;

    typename Digraph::template NodeMap<std::vector<Arc> > _out_arcs;

    // Data for the found cycles

    bool _curr_found, _best_found;

    LargeValue _curr_length, _best_length;

    int _curr_size, _best_size;

    Node _curr_node, _best_node;

    int _curr_level, _best_level;

    Path *_cycle_path;

    bool _local_path;

    // Node map for storing path data

    PathDataNodeMap _data;

    // The processed nodes in the last round

    std::vector<Node> _process;

    Tolerance _tolerance;

    // Infinite constant

    const LargeValue INF;

  public:

    /// \name Named Template Parameters

    /// @{

    template <typename T>

    struct SetLargeValueTraits : public Traits {

      typedef T LargeValue;

      typedef lemon::Tolerance<T> Tolerance;

    };

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

    /// \c LargeValue type.

    ///

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

    /// type. It is used for internal computations in the algorithm.

    template <typename T>

    struct SetLargeValue

      : public HartmannOrlin<GR, LEN, SetLargeValueTraits<T> > {

      typedef HartmannOrlin<GR, LEN, SetLargeValueTraits<T> > Create;

    };

    template <typename T>

    struct SetPathTraits : public Traits {

      typedef T Path;

    };

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

    /// \c %Path type.

    ///

    /// \ref named-templ-param "Named parameter" for setting the \c %Path

    /// type of the found cycles.

    /// It must conform to the \ref lemon::concepts::Path "Path" concept

    /// and it must have an \c addFront() function.

    template <typename T>

    struct SetPath

      : public HartmannOrlin<GR, LEN, SetPathTraits<T> > {

      typedef HartmannOrlin<GR, LEN, SetPathTraits<T> > Create;

    };

    /// @}

  public:

    /// \brief Constructor.

    ///

    /// The constructor of the class.

    ///

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

    /// \param length The lengths (costs) of the arcs.

    HartmannOrlin( const Digraph &digraph,

                   const LengthMap &length ) :

      _gr(digraph), _length(length), _comp(digraph), _out_arcs(digraph),

      _best_found(false), _best_length(0), _best_size(1),

      _cycle_path(NULL), _local_path(false), _data(digraph),

      INF(std::numeric_limits<LargeValue>::has_infinity ?

          std::numeric_limits<LargeValue>::infinity() :

          std::numeric_limits<LargeValue>::max())

    {}

    /// Destructor.

    ~HartmannOrlin() {

      if (_local_path) delete _cycle_path;

    }

    /// \brief Set the path structure for storing the found cycle.

    ///

    /// This function sets an external path structure for storing the

    /// found cycle.

    ///

    /// If you don't call this function before calling \ref run() or

    /// \ref findMinMean(), it will allocate a local \ref Path "path"

    /// structure. The destuctor deallocates this automatically

    /// allocated object, of course.

    ///

    /// \note The algorithm calls only the \ref lemon::Path::addFront()

    /// "addFront()" function of the given path structure.

    ///

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

    HartmannOrlin& cycle(Path &path) {

      if (_local_path) {

        delete _cycle_path;

        _local_path = false;

      }

      _cycle_path = &path;

      return *this;

    }

    /// \brief Set the tolerance used by the algorithm.

    ///

    /// This function sets the tolerance object used by the algorithm.

    ///

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

    HartmannOrlin& tolerance(const Tolerance& tolerance) {

      _tolerance = tolerance;

      return *this;

    }

    /// \brief Return a const reference to the tolerance.

    ///

    /// This function returns a const reference to the tolerance object

    /// used by the algorithm.

    const Tolerance& tolerance() const {

      return _tolerance;

    }

    /// \name Execution control

    /// The simplest way to execute the algorithm is to call the \ref run()

    /// function.\n

    /// If you only need the minimum mean length, you may call

    /// \ref findMinMean().

    /// @{

    /// \brief Run the algorithm.

    ///

    /// This function runs the algorithm.

    /// It can be called more than once (e.g. if the underlying digraph

    /// and/or the arc lengths have been modified).

    ///

    /// \return \c true if a directed cycle exists in the digraph.

    ///

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

    /// \code

    ///   return mmc.findMinMean() && mmc.findCycle();

    /// \endcode

    bool run() {

      return findMinMean() && findCycle();

    }

    /// \brief Find the minimum cycle mean.

    ///

    /// This function finds the minimum mean length of the directed

    /// cycles in the digraph.

    ///

    /// \return \c true if a directed cycle exists in the digraph.

    bool findMinMean() {

      // Initialization and find strongly connected components

      init();

      findComponents();

      // Find the minimum cycle mean in the components

      for (int comp = 0; comp < _comp_num; ++comp) {

        if (!initComponent(comp)) continue;

        processRounds();

        // Update the best cycle (global minimum mean cycle)

        if ( _curr_found && (!_best_found || 

             _curr_length * _best_size < _best_length * _curr_size) ) {

          _best_found = true;

          _best_length = _curr_length;

          _best_size = _curr_size;

          _best_node = _curr_node;

          _best_level = _curr_level;

        }

      }

      return _best_found;

    }

    /// \brief Find a minimum mean directed cycle.

    ///

    /// This function finds a directed cycle of minimum mean length

    /// in the digraph using the data computed by findMinMean().

    ///

    /// \return \c true if a directed cycle exists in the digraph.

    ///

    /// \pre \ref findMinMean() must be called before using this function.

    bool findCycle() {

      if (!_best_found) return false;

      IntNodeMap reached(_gr, -1);

      int r = _best_level + 1;

      Node u = _best_node;

      while (reached[u] < 0) {

        reached[u] = --r;

        u = _gr.source(_data[u][r].pred);

      }

      r = reached[u];

      Arc e = _data[u][r].pred;

      _cycle_path->addFront(e);

      _best_length = _length[e];

      _best_size = 1;

      Node v;

      while ((v = _gr.source(e)) != u) {

        e = _data[v][--r].pred;

        _cycle_path->addFront(e);

        _best_length += _length[e];

        ++_best_size;

      }

      return true;

    }

    /// @}

    /// \name Query Functions

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

    /// functions.\n

    /// The algorithm should be executed before using them.

    /// @{

    /// \brief Return the total length of the found cycle.

    ///

    /// This function returns the total length of the found cycle.

    ///

    /// \pre \ref run() or \ref findMinMean() must be called before

    /// using this function.

    LargeValue cycleLength() const {

      return _best_length;

    }

    /// \brief Return the number of arcs on the found cycle.

    ///

    /// This function returns the number of arcs on the found cycle.

    ///

    /// \pre \ref run() or \ref findMinMean() must be called before

    /// using this function.

    int cycleArcNum() const {

      return _best_size;

    }

    /// \brief Return the mean length of the found cycle.

    ///

    /// This function returns the mean length of the found cycle.

    ///

    /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the

    /// following code.

    /// \code

    ///   return static_cast<double>(alg.cycleLength()) / alg.cycleArcNum();

    /// \endcode

    ///

    /// \pre \ref run() or \ref findMinMean() must be called before

    /// using this function.

    double cycleMean() const {

      return static_cast<double>(_best_length) / _best_size;

    }

    /// \brief Return the found cycle.

    ///

    /// This function returns a const reference to the path structure

    /// storing the found cycle.

    ///

    /// \pre \ref run() or \ref findCycle() must be called before using

    /// this function.

    const Path& cycle() const {

      return *_cycle_path;

    }

    ///@}

  private:

    // Initialization

    void init() {

      if (!_cycle_path) {

        _local_path = true;

        _cycle_path = new Path;

      }

      _cycle_path->clear();

      _best_found = false;

      _best_length = 0;

      _best_size = 1;

      _cycle_path->clear();

      for (NodeIt u(_gr); u != INVALID; ++u)

        _data[u].clear();

    }

    // Find strongly connected components and initialize _comp_nodes

    // and _out_arcs

    void findComponents() {

      _comp_num = stronglyConnectedComponents(_gr, _comp);

      _comp_nodes.resize(_comp_num);

      if (_comp_num == 1) {

        _comp_nodes[0].clear();

        for (NodeIt n(_gr); n != INVALID; ++n) {

          _comp_nodes[0].push_back(n);

          _out_arcs[n].clear();

          for (OutArcIt a(_gr, n); a != INVALID; ++a) {

            _out_arcs[n].push_back(a);

          }

        }

      } else {

        for (int i = 0; i < _comp_num; ++i)

          _comp_nodes[i].clear();

        for (NodeIt n(_gr); n != INVALID; ++n) {

          int k = _comp[n];

          _comp_nodes[k].push_back(n);

          _out_arcs[n].clear();

          for (OutArcIt a(_gr, n); a != INVALID; ++a) {

            if (_comp[_gr.target(a)] == k) _out_arcs[n].push_back(a);

          }

        }

      }

    }

    // Initialize path data for the current component

    bool initComponent(int comp) {

      _nodes = &(_comp_nodes[comp]);

      int n = _nodes->size();

      if (n < 1 || (n == 1 && _out_arcs[(*_nodes)[0]].size() == 0)) {

        return false;

      }      

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

        _data[(*_nodes)[i]].resize(n + 1, PathData(INF));

      }

      return true;

    }

    // Process all rounds of computing path data for the current component.

    // _data[v][k] is the length of a shortest directed walk from the root

    // node to node v containing exactly k arcs.

    void processRounds() {

      Node start = (*_nodes)[0];

      _data[start][0] = PathData(0);

      _process.clear();

      _process.push_back(start);

      int k, n = _nodes->size();

      int next_check = 4;

      bool terminate = false;

      for (k = 1; k <= n && int(_process.size()) < n && !terminate; ++k) {

        processNextBuildRound(k);

        if (k == next_check || k == n) {

          terminate = checkTermination(k);

          next_check = next_check * 3 / 2;

        }

      }

      for ( ; k <= n && !terminate; ++k) {

        processNextFullRound(k);

        if (k == next_check || k == n) {

          terminate = checkTermination(k);

          next_check = next_check * 3 / 2;

        }

      }

    }

    // Process one round and rebuild _process

    void processNextBuildRound(int k) {

      std::vector<Node> next;

      Node u, v;

      Arc e;

      LargeValue d;

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

        u = _process[i];

        for (int j = 0; j < int(_out_arcs[u].size()); ++j) {

          e = _out_arcs[u][j];

          v = _gr.target(e);

          d = _data[u][k-1].dist + _length[e];

          if (_tolerance.less(d, _data[v][k].dist)) {

            if (_data[v][k].dist == INF) next.push_back(v);

            _data[v][k] = PathData(d, e);

          }

        }

      }

      _process.swap(next);

    }

    // Process one round using _nodes instead of _process

    void processNextFullRound(int k) {

      Node u, v;

      Arc e;

      LargeValue d;

      for (int i = 0; i < int(_nodes->size()); ++i) {

        u = (*_nodes)[i];

        for (int j = 0; j < int(_out_arcs[u].size()); ++j) {

          e = _out_arcs[u][j];

          v = _gr.target(e);

          d = _data[u][k-1].dist + _length[e];

          if (_tolerance.less(d, _data[v][k].dist)) {

            _data[v][k] = PathData(d, e);

          }

        }

      }

    }

    // Check early termination

    bool checkTermination(int k) {

      typedef std::pair<int, int> Pair;

      typename GR::template NodeMap<Pair> level(_gr, Pair(-1, 0));

      typename GR::template NodeMap<LargeValue> pi(_gr);

      int n = _nodes->size();

      LargeValue length;

      int size;

      Node u;

      // Search for cycles that are already found

      _curr_found = false;

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

        u = (*_nodes)[i];

        if (_data[u][k].dist == INF) continue;

        for (int j = k; j >= 0; --j) {

          if (level[u].first == i && level[u].second > 0) {

            // A cycle is found

            length = _data[u][level[u].second].dist - _data[u][j].dist;

            size = level[u].second - j;

            if (!_curr_found || length * _curr_size < _curr_length * size) {

              _curr_length = length;

              _curr_size = size;

              _curr_node = u;

              _curr_level = level[u].second;

              _curr_found = true;

            }

          }

          level[u] = Pair(i, j);

          u = _gr.source(_data[u][j].pred);

        }

      }

      // If at least one cycle is found, check the optimality condition

      LargeValue d;

      if (_curr_found && k < n) {

        // Find node potentials

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

          u = (*_nodes)[i];

          pi[u] = INF;

          for (int j = 0; j <= k; ++j) {

            if (_data[u][j].dist < INF) {

              d = _data[u][j].dist * _curr_size - j * _curr_length;

              if (_tolerance.less(d, pi[u])) pi[u] = d;

            }

          }

        }

        // Check the optimality condition for all arcs

        bool done = true;

        for (ArcIt a(_gr); a != INVALID; ++a) {

          if (_tolerance.less(_length[a] * _curr_size - _curr_length,

                              pi[_gr.target(a)] - pi[_gr.source(a)]) ) {

            done = false;

            break;

          }

        }

        return done;

      }

      return (k == n);

    }

  }; //class HartmannOrlin

  ///@}

} //namespace lemon

#endif //LEMON_HARTMANN_ORLIN_H

/* -*- 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_HOWARD_H

#define LEMON_HOWARD_H

/// \ingroup min_mean_cycle

///

/// \file

/// \brief Howard's algorithm for finding a minimum mean cycle.

#include <vector>

#include <limits>

#include <lemon/core.h>

#include <lemon/path.h>

#include <lemon/tolerance.h>

#include <lemon/connectivity.h>

namespace lemon {

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

  ///

  /// Default traits class of Howard class.

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

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

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

#ifdef DOXYGEN

  template <typename GR, typename LEN>

#else

  template <typename GR, typename LEN,

    bool integer = std::numeric_limits<typename LEN::Value>::is_integer>

#endif

  struct HowardDefaultTraits

  {

    /// The type of the digraph

    typedef GR Digraph;

    /// The type of the length map

    typedef LEN LengthMap;

    /// The type of the arc lengths

    typedef typename LengthMap::Value Value;

    /// \brief The large value type used for internal computations

    ///

    /// The large value type used for internal computations.

    /// It is \c long \c long if the \c Value type is integer,

    /// otherwise it is \c double.

    /// \c Value must be convertible to \c LargeValue.

    typedef double LargeValue;

    /// The tolerance type used for internal computations

    typedef lemon::Tolerance<LargeValue> Tolerance;

    /// \brief The path type of the found cycles

    ///

    /// The path type of the found cycles.

    /// It must conform to the \ref lemon::concepts::Path "Path" concept

    /// and it must have an \c addBack() function.

    typedef lemon::Path<Digraph> Path;

  };

  // Default traits class for integer value types

  template <typename GR, typename LEN>

  struct HowardDefaultTraits<GR, LEN, true>

  {

    typedef GR Digraph;

    typedef LEN LengthMap;

    typedef typename LengthMap::Value Value;

#ifdef LEMON_HAVE_LONG_LONG

    typedef long long LargeValue;

#else

    typedef long LargeValue;

#endif

    typedef lemon::Tolerance<LargeValue> Tolerance;

    typedef lemon::Path<Digraph> Path;

  };

  /// \addtogroup min_mean_cycle

  /// @{

  /// \brief Implementation of Howard's algorithm for finding a minimum

  /// mean cycle.

  ///

  /// This class implements Howard's policy iteration algorithm for finding

  /// a directed cycle of minimum mean length (cost) in a digraph

  /// \ref amo93networkflows, \ref dasdan98minmeancycle.

  /// This class provides the most efficient algorithm for the

  /// minimum mean cycle problem, though the best known theoretical

  /// bound on its running time is exponential.

  ///

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

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

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

             typename TR = HowardDefaultTraits<GR, LEN> >

#endif

  class Howard

  {

  public: