/* -*- 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