/* -*- C++ -*-

  * lemon/prim.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_PRIM_H

 #define LEMON_PRIM_H

 ///\ingroup spantree

 ///\file

 ///\brief Prim algorithm to compute minimum spanning tree.

 #include <lemon/list_graph.h>

 #include <lemon/bin_heap.h>

 #include <lemon/invalid.h>

 #include <lemon/error.h>

 #include <lemon/maps.h>

 #include <lemon/traits.h>

 #include <lemon/concept/ugraph.h>

 namespace lemon {

   ///Default traits class of Prim class.

   ///Default traits class of Prim class.

   ///\param GR Graph type.

   ///\param LM Type of cost map.

   template<class GR, class LM>

   struct PrimDefaultTraits{

     ///The graph type the algorithm runs on. 

     typedef GR UGraph;

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

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

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

     typedef LM CostMap;

     //The type of the cost of the edges.

     typedef typename LM::Value Value;

     /// The cross reference type used by heap.

     /// The cross reference type used by heap.

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

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

     ///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 GR &_graph){

       return new HeapCrossRef(_graph);

     }

     ///The heap type used by Prim algorithm.

     ///The heap type used by Prim algorithm.

     ///

     ///\sa BinHeap

     ///\sa Prim

     typedef BinHeap<typename UGraph::Node, typename LM::Value,

 		    HeapCrossRef, std::less<Value> > Heap;

     static Heap *createHeap(HeapCrossRef& _ref){

       return new Heap(_ref);

     }

     ///\brief The type of the map that stores the last

     ///edges of the minimum spanning tree.

     /// 

     ///The type of the map that stores the last

     ///edges of the minimum spanning tree.

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

     ///

     typedef typename UGraph::template NodeMap<typename GR::UEdge> PredMap;

     ///Instantiates a PredMap.

     ///This function instantiates a \ref PredMap. 

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

     static PredMap *createPredMap(const GR &_graph){

       return new PredMap(_graph);

     }

     ///The type of the map that stores whether an edge is in the

     ///spanning tree or not.

     ///The type of the map that stores whether an edge is in the

     ///spanning tree or not.

     ///By default it is a NullMap.

     typedef NullMap<typename UGraph::UEdge,bool> TreeMap;

     ///Instantiates a TreeMap.

     ///This function instantiates a \ref TreeMap.

     ///

     ///The first parameter is the graph, to which

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

     static TreeMap *createTreeMap(const GR &){

       return new TreeMap();

     }

     ///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 NodeMap<bool>.

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

     ///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 GR &_graph)

 #else

     static ProcessedMap *createProcessedMap(const GR &)

 #endif

     {

       return new ProcessedMap();

     }

   };

   ///%Prim algorithm class to find a minimum spanning tree.

   /// \ingroup spantree

   ///This class provides an efficient implementation of %Prim algorithm.

   ///

   ///The running time is O(e*log n) where e is the number of edges and

   ///n is the number of nodes in the graph.

   ///

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

   ///\ref concept::ReadMap "ReadMap",

   ///so it is easy to change it to any kind of cost.

   ///

   ///The type of the cost is determined by the

   ///\ref concept::ReadMap::Value "Value" of the cost map.

   ///

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

   ///

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

   ///is \ref ListUGraph. The value of GR is not used directly by

   ///Prim, it is only passed to \ref PrimDefaultTraits.

   ///

   ///\param LM This read-only UEdgeMap determines the costs of the

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

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

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

   ///concept::UGraph::UEdgeMap "UGraph::UEdgeMap<int>".  The value

   ///of LM is not used directly by Prim, it is only passed to \ref

   ///PrimDefaultTraits.

   ///

   ///\param TR Traits class to set

   ///various data types used by the algorithm.  The default traits

   ///class is \ref PrimDefaultTraits

   ///"PrimDefaultTraits<GR,LM>".  See \ref

   ///PrimDefaultTraits for the documentation of a Prim traits

   ///class.

   ///

   ///\author Balazs Attila Mihaly

 #ifdef DOXYGEN

   template <typename GR,

 	    typename LM,

 	    typename TR>

 #else

   template <typename GR=ListUGraph,

 	    typename LM=typename GR::template UEdgeMap<int>,

 	    typename TR=PrimDefaultTraits<GR,LM> >

 #endif

   class Prim {

   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* exceptionName() const {

 	return "lemon::Prim::UninitializedParameter";

       }

     };

     typedef TR Traits;

     ///The type of the underlying graph.

     typedef typename TR::UGraph UGraph;

     ///\e

     typedef typename UGraph::Node Node;

     ///\e

     typedef typename UGraph::NodeIt NodeIt;

     ///\e

     typedef typename UGraph::UEdge UEdge;

     ///\e

     typedef typename UGraph::IncEdgeIt IncEdgeIt;

     ///The type of the cost of the edges.

     typedef typename TR::CostMap::Value Value;

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

     typedef typename TR::CostMap CostMap;

     ///\brief The type of the map that stores the last

     ///predecessor edges of the spanning tree.

     typedef typename TR::PredMap PredMap;

     ///Edges of the spanning tree.

     typedef typename TR::TreeMap TreeMap;

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

     typedef typename TR::ProcessedMap ProcessedMap;

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

     typedef typename TR::HeapCrossRef HeapCrossRef;

     ///The heap type used by the prim algorithm.

     typedef typename TR::Heap Heap;

   private:

     /// Pointer to the underlying graph.

     const UGraph *graph;

     /// Pointer to the cost map

     const CostMap *cost;

     ///Pointer to the map of predecessors edges.

     PredMap *_pred;

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

     bool local_pred;

     ///Pointer to the map of tree edges.

     TreeMap *_tree;

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

     bool local_tree;

     ///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;

     ///Creates the maps if necessary.

     void create_maps(){

       if(!_pred) {

 	local_pred = true;

 	_pred = Traits::createPredMap(*graph);

       }

       if(!_tree) {

 	local_tree = true;

 	_tree = Traits::createTreeMap(*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);

       }

     }

   public :

     typedef Prim Create;

     ///\name Named template parameters

     ///@{

     template <class T>

     struct DefPredMapTraits : public Traits {

       typedef T PredMap;

       static PredMap *createPredMap(const UGraph &_graph){

 	throw UninitializedParameter();

       }

     };

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

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

     ///

     template <class T>

     struct DefPredMap 

       : public Prim< UGraph, CostMap, DefPredMapTraits<T> > {

       typedef Prim< UGraph, CostMap, DefPredMapTraits<T> > Create;

     };

     template <class T>

     struct DefProcessedMapTraits : public Traits {

       typedef T ProcessedMap;

       static ProcessedMap *createProcessedMap(const UGraph &_graph){

 	throw UninitializedParameter();

       }

     };

     ///\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 Prim< UGraph, CostMap, DefProcessedMapTraits<T> > { 

       typedef Prim< UGraph, CostMap, DefProcessedMapTraits<T> > Create;

     };

     struct DefGraphProcessedMapTraits : public Traits {

       typedef typename UGraph::template NodeMap<bool> ProcessedMap;

       static ProcessedMap *createProcessedMap(const UGraph &_graph){

 	return new ProcessedMap(_graph);

       }

     };

     template <class H, class CR>

     struct DefHeapTraits : public Traits {

       typedef CR HeapCrossRef;

       typedef H Heap;

       static HeapCrossRef *createHeapCrossRef(const UGraph &) {

 	throw UninitializedParameter();

       }

       static Heap *createHeap(HeapCrossRef &){

 	return UninitializedParameter();

       }

     };

     ///\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 UGraph::template NodeMap<int> >

     struct DefHeap

       : public Prim< UGraph, CostMap, DefHeapTraits<H, CR> > {

       typedef Prim< UGraph, CostMap, DefHeapTraits<H, CR> > Create;

     };

     template <class H, class CR>

     struct DefStandardHeapTraits : public Traits {

       typedef CR HeapCrossRef;

       typedef H Heap;

       static HeapCrossRef *createHeapCrossRef(const UGraph &_graph) {

 	return new HeapCrossRef(_graph);

       }

       static Heap *createHeap(HeapCrossRef &ref){

 	return new Heap(ref);

       }

     };

     ///\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 UGraph::template NodeMap<int> >

     struct DefStandardHeap

       : public Prim< UGraph, CostMap, DefStandardHeapTraits<H, CR> > { 

       typedef Prim< UGraph, CostMap, DefStandardHeapTraits<H, CR> > 

       Create;

     };

     template <class TM>

     struct DefTreeMapTraits : public Traits {

       typedef TM TreeMap;

       static TreeMap *createTreeMap(const UGraph &) {

         throw UninitializedParameter();

       }

     };

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

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

     ///

     template <class TM>

     struct DefTreeMap

       : public Prim< UGraph, CostMap, DefTreeMapTraits<TM> > {

       typedef Prim< UGraph, CostMap, DefTreeMapTraits<TM> > Create;

     };    

     struct DefGraphTreeMapTraits : public Traits {

       typedef typename UGraph::template NodeMap<bool> TreeMap;

       static TreeMap *createTreeMap(const UGraph &_graph){

 	return new TreeMap(_graph);

       }

     };

     ///@}

   protected:

     Prim() {}

   public:      

     ///Constructor.

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

     ///\param _cost the cost map used by the algorithm.

     Prim(const UGraph& _graph, const CostMap& _cost) :

       graph(&_graph), cost(&_cost),

       _pred(NULL), local_pred(false),

       _tree(NULL), local_tree(false),

       _processed(NULL), local_processed(false),

       _heap_cross_ref(NULL), local_heap_cross_ref(false),

       _heap(NULL), local_heap(false)

     {

       checkConcept<concept::UGraph, UGraph>();

     }

     ///Destructor.

     ~Prim(){

       if(local_pred) delete _pred;

       if(local_tree) delete _tree;

       if(local_processed) delete _processed;

       if(local_heap_cross_ref) delete _heap_cross_ref;

       if(local_heap) delete _heap;

     }

     ///\brief Sets the cost map.

     ///Sets the cost map.

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

     Prim &costMap(const CostMap &m){

       cost = &m;

       return *this;

     }

     ///\brief Sets the map storing the predecessor edges.

     ///Sets the map storing the predecessor edges.

     ///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>

     Prim &predMap(PredMap &m){

       if(local_pred) {

 	delete _pred;

 	local_pred=false;

       }

       _pred = &m;

       return *this;

     }

     ///\brief Sets the map storing the tree edges.

     ///Sets the map storing the tree edges.

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

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

     ///automatically allocated map, of course.

     ///By default this is a NullMap.

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

     Prim &treeMap(TreeMap &m){

       if(local_tree) {

 	delete _tree;

 	local_tree=false;

       }

       _tree = &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>

     Prim &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;

     }

   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 u(*graph) ; u!=INVALID ; ++u ) {

 	_pred->set(u,INVALID);

 	_processed->set(u,false);

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

       }

     }

     ///\brief Adds a new source node.

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

     ///

     ///It checks if the node has already been added to the heap and

     ///it is pushed to the heap only if it was not in the heap.

     void addSource(Node s){

       if(_heap->state(s) != Heap::IN_HEAP) {

 	_heap->push(s,Value());

       }

     }

     ///\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 v=_heap->top(); 

       _heap->pop();

       _processed->set(v,true);

       for(IncEdgeIt e(*graph,v); e!=INVALID; ++e) {

 	Node w=graph->oppositeNode(v,e);

 	switch(_heap->state(w)) {

 	case Heap::PRE_HEAP:

 	  _heap->push(w,(*cost)[e]);

 	  _pred->set(w,e);

 	  break;

 	case Heap::IN_HEAP:

 	  if ( (*cost)[e] < (*_heap)[w] ) {

 	    _heap->decrease(w,(*cost)[e]); 

 	    _pred->set(w,e);

 	  }

 	  break;

 	case Heap::POST_HEAP:

 	  break;

 	}

       }

       if ((*_pred)[v]!=INVALID)_tree->set((*_pred)[v],true);

       return v;

     }

     ///\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 %Prim algorithm from the node(s)

     ///in order to compute the

     ///minimum spanning tree.

     ///

     void start(){

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

     }

     ///\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<class NodeBoolMap>

     void start(const NodeBoolMap &nm){

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

       if ( !_heap->empty() ) _processed->set(_heap->top(),true);

     }

     ///\brief Runs %Prim algorithm.

     ///This method runs the %Prim algorithm

     ///in order to compute the

     ///minimum spanning tree (or minimum spanning forest).

     ///The method also works on graphs that has more than one components.

     ///In this case it computes the minimum spanning forest.

     void run() {

       init();

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

 	if(!processed(it)){

 	  addSource(it);

 	  start();

 	}

       }

     }

     ///\brief Runs %Prim algorithm from node \c s.

     ///This method runs the %Prim algorithm from node \c s

     ///in order to

     ///compute the

     ///minimun spanning tree

     ///

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

     ///\code

     ///  d.init();

     ///  d.addSource(s);

     ///  d.start();

     ///\endcode

     ///\note If the graph has more than one components, the method

     ///will compute the minimun spanning tree for only one component.

     ///

     ///See \ref run() if you want to compute the minimal spanning forest.

     void run(Node s){

       init();

       addSource(s);

       start();

     }

     ///@}

     ///\name Query Functions

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

     ///functions.\n

     ///Before the use of these functions,

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

     ///@{

     ///\brief Returns the 'previous edge' of the minimum spanning tree.

     ///For a node \c v it returns the 'previous edge' of the minimum spanning tree,

     ///i.e. it returns the edge from where \c v was reached. For a source node

     ///or an unreachable node it is \ref INVALID.

     ///The minimum spanning tree used here is equal to the minimum spanning tree used

     ///in \ref predNode().  \pre \ref run() or \ref start() must be called before

     ///using this function.

     UEdge predEdge(Node v) const { return (*_pred)[v]; }

     ///\brief Returns the 'previous node' of the minimum spanning tree.

     ///For a node \c v it returns the 'previous node' of the minimum spanning tree,

     ///i.e. it returns the node from where \c v was reached. For a source node

     ///or an unreachable node it is \ref INVALID.

     //The minimum spanning tree used here is equal to the minimum spanning

     ///tree used in \ref predEdge().  \pre \ref run() or \ref start() must be called

     ///before using this function.

     Node predNode(Node v) const { return (*_pred)[v]==INVALID ? INVALID:

 				  graph->source((*_pred)[v]); }

     ///\brief Returns a reference to the NodeMap of the edges of the minimum spanning tree.

     ///Returns a reference to the NodeMap of the edges of the

     ///minimum spanning tree.

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

     const PredMap &predMap() const { return *_pred;}

     ///\brief Returns a reference to the tree edges map.

     ///Returns a reference to the TreeEdgeMap of the edges of the

     ///minimum spanning tree. The value of the map is \c true only if the edge is in

     ///the minimum spanning tree.

     ///\warning By default, the TreeEdgeMap is a NullMap.

     ///

     ///If it is not set before the execution of the algorithm, use the \ref

     ///treeMap(TreeMap&) function (after the execution) to set an UEdgeMap with the

     ///edges of the minimum spanning tree in O(n) time where n is the number of

     ///nodes in the graph.

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

     const TreeMap &treeMap() const { return *_tree;}

     ///\brief Sets the tree edges map.

     ///Sets the TreeMap of the edges of the minimum spanning tree.

     ///The map values belonging to the edges of the minimum

     ///spanning tree are set to \c tree_edge_value or \c true by default,

     ///the other map values remain untouched.

     ///

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

     template<class TreeMap>

     void quickTreeEdges(

         TreeMap& tree,

         const typename TreeMap::Value& tree_edge_value=true) const {

       for(NodeIt i(*graph);i!=INVALID;++i){

         if((*_pred)[i]!=INVALID) tree.set((*_pred)[i],tree_edge_value);

       }

     }

     ///\brief Sets the tree edges map.

     ///Sets the TreeMap of the edges of the minimum spanning tree.

     ///The map values belonging to the edges of the minimum

     ///spanning tree are set to \c tree_edge_value or \c true by default while

     ///the edge values not belonging to the minimum spanning tree are set to

     ///\c tree_default_value or \c false by default.

     ///

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

     template<class TreeMap>

     void treeEdges(

         TreeMap& tree,

         const typename TreeMap::Value& tree_edge_value=true,

         const typename TreeMap::Value& tree_default_value=false) const {

       for(typename ItemSetTraits<UGraph,UEdge>::ItemIt i(*graph);i!=INVALID;++i)

 	tree.set(i,tree_default_value);

       for(NodeIt i(*graph);i!=INVALID;++i){

         if((*_pred)[i]!=INVALID) tree.set((*_pred)[i],tree_edge_value);

       }

     }

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

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

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

     ///\pre \ref run() or \ref start() 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. \c v is already connencted to the

     ///minimum spanning tree.

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

     ///

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

     ///\brief Checks if an edge is in the spanning tree or not.

     ///Checks if an edge is in the spanning tree or not.

     ///\param e is the edge that will be checked

     ///\return \c true if e is in the spanning tree, \c false otherwise

     bool tree(UEdge e){

       return (*_pred)[*graph.source(e)]==e || (*_pred)[*graph.target(e)]==e;

     }

     ///@}

   };

   /// \ingroup spantree

   ///

   /// \brief Function type interface for Prim algorithm.

   ///

   /// Function type interface for Prim algorithm.

   /// \param graph the UGraph that the algorithm runs on

   /// \param cost the CostMap of the edges

   /// \retval tree the EdgeMap that contains whether an edge is in 

   /// the spanning tree or not

   ///

   ///\sa Prim

   template<class Graph,class CostMap,class TreeMap>

   void prim(const Graph& graph, const CostMap& cost,TreeMap& tree){

     typename Prim<Graph,CostMap>::template DefTreeMap<TreeMap>::

       Create prm(graph,cost);

     prm.treeMap(tree);

     prm.run();

   };

 } //END OF NAMESPACE LEMON

 #endif