/* -*- C++ -*-

  * lemon/min_cut.h - Part of LEMON, a generic C++ optimization library

  *

  * Copyright (C) 2005 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_CUT_H

 #define LEMON_MIN_CUT_H

 /// \ingroup topology

 /// \file

 /// \brief Maximum cardinality search and min cut in undirected graphs.

 #include <lemon/list_graph.h>

 #include <lemon/bin_heap.h>

 #include <lemon/bucket_heap.h>

 #include <lemon/bits/invalid.h>

 #include <lemon/error.h>

 #include <lemon/maps.h>

 #include <functional>

 namespace lemon {

   namespace _min_cut_bits {

     template <typename CapacityMap>

     struct HeapSelector {

       template <typename Key, typename Value, typename Ref>

       struct Selector {

         typedef BinHeap<Key, Value, Ref, std::greater<Value> > Heap;

       };

     };

     template <typename CapacityKey>

     struct HeapSelector<ConstMap<CapacityKey, Const<int, 1> > > {

       template <typename Key, typename Value, typename Ref>

       struct Selector {

         typedef BucketHeap<Key, Ref, false > Heap;

       };

     };

   }

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

   ///

   /// Default traits class of MaxCardinalitySearch class.

   /// \param Graph Graph type.

   /// \param CapacityMap Type of length map.

   template <typename _Graph, typename _CapacityMap>

   struct MaxCardinalitySearchDefaultTraits {

     /// The graph type the algorithm runs on. 

     typedef _Graph Graph;

     /// \brief The type of the map that stores the edge capacities.

     ///

     /// The type of the map that stores the edge capacities.

     /// It must meet the \ref concept::ReadMap "ReadMap" concept.

     typedef _CapacityMap CapacityMap;

     /// \brief The type of the capacity of the edges.

     typedef typename CapacityMap::Value Value;

     /// \brief The cross reference type used by heap.

     ///

     /// The cross reference type used by heap.

     /// Usually it is \c Graph::NodeMap<int>.

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

     /// \brief Instantiates a HeapCrossRef.

     ///

     /// This function instantiates a \ref HeapCrossRef. 

     /// \param graph is the graph, to which we would like to define the 

     /// HeapCrossRef.

     static HeapCrossRef *createHeapCrossRef(const Graph &graph) {

       return new HeapCrossRef(graph);

     }

     /// \brief The heap type used by MaxCardinalitySearch algorithm.

     ///

     /// The heap type used by MaxCardinalitySearch algorithm. It should

     /// maximalize the priorities. The default heap type is

     /// the \ref BinHeap, but it is specialized when the

     /// CapacityMap is ConstMap<Graph::Node, Const<int, 1> >

     /// to BucketHeap.

     ///

     /// \sa MaxCardinalitySearch

     typedef typename _min_cut_bits

     ::HeapSelector<CapacityMap>

     ::template Selector<typename Graph::Node, Value, HeapCrossRef>

     ::Heap Heap;

     /// \brief Instantiates a Heap.

     ///

     /// This function instantiates a \ref Heap. 

     /// \param crossref The cross reference of the heap.

     static Heap *createHeap(HeapCrossRef& crossref) {

       return new Heap(crossref);

     }

     /// \brief The type of the map that stores whether a nodes is processed.

     ///

     /// The type of the map that stores whether a nodes is processed.

     /// It must meet the \ref concept::WriteMap "WriteMap" concept.

     /// By default it is a NullMap.

     typedef NullMap<typename Graph::Node, bool> ProcessedMap;

     /// \brief Instantiates a ProcessedMap.

     ///

     /// This function instantiates a \ref ProcessedMap. 

     /// \param graph is the graph, to which

     /// we would like to define the \ref ProcessedMap

 #ifdef DOXYGEN

     static ProcessedMap *createProcessedMap(const Graph &graph)

 #else

     static ProcessedMap *createProcessedMap(const Graph &)

 #endif

     {

       return new ProcessedMap();

     }

     /// \brief The type of the map that stores the cardinalties of the nodes.

     /// 

     /// The type of the map that stores the cardinalities of the nodes.

     /// It must meet the \ref concept::WriteMap "WriteMap" concept.

     typedef typename Graph::template NodeMap<Value> CardinalityMap;

     /// \brief Instantiates a CardinalityMap.

     ///

     /// This function instantiates a \ref CardinalityMap. 

     /// \param graph is the graph, to which we would like to define the \ref 

     /// CardinalityMap

     static CardinalityMap *createCardinalityMap(const Graph &graph) {

       return new CardinalityMap(graph);

     }

   };

   /// \ingroup topology

   ///

   /// \brief Maximum Cardinality Search algorithm class.

   ///

   /// This class provides an efficient implementation of Maximum Cardinality 

   /// Search algorithm. The maximum cardinality search chooses first time any 

   /// node of the graph. Then every time it chooses the node which is connected

   /// to the processed nodes at most in the sum of capacities on the out 

   /// edges. If there is a cut in the graph the algorithm should choose

   /// again any unprocessed node of the graph. Each node cardinality is

   /// the sum of capacities on the out edges to the nodes which are processed

   /// before the given node.

   ///

   /// The edge capacities are passed to the algorithm using a

   /// \ref concept::ReadMap "ReadMap", so it is easy to change it to any 

   /// kind of capacity.

   ///

   /// The type of the capacity is determined by the \ref 

   /// concept::ReadMap::Value "Value" of the capacity map.

   ///

   /// It is also possible to change the underlying priority heap.

   ///

   ///

   /// \param _Graph The graph type the algorithm runs on. The default value

   /// is \ref ListGraph. The value of Graph is not used directly by

   /// the search algorithm, it is only passed to 

   /// \ref MaxCardinalitySearchDefaultTraits.

   /// \param _CapacityMap This read-only EdgeMap determines the capacities of 

   /// the edges. It is read once for each edge, so the map may involve in

   /// relatively time consuming process to compute the edge capacity if

   /// it is necessary. The default map type is \ref

   /// concept::Graph::EdgeMap "Graph::EdgeMap<int>".  The value

   /// of CapacityMap is not used directly by search algorithm, it is only 

   /// passed to \ref MaxCardinalitySearchDefaultTraits.  

   /// \param _Traits Traits class to set various data types used by the 

   /// algorithm.  The default traits class is 

   /// \ref MaxCardinalitySearchDefaultTraits 

   /// "MaxCardinalitySearchDefaultTraits<_Graph, _CapacityMap>".  

   /// See \ref MaxCardinalitySearchDefaultTraits 

   /// for the documentation of a MaxCardinalitySearch traits class.

   ///

   /// \author Balazs Dezso

 #ifdef DOXYGEN

   template <typename _Graph, typename _CapacityMap, typename _Traits>

 #else

   template <typename _Graph = ListUGraph,

 	    typename _CapacityMap = typename _Graph::template EdgeMap<int>,

 	    typename _Traits = 

             MaxCardinalitySearchDefaultTraits<_Graph, _CapacityMap> >

 #endif

   class MaxCardinalitySearch {

   public:

     /// \brief \ref Exception for uninitialized parameters.

     ///

     /// This error represents problems in the initialization

     /// of the parameters of the algorithms.

     class UninitializedParameter : public lemon::UninitializedParameter {

     public:

       virtual const char* what() const throw() {

 	return "lemon::MaxCardinalitySearch::UninitializedParameter";

       }

     };

     typedef _Traits Traits;

     ///The type of the underlying graph.

     typedef typename Traits::Graph Graph;

     ///The type of the capacity of the edges.

     typedef typename Traits::CapacityMap::Value Value;

     ///The type of the map that stores the edge capacities.

     typedef typename Traits::CapacityMap CapacityMap;

     ///The type of the map indicating if a node is processed.

     typedef typename Traits::ProcessedMap ProcessedMap;

     ///The type of the map that stores the cardinalities of the nodes.

     typedef typename Traits::CardinalityMap CardinalityMap;

     ///The cross reference type used for the current heap.

     typedef typename Traits::HeapCrossRef HeapCrossRef;

     ///The heap type used by the algorithm. It maximize the priorities.

     typedef typename Traits::Heap Heap;

   private:

     /// Pointer to the underlying graph.

     const Graph *_graph;

     /// Pointer to the capacity map

     const CapacityMap *_capacity;

     ///Pointer to the map of cardinality.

     CardinalityMap *_cardinality;

     ///Indicates if \ref _cardinality is locally allocated (\c true) or not.

     bool local_cardinality;

     ///Pointer to the map of processed status of the nodes.

     ProcessedMap *_processed;

     ///Indicates if \ref _processed is locally allocated (\c true) or not.

     bool local_processed;

     ///Pointer to the heap cross references.

     HeapCrossRef *_heap_cross_ref;

     ///Indicates if \ref _heap_cross_ref is locally allocated (\c true) or not.

     bool local_heap_cross_ref;

     ///Pointer to the heap.

     Heap *_heap;

     ///Indicates if \ref _heap is locally allocated (\c true) or not.

     bool local_heap;

   public :

     typedef MaxCardinalitySearch Create;

     ///\name Named template parameters

     ///@{

     template <class T>

     struct DefCardinalityMapTraits : public Traits {

       typedef T CardinalityMap;

       static CardinalityMap *createCardinalityMap(const Graph &) 

       {

 	throw UninitializedParameter();

       }

     };

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

     /// CardinalityMap type

     ///

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

     /// type

     template <class T>

     struct DefCardinalityMap 

       : public MaxCardinalitySearch<Graph, CapacityMap, 

                                     DefCardinalityMapTraits<T> > { 

       typedef MaxCardinalitySearch<Graph, CapacityMap, 

                                    DefCardinalityMapTraits<T> > Create;

     };

     template <class T>

     struct DefProcessedMapTraits : public Traits {

       typedef T ProcessedMap;

       static ProcessedMap *createProcessedMap(const Graph &) {

 	throw UninitializedParameter();

       }

     };

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

     /// ProcessedMap type

     ///

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

     ///

     template <class T>

     struct DefProcessedMap 

       : public MaxCardinalitySearch<Graph, CapacityMap, 

                                     DefProcessedMapTraits<T> > { 

       typedef MaxCardinalitySearch<Graph, CapacityMap, 

                                    DefProcessedMapTraits<T> > Create;

     };

     template <class H, class CR>

     struct DefHeapTraits : public Traits {

       typedef CR HeapCrossRef;

       typedef H Heap;

       static HeapCrossRef *createHeapCrossRef(const Graph &) {

 	throw UninitializedParameter();

       }

       static Heap *createHeap(HeapCrossRef &) {

 	throw UninitializedParameter();

       }

     };

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

     /// and cross reference type

     ///

     /// \ref named-templ-param "Named parameter" for setting heap and cross 

     /// reference type

     template <class H, class CR = typename Graph::template NodeMap<int> >

     struct DefHeap

       : public MaxCardinalitySearch<Graph, CapacityMap, 

                                     DefHeapTraits<H, CR> > { 

       typedef MaxCardinalitySearch< Graph, CapacityMap, 

                                     DefHeapTraits<H, CR> > Create;

     };

     template <class H, class CR>

     struct DefStandardHeapTraits : public Traits {

       typedef CR HeapCrossRef;

       typedef H Heap;

       static HeapCrossRef *createHeapCrossRef(const Graph &graph) {

 	return new HeapCrossRef(graph);

       }

       static Heap *createHeap(HeapCrossRef &crossref) {

 	return new Heap(crossref);

       }

     };

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

     /// cross reference type with automatic allocation

     ///

     /// \ref named-templ-param "Named parameter" for setting heap and cross 

     /// reference type. It can allocate the heap and the cross reference 

     /// object if the cross reference's constructor waits for the graph as 

     /// parameter and the heap's constructor waits for the cross reference.

     template <class H, class CR = typename Graph::template NodeMap<int> >

     struct DefStandardHeap

       : public MaxCardinalitySearch<Graph, CapacityMap, 

                                     DefStandardHeapTraits<H, CR> > { 

       typedef MaxCardinalitySearch<Graph, CapacityMap, 

                                    DefStandardHeapTraits<H, CR> > 

       Create;

     };

     ///@}

   protected:

     MaxCardinalitySearch() {}

   public:      

     /// \brief Constructor.

     ///

     ///\param graph the graph the algorithm will run on.

     ///\param capacity the capacity map used by the algorithm.

     MaxCardinalitySearch(const Graph& graph, const CapacityMap& capacity) :

       _graph(&graph), _capacity(&capacity),

       _cardinality(0), local_cardinality(false),

       _processed(0), local_processed(false),

       _heap_cross_ref(0), local_heap_cross_ref(false),

       _heap(0), local_heap(false)

     { }

     /// \brief Destructor.

     ~MaxCardinalitySearch() {

       if(local_cardinality) delete _cardinality;

       if(local_processed) delete _processed;

       if(local_heap_cross_ref) delete _heap_cross_ref;

       if(local_heap) delete _heap;

     }

     /// \brief Sets the capacity map.

     ///

     /// Sets the capacity map.

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

     MaxCardinalitySearch &capacityMap(const CapacityMap &m) {

       _capacity = &m;

       return *this;

     }

     /// \brief Sets the map storing the cardinalities calculated by the 

     /// algorithm.

     ///

     /// Sets the map storing the cardinalities calculated by the algorithm.

     /// If you don't use this function before calling \ref run(),

     /// it will allocate one. The destuctor deallocates this

     /// automatically allocated map, of course.

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

     MaxCardinalitySearch &cardinalityMap(CardinalityMap &m) {

       if(local_cardinality) {

 	delete _cardinality;

 	local_cardinality=false;

       }

       _cardinality = &m;

       return *this;

     }

     /// \brief Sets the map storing the processed nodes.

     ///

     /// Sets the map storing the processed nodes.

     /// If you don't use this function before calling \ref run(),

     /// it will allocate one. The destuctor deallocates this

     /// automatically allocated map, of course.

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

     MaxCardinalitySearch &processedMap(ProcessedMap &m) 

     {

       if(local_processed) {

 	delete _processed;

 	local_processed=false;

       }

       _processed = &m;

       return *this;

     }

     /// \brief Sets the heap and the cross reference used by algorithm.

     ///

     /// Sets the heap and the cross reference used by algorithm.

     /// If you don't use this function before calling \ref run(),

     /// it will allocate one. The destuctor deallocates this

     /// automatically allocated map, of course.

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

     MaxCardinalitySearch &heap(Heap& heap, HeapCrossRef &crossRef) {

       if(local_heap_cross_ref) {

 	delete _heap_cross_ref;

 	local_heap_cross_ref = false;

       }

       _heap_cross_ref = &crossRef;

       if(local_heap) {

 	delete _heap;

 	local_heap = false;

       }

       _heap = &heap;

       return *this;

     }

   private:

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Edge Edge;

     typedef typename Graph::InEdgeIt InEdgeIt;

     void create_maps() {

       if(!_cardinality) {

 	local_cardinality = true;

 	_cardinality = Traits::createCardinalityMap(*_graph);

       }

       if(!_processed) {

 	local_processed = true;

 	_processed = Traits::createProcessedMap(*_graph);

       }

       if (!_heap_cross_ref) {

 	local_heap_cross_ref = true;

 	_heap_cross_ref = Traits::createHeapCrossRef(*_graph);

       }

       if (!_heap) {

 	local_heap = true;

 	_heap = Traits::createHeap(*_heap_cross_ref);

       }

     }

     void finalizeNodeData(Node node, Value capacity) {

       _processed->set(node, true);

       _cardinality->set(node, capacity);

     }

   public:

     /// \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 more control on the execution,

     /// first you must call \ref init(), then you can add several source nodes

     /// with \ref addSource().

     /// Finally \ref start() will perform the actual path

     /// computation.

     ///@{

     /// \brief Initializes the internal data structures.

     ///

     /// Initializes the internal data structures.

     void init() {

       create_maps();

       _heap->clear();

       for (NodeIt it(*_graph) ; it != INVALID ; ++it) {

 	_processed->set(it, false);

 	_heap_cross_ref->set(it, Heap::PRE_HEAP);

       }

     }

     /// \brief Adds a new source node.

     /// 

     /// Adds a new source node to the priority heap.

     ///

     /// It checks if the node has not yet been added to the heap.

     void addSource(Node source, Value capacity = 0) {

       if(_heap->state(source) == Heap::PRE_HEAP) {

 	_heap->push(source, capacity);

       } 

     }

     /// \brief Processes the next node in the priority heap

     ///

     /// Processes the next node in the priority heap.

     ///

     /// \return The processed node.

     ///

     /// \warning The priority heap must not be empty!

     Node processNextNode() {

       Node node = _heap->top(); 

       finalizeNodeData(node, _heap->prio());

       _heap->pop();

       for (InEdgeIt it(*_graph, node); it != INVALID; ++it) {

 	Node source = _graph->source(it); 

 	switch (_heap->state(source)) {

 	case Heap::PRE_HEAP:

 	  _heap->push(source, (*_capacity)[it]); 

 	  break;

 	case Heap::IN_HEAP:

 	  _heap->decrease(source, (*_heap)[source] + (*_capacity)[it]); 

 	  break;

 	case Heap::POST_HEAP:

 	  break;

 	}

       }

       return node;

     }

     /// \brief Next node to be processed.

     ///

     /// Next node to be processed.

     ///

     /// \return The next node to be processed or INVALID if the 

     /// priority heap is empty.

     Node nextNode() { 

       return _heap->empty() ? _heap->top() : INVALID;

     }

     /// \brief Returns \c false if there are nodes

     /// to be processed in the priority heap

     ///

     /// Returns \c false if there are nodes

     /// to be processed in the priority heap

     bool emptyQueue() { return _heap->empty(); }

     /// \brief Returns the number of the nodes to be processed 

     /// in the priority heap

     ///

     /// Returns the number of the nodes to be processed in the priority heap

     int queueSize() { return _heap->size(); }

     /// \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.

     ///

     /// This method runs the Maximum Cardinality Search algorithm from the 

     /// source node(s).

     void start() {

       while ( !_heap->empty() ) processNextNode();

     }

     /// \brief Executes the algorithm until \c dest is reached.

     ///

     /// Executes the algorithm until \c dest is reached.

     ///

     /// \pre init() must be called and at least one node should be added

     /// with addSource() before using this function.

     ///

     /// This method runs the %MaxCardinalitySearch algorithm from the source 

     /// nodes.

     void start(Node dest) {

       while ( !_heap->empty() && _heap->top()!=dest ) processNextNode();

       if ( !_heap->empty() ) finalizeNodeData(_heap->top(), _heap->prio());

     }

     /// \brief Executes the algorithm until a condition is met.

     ///

     /// Executes the algorithm until a condition is met.

     ///

     /// \pre init() must be called and at least one node should be added

     /// with addSource() before using this function.

     ///

     /// \param nm must be a bool (or convertible) node map. The algorithm

     /// will stop when it reaches a node \c v with <tt>nm[v]==true</tt>.

     template <typename NodeBoolMap>

     void start(const NodeBoolMap &nm) {

       while ( !_heap->empty() && !nm[_heap->top()] ) processNextNode();

       if ( !_heap->empty() ) finalizeNodeData(_heap->top(),_heap->prio());

     }

     /// \brief Runs the maximal cardinality search algorithm from node \c s.

     ///

     /// This method runs the %MaxCardinalitySearch algorithm from a root 

     /// node \c s.

     ///

     ///\note d.run(s) is just a shortcut of the following code.

     ///\code

     ///  d.init();

     ///  d.addSource(s);

     ///  d.start();

     ///\endcode

     void run(Node s) {

       init();

       addSource(s);

       start();

     }

     /// \brief Runs the maximal cardinality search algorithm for the 

     /// whole graph.

     ///

     /// This method runs the %MaxCardinalitySearch algorithm from all 

     /// unprocessed node of the graph.

     ///

     ///\note d.run(s) is just a shortcut of the following code.

     ///\code

     ///  d.init();

     ///  for (NodeIt it(graph); it != INVALID; ++it) {

     ///    if (!d.reached(it)) {

     ///      d.addSource(s);

     ///      d.start();

     ///    }

     ///  }

     ///\endcode

     void run() {

       init();

       for (NodeIt it(*_graph); it != INVALID; ++it) {

         if (!reached(it)) {

           addSource(it);

           start();

         }

       }

     }

     ///@}

     /// \name Query Functions

     /// The result of the maximum cardinality search algorithm can be 

     /// obtained using these functions.

     /// \n

     /// Before the use of these functions, either run() or start() must be 

     /// called.

     ///@{

     /// \brief The cardinality of a node.

     ///

     /// Returns the cardinality of a node.

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

     /// \warning If node \c v in unreachable from the root the return value

     /// of this funcion is undefined.

     Value cardinality(Node node) const { return (*_cardinality)[node]; }

     /// \brief Returns a reference to the NodeMap of cardinalities.

     ///

     /// Returns a reference to the NodeMap of cardinalities. \pre \ref run() 

     /// must be called before using this function.

     const CardinalityMap &cardinalityMap() const { return *_cardinality;}

     /// \brief Checks if a node is reachable from the root.

     ///

     /// Returns \c true if \c v is reachable from the root.

     /// \warning The source nodes are inditated as unreached.

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

     bool reached(Node v) { return (*_heap_cross_ref)[v] != Heap::PRE_HEAP; }

     /// \brief Checks if a node is processed.

     ///

     /// Returns \c true if \c v is processed, i.e. the shortest

     /// path to \c v has already found.

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

     bool processed(Node v) { return (*_heap_cross_ref)[v] == Heap::POST_HEAP; }

     ///@}

   };

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

   ///

   /// Default traits class of MinCut class.

   /// \param Graph Graph type.

   /// \param CapacityMap Type of length map.

   template <typename _Graph, typename _CapacityMap>

   struct MinCutDefaultTraits {

     /// \brief The type of the capacity of the edges.

     typedef typename _CapacityMap::Value Value;

     /// The graph type the algorithm runs on. 

     typedef _Graph Graph;

     /// The AuxGraph type which is an Graph

     typedef ListUGraph AuxGraph;

     /// \brief Instantiates a AuxGraph.

     ///

     /// This function instantiates a \ref AuxGraph. 

     static AuxGraph *createAuxGraph() {

       return new AuxGraph();

     }

     /// \brief The type of the map that stores the edge capacities.

     ///

     /// The type of the map that stores the edge capacities.

     /// It must meet the \ref concept::ReadMap "ReadMap" concept.

     typedef _CapacityMap CapacityMap;

     /// \brief Instantiates a CapacityMap.

     ///

     /// This function instantiates a \ref CapacityMap.

 #ifdef DOXYGEN

     static CapacityMap *createCapacityMap(const Graph& graph) 

 #else

     static CapacityMap *createCapacityMap(const Graph&)

 #endif

     {

       throw UninitializedParameter();

     }

     /// \brief The AuxCapacityMap type

     ///

     /// The type of the map that stores the auxing edge capacities.

     typedef AuxGraph::UEdgeMap<Value> AuxCapacityMap;

     /// \brief Instantiates a AuxCapacityMap.

     ///

     /// This function instantiates a \ref AuxCapacityMap. 

     static AuxCapacityMap *createAuxCapacityMap(const AuxGraph& graph) {

       return new AuxCapacityMap(graph);

     }

     /// \brief The cross reference type used by heap.

     ///

     /// The cross reference type used by heap.

     /// Usually it is \c Graph::NodeMap<int>.

     typedef AuxGraph::NodeMap<int> HeapCrossRef;

     /// \brief Instantiates a HeapCrossRef.

     ///

     /// This function instantiates a \ref HeapCrossRef. 

     /// \param graph is the graph, to which we would like to define the 

     /// HeapCrossRef.

     static HeapCrossRef *createHeapCrossRef(const AuxGraph &graph) {

       return new HeapCrossRef(graph);

     }

     /// \brief The heap type used by MinCut algorithm.

     ///

     /// The heap type used by MinCut algorithm. It should

     /// maximalize the priorities and the heap's key type is

     /// the aux graph's node.

     ///

     /// \sa BinHeap

     /// \sa MinCut

     typedef typename _min_cut_bits

     ::HeapSelector<CapacityMap>

     ::template Selector<typename AuxGraph::Node, Value, HeapCrossRef>

     ::Heap Heap;

     /// \brief Instantiates a Heap.

     ///

     /// This function instantiates a \ref Heap. 

     /// \param crossref The cross reference of the heap.

     static Heap *createHeap(HeapCrossRef& crossref) {

       return new Heap(crossref);

     }

     /// \brief Map from the AuxGraph's node type to the Graph's node type.

     ///

     /// Map from the AuxGraph's node type to the Graph's node type.

     typedef typename AuxGraph

     ::template NodeMap<typename Graph::Node> NodeRefMap;

     /// \brief Instantiates a NodeRefMap.

     ///

     /// This function instantiates a \ref NodeRefMap. 

     static NodeRefMap *createNodeRefMap(const AuxGraph& graph) {

       return new NodeRefMap(graph);

     }

     /// \brief Map from the Graph's node type to the Graph's node type.

     ///

     /// Map from the Graph's node type to the Graph's node type.

     typedef typename Graph

     ::template NodeMap<typename Graph::Node> ListRefMap;

     /// \brief Instantiates a ListRefMap.

     ///

     /// This function instantiates a \ref ListRefMap. 

     static ListRefMap *createListRefMap(const Graph& graph) {

       return new ListRefMap(graph);

     }

   };

   namespace _min_cut_bits {

     template <typename _Key>

     class LastTwoMap {

     public:

       typedef _Key Key;

       typedef bool Value;

       LastTwoMap(int _num) : num(_num) {}

       void set(const Key& key, bool val) {

         if (!val) return;

         --num;

         if (num > 1) return;

         keys[num] = key;

       }

       Key operator[](int index) const { return keys[index]; }

     private:

       Key keys[2];

       int num;

     };

   }

   /// \ingroup topology

   ///

   /// \brief Calculates the minimum cut in an undirected graph.

   ///

   /// Calculates the minimum cut in an undirected graph with the

   /// Nagamochi-Ibaraki algorithm. The algorithm separates the graph's

   /// nodes into two partitions with the minimum sum of edge capacities

   /// between the two partitions. The algorithm can be used to test

   /// the network reliability specifically to test how many links have

   /// to be destroyed in the network to split it at least two

   /// distinict subnetwork.

   ///

   /// The complexity of the algorithm is \f$ O(ne\log(n)) \f$ but with

   /// Fibonacci heap it can be decreased to \f$ O(ne+n^2\log(n))

   /// \f$. When capacity map is neutral then it uses BucketHeap which

   /// results \f$ O(ne) \f$ time complexity.

 #ifdef DOXYGEN

   template <typename _Graph, typename _CapacityMap, typename _Traits>

 #else

   template <typename _Graph = ListUGraph, 

 	    typename _CapacityMap = typename _Graph::template UEdgeMap<int>, 

 	    typename _Traits = MinCutDefaultTraits<_Graph, _CapacityMap> >

 #endif

   class MinCut {

   public:

     /// \brief \ref Exception for uninitialized parameters.

     ///

     /// This error represents problems in the initialization

     /// of the parameters of the algorithms.

     class UninitializedParameter : public lemon::UninitializedParameter {

     public:

       virtual const char* what() const throw() {

 	return "lemon::MinCut::UninitializedParameter";

       }

     };

   private:

     typedef _Traits Traits;

     /// The type of the underlying graph.

     typedef typename Traits::Graph Graph;

     /// The type of the capacity of the edges.

     typedef typename Traits::CapacityMap::Value Value;

     /// The type of the map that stores the edge capacities.

     typedef typename Traits::CapacityMap CapacityMap;

     /// The type of the aux graph

     typedef typename Traits::AuxGraph AuxGraph;

     /// The type of the aux capacity map

     typedef typename Traits::AuxCapacityMap AuxCapacityMap;

     /// The cross reference type used for the current heap.

     typedef typename Traits::HeapCrossRef HeapCrossRef;

     /// The heap type used by the max cardinality algorithm.

     typedef typename Traits::Heap Heap;

     /// The node refrefernces between the original and aux graph type.

     typedef typename Traits::NodeRefMap NodeRefMap;

     /// The list node refrefernces in the original graph type.

     typedef typename Traits::ListRefMap ListRefMap;

   public:

     ///\name Named template parameters

     ///@{

     struct DefNeutralCapacityTraits : public Traits {

       typedef ConstMap<typename Graph::UEdge, Const<int, 1> > CapacityMap;

       static CapacityMap *createCapacityMap(const Graph&) {

 	return new CapacityMap();

       }

     };

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

     /// the capacity type to constMap<UEdge, int, 1>()

     ///

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

     /// the capacity type to constMap<UEdge, int, 1>()

     struct DefNeutralCapacity

       : public MinCut<Graph, CapacityMap, DefNeutralCapacityTraits> { 

       typedef MinCut<Graph, CapacityMap, DefNeutralCapacityTraits> Create;

     };

     template <class H, class CR>

     struct DefHeapTraits : public Traits {

       typedef CR HeapCrossRef;

       typedef H Heap;

       static HeapCrossRef *createHeapCrossRef(const AuxGraph &) {

 	throw UninitializedParameter();

       }

       static Heap *createHeap(HeapCrossRef &) {

 	throw UninitializedParameter();

       }

     };

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

     /// and cross reference type

     ///

     /// \ref named-templ-param "Named parameter" for setting heap and cross 

     /// reference type

     template <class H, class CR = typename Graph::template NodeMap<int> >

     struct DefHeap

       : public MinCut<Graph, CapacityMap, DefHeapTraits<H, CR> > { 

       typedef MinCut< Graph, CapacityMap, DefHeapTraits<H, CR> > Create;

     };

     template <class H, class CR>

     struct DefStandardHeapTraits : public Traits {

       typedef CR HeapCrossRef;

       typedef H Heap;

       static HeapCrossRef *createHeapCrossRef(const AuxGraph &graph) {

 	return new HeapCrossRef(graph);

       }

       static Heap *createHeap(HeapCrossRef &crossref) {

 	return new Heap(crossref);

       }

     };

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

     /// cross reference type with automatic allocation

     ///

     /// \ref named-templ-param "Named parameter" for setting heap and cross 

     /// reference type. It can allocate the heap and the cross reference 

     /// object if the cross reference's constructor waits for the graph as 

     /// parameter and the heap's constructor waits for the cross reference.

     template <class H, class CR = typename Graph::template NodeMap<int> >

     struct DefStandardHeap

       : public MinCut<Graph, CapacityMap, DefStandardHeapTraits<H, CR> > { 

       typedef MinCut<Graph, CapacityMap, DefStandardHeapTraits<H, CR> > 

       Create;

     };

     ///@}

   private:

     /// Pointer to the underlying graph.

     const Graph *_graph;

     /// Pointer to the capacity map

     const CapacityMap *_capacity;

     /// \brief Indicates if \ref _capacity is locally allocated 

     /// (\c true) or not.

     bool local_capacity;

     /// Pointer to the aux graph.

     AuxGraph *_aux_graph;

     /// \brief Indicates if \ref _aux_graph is locally allocated 

     /// (\c true) or not.

     bool local_aux_graph;

     /// Pointer to the aux capacity map

     AuxCapacityMap *_aux_capacity;

     /// \brief Indicates if \ref _aux_capacity is locally allocated 

     /// (\c true) or not.

     bool local_aux_capacity;

     /// Pointer to the heap cross references.

     HeapCrossRef *_heap_cross_ref;

     /// \brief Indicates if \ref _heap_cross_ref is locally allocated 

     /// (\c true) or not.

     bool local_heap_cross_ref;

     /// Pointer to the heap.

     Heap *_heap;

     /// Indicates if \ref _heap is locally allocated (\c true) or not.

     bool local_heap;

     /// The min cut value.

     Value _min_cut;

     /// The number of the nodes of the aux graph.

     int _node_num;

     /// The first and last node of the min cut in the next list;

     typename Graph::Node _first_node, _last_node;

     /// \brief The first and last element in the list associated

     /// to the aux graph node.

     NodeRefMap *_first, *_last;

     /// \brief The next node in the node lists.

     ListRefMap *_next;

     void create_structures() {

       if (!_capacity) {

         local_capacity = true;

         _capacity = Traits::createCapacityMap(*_graph);

       }

       if(!_aux_graph) {

 	local_aux_graph = true;

 	_aux_graph = Traits::createAuxGraph();

       }

       if(!_aux_capacity) {

 	local_aux_capacity = true;

 	_aux_capacity = Traits::createAuxCapacityMap(*_aux_graph);

       }

       _first = Traits::createNodeRefMap(*_aux_graph);

       _last = Traits::createNodeRefMap(*_aux_graph);

       _next = Traits::createListRefMap(*_graph);

       typename Graph::template NodeMap<typename AuxGraph::Node> ref(*_graph);

       for (typename Graph::NodeIt it(*_graph); it != INVALID; ++it) {

         _next->set(it, INVALID);

         typename AuxGraph::Node node = _aux_graph->addNode();

         _first->set(node, it);

         _last->set(node, it);

         ref.set(it, node);

       }

       for (typename Graph::UEdgeIt it(*_graph); it != INVALID; ++it) {

         if (_graph->source(it) == _graph->target(it)) continue;

         typename AuxGraph::UEdge uedge = 

           _aux_graph->addEdge(ref[_graph->source(it)], 

                                ref[_graph->target(it)]);

         _aux_capacity->set(uedge, (*_capacity)[it]);

       }

       if (!_heap_cross_ref) {

 	local_heap_cross_ref = true;

 	_heap_cross_ref = Traits::createHeapCrossRef(*_aux_graph);

       }

       if (!_heap) {

 	local_heap = true;

 	_heap = Traits::createHeap(*_heap_cross_ref);

       }

     }

   public :

     typedef MinCut Create;

     /// \brief Constructor.

     ///

     ///\param graph the graph the algorithm will run on.

     ///\param capacity the capacity map used by the algorithm.

     MinCut(const Graph& graph, const CapacityMap& capacity) 

       : _graph(&graph), 

         _capacity(&capacity), local_capacity(false),

         _aux_graph(0), local_aux_graph(false),

         _aux_capacity(0), local_aux_capacity(false),

         _heap_cross_ref(0), local_heap_cross_ref(false),

         _heap(0), local_heap(false),

         _first(0), _last(0), _next(0) {}

     /// \brief Constructor.

     ///

     /// This constructor can be used only when the Traits class

     /// defines how can we instantiate a local capacity map.

     /// If the DefNeutralCapacity used the algorithm automatically

     /// construct the capacity map.

     ///

     ///\param graph the graph the algorithm will run on.

     MinCut(const Graph& graph) 

       : _graph(&graph), 

         _capacity(0), local_capacity(false),

         _aux_graph(0), local_aux_graph(false),

         _aux_capacity(0), local_aux_capacity(false),

         _heap_cross_ref(0), local_heap_cross_ref(false),

         _heap(0), local_heap(false),

         _first(0), _last(0), _next(0) {}

     /// \brief Destructor.

     ///

     /// Destructor.

     ~MinCut() {

       if (local_heap) delete _heap;

       if (local_heap_cross_ref) delete _heap_cross_ref;

       if (_first) delete _first;

       if (_last) delete _last;

       if (_next) delete _next;

       if (local_aux_capacity) delete _aux_capacity;

       if (local_aux_graph) delete _aux_graph;

       if (local_capacity) delete _capacity;

     }

     /// \brief Sets the heap and the cross reference used by algorithm.

     ///

     /// Sets the heap and the cross reference used by algorithm.

     /// If you don't use this function before calling \ref run(),

     /// it will allocate one. The destuctor deallocates this

     /// automatically allocated heap and cross reference, of course.

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

     MinCut &heap(Heap& heap, HeapCrossRef &crossRef)

     {

       if (local_heap_cross_ref) {

 	delete _heap_cross_ref;

 	local_heap_cross_ref=false;

       }

       _heap_cross_ref = &crossRef;

       if (local_heap) {

 	delete _heap;

 	local_heap=false;

       }

       _heap = &heap;

       return *this;

     }

     /// \brief Sets the aux graph.

     ///

     /// Sets the aux graph used by algorithm.

     /// If you don't use this function before calling \ref run(),

     /// it will allocate one. The destuctor deallocates this

     /// automatically allocated graph, of course.

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

     MinCut &auxGraph(AuxGraph& aux_graph)

     {

       if(local_aux_graph) {

 	delete _aux_graph;

 	local_aux_graph=false;

       }

       _aux_graph = &aux_graph;

       return *this;

     }

     /// \brief Sets the aux capacity map.

     ///

     /// Sets the aux capacity map used by algorithm.

     /// If you don't use this function before calling \ref run(),

     /// it will allocate one. The destuctor deallocates this

     /// automatically allocated graph, of course.

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

     MinCut &auxCapacityMap(AuxCapacityMap& aux_capacity_map)

     {

       if(local_aux_capacity) {

 	delete _aux_capacity;

 	local_aux_capacity=false;

       }

       _aux_capacity = &aux_capacity_map;

       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 more control on the execution,

     /// first you must call \ref init() and then call the start()

     /// or proper times the processNextPhase() member functions.

     ///@{

     /// \brief Initializes the internal data structures.

     ///

     /// Initializes the internal data structures.

     void init() {

       create_structures();

       _first_node = _last_node = INVALID;

       _node_num = countNodes(*_graph);

     }

     /// \brief Processes the next phase

     ///

     /// Processes the next phase in the algorithm. The function

     /// should be called countNodes(graph) - 1 times to get

     /// surely the min cut in the graph. The  

     ///

     ///\return %True when the algorithm finished.

     bool processNextPhase() {

       if (_node_num <= 1) return true;

       using namespace _min_cut_bits;

       typedef typename AuxGraph::Node Node;

       typedef typename AuxGraph::NodeIt NodeIt;

       typedef typename AuxGraph::UEdge UEdge;

       typedef typename AuxGraph::IncEdgeIt IncEdgeIt;

       typedef typename MaxCardinalitySearch<AuxGraph, AuxCapacityMap>::

       template DefHeap<Heap, HeapCrossRef>::

       template DefCardinalityMap<NullMap<Node, Value> >::

       template DefProcessedMap<LastTwoMap<Node> >::

       Create MaxCardinalitySearch;

       MaxCardinalitySearch mcs(*_aux_graph, *_aux_capacity);

       for (NodeIt it(*_aux_graph); it != INVALID; ++it) {

         _heap_cross_ref->set(it, Heap::PRE_HEAP);

       }

       mcs.heap(*_heap, *_heap_cross_ref);

       LastTwoMap<Node> last_two_nodes(_node_num);

       mcs.processedMap(last_two_nodes);

       NullMap<Node, Value> cardinality;

       mcs.cardinalityMap(cardinality);

       mcs.run();

       Node new_node = _aux_graph->addNode();

       typename AuxGraph::template NodeMap<UEdge> edges(*_aux_graph, INVALID);

       Node first_node = last_two_nodes[0];

       Node second_node = last_two_nodes[1];

       _next->set((*_last)[first_node], (*_first)[second_node]);

       _first->set(new_node, (*_first)[first_node]);

       _last->set(new_node, (*_last)[second_node]);

       Value current_cut = 0;      

       for (IncEdgeIt it(*_aux_graph, first_node); it != INVALID; ++it) {

         Node node = _aux_graph->runningNode(it);

         current_cut += (*_aux_capacity)[it];

         if (node == second_node) continue;

         if (edges[node] == INVALID) {

           edges[node] = _aux_graph->addEdge(new_node, node);

           (*_aux_capacity)[edges[node]] = (*_aux_capacity)[it];

         } else {

           (*_aux_capacity)[edges[node]] += (*_aux_capacity)[it];          

         }

       }

       if (_first_node == INVALID || current_cut < _min_cut) {

         _first_node = (*_first)[first_node];

         _last_node = (*_last)[first_node];

         _min_cut = current_cut;

       }

       _aux_graph->erase(first_node);

       for (IncEdgeIt it(*_aux_graph, second_node); it != INVALID; ++it) {

         Node node = _aux_graph->runningNode(it);

         if (edges[node] == INVALID) {

           edges[node] = _aux_graph->addEdge(new_node, node);

           (*_aux_capacity)[edges[node]] = (*_aux_capacity)[it];

         } else {

           (*_aux_capacity)[edges[node]] += (*_aux_capacity)[it];          

         }

       }

       _aux_graph->erase(second_node);

       --_node_num;

       return _node_num == 1;

     }

     /// \brief Executes the algorithm.

     ///

     /// Executes the algorithm.

     ///

     /// \pre init() must be called

     void start() {

       while (!processNextPhase());

     }

     /// \brief Runs %MinCut algorithm.

     ///

     /// This method runs the %Min cut algorithm

     ///

     /// \note mc.run(s) is just a shortcut of the following code.

     ///\code

     ///  mc.init();

     ///  mc.start();

     ///\endcode

     void run() {

       init();

       start();

     }

     ///@}

     /// \name Query Functions

     /// The result of the %MinCut algorithm can be obtained using these

     /// functions.\n

     /// Before the use of these functions,

     /// either run() or start() must be called.

     ///@{

     /// \brief Returns the min cut value.

     ///

     /// Returns the min cut value if the algorithm finished.

     /// After the first processNextPhase() it is a value of a

     /// valid cut in the graph.

     Value minCut() const {

       return _min_cut;

     }

     /// \brief Returns a min cut in a NodeMap.

     ///

     /// It sets the nodes of one of the two partitions to true in

     /// the given BoolNodeMap. The map contains a valid cut if the

     /// map have been set false previously. 

     template <typename NodeMap>

     Value quickMinCut(NodeMap& nodeMap) const { 

       for (typename Graph::Node it = _first_node; 

            it != _last_node; it = (*_next)[it]) {

              nodeMap.set(it, true);

            }

       nodeMap.set(_last_node, true);

       return minCut();

     }

     /// \brief Returns a min cut in a NodeMap.

     ///

     /// It sets the nodes of one of the two partitions to true and

     /// the other partition to false. The function first set all of the

     /// nodes to false and after it call the quickMinCut() member.

     template <typename NodeMap>

     Value minCut(NodeMap& nodeMap) const { 

       for (typename Graph::NodeIt it(*_graph); it != INVALID; ++it) {

         nodeMap.set(it, false);      

       }

       quickMinCut(nodeMap);

       return minCut();

     }

     /// \brief Returns a min cut in an EdgeMap.

     ///

     /// If an undirected edge is in a min cut then it will be

     /// set to true and the others will be set to false in the given map.

     template <typename EdgeMap>

     Value cutEdges(EdgeMap& edgeMap) const {

       typename Graph::template NodeMap<bool> cut(*_graph, false);

       quickMinCut(cut);

       for (typename Graph::EdgeIt it(*_graph); it != INVALID; ++it) {

         edgeMap.set(it, cut[_graph->source(it)] ^ cut[_graph->target(it)]);

       }

       return minCut();

     }

     ///@}

   };

 }

 #endif