/* -*- 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>".

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

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

  /// "HowardDefaultTraits<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,

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

             typename TR = HowardDefaultTraits<GR, LEN> >

#endif

  class Howard

  {

  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.

    /// By default, 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 HowardDefaultTraits "default traits class",

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

    typedef typename TR::Path Path;

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

    typedef TR Traits;

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    // The digraph the algorithm runs on

    const Digraph &_gr;

    // The length of the arcs

    const LengthMap &_length;

    // 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;

    Path *_cycle_path;

    bool _local_path;

    // Internal data used by the algorithm

    typename Digraph::template NodeMap<Arc> _policy;

    typename Digraph::template NodeMap<bool> _reached;

    typename Digraph::template NodeMap<int> _level;

    typename Digraph::template NodeMap<LargeValue> _dist;

    // 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> > _in_arcs;

    // Queue used for BFS search

    std::vector<Node> _queue;

    int _qfront, _qback;

    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 Howard<GR, LEN, SetLargeValueTraits<T> > {

      typedef Howard<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 addBack() function.

    template <typename T>

    struct SetPath

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

      typedef Howard<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.

    Howard( const Digraph &digraph,

            const LengthMap &length ) :

      _gr(digraph), _length(length), _best_found(false),

      _best_length(0), _best_size(1), _cycle_path(NULL), _local_path(false),

      _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph),

      _comp(digraph), _in_arcs(digraph),

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

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

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

    {}

    /// Destructor.

    ~Howard() {

      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::addBack()

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

    ///

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

    Howard& 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>

    Howard& 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() {

      // Initialize and find strongly connected components

      init();

      findComponents();

      // Find the minimum cycle mean in the components

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

        // Find the minimum mean cycle in the current component

        if (!buildPolicyGraph(comp)) continue;

        while (true) {

          findPolicyCycle();

          if (!computeNodeDistances()) break;

        }

        // 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;

        }

      }

      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;

      _cycle_path->addBack(_policy[_best_node]);

      for ( Node v = _best_node;

            (v = _gr.target(_policy[v])) != _best_node; ) {

        _cycle_path->addBack(_policy[v]);

      }

      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.

    Value cycleLength() const {

      return static_cast<Value>(_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:

    // Initialize

    void init() {

      if (!_cycle_path) {

        _local_path = true;

        _cycle_path = new Path;

      }

      _queue.resize(countNodes(_gr));

      _best_found = false;

      _best_length = 0;

      _best_size = 1;

      _cycle_path->clear();

    }

    // Find strongly connected components and initialize _comp_nodes

    // and _in_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);

          _in_arcs[n].clear();

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

            _in_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);

          _in_arcs[n].clear();

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

            if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a);

          }

        }

      }

    }

    // Build the policy graph in the given strongly connected component

    // (the out-degree of every node is 1)

    bool buildPolicyGraph(int comp) {

      _nodes = &(_comp_nodes[comp]);

      if (_nodes->size() < 1 ||

          (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) {

        return false;

      }

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

        _dist[(*_nodes)[i]] = INF;

      }

      Node u, v;

      Arc e;

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

        v = (*_nodes)[i];

        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {

          e = _in_arcs[v][j];

          u = _gr.source(e);

          if (_length[e] < _dist[u]) {

            _dist[u] = _length[e];

            _policy[u] = e;

          }

        }

      }

      return true;

    }

    // Find the minimum mean cycle in the policy graph

    void findPolicyCycle() {

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

        _level[(*_nodes)[i]] = -1;

      }

      LargeValue clength;

      int csize;

      Node u, v;

      _curr_found = false;

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

        u = (*_nodes)[i];

        if (_level[u] >= 0) continue;

        for (; _level[u] < 0; u = _gr.target(_policy[u])) {

          _level[u] = i;

        }

        if (_level[u] == i) {

          // A cycle is found

          clength = _length[_policy[u]];

          csize = 1;

          for (v = u; (v = _gr.target(_policy[v])) != u; ) {

            clength += _length[_policy[v]];

            ++csize;

          }

          if ( !_curr_found ||

               (clength * _curr_size < _curr_length * csize) ) {

            _curr_found = true;

            _curr_length = clength;

            _curr_size = csize;

            _curr_node = u;

          }

        }

      }

    }

    // Contract the policy graph and compute node distances

    bool computeNodeDistances() {

      // Find the component of the main cycle and compute node distances

      // using reverse BFS

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

        _reached[(*_nodes)[i]] = false;

      }

      _qfront = _qback = 0;

      _queue[0] = _curr_node;

      _reached[_curr_node] = true;

      _dist[_curr_node] = 0;

      Node u, v;

      Arc e;

      while (_qfront <= _qback) {

        v = _queue[_qfront++];

        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {

          e = _in_arcs[v][j];

          u = _gr.source(e);

          if (_policy[u] == e && !_reached[u]) {

            _reached[u] = true;

            _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;

            _queue[++_qback] = u;

          }

        }

      }

      // Connect all other nodes to this component and compute node

      // distances using reverse BFS

      _qfront = 0;

      while (_qback < int(_nodes->size())-1) {

        v = _queue[_qfront++];

        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {

          e = _in_arcs[v][j];

          u = _gr.source(e);

          if (!_reached[u]) {

            _reached[u] = true;

            _policy[u] = e;

            _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;

            _queue[++_qback] = u;

          }

        }

      }

      // Improve node distances

      bool improved = false;

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

        v = (*_nodes)[i];

        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {

          e = _in_arcs[v][j];

          u = _gr.source(e);

          LargeValue delta = _dist[v] + _length[e] * _curr_size - _curr_length;

          if (_tolerance.less(delta, _dist[u])) {

            _dist[u] = delta;

            _policy[u] = e;

            improved = true;

          }

        }

      }

      return improved;

    }

  }; //class Howard

  ///@}

} //namespace lemon

#endif //LEMON_HOWARD_H