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				/* -- mode: C++; indent-tabs-mode: nil; --
				*
				* This file is a part of LEMON, a generic C++ optimization library.
				*
				* Copyright (C) 2003-2013
				* 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_MMC_H
				#define LEMON_HOWARD_MMC_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 HowardMmc class.
				///
				/// Default traits class of HowardMmc class.
				/// \tparam GR The type of the digraph.
				/// \tparam CM The type of the cost map.
				/// It must conform to the \ref concepts::ReadMap "ReadMap" concept.
				#ifdef DOXYGEN
				template <typename GR, typename CM>
				#else
				template <typename GR, typename CM,
				bool integer = std::numeric_limits<typename CM::Value>::is_integer>
				#endif
				struct HowardMmcDefaultTraits
				{
				/// The type of the digraph
				typedef GR Digraph;
				/// The type of the cost map
				typedef CM CostMap;
				/// The type of the arc costs
				typedef typename CostMap::Value Cost;

				/// \brief The large cost type used for internal computations
				///
				/// The large cost type used for internal computations.
				/// It is \c long \c long if the \c Cost type is integer,
				/// otherwise it is \c double.
				/// \c Cost must be convertible to \c LargeCost.
				typedef double LargeCost;

				/// The tolerance type used for internal computations
				typedef lemon::Tolerance<LargeCost> 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 cost types
				template <typename GR, typename CM>
				struct HowardMmcDefaultTraits<GR, CM, true>
				{
				typedef GR Digraph;
				typedef CM CostMap;
				typedef typename CostMap::Value Cost;
				#ifdef LEMON_HAVE_LONG_LONG
				typedef long long LargeCost;
				#else
				typedef long LargeCost;
				#endif
				typedef lemon::Tolerance<LargeCost> 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 cost in a digraph
				/// \cite dasdan98minmeancycle, \cite dasdan04experimental.
				/// 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 CM The type of the cost 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 HowardMmcDefaultTraits
				/// "HowardMmcDefaultTraits<GR, CM>".
				/// In most cases, this parameter should not be set directly,
				/// consider to use the named template parameters instead.
				#ifdef DOXYGEN
				template <typename GR, typename CM, typename TR>
				#else
				template < typename GR,
				typename CM = typename GR::template ArcMap<int>,
				typename TR = HowardMmcDefaultTraits<GR, CM> >
				#endif
				class HowardMmc
				{
				public:

				/// The type of the digraph
				typedef typename TR::Digraph Digraph;
				/// The type of the cost map
				typedef typename TR::CostMap CostMap;
				/// The type of the arc costs
				typedef typename TR::Cost Cost;

				/// \brief The large cost type
				///
				/// The large cost type used for internal computations.
				/// By default, it is \c long \c long if the \c Cost type is integer,
				/// otherwise it is \c double.
				typedef typename TR::LargeCost LargeCost;

				/// 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 lemon::HowardMmcDefaultTraits "default traits class",
				/// it is \ref lemon::Path "Path<Digraph>".
				typedef typename TR::Path Path;

				/// The \ref lemon::HowardMmcDefaultTraits "traits class" of the algorithm
				typedef TR Traits;

				/// \brief Constants for the causes of search termination.
				///
				/// Enum type containing constants for the different causes of search
				/// termination. The \ref findCycleMean() function returns one of
				/// these values.
				enum TerminationCause {

				/// No directed cycle can be found in the digraph.
				NO_CYCLE = 0,

				/// Optimal solution (minimum cycle mean) is found.
				OPTIMAL = 1,

				/// The iteration count limit is reached.
				ITERATION_LIMIT
				};

				private:

				TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

				// The digraph the algorithm runs on
				const Digraph &_gr;
				// The cost of the arcs
				const CostMap &_cost;

				// Data for the found cycles
				bool _curr_found, _best_found;
				LargeCost _curr_cost, _best_cost;
				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<LargeCost> _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 LargeCost INF;

				public:

				/// \name Named Template Parameters
				/// @{

				template <typename T>
				struct SetLargeCostTraits : public Traits {
				typedef T LargeCost;
				typedef lemon::Tolerance<T> Tolerance;
				};

				/// \brief \ref named-templ-param "Named parameter" for setting
				/// \c LargeCost type.
				///
				/// \ref named-templ-param "Named parameter" for setting \c LargeCost
				/// type. It is used for internal computations in the algorithm.
				template <typename T>
				struct SetLargeCost
				: public HowardMmc<GR, CM, SetLargeCostTraits<T> > {
				typedef HowardMmc<GR, CM, SetLargeCostTraits<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 HowardMmc<GR, CM, SetPathTraits<T> > {
				typedef HowardMmc<GR, CM, SetPathTraits<T> > Create;
				};

				/// @}

				protected:

				HowardMmc() {}

				public:

				/// \brief Constructor.
				///
				/// The constructor of the class.
				///
				/// \param digraph The digraph the algorithm runs on.
				/// \param cost The costs of the arcs.
				HowardMmc( const Digraph &digraph,
				const CostMap &cost ) :
				_gr(digraph), _cost(cost), _best_found(false),
				_best_cost(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<LargeCost>::has_infinity ?
				std::numeric_limits<LargeCost>::infinity() :
				std::numeric_limits<LargeCost>::max())
				{}

				/// Destructor.
				~HowardMmc() {
				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 findCycleMean(), a local \ref Path "path" structure
				/// will be allocated. 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>
				HowardMmc& 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>
				HowardMmc& 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 cost, you may call
				/// \ref findCycleMean().

				/// @{

				/// \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 costs 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.findCycleMean() && mmc.findCycle();
				/// \endcode
				bool run() {
				return findCycleMean() && findCycle();
				}

				/// \brief Find the minimum cycle mean (or an upper bound).
				///
				/// This function finds the minimum mean cost of the directed
				/// cycles in the digraph (or an upper bound for it).
				///
				/// By default, the function finds the exact minimum cycle mean,
				/// but an optional limit can also be specified for the number of
				/// iterations performed during the search process.
				/// The return value indicates if the optimal solution is found
				/// or the iteration limit is reached. In the latter case, an
				/// approximate solution is provided, which corresponds to a directed
				/// cycle whose mean cost is relatively small, but not necessarily
				/// minimal.
				///
				/// \param limit The maximum allowed number of iterations during
				/// the search process. Its default value implies that the algorithm
				/// runs until it finds the exact optimal solution.
				///
				/// \return The termination cause of the search process.
				/// For more information, see \ref TerminationCause.
				TerminationCause findCycleMean(int limit =
				std::numeric_limits<int>::max()) {
				// Initialize and find strongly connected components
				init();
				findComponents();

				// Find the minimum cycle mean in the components
				int iter_count = 0;
				bool iter_limit_reached = false;
				for (int comp = 0; comp < _comp_num; ++comp) {
				// Find the minimum mean cycle in the current component
				if (!buildPolicyGraph(comp)) continue;
				while (true) {
				if (++iter_count > limit) {
				iter_limit_reached = true;
				break;
				}
				findPolicyCycle();
				if (!computeNodeDistances()) break;
				}

				// Update the best cycle (global minimum mean cycle)
				if ( _curr_found && (!_best_found \|\|
				_curr_cost * _best_size < _best_cost * _curr_size) ) {
				_best_found = true;
				_best_cost = _curr_cost;
				_best_size = _curr_size;
				_best_node = _curr_node;
				}

				if (iter_limit_reached) break;
				}

				if (iter_limit_reached) {
				return ITERATION_LIMIT;
				} else {
				return _best_found ? OPTIMAL : NO_CYCLE;
				}
				}

				/// \brief Find a minimum mean directed cycle.
				///
				/// This function finds a directed cycle of minimum mean cost
				/// in the digraph using the data computed by findCycleMean().
				///
				/// \return \c true if a directed cycle exists in the digraph.
				///
				/// \pre \ref findCycleMean() 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 cost of the found cycle.
				///
				/// This function returns the total cost of the found cycle.
				///
				/// \pre \ref run() or \ref findCycleMean() must be called before
				/// using this function.
				Cost cycleCost() const {
				return static_cast<Cost>(_best_cost);
				}

				/// \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 findCycleMean() must be called before
				/// using this function.
				int cycleSize() const {
				return _best_size;
				}

				/// \brief Return the mean cost of the found cycle.
				///
				/// This function returns the mean cost of the found cycle.
				///
				/// \note <tt>alg.cycleMean()</tt> is just a shortcut of the
				/// following code.
				/// \code
				/// return static_cast<double>(alg.cycleCost()) / alg.cycleSize();
				/// \endcode
				///
				/// \pre \ref run() or \ref findCycleMean() must be called before
				/// using this function.
				double cycleMean() const {
				return static_cast<double>(_best_cost) / _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_cost = 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 (_cost[e] < _dist[u]) {
				_dist[u] = _cost[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;
				}
				LargeCost ccost;
				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
				ccost = _cost[_policy[u]];
				csize = 1;
				for (v = u; (v = _gr.target(_policy[v])) != u; ) {
				ccost += _cost[_policy[v]];
				++csize;
				}
				if ( !_curr_found \|\|
				(ccost * _curr_size < _curr_cost * csize) ) {
				_curr_found = true;
				_curr_cost = ccost;
				_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] + _cost[e] * _curr_size - _curr_cost;
				_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] + _cost[e] * _curr_size - _curr_cost;
				_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);
				LargeCost delta = _dist[v] + _cost[e] * _curr_size - _curr_cost;
				if (_tolerance.less(delta, _dist[u])) {
				_dist[u] = delta;
				_policy[u] = e;
				improved = true;
				}
				}
				}
				return improved;
				}

				}; //class HowardMmc

				///@}

				} //namespace lemon

				#endif //LEMON_HOWARD_MMC_H