/* -*- C++ -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

#ifndef LEMON_DAG_SHORTEST_PATH_H

#define LEMON_DAG_SHORTEST_PATH_H

///\ingroup flowalgs

/// \file

/// \brief DagShortestPath algorithm.

///

#include <lemon/list_graph.h>

#include <lemon/bits/invalid.h>

#include <lemon/error.h>

#include <lemon/maps.h>

#include <lemon/topology.h>

#include <limits>

namespace lemon {

  /// \brief Default OperationTraits for the DagShortestPath algorithm class.

  ///  

  /// It defines all computational operations and constants which are

  /// used in the dag shortest path algorithm. The default implementation

  /// is based on the numeric_limits class. If the numeric type does not

  /// have infinity value then the maximum value is used as extremal

  /// infinity value.

  template <

    typename Value, 

    bool has_infinity = std::numeric_limits<Value>::has_infinity>

  struct DagShortestPathDefaultOperationTraits {

    /// \brief Gives back the zero value of the type.

    static Value zero() {

      return static_cast<Value>(0);

    }

    /// \brief Gives back the positive infinity value of the type.

    static Value infinity() {

      return std::numeric_limits<Value>::infinity();

    }

    /// \brief Gives back the sum of the given two elements.

    static Value plus(const Value& left, const Value& right) {

      return left + right;

    }

    /// \brief Gives back true only if the first value less than the second.

    static bool less(const Value& left, const Value& right) {

      return left < right;

    }

  };

  template <typename Value>

  struct DagShortestPathDefaultOperationTraits<Value, false> {

    static Value zero() {

      return static_cast<Value>(0);

    }

    static Value infinity() {

      return std::numeric_limits<Value>::max();

    }

    static Value plus(const Value& left, const Value& right) {

      if (left == infinity() || right == infinity()) return infinity();

      return left + right;

    }

    static bool less(const Value& left, const Value& right) {

      return left < right;

    }

  };

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

  ///

  /// Default traits class of DagShortestPath class.

  /// \param _Graph Graph type.

  /// \param _LegthMap Type of length map.

  template<class _Graph, class _LengthMap>

  struct DagShortestPathDefaultTraits {

    /// The graph type the algorithm runs on. 

    typedef _Graph Graph;

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

    ///

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

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

    typedef _LengthMap LengthMap;

    // The type of the length of the edges.

    typedef typename _LengthMap::Value Value;

    /// \brief Operation traits for dag shortest path algorithm.

    ///

    /// It defines the infinity type on the given Value type

    /// and the used operation.

    /// \see DagShortestPathDefaultOperationTraits

    typedef DagShortestPathDefaultOperationTraits<Value> OperationTraits;

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

    /// shortest paths.

    /// 

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

    /// edges of the shortest paths.

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

    ///

    typedef typename Graph::template NodeMap<typename _Graph::Edge> PredMap;

    /// \brief Instantiates a PredMap.

    /// 

    /// This function instantiates a \ref PredMap. 

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

    /// like to define the PredMap.

    /// \todo The graph alone may be insufficient for the initialization

    static PredMap *createPredMap(const _Graph& graph) {

      return new PredMap(graph);

    }

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

    ///

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

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

    ///

    typedef typename Graph::template NodeMap<typename _LengthMap::Value> 

    DistMap;

    /// \brief Instantiates a DistMap.

    ///

    /// This function instantiates a \ref DistMap. 

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

    /// \ref DistMap

    static DistMap *createDistMap(const _Graph& graph) {

      return new DistMap(graph);

    }

  };

  /// \brief Inverse OperationTraits for the DagShortestPath algorithm class.

  /// 

  /// It defines all computational operations and constants which are

  /// used in the dag shortest path algorithm. It is the inverse of

  /// \ref DagShortestPathDefaultOperationTraits, so it can be used to

  /// calculate the longest path. The default implementation

  /// is based on the numeric_limits class. If the numeric type does not

  /// have infinity value then the minimum value is used as extremal

  /// infinity value.

  template <

    typename Value, 

    bool has_infinity = std::numeric_limits<Value>::has_infinity>

  struct DagLongestPathOperationTraits {

    /// \brief Gives back the zero value of the type.

    static Value zero() {

      return static_cast<Value>(0);

    }

    /// \brief Gives back the negative infinity value of the type.

    static Value infinity() {

      return -(std::numeric_limits<Value>::infinity());

    }

    /// \brief Gives back the sum of the given two elements.

    static Value plus(const Value& left, const Value& right) {

      return left + right;

    }

    /// \brief Gives back true only if the first value less than the second.

    static bool less(const Value& left, const Value& right) {

      return left > right;

    }

  };

  template <typename Value>

  struct DagLongestPathOperationTraits<Value, false> {

    static Value zero() {

      return static_cast<Value>(0);

    }

    static Value infinity() {

      return std::numeric_limits<Value>::min();

    }

    static Value plus(const Value& left, const Value& right) {

      if (left == infinity() || right == infinity()) return infinity();

      return left + right;

    }

    static bool less(const Value& left, const Value& right) {

      return left > right;

    }

  };

  /// \brief Inverse traits class of DagShortestPath class.

  ///

  /// Inverse traits class of DagShortestPath class.

  /// \param _Graph Graph type.

  /// \param _LegthMap Type of length map.

  template<class _Graph, class _LengthMap>

  struct DagLongestPathTraits {

    /// The graph type the algorithm runs on. 

    typedef _Graph Graph;

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

    ///

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

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

    typedef _LengthMap LengthMap;

    // The type of the length of the edges.

    typedef typename _LengthMap::Value Value;

    /// \brief Inverse operation traits for dag shortest path algorithm.

    ///

    /// It defines the infinity type on the given Value type

    /// and the used operation.

    /// \see DagLongestPathOperationTraits

    typedef DagLongestPathOperationTraits<Value> OperationTraits;

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

    /// longest paths.

    /// 

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

    /// edges of the longest paths.

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

    ///

    typedef typename Graph::template NodeMap<typename _Graph::Edge> PredMap;

    /// \brief Instantiates a PredMap.

    /// 

    /// This function instantiates a \ref PredMap. 

    /// \param graph is the graph,

    /// to which we would like to define the PredMap.

    /// \todo The graph alone may be insufficient for the initialization

    static PredMap *createPredMap(const _Graph& graph) {

      return new PredMap(graph);

    }

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

    ///

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

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

    ///

    typedef typename Graph::template NodeMap<typename _LengthMap::Value> 

    DistMap;

    /// \brief Instantiates a DistMap.

    ///

    /// This function instantiates a \ref DistMap. 

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

    /// \ref DistMap

    static DistMap *createDistMap(const _Graph& graph) {

      return new DistMap(graph);

    }

  };

  /// \brief %DagShortestPath algorithm class.

  ///

  /// \ingroup flowalgs

  /// This class provides an efficient implementation of a Dag sortest path

  /// searching algorithm. The edge lengths are passed to the algorithm

  /// using a \ref concepts::ReadMap "ReadMap", so it is easy to change it

  /// to any kind of length.

  ///

  /// The complexity of the algorithm is O(n + e).

  ///

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

  /// \ref concepts::ReadMap::Value "Value" of the length map.

  ///

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

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

  /// DagShortestPath, it is only passed to \ref DagShortestPathDefaultTraits.

  /// \param _LengthMap This read-only EdgeMap determines the lengths of the

  /// edges. The default map type is \ref concepts::Graph::EdgeMap 

  /// "Graph::EdgeMap<int>".  The value of _LengthMap is not used directly 

  /// by DagShortestPath, it is only passed to \ref DagShortestPathDefaultTraits.  

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

  /// algorithm.  The default traits class is \ref DagShortestPathDefaultTraits

  /// "DagShortestPathDefaultTraits<_Graph,_LengthMap>".  See \ref

  /// DagShortestPathDefaultTraits for the documentation of a DagShortestPath traits

  /// class.

  ///

  /// \author Balazs Attila Mihaly (based on Balazs Dezso's work)

#ifdef DOXYGEN

  template <typename _Graph, typename _LengthMap, typename _Traits>

#else

  template <typename _Graph=ListGraph,

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

	    typename _Traits=DagShortestPathDefaultTraits<_Graph,_LengthMap> >

#endif

  class DagShortestPath {

  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::DagShortestPath::UninitializedParameter";

      }

    };

    typedef _Traits Traits;

    ///The type of the underlying graph.

    typedef typename _Traits::Graph Graph;

    typedef typename Graph::Node Node;

    typedef typename Graph::NodeIt NodeIt;

    typedef typename Graph::Edge Edge;

    typedef typename Graph::EdgeIt EdgeIt;

    typedef typename Graph::OutEdgeIt OutEdgeIt;

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

    typedef typename _Traits::LengthMap::Value Value;

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

    typedef typename _Traits::LengthMap LengthMap;

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

    /// edges of the shortest paths.

    typedef typename _Traits::PredMap PredMap;

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

    typedef typename _Traits::DistMap DistMap;

    /// \brief The operation traits.

    typedef typename _Traits::OperationTraits OperationTraits;

    /// \brief The Node weight map.

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

  private:

    /// Pointer to the underlying graph

    const Graph *graph;

    /// Pointer to the length map

    const LengthMap *length;

    ///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 distances

    DistMap *_dist;

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

    bool local_dist;

    ///Process step counter

    unsigned int _process_step;

    std::vector<Node> _node_order;

    /// Creates the maps if necessary.

    void create_maps() {

      if(!_pred) {

	local_pred = true;

	_pred = Traits::createPredMap(*graph);

      }

      if(!_dist) {

	local_dist = true;

	_dist = Traits::createDistMap(*graph);

      }

    }

  public :

    typedef DagShortestPath Create;

    /// \name Named template parameters

    ///@{

    template <class T>

    struct DefPredMapTraits : public Traits {

      typedef T PredMap;

      static PredMap *createPredMap(const Graph&) {

	throw UninitializedParameter();

      }

    };

    /// \brief \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 {

      typedef DagShortestPath< Graph, LengthMap, DefPredMapTraits<T> > Create;

    };

    template <class T>

    struct DefDistMapTraits : public Traits {

      typedef T DistMap;

      static DistMap *createDistMap(const Graph& graph) {

	throw UninitializedParameter();

      }

    };

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

    /// type

    ///

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

    ///

    template <class T>

    struct DefDistMap 

      : public DagShortestPath< Graph, LengthMap, DefDistMapTraits<T> > {

      typedef DagShortestPath< Graph, LengthMap, DefDistMapTraits<T> > Create;

    };

    template <class T>

    struct DefOperationTraitsTraits : public Traits {

      typedef T OperationTraits;

    };

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

    /// OperationTraits type

    ///

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

    /// type

    template <class T>

    struct DefOperationTraits

      : public DagShortestPath< Graph, LengthMap, DefOperationTraitsTraits<T> > {

      typedef DagShortestPath< Graph, LengthMap, DefOperationTraitsTraits<T> >

      Create;

    };

    ///@}

  protected:

    DagShortestPath() {}

  public:      

    /// \brief Constructor.

    ///

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

    /// \param _length the length map used by the algorithm.

    DagShortestPath(const Graph& _graph, const LengthMap& _length) :

      graph(&_graph), length(&_length),

      _pred(0), local_pred(false),

      _dist(0), local_dist(false){}

    /// \brief Constructor with node weight map.

    ///

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

    /// \param _length the length map used by the algorithm.

    /// The NodeMap _length will be used as the weight map.

    /// Each edge will have the weight of its target node.

    DagShortestPath(const Graph& _graph, const WeightMap& _length) :

      graph(&_graph),

      _pred(0), local_pred(false),

      _dist(0), local_dist(false){

      length=new LengthMap(_graph);

      _dist=new DistMap(_graph);

      for(EdgeIt eit(_graph);eit!=INVALID;++eit)

	(const_cast<LengthMap*>(length))->set(eit,_length[_graph.target(eit)]);

      }

    ///Destructor.

    ~DagShortestPath() {

      if(local_pred) delete _pred;

      if(local_dist) delete _dist;

    }

    /// \brief Sets the length map.

    ///

    /// Sets the length map.

    /// \return \c (*this)

    DagShortestPath &lengthMap(const LengthMap &m) {

      length = &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 \c (*this)

    DagShortestPath &predMap(PredMap &m) {

      if(local_pred) {

	delete _pred;

	local_pred=false;

      }

      _pred = &m;

      return *this;

    }

    /// \brief Sets the map storing the distances calculated by the algorithm.

    ///

    /// Sets the map storing the distances 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 \c (*this)

    DagShortestPath &distMap(DistMap &m) {

      if(local_dist) {

	delete _dist;

	local_dist=false;

      }

      _dist = &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 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.

    /// Some functions have an alternative form (\ref checkedInit(...),

    /// \ref checkedRun(...)) which also verifies if the graph given in the

    /// constructor is a dag.

    ///@{

    /// \brief Initializes the internal data structures.

    ///

    /// Initializes the internal data structures.

    void init(const Value value = OperationTraits::infinity()) {

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

      _process_step=0;

      NodeOrderMap node_order(*graph);

      topologicalSort(*graph,node_order);

      _node_order.resize(countNodes(*graph),INVALID);

      create_maps();

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

        _node_order[node_order[it]]=it;

        _pred->set(it, INVALID);

        _dist->set(it, value);

      }

    }

    /// \brief Initializes the internal data structures

    /// with a given topological sort (NodeMap).

    ///

    /// Initializes the internal data structures

    /// with a given topological sort (NodeMap).

    void init(const typename Graph::template NodeMap<int>& node_order,

         const Value value = OperationTraits::infinity()) {

      _process_step=0;

      _node_order.resize(countNodes(*graph),INVALID);

      create_maps();

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

        _node_order[node_order[it]]=it;

        _pred->set(it, INVALID);

        _dist->set(it, value);

      }

    }

    /// \brief Initializes the internal data structures

    /// with a given topological sort (std::vector).

    ///

    /// Initializes the internal data structures

    /// with a given topological sort (std::vector).

    void init(const std::vector<Node>& node_order,

        const Value value = OperationTraits::infinity()) {

      _process_step=0;

      _node_order=node_order;

      create_maps();

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

        _pred->set(it, INVALID);

        _dist->set(it, value);

      }

    }

    /// \brief Initializes the internal data structures. It also checks if the graph is dag.

    ///

    /// Initializes the internal data structures. It also checks if the graph is dag.

    /// \return true if the graph (given in the constructor) is dag, false otherwise.

    bool checkedInit(const Value value = OperationTraits::infinity()) {

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

      NodeOrderMap node_order(*graph);

      if(!checkedTopologicalSort(*graph,node_order))return false;

      init(node_order,value);

      return true;

    }

    /// \brief Initializes the internal data structures with a given

    /// topological sort (NodeMap). It also checks if the graph is dag.

    ///

    /// Initializes the internal data structures with a given

    /// topological sort (NodeMap). It also checks if the graph is dag.

    /// \return true if the graph (given in the constructor) is dag, false otherwise.

    bool checkedInit(const typename Graph::template NodeMap<int>& node_order, 

                     const Value value = OperationTraits::infinity()) {

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

        for(OutEdgeIt oeit(*graph,it);oeit!=INVALID;++oeit){

          if(node_order[graph->target(oeit)]<node_order[it])return false;

        }

      }

      init(node_order,value);

      return true;

    }

    /// \brief Initializes the internal data structures with a given

    /// topological sort (std::vector). It also checks if the graph is dag.

    ///

    /// Initializes the internal data structures with a given

    /// topological sort (std::vector). It also checks if the graph is dag.

    /// \return true if the graph (given in the constructor) is dag, false otherwise.

    bool checkedInit(const std::vector<Node>& node_order, 

                     const Value value = OperationTraits::infinity()) {

      typedef typename Graph::template NodeMap<bool> BoolNodeMap;

      BoolNodeMap _processed(*graph,false);

      for(unsigned int i=0;i<_node_order.size();++i){

        _processed[node_order[i]]=true;

        for(OutEdgeIt oeit(*graph,node_order[i]);oeit!=INVALID;++oeit){

          if(_processed[graph->target(oeit)])return false;

        }

      }

      init(node_order,value);

      return true;

    }

    /// \brief Adds a new source node.

    ///

    /// The optional second parameter is the initial distance of the node.

    /// It just sets the distance of the node to the given value.

    void addSource(Node source, Value dst = OperationTraits::zero()) {

      if((*_dist)[source] != dst){

        _dist->set(source, dst);

      }

    }

    /// \brief Executes one step from the dag shortest path algorithm.

    ///

    /// If the algoritm calculated the distances in the previous step 

    /// strictly for all at most k length paths then it will calculate the 

    /// distances strictly for all at most k + 1 length paths. With k

    /// iteration this function calculates the at most k length paths.

    ///\pre the queue is not empty

    ///\return the currently processed node

    Node processNextNode() {

      if(reached(_node_order[_process_step])){

        for (OutEdgeIt it(*graph, _node_order[_process_step]); it != INVALID; ++it) {

	  Node target = graph->target(it);

	  Value relaxed =

	    OperationTraits::plus((*_dist)[_node_order[_process_step]], (*length)[it]);

	  if (OperationTraits::less(relaxed, (*_dist)[target])) {

	    _pred->set(target, it);

	    _dist->set(target, relaxed);

	  }

        }

      }

      ++_process_step;

      return _node_order[_process_step-1];

    }

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

    ///to be processed in the queue

    ///

    ///Returns \c false if there are nodes

    ///to be processed in the queue

    bool emptyQueue() { return _node_order.size()-1==_process_step; }

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

    ///

    ///Returns the number of the nodes to be processed in the queue.

    int queueSize() { return _node_order.size()-1-_process_step; }

    /// \brief 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 %DagShortestPath algorithm from the root node(s)

    /// in order to compute the shortest path to each node. The algorithm 

    /// computes 

    /// - The shortest path tree.

    /// - The distance of each node from the root(s).

    void start() {

      while(!emptyQueue()) {

	processNextNode();

      }

    }

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

    ///    

    /// This method runs the %DagShortestPath algorithm from a root node \c s

    /// in order to compute the shortest path to each node. The algorithm 

    /// computes

    /// - The shortest path tree.

    /// - The distance of each node from the root.

    ///

    /// \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 %DagShortestPath algorithm from node \c s.

    /// It also checks if the graph is a dag.

    ///    

    /// This method runs the %DagShortestPath algorithm from a root node \c s

    /// in order to compute the shortest path to each node. The algorithm 

    /// computes

    /// - The shortest path tree.

    /// - The distance of each node from the root.

    /// The algorithm checks if the graph given int the constructor is a dag.

    bool checkedRun(Node s) {

      if(!checkedInit())return false;

      addSource(s);

      start();

      return true;

    }

    ///@}

    /// \name Query Functions

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

    /// functions.\n

    /// Before the use of these functions,

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

    ///@{

    typedef PredMapPath<Graph, PredMap> Path;

    ///Gives back the shortest path.

    ///Gives back the shortest path.

    ///\pre The \c t should be reachable from the source.

    Path path(Node t) 

    {

      return Path(*graph, *_pred, t);

    }

    /// \brief The distance of a node from the root.

    ///

    /// Returns the distance of a node from the root.

    /// \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 dist(Node v) const { return (*_dist)[v]; }

    /// \brief Returns the 'previous edge' of the shortest path tree.

    ///

    /// For a node \c v it returns the 'previous edge' of the shortest path 

    /// tree, i.e. it returns the last edge of a shortest path from the root 

    /// to \c v. It is \ref INVALID if \c v is unreachable from the root or 

    /// if \c v=s. The shortest path tree used here is equal to the shortest 

    /// path tree used in \ref predNode(). 

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

    /// this function.

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

    /// \brief Returns the 'previous node' of the shortest path tree.

    ///

    /// For a node \c v it returns the 'previous node' of the shortest path 

    /// tree, i.e. it returns the last but one node from a shortest path from 

    /// the root to \c /v. It is INVALID if \c v is unreachable from the root 

    /// or if \c v=s. The shortest path tree used here is equal to the 

    /// shortest path tree used in \ref predEdge().  \pre \ref run() 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 distances.

    ///

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

    /// be called before using this function.

    const DistMap &distMap() const { return *_dist;}

    /// \brief Returns a reference to the shortest path tree map.

    ///

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

    /// shortest path tree.

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

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

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

    ///

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

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

    ///

    bool reached(Node v) { return (*_dist)[v] != OperationTraits::infinity(); }

    ///@}

  };

  /// \brief Default traits class of DagShortestPath function.

  ///

  /// Default traits class of DagShortestPath function.

  /// \param _Graph Graph type.

  /// \param _LengthMap Type of length map.

  template <typename _Graph, typename _LengthMap>

  struct DagShortestPathWizardDefaultTraits {

    /// \brief The graph type the algorithm runs on. 

    typedef _Graph Graph;

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

    ///

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

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

    typedef _LengthMap LengthMap;

    /// \brief The value type of the length map.

    typedef typename _LengthMap::Value Value;

    /// \brief Operation traits for dag shortest path algorithm.

    ///

    /// It defines the infinity type on the given Value type

    /// and the used operation.

    /// \see DagShortestPathDefaultOperationTraits

    typedef DagShortestPathDefaultOperationTraits<Value> OperationTraits;

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

    /// edges of the shortest paths.

    /// 

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

    /// edges of the shortest paths.

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

    typedef NullMap <typename _Graph::Node,typename _Graph::Edge> PredMap;

    /// \brief Instantiates a PredMap.

    /// 

    /// This function instantiates a \ref PredMap. 

    static PredMap *createPredMap(const _Graph &) {

      return new PredMap();

    }

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

    ///

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

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

    typedef NullMap<typename Graph::Node, Value> DistMap;

    /// \brief Instantiates a DistMap.

    ///

    /// This function instantiates a \ref DistMap. 

    static DistMap *createDistMap(const _Graph &) {

      return new DistMap();

    }

  };

  /// \brief Default traits used by \ref DagShortestPathWizard

  ///

  /// To make it easier to use DagShortestPath algorithm

  /// we have created a wizard class.

  /// This \ref DagShortestPathWizard class needs default traits,

  /// as well as the \ref DagShortestPath class.

  /// The \ref DagShortestPathWizardBase is a class to be the default traits of the

  /// \ref DagShortestPathWizard class.

  /// \todo More named parameters are required...

  template<class _Graph,class _LengthMap>

  class DagShortestPathWizardBase 

    : public DagShortestPathWizardDefaultTraits<_Graph,_LengthMap> {

    typedef DagShortestPathWizardDefaultTraits<_Graph,_LengthMap> Base;

  protected:

    /// Type of the nodes in the graph.

    typedef typename Base::Graph::Node Node;

    /// Pointer to the underlying graph.

    void *_graph;

    /// Pointer to the length map

    void *_length;

    ///Pointer to the map of predecessors edges.

    void *_pred;

    ///Pointer to the map of distances.

    void *_dist;

    ///Pointer to the source node.

    Node _source;

    public:

    /// Constructor.

    /// This constructor does not require parameters, therefore it initiates

    /// all of the attributes to default values (0, INVALID).

    DagShortestPathWizardBase() : _graph(0), _length(0), _pred(0),

			   _dist(0), _source(INVALID) {}

    /// Constructor.

    /// This constructor requires some parameters,

    /// listed in the parameters list.

    /// Others are initiated to 0.

    /// \param graph is the initial value of  \ref _graph

    /// \param length is the initial value of  \ref _length

    /// \param source is the initial value of  \ref _source

    DagShortestPathWizardBase(const _Graph& graph, 

			  const _LengthMap& length, 

			  Node source = INVALID) :

      _graph((void *)&graph), _length((void *)&length), _pred(0),

      _dist(0), _source(source) {}

  };

  /// A class to make the usage of DagShortestPath algorithm easier

  /// This class is created to make it easier to use DagShortestPath algorithm.

  /// It uses the functions and features of the plain \ref DagShortestPath,

  /// but it is much simpler to use it.

  ///

  /// Simplicity means that the way to change the types defined

  /// in the traits class is based on functions that returns the new class

  /// and not on templatable built-in classes.

  /// When using the plain \ref DagShortestPath

  /// the new class with the modified type comes from

  /// the original class by using the ::

  /// operator. In the case of \ref DagShortestPathWizard only

  /// a function have to be called and it will

  /// return the needed class.

  ///

  /// It does not have own \ref run method. When its \ref run method is called

  /// it initiates a plain \ref DagShortestPath class, and calls the \ref 

  /// DagShortestPath::run() method of it.

  template<class _Traits>

  class DagShortestPathWizard : public _Traits {

    typedef _Traits Base;

    ///The type of the underlying graph.

    typedef typename _Traits::Graph Graph;

    typedef typename Graph::Node Node;

    typedef typename Graph::NodeIt NodeIt;

    typedef typename Graph::Edge Edge;

    typedef typename Graph::OutEdgeIt EdgeIt;

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

    typedef typename _Traits::LengthMap LengthMap;

    ///The type of the length of the edges.

    typedef typename LengthMap::Value Value;

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

    ///edges of the shortest paths.

    typedef typename _Traits::PredMap PredMap;

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

    typedef typename _Traits::DistMap DistMap;

  public:

    /// Constructor.

    DagShortestPathWizard() : _Traits() {}

    /// \brief Constructor that requires parameters.

    ///

    /// Constructor that requires parameters.

    /// These parameters will be the default values for the traits class.

    DagShortestPathWizard(const Graph& graph, const LengthMap& length, 

		      Node source = INVALID) 

      : _Traits(graph, length, source) {}

    /// \brief Copy constructor

    DagShortestPathWizard(const _Traits &b) : _Traits(b) {}

    ~DagShortestPathWizard() {}

    /// \brief Runs DagShortestPath algorithm from a given node.

    ///    

    /// Runs DagShortestPath algorithm from a given node.

    /// The node can be given by the \ref source function.

    void run() {

      if(Base::_source == INVALID) throw UninitializedParameter();

      DagShortestPath<Graph,LengthMap,_Traits> 

	bf(*(Graph*)Base::_graph, *(LengthMap*)Base::_length);

      if (Base::_pred) bf.predMap(*(PredMap*)Base::_pred);

      if (Base::_dist) bf.distMap(*(DistMap*)Base::_dist);

      bf.run(Base::_source);

    }

    /// \brief Runs DagShortestPath algorithm from the given node.

    ///

    /// Runs DagShortestPath algorithm from the given node.

    /// \param source is the given source.

    void run(Node source) {

      Base::_source = source;

      run();

    }

    template<class T>

    struct DefPredMapBase : public Base {

      typedef T PredMap;

      static PredMap *createPredMap(const Graph &) { return 0; };

      DefPredMapBase(const _Traits &b) : _Traits(b) {}

    };

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

    ///function for setting PredMap type

    ///

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

    ///function for setting PredMap type

    ///

    template<class T>

    DagShortestPathWizard<DefPredMapBase<T> > predMap(const T &t) 

    {