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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_MIN_COST_ARBORESCENCE_H
				#define LEMON_MIN_COST_ARBORESCENCE_H

				///\ingroup spantree
				///\file
				///\brief Minimum Cost Arborescence algorithm.

				#include <vector>

				#include <lemon/list_graph.h>
				#include <lemon/bin_heap.h>
				#include <lemon/assert.h>

				namespace lemon {


				/// \brief Default traits class for MinCostArborescence class.
				///
				/// Default traits class for MinCostArborescence class.
				/// \param GR Digraph type.
				/// \param CM Type of the cost map.
				template <class GR, class CM>
				struct MinCostArborescenceDefaultTraits{

				/// \brief The digraph type the algorithm runs on.
				typedef GR Digraph;

				/// \brief The type of the map that stores the arc costs.
				///
				/// The type of the map that stores the arc costs.
				/// It must conform to the \ref concepts::ReadMap "ReadMap" concept.
				typedef CM CostMap;

				/// \brief The value type of the costs.
				///
				/// The value type of the costs.
				typedef typename CostMap::Value Value;

				/// \brief The type of the map that stores which arcs are in the
				/// arborescence.
				///
				/// The type of the map that stores which arcs are in the
				/// arborescence. It must conform to the \ref concepts::WriteMap
				/// "WriteMap" concept, and its value type must be \c bool
				/// (or convertible). Initially it will be set to \c false on each
				/// arc, then it will be set on each arborescence arc once.
				typedef typename Digraph::template ArcMap<bool> ArborescenceMap;

				/// \brief Instantiates a \c ArborescenceMap.
				///
				/// This function instantiates a \c ArborescenceMap.
				/// \param digraph The digraph to which we would like to calculate
				/// the \c ArborescenceMap.
				static ArborescenceMap *createArborescenceMap(const Digraph &digraph){
				return new ArborescenceMap(digraph);
				}

				/// \brief The type of the \c PredMap
				///
				/// The type of the \c PredMap. It must confrom to the
				/// \ref concepts::WriteMap "WriteMap" concept, and its value type
				/// must be the \c Arc type of the digraph.
				typedef typename Digraph::template NodeMap<typename Digraph::Arc> PredMap;

				/// \brief Instantiates a \c PredMap.
				///
				/// This function instantiates a \c PredMap.
				/// \param digraph The digraph to which we would like to define the
				/// \c PredMap.
				static PredMap *createPredMap(const Digraph &digraph){
				return new PredMap(digraph);
				}

				};

				/// \ingroup spantree
				///
				/// \brief Minimum Cost Arborescence algorithm class.
				///
				/// This class provides an efficient implementation of the
				/// Minimum Cost Arborescence algorithm. The arborescence is a tree
				/// which is directed from a given source node of the digraph. One or
				/// more sources should be given to the algorithm and it will calculate
				/// the minimum cost subgraph that is the union of arborescences with the
				/// given sources and spans all the nodes which are reachable from the
				/// sources. The time complexity of the algorithm is O(n<sup>2</sup>+m).
				///
				/// The algorithm also provides an optimal dual solution, therefore
				/// the optimality of the solution can be checked.
				///
				/// \param GR The digraph type the algorithm runs on.
				/// \param CM A read-only arc map storing the costs of the
				/// arcs. It is read once for each arc, so the map may involve in
				/// relatively time consuming process to compute the arc costs if
				/// it is necessary. The default map type is \ref
				/// concepts::Digraph::ArcMap "Digraph::ArcMap<int>".
				/// \tparam TR The traits class that defines various types used by the
				/// algorithm. By default, it is \ref MinCostArborescenceDefaultTraits
				/// "MinCostArborescenceDefaultTraits<GR, CM>".
				/// In most cases, this parameter should not be set directly,
				/// consider to use the named template parameters instead.
				#ifndef DOXYGEN
				template <typename GR,
				typename CM = typename GR::template ArcMap<int>,
				typename TR =
				MinCostArborescenceDefaultTraits<GR, CM> >
				#else
				template <typename GR, typename CM, typename TR>
				#endif
				class MinCostArborescence {
				public:

				/// \brief The \ref lemon::MinCostArborescenceDefaultTraits "traits class"
				/// of the algorithm.
				typedef TR Traits;
				/// The type of the underlying digraph.
				typedef typename Traits::Digraph Digraph;
				/// The type of the map that stores the arc costs.
				typedef typename Traits::CostMap CostMap;
				///The type of the costs of the arcs.
				typedef typename Traits::Value Value;
				///The type of the predecessor map.
				typedef typename Traits::PredMap PredMap;
				///The type of the map that stores which arcs are in the arborescence.
				typedef typename Traits::ArborescenceMap ArborescenceMap;

				typedef MinCostArborescence Create;

				private:

				TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

				struct CostArc {

				Arc arc;
				Value value;

				CostArc() {}
				CostArc(Arc _arc, Value _value) : arc(_arc), value(_value) {}

				};

				const Digraph *_digraph;
				const CostMap *_cost;

				PredMap *_pred;
				bool local_pred;

				ArborescenceMap *_arborescence;
				bool local_arborescence;

				typedef typename Digraph::template ArcMap<int> ArcOrder;
				ArcOrder *_arc_order;

				typedef typename Digraph::template NodeMap<int> NodeOrder;
				NodeOrder *_node_order;

				typedef typename Digraph::template NodeMap<CostArc> CostArcMap;
				CostArcMap *_cost_arcs;

				struct StackLevel {

				std::vector<CostArc> arcs;
				int node_level;

				};

				std::vector<StackLevel> level_stack;
				std::vector<Node> queue;

				typedef std::vector<typename Digraph::Node> DualNodeList;

				DualNodeList _dual_node_list;

				struct DualVariable {
				int begin, end;
				Value value;

				DualVariable(int _begin, int _end, Value _value)
				: begin(_begin), end(_end), value(_value) {}

				};

				typedef std::vector<DualVariable> DualVariables;

				DualVariables _dual_variables;

				typedef typename Digraph::template NodeMap<int> HeapCrossRef;

				HeapCrossRef *_heap_cross_ref;

				typedef BinHeap<int, HeapCrossRef> Heap;

				Heap *_heap;

				protected:

				MinCostArborescence() {}

				private:

				void createStructures() {
				if (!_pred) {
				local_pred = true;
				_pred = Traits::createPredMap(*_digraph);
				}
				if (!_arborescence) {
				local_arborescence = true;
				_arborescence = Traits::createArborescenceMap(*_digraph);
				}
				if (!_arc_order) {
				_arc_order = new ArcOrder(*_digraph);
				}
				if (!_node_order) {
				_node_order = new NodeOrder(*_digraph);
				}
				if (!_cost_arcs) {
				_cost_arcs = new CostArcMap(*_digraph);
				}
				if (!_heap_cross_ref) {
				_heap_cross_ref = new HeapCrossRef(*_digraph, -1);
				}
				if (!_heap) {
				_heap = new Heap(*_heap_cross_ref);
				}
				}

				void destroyStructures() {
				if (local_arborescence) {
				delete _arborescence;
				}
				if (local_pred) {
				delete _pred;
				}
				if (_arc_order) {
				delete _arc_order;
				}
				if (_node_order) {
				delete _node_order;
				}
				if (_cost_arcs) {
				delete _cost_arcs;
				}
				if (_heap) {
				delete _heap;
				}
				if (_heap_cross_ref) {
				delete _heap_cross_ref;
				}
				}

				Arc prepare(Node node) {
				std::vector<Node> nodes;
				(*_node_order)[node] = _dual_node_list.size();
				StackLevel level;
				level.node_level = _dual_node_list.size();
				_dual_node_list.push_back(node);
				for (InArcIt it(*_digraph, node); it != INVALID; ++it) {
				Arc arc = it;
				Node source = _digraph->source(arc);
				Value value = (*_cost)[it];
				if (source == node \|\| (*_node_order)[source] == -3) continue;
				if ((*_cost_arcs)[source].arc == INVALID) {
				(*_cost_arcs)[source].arc = arc;
				(*_cost_arcs)[source].value = value;
				nodes.push_back(source);
				} else {
				if ((*_cost_arcs)[source].value > value) {
				(*_cost_arcs)[source].arc = arc;
				(*_cost_arcs)[source].value = value;
				}
				}
				}
				CostArc minimum = (*_cost_arcs)[nodes[0]];
				for (int i = 1; i < int(nodes.size()); ++i) {
				if ((*_cost_arcs)[nodes[i]].value < minimum.value) {
				minimum = (*_cost_arcs)[nodes[i]];
				}
				}
				(*_arc_order)[minimum.arc] = _dual_variables.size();
				DualVariable var(_dual_node_list.size() - 1,
				_dual_node_list.size(), minimum.value);
				_dual_variables.push_back(var);
				for (int i = 0; i < int(nodes.size()); ++i) {
				(*_cost_arcs)[nodes[i]].value -= minimum.value;
				level.arcs.push_back((*_cost_arcs)[nodes[i]]);
				(*_cost_arcs)[nodes[i]].arc = INVALID;
				}
				level_stack.push_back(level);
				return minimum.arc;
				}

				Arc contract(Node node) {
				int node_bottom = bottom(node);
				std::vector<Node> nodes;
				while (!level_stack.empty() &&
				level_stack.back().node_level >= node_bottom) {
				for (int i = 0; i < int(level_stack.back().arcs.size()); ++i) {
				Arc arc = level_stack.back().arcs[i].arc;
				Node source = _digraph->source(arc);
				Value value = level_stack.back().arcs[i].value;
				if ((*_node_order)[source] >= node_bottom) continue;
				if ((*_cost_arcs)[source].arc == INVALID) {
				(*_cost_arcs)[source].arc = arc;
				(*_cost_arcs)[source].value = value;
				nodes.push_back(source);
				} else {
				if ((*_cost_arcs)[source].value > value) {
				(*_cost_arcs)[source].arc = arc;
				(*_cost_arcs)[source].value = value;
				}
				}
				}
				level_stack.pop_back();
				}
				CostArc minimum = (*_cost_arcs)[nodes[0]];
				for (int i = 1; i < int(nodes.size()); ++i) {
				if ((*_cost_arcs)[nodes[i]].value < minimum.value) {
				minimum = (*_cost_arcs)[nodes[i]];
				}
				}
				(*_arc_order)[minimum.arc] = _dual_variables.size();
				DualVariable var(node_bottom, _dual_node_list.size(), minimum.value);
				_dual_variables.push_back(var);
				StackLevel level;
				level.node_level = node_bottom;
				for (int i = 0; i < int(nodes.size()); ++i) {
				(*_cost_arcs)[nodes[i]].value -= minimum.value;
				level.arcs.push_back((*_cost_arcs)[nodes[i]]);
				(*_cost_arcs)[nodes[i]].arc = INVALID;
				}
				level_stack.push_back(level);
				return minimum.arc;
				}

				int bottom(Node node) {
				int k = level_stack.size() - 1;
				while (level_stack[k].node_level > (*_node_order)[node]) {
				--k;
				}
				return level_stack[k].node_level;
				}

				void finalize(Arc arc) {
				Node node = _digraph->target(arc);
				_heap->push(node, (*_arc_order)[arc]);
				_pred->set(node, arc);
				while (!_heap->empty()) {
				Node source = _heap->top();
				_heap->pop();
				(*_node_order)[source] = -1;
				for (OutArcIt it(*_digraph, source); it != INVALID; ++it) {
				if ((*_arc_order)[it] < 0) continue;
				Node target = _digraph->target(it);
				switch(_heap->state(target)) {
				case Heap::PRE_HEAP:
				_heap->push(target, (*_arc_order)[it]);
				_pred->set(target, it);
				break;
				case Heap::IN_HEAP:
				if ((_arc_order)[it] < (_heap)[target]) {
				_heap->decrease(target, (*_arc_order)[it]);
				_pred->set(target, it);
				}
				break;
				case Heap::POST_HEAP:
				break;
				}
				}
				_arborescence->set((*_pred)[source], true);
				}
				}


				public:

				/// \name Named Template Parameters

				/// @{

				template <class T>
				struct SetArborescenceMapTraits : public Traits {
				typedef T ArborescenceMap;
				static ArborescenceMap *createArborescenceMap(const Digraph &)
				{
				LEMON_ASSERT(false, "ArborescenceMap is not initialized");
				return 0; // ignore warnings
				}
				};

				/// \brief \ref named-templ-param "Named parameter" for
				/// setting \c ArborescenceMap type
				///
				/// \ref named-templ-param "Named parameter" for setting
				/// \c ArborescenceMap type.
				/// It must conform to the \ref concepts::WriteMap "WriteMap" concept,
				/// and its value type must be \c bool (or convertible).
				/// Initially it will be set to \c false on each arc,
				/// then it will be set on each arborescence arc once.
				template <class T>
				struct SetArborescenceMap
				: public MinCostArborescence<Digraph, CostMap,
				SetArborescenceMapTraits<T> > {
				};

				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 meet the \ref concepts::WriteMap "WriteMap" concept,
				/// and its value type must be the \c Arc type of the digraph.
				template <class T>
				struct SetPredMap
				: public MinCostArborescence<Digraph, CostMap, SetPredMapTraits<T> > {
				};

				/// @}

				/// \brief Constructor.
				///
				/// \param digraph The digraph the algorithm will run on.
				/// \param cost The cost map used by the algorithm.
				MinCostArborescence(const Digraph& digraph, const CostMap& cost)
				: _digraph(&digraph), _cost(&cost), _pred(0), local_pred(false),
				_arborescence(0), local_arborescence(false),
				_arc_order(0), _node_order(0), _cost_arcs(0),
				_heap_cross_ref(0), _heap(0) {}

				/// \brief Destructor.
				~MinCostArborescence() {
				destroyStructures();
				}

				/// \brief Sets the arborescence map.
				///
				/// Sets the arborescence map.
				/// \return <tt>(*this)</tt>
				MinCostArborescence& arborescenceMap(ArborescenceMap& m) {
				if (local_arborescence) {
				delete _arborescence;
				}
				local_arborescence = false;
				_arborescence = &m;
				return *this;
				}

				/// \brief Sets the predecessor map.
				///
				/// Sets the predecessor map.
				/// \return <tt>(*this)</tt>
				MinCostArborescence& predMap(PredMap& m) {
				if (local_pred) {
				delete _pred;
				}
				local_pred = false;
				_pred = &m;
				return *this;
				}

				/// \name Execution Control
				/// The simplest way to execute the algorithm is to use
				/// one of the member functions called \c 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 \ref start() will perform the arborescence
				/// computation.

				///@{

				/// \brief Initializes the internal data structures.
				///
				/// Initializes the internal data structures.
				///
				void init() {
				createStructures();
				_heap->clear();
				for (NodeIt it(*_digraph); it != INVALID; ++it) {
				(*_cost_arcs)[it].arc = INVALID;
				(*_node_order)[it] = -3;
				(*_heap_cross_ref)[it] = Heap::PRE_HEAP;
				_pred->set(it, INVALID);
				}
				for (ArcIt it(*_digraph); it != INVALID; ++it) {
				_arborescence->set(it, false);
				(*_arc_order)[it] = -1;
				}
				_dual_node_list.clear();
				_dual_variables.clear();
				}

				/// \brief Adds a new source node.
				///
				/// Adds a new source node to the algorithm.
				void addSource(Node source) {
				std::vector<Node> nodes;
				nodes.push_back(source);
				while (!nodes.empty()) {
				Node node = nodes.back();
				nodes.pop_back();
				for (OutArcIt it(*_digraph, node); it != INVALID; ++it) {
				Node target = _digraph->target(it);
				if ((*_node_order)[target] == -3) {
				(*_node_order)[target] = -2;
				nodes.push_back(target);
				queue.push_back(target);
				}
				}
				}
				(*_node_order)[source] = -1;
				}

				/// \brief Processes the next node in the priority queue.
				///
				/// Processes the next node in the priority queue.
				///
				/// \return The processed node.
				///
				/// \warning The queue must not be empty.
				Node processNextNode() {
				Node node = queue.back();
				queue.pop_back();
				if ((*_node_order)[node] == -2) {
				Arc arc = prepare(node);
				Node source = _digraph->source(arc);
				while ((*_node_order)[source] != -1) {
				if ((*_node_order)[source] >= 0) {
				arc = contract(source);
				} else {
				arc = prepare(source);
				}
				source = _digraph->source(arc);
				}
				finalize(arc);
				level_stack.clear();
				}
				return node;
				}

				/// \brief Returns the number of the nodes to be processed.
				///
				/// Returns the number of the nodes to be processed in the priority
				/// queue.
				int queueSize() const {
				return queue.size();
				}

				/// \brief Returns \c false if there are nodes to be processed.
				///
				/// Returns \c false if there are nodes to be processed.
				bool emptyQueue() const {
				return queue.empty();
				}

				/// \brief Executes the algorithm.
				///
				/// Executes the algorithm.
				///
				/// \pre init() must be called and at least one node should be added
				/// with addSource() before using this function.
				///
				///\note mca.start() is just a shortcut of the following code.
				///\code
				///while (!mca.emptyQueue()) {
				/// mca.processNextNode();
				///}
				///\endcode
				void start() {
				while (!emptyQueue()) {
				processNextNode();
				}
				}

				/// \brief Runs %MinCostArborescence algorithm from node \c s.
				///
				/// This method runs the %MinCostArborescence algorithm from
				/// a root node \c s.
				///
				/// \note mca.run(s) is just a shortcut of the following code.
				/// \code
				/// mca.init();
				/// mca.addSource(s);
				/// mca.start();
				/// \endcode
				void run(Node s) {
				init();
				addSource(s);
				start();
				}

				///@}

				/// \name Query Functions
				/// The result of the %MinCostArborescence algorithm can be obtained
				/// using these functions.\n
				/// Either run() or start() must be called before using them.

				/// @{

				/// \brief Returns the cost of the arborescence.
				///
				/// Returns the cost of the arborescence.
				Value arborescenceCost() const {
				Value sum = 0;
				for (ArcIt it(*_digraph); it != INVALID; ++it) {
				if (arborescence(it)) {
				sum += (*_cost)[it];
				}
				}
				return sum;
				}

				/// \brief Returns \c true if the arc is in the arborescence.
				///
				/// Returns \c true if the given arc is in the arborescence.
				/// \param arc An arc of the digraph.
				/// \pre \ref run() must be called before using this function.
				bool arborescence(Arc arc) const {
				return (*_pred)[_digraph->target(arc)] == arc;
				}

				/// \brief Returns a const reference to the arborescence map.
				///
				/// Returns a const reference to the arborescence map.
				/// \pre \ref run() must be called before using this function.
				const ArborescenceMap& arborescenceMap() const {
				return *_arborescence;
				}

				/// \brief Returns the predecessor arc of the given node.
				///
				/// Returns the predecessor arc of the given node.
				/// \pre \ref run() must be called before using this function.
				Arc pred(Node node) const {
				return (*_pred)[node];
				}

				/// \brief Returns a const reference to the pred map.
				///
				/// Returns a const reference to the pred map.
				/// \pre \ref run() must be called before using this function.
				const PredMap& predMap() const {
				return *_pred;
				}

				/// \brief Indicates that a node is reachable from the sources.
				///
				/// Indicates that a node is reachable from the sources.
				bool reached(Node node) const {
				return (*_node_order)[node] != -3;
				}

				/// \brief Indicates that a node is processed.
				///
				/// Indicates that a node is processed. The arborescence path exists
				/// from the source to the given node.
				bool processed(Node node) const {
				return (*_node_order)[node] == -1;
				}

				/// \brief Returns the number of the dual variables in basis.
				///
				/// Returns the number of the dual variables in basis.
				int dualNum() const {
				return _dual_variables.size();
				}

				/// \brief Returns the value of the dual solution.
				///
				/// Returns the value of the dual solution. It should be
				/// equal to the arborescence value.
				Value dualValue() const {
				Value sum = 0;
				for (int i = 0; i < int(_dual_variables.size()); ++i) {
				sum += _dual_variables[i].value;
				}
				return sum;
				}

				/// \brief Returns the number of the nodes in the dual variable.
				///
				/// Returns the number of the nodes in the dual variable.
				int dualSize(int k) const {
				return _dual_variables[k].end - _dual_variables[k].begin;
				}

				/// \brief Returns the value of the dual variable.
				///
				/// Returns the the value of the dual variable.
				Value dualValue(int k) const {
				return _dual_variables[k].value;
				}

				/// \brief LEMON iterator for getting a dual variable.
				///
				/// This class provides a common style LEMON iterator for getting a
				/// dual variable of \ref MinCostArborescence algorithm.
				/// It iterates over a subset of the nodes.
				class DualIt {
				public:

				/// \brief Constructor.
				///
				/// Constructor for getting the nodeset of the dual variable
				/// of \ref MinCostArborescence algorithm.
				DualIt(const MinCostArborescence& algorithm, int variable)
				: _algorithm(&algorithm)
				{
				_index = _algorithm->_dual_variables[variable].begin;
				_last = _algorithm->_dual_variables[variable].end;
				}

				/// \brief Conversion to \c Node.
				///
				/// Conversion to \c Node.
				operator Node() const {
				return _algorithm->_dual_node_list[_index];
				}

				/// \brief Increment operator.
				///
				/// Increment operator.
				DualIt& operator++() {
				++_index;
				return *this;
				}

				/// \brief Validity checking
				///
				/// Checks whether the iterator is invalid.
				bool operator==(Invalid) const {
				return _index == _last;
				}

				/// \brief Validity checking
				///
				/// Checks whether the iterator is valid.
				bool operator!=(Invalid) const {
				return _index != _last;
				}

				private:
				const MinCostArborescence* _algorithm;
				int _index, _last;
				};

				/// @}

				};

				/// \ingroup spantree
				///
				/// \brief Function type interface for MinCostArborescence algorithm.
				///
				/// Function type interface for MinCostArborescence algorithm.
				/// \param digraph The digraph the algorithm runs on.
				/// \param cost An arc map storing the costs.
				/// \param source The source node of the arborescence.
				/// \retval arborescence An arc map with \c bool (or convertible) value
				/// type that stores the arborescence.
				/// \return The total cost of the arborescence.
				///
				/// \sa MinCostArborescence
				template <typename Digraph, typename CostMap, typename ArborescenceMap>
				typename CostMap::Value minCostArborescence(const Digraph& digraph,
				const CostMap& cost,
				typename Digraph::Node source,
				ArborescenceMap& arborescence) {
				typename MinCostArborescence<Digraph, CostMap>
				::template SetArborescenceMap<ArborescenceMap>
				::Create mca(digraph, cost);
				mca.arborescenceMap(arborescence);
				mca.run(source);
				return mca.arborescenceCost();
				}

				}

				#endif