RhodeCode

LEMON provides flexible data structures to work with paths.

All of them have similar interfaces and they can be copied easily with

assignment operators and copy constructors. This makes it easy and

efficient to have e.g. the Dijkstra algorithm to store its result in

any kind of path structure.

\sa \ref concepts::Path "Path concept"

*/

/**

@defgroup heaps Heap Structures

@ingroup datas

\brief %Heap structures implemented in LEMON.

This group contains the heap structures implemented in LEMON.

LEMON provides several heap classes. They are efficient implementations

of the abstract data type \e priority \e queue. They store items with

specified values called \e priorities in such a way that finding and

removing the item with minimum priority are efficient.

The basic operations are adding and erasing items, changing the priority

of an item, etc.

Heaps are crucial in several algorithms, such as Dijkstra and Prim.

The heap implementations have the same interface, thus any of them can be

used easily in such algorithms.

\sa \ref concepts::Heap "Heap concept"

*/

/**

@defgroup matrices Matrices

@ingroup datas

\brief Two dimensional data storages implemented in LEMON.

This group contains two dimensional data storages implemented in LEMON.

*/

/**

@defgroup auxdat Auxiliary Data Structures

@ingroup datas

\brief Auxiliary data structures implemented in LEMON.

This group contains some data structures implemented in LEMON in

order to make it easier to implement combinatorial algorithms.

*/

/**

@defgroup algs Algorithms

\brief This group contains the several algorithms

implemented in LEMON.

This group contains the several algorithms

implemented in LEMON.

*/

/**

@defgroup search Graph Search

@ingroup algs

\brief Common graph search algorithms.

This group contains the common graph search algorithms, namely

\e breadth-first \e search (BFS) and \e depth-first \e search (DFS).

*/

/**

@defgroup shortest_path Shortest Path Algorithms

@ingroup algs

\brief Algorithms for finding shortest paths.

This group contains the algorithms for finding shortest paths in digraphs.

 - \ref Dijkstra algorithm for finding shortest paths from a source node

   when all arc lengths are non-negative.

 - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths

   from a source node when arc lenghts can be either positive or negative,

   but the digraph should not contain directed cycles with negative total

   length.

 - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms

   for solving the \e all-pairs \e shortest \e paths \e problem when arc

   lenghts can be either positive or negative, but the digraph should

   not contain directed cycles with negative total length.

 - \ref Suurballe A successive shortest path algorithm for finding

   arc-disjoint paths between two nodes having minimum total length.

*/

/**

@defgroup max_flow Maximum Flow Algorithms

@ingroup algs

\brief Algorithms for finding maximum flows.

This group contains the algorithms for finding maximum flows and

feasible circulations.

The \e maximum \e flow \e problem is to find a flow of maximum value between

a single source and a single target. Formally, there is a \f$G=(V,A)\f$

digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and

\f$s, t \in V\f$ source and target nodes.

A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the

following optimization problem.

\f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]

\f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)

    \quad \forall u\in V\setminus\{s,t\} \f]

\f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]

LEMON contains several algorithms for solving maximum flow problems:

- \ref EdmondsKarp Edmonds-Karp algorithm.

- \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm.

- \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees.

- \ref GoldbergTarjan Preflow push-relabel algorithm with dynamic trees.

In most cases the \ref Preflow "Preflow" algorithm provides the

fastest method for computing a maximum flow. All implementations

also provide functions to query the minimum cut, which is the dual

problem of maximum flow.

\ref Circulation is a preflow push-relabel algorithm implemented directly 

for finding feasible circulations, which is a somewhat different problem,

but it is strongly related to maximum flow.

For more information, see \ref Circulation.

*/

/**

@defgroup min_cost_flow_algs Minimum Cost Flow Algorithms

@ingroup algs

\brief Algorithms for finding minimum cost flows and circulations.

This group contains the algorithms for finding minimum cost flows and

circulations. For more information about this problem and its dual

solution see \ref min_cost_flow "Minimum Cost Flow Problem".

LEMON contains several algorithms for this problem.

 - \ref NetworkSimplex Primal Network Simplex algorithm with various

   pivot strategies.

 - \ref CostScaling Push-Relabel and Augment-Relabel algorithms based on

   cost scaling.

 - \ref CapacityScaling Successive Shortest %Path algorithm with optional

   capacity scaling.

 - \ref CancelAndTighten The Cancel and Tighten algorithm.

 - \ref CycleCanceling Cycle-Canceling algorithms.

In general NetworkSimplex is the most efficient implementation,

but in special cases other algorithms could be faster.

For example, if the total supply and/or capacities are rather small,

CapacityScaling is usually the fastest algorithm (without effective scaling).

*/

/**

@defgroup min_cut Minimum Cut Algorithms

@ingroup algs

\brief Algorithms for finding minimum cut in graphs.

This group contains the algorithms for finding minimum cut in graphs.

The \e minimum \e cut \e problem is to find a non-empty and non-complete

\f$X\f$ subset of the nodes with minimum overall capacity on

outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a

\f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum

cut is the \f$X\f$ solution of the next optimization problem:

\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}

LEMON contains several algorithms related to minimum cut problems:

- \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut

  in directed graphs.

- \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for

  calculating minimum cut in undirected graphs.

- \ref GomoryHu "Gomory-Hu tree computation" for calculating

  all-pairs minimum cut in undirected graphs.

If you want to find minimum cut just between two distinict nodes,

see the \ref max_flow "maximum flow problem".

*/

/**

@defgroup matching Matching Algorithms

@ingroup algs

\brief Algorithms for finding matchings in graphs and bipartite graphs.

This group contains the algorithms for calculating

matchings in graphs and bipartite graphs. The general matching problem is

finding a subset of the edges for which each node has at most one incident

edge.

There are several different algorithms for calculate matchings in

graphs.  The matching problems in bipartite graphs are generally

easier than in general graphs. The goal of the matching optimization

can be finding maximum cardinality, maximum weight or minimum cost

matching. The search can be constrained to find perfect or

maximum cardinality matching.

The matching algorithms implemented in LEMON:

- \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm

  for calculating maximum cardinality matching in bipartite graphs.

- \ref PrBipartiteMatching Push-relabel algorithm

  for calculating maximum cardinality matching in bipartite graphs.

- \ref MaxWeightedBipartiteMatching

  Successive shortest path algorithm for calculating maximum weighted

  matching and maximum weighted bipartite matching in bipartite graphs.

- \ref MinCostMaxBipartiteMatching

  Successive shortest path algorithm for calculating minimum cost maximum

  matching in bipartite graphs.

- \ref MaxMatching Edmond's blossom shrinking algorithm for calculating

  maximum cardinality matching in general graphs.

- \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating

  maximum weighted matching in general graphs.

- \ref MaxWeightedPerfectMatching

  Edmond's blossom shrinking algorithm for calculating maximum weighted

  perfect matching in general graphs.

\image html bipartite_matching.png

\image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth

*/

/**

@ingroup algs

*/

/**

@defgroup auxalg Auxiliary Algorithms

@ingroup algs

\brief Auxiliary algorithms implemented in LEMON.

This group contains some algorithms implemented in LEMON

in order to make it easier to implement complex algorithms.

*/

/**

@defgroup gen_opt_group General Optimization Tools

\brief This group contains some general optimization frameworks

implemented in LEMON.

This group contains some general optimization frameworks

implemented in LEMON.

*/

/**

@defgroup lp_group Lp and Mip Solvers

@ingroup gen_opt_group

\brief Lp and Mip solver interfaces for LEMON.

This group contains Lp and Mip solver interfaces for LEMON. The

various LP solvers could be used in the same manner with this

interface.

*/

/**

@defgroup lp_utils Tools for Lp and Mip Solvers

@ingroup lp_group

\brief Helper tools to the Lp and Mip solvers.

This group adds some helper tools to general optimization framework

implemented in LEMON.

*/

/**

@defgroup metah Metaheuristics

@ingroup gen_opt_group

\brief Metaheuristics for LEMON library.

This group contains some metaheuristic optimization tools.

*/

/**

@defgroup utils Tools and Utilities

\brief Tools and utilities for programming in LEMON

Tools and utilities for programming in LEMON.

*/

/**

@defgroup gutils Basic Graph Utilities

@ingroup utils

\brief Simple basic graph utilities.

This group contains some simple basic graph utilities.

*/

/**

@defgroup timecount Time Measuring and Counting

@ingroup misc

\brief Simple tools for measuring the performance of algorithms.

This group contains simple tools for measuring the performance

of algorithms.

*/

/**

@defgroup exceptions Exceptions

@ingroup utils

\brief Exceptions defined in LEMON.

This group contains the exceptions defined in LEMON.

*/

/**

@defgroup io_group Input-Output

\brief Graph Input-Output methods

This group contains the tools for importing and exporting graphs

and graph related data. Now it supports the \ref lgf-format

"LEMON Graph Format", the \c DIMACS format and the encapsulated

postscript (EPS) format.

*/

/**

@defgroup lemon_io LEMON Graph Format

@ingroup io_group

\brief Reading and writing LEMON Graph Format.

This group contains methods for reading and writing

\ref lgf-format "LEMON Graph Format".

*/

/**

@defgroup eps_io Postscript Exporting

@ingroup io_group

\brief General \c EPS drawer and graph exporter

This group contains general \c EPS drawing methods and special

graph exporting tools.

*/

/**

@ingroup io_group

\brief Read and write files in DIMACS format

Tools to read a digraph from or write it to a file in DIMACS format data.

*/

/**

@defgroup nauty_group NAUTY Format

@ingroup io_group

\brief Read \e Nauty format

Tool to read graphs from \e Nauty format data.

*/

/**

@defgroup concept Concepts

\brief Skeleton classes and concept checking classes

This group contains the data/algorithm skeletons and concept checking

classes implemented in LEMON.

The purpose of the classes in this group is fourfold.

- These classes contain the documentations of the %concepts. In order

  to avoid document multiplications, an implementation of a concept

  simply refers to the corresponding concept class.

- These classes declare every functions, <tt>typedef</tt>s etc. an

  implementation of the %concepts should provide, however completely

  without implementations and real data structures behind the

  interface. On the other hand they should provide nothing else. All

  the algorithms working on a data structure meeting a certain concept

  should compile with these classes. (Though it will not run properly,

  of course.) In this way it is easily to check if an algorithm

  doesn't use any extra feature of a certain implementation.

- The concept descriptor classes also provide a <em>checker class</em>

  that makes it possible to check whether a certain implementation of a

  concept indeed provides all the required features.

- Finally, They can serve as a skeleton of a new implementation of a concept.

*/

/**

@defgroup graph_concepts Graph Structure Concepts

@ingroup concept

\brief Skeleton and concept checking classes for graph structures

This group contains the skeletons and concept checking classes of LEMON's

graph structures and helper classes used to implement these.

*/

/**

@defgroup map_concepts Map Concepts

@ingroup concept

\brief Skeleton and concept checking classes for maps

This group contains the skeletons and concept checking classes of maps.

*/

/**

\anchor demoprograms

@defgroup demos Demo Programs

Some demo programs are listed here. Their full source codes can be found in

the \c demo subdirectory of the source tree.

In order to compile them, use the <tt>make demo</tt> or the

<tt>make check</tt> commands.

*/

}

 *

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

 *

 * Copyright (C) 2003-2009

 * 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_BFS_H

#define LEMON_BFS_H

///\ingroup search

///\file

///\brief BFS algorithm.

#include <lemon/list_graph.h>

#include <lemon/bits/path_dump.h>

#include <lemon/core.h>

#include <lemon/error.h>

#include <lemon/maps.h>

#include <lemon/path.h>

namespace lemon {

  ///Default traits class of Bfs class.

  ///Default traits class of Bfs class.

  ///\tparam GR Digraph type.

  template<class GR>

  struct BfsDefaultTraits

  {

    ///The type of the digraph the algorithm runs on.

    typedef GR Digraph;

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

    ///arcs of the shortest paths.

    ///

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

    ///arcs of the shortest paths.

    typedef typename Digraph::template NodeMap<typename Digraph::Arc> PredMap;

    ///Instantiates a \c PredMap.

    ///This function instantiates a \ref PredMap.

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

    ///\ref PredMap.

    static PredMap *createPredMap(const Digraph &g)

    {

      return new PredMap(g);

    }

    ///The type of the map that indicates which nodes are processed.

    ///The type of the map that indicates which nodes are processed.

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

    ///Instantiates a \c ProcessedMap.

    ///This function instantiates a \ref ProcessedMap.

    ///\param g is the digraph, to which

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

#ifdef DOXYGEN

    static ProcessedMap *createProcessedMap(const Digraph &g)

#else

    static ProcessedMap *createProcessedMap(const Digraph &)

#endif

    {

      return new ProcessedMap();

    }

    ///The type of the map that indicates which nodes are reached.

    ///The type of the map that indicates which nodes are reached.

    typedef typename Digraph::template NodeMap<bool> ReachedMap;

    ///Instantiates a \c ReachedMap.

    ///This function instantiates a \ref ReachedMap.

    ///\param g is the digraph, to which

    ///we would like to define the \ref ReachedMap.

    static ReachedMap *createReachedMap(const Digraph &g)

    {

      return new ReachedMap(g);

    }

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

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

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

    ///Instantiates a \c DistMap.

    ///This function instantiates a \ref DistMap.

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

    ///\ref DistMap.

    static DistMap *createDistMap(const Digraph &g)

    {

      return new DistMap(g);

    }

  };

  ///%BFS algorithm class.

  ///\ingroup search

  ///This class provides an efficient implementation of the %BFS algorithm.

  ///

  ///There is also a \ref bfs() "function-type interface" for the BFS

  ///algorithm, which is convenient in the simplier cases and it can be

  ///used easier.

  ///

  ///\tparam GR The type of the digraph the algorithm runs on.

  ///The default type is \ref ListDigraph.

#ifdef DOXYGEN

  template <typename GR,

            typename TR>

#else

  template <typename GR=ListDigraph,

            typename TR=BfsDefaultTraits<GR> >

#endif

  class Bfs {

  public:

    ///The type of the digraph the algorithm runs on.

    typedef typename TR::Digraph Digraph;

    ///\brief The type of the map that stores the predecessor arcs of the

    ///shortest paths.

    typedef typename TR::PredMap PredMap;

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

    typedef typename TR::DistMap DistMap;

    ///The type of the map that indicates which nodes are reached.

    typedef typename TR::ReachedMap ReachedMap;

    ///The type of the map that indicates which nodes are processed.

    typedef typename TR::ProcessedMap ProcessedMap;

    ///The type of the paths.

    typedef PredMapPath<Digraph, PredMap> Path;

    //Creates the maps if necessary.

    void create_maps()

    {

      if(!_pred) {

        local_pred = true;

        _pred = Traits::createPredMap(*G);

      }

      if(!_dist) {

        local_dist = true;

        _dist = Traits::createDistMap(*G);

      }

      if(!_reached) {

        local_reached = true;

        _reached = Traits::createReachedMap(*G);

      }

      if(!_processed) {

        local_processed = true;

        _processed = Traits::createProcessedMap(*G);

      }

    }

  protected:

    Bfs() {}

  public:

    typedef Bfs Create;

    ///\name Named Template Parameters

    ///@{

    template <class T>

    struct SetPredMapTraits : public Traits {

      typedef T PredMap;

      static PredMap *createPredMap(const Digraph &)

      {

        LEMON_ASSERT(false, "PredMap is not initialized");

        return 0; // ignore warnings

      }

    };

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

    ///\c PredMap type.

    ///

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

    ///\c PredMap type.

    template <class T>

    struct SetPredMap : public Bfs< Digraph, SetPredMapTraits<T> > {

      typedef Bfs< Digraph, SetPredMapTraits<T> > Create;

    };

    template <class T>

    struct SetDistMapTraits : public Traits {

      typedef T DistMap;

      static DistMap *createDistMap(const Digraph &)

      {

        LEMON_ASSERT(false, "DistMap is not initialized");

        return 0; // ignore warnings

      }

    };

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

    ///\c DistMap type.

    ///

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

    ///\c DistMap type.

    template <class T>

    struct SetDistMap : public Bfs< Digraph, SetDistMapTraits<T> > {

      typedef Bfs< Digraph, SetDistMapTraits<T> > Create;

    };

    template <class T>

    struct SetReachedMapTraits : public Traits {

      typedef T ReachedMap;

      static ReachedMap *createReachedMap(const Digraph &)

      {

        LEMON_ASSERT(false, "ReachedMap is not initialized");

        return 0; // ignore warnings

      }

    };

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

    ///\c ReachedMap type.

    ///

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

    ///\c ReachedMap type.

    template <class T>

    struct SetReachedMap : public Bfs< Digraph, SetReachedMapTraits<T> > {

      typedef Bfs< Digraph, SetReachedMapTraits<T> > Create;

    };

    template <class T>

    struct SetProcessedMapTraits : public Traits {

      typedef T ProcessedMap;

      static ProcessedMap *createProcessedMap(const Digraph &)

      {

        LEMON_ASSERT(false, "ProcessedMap is not initialized");

        return 0; // ignore warnings

      }

    };

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

    ///\c ProcessedMap type.

    ///

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

    ///\c ProcessedMap type.

    template <class T>

    struct SetProcessedMap : public Bfs< Digraph, SetProcessedMapTraits<T> > {

      typedef Bfs< Digraph, SetProcessedMapTraits<T> > Create;

    };

    struct SetStandardProcessedMapTraits : public Traits {

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

      static ProcessedMap *createProcessedMap(const Digraph &g)

      {

        return new ProcessedMap(g);

        return 0; // ignore warnings

      }

    };

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

    ///\c ProcessedMap type to be <tt>Digraph::NodeMap<bool></tt>.

    ///

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

    ///\c ProcessedMap type to be <tt>Digraph::NodeMap<bool></tt>.

    ///If you don't set it explicitly, it will be automatically allocated.

    struct SetStandardProcessedMap :

      public Bfs< Digraph, SetStandardProcessedMapTraits > {

      typedef Bfs< Digraph, SetStandardProcessedMapTraits > Create;

    };

    ///@}

  public:

    ///Constructor.

    ///Constructor.

    ///\param g The digraph the algorithm runs on.

    Bfs(const Digraph &g) :

      G(&g),

      _pred(NULL), local_pred(false),

      _dist(NULL), local_dist(false),

      _reached(NULL), local_reached(false),

      _processed(NULL), local_processed(false)

    { }

    ///Destructor.

    ~Bfs()

    {

      if(local_pred) delete _pred;

      if(local_dist) delete _dist;

      if(local_reached) delete _reached;

      if(local_processed) delete _processed;

    }

        local_reached=false;

      }

      _reached = &m;

      return *this;

    }

    ///Sets the map that indicates which nodes are processed.

    ///Sets the map that indicates which nodes are processed.

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

    ///or \ref init(), an instance will be allocated automatically.

    ///The destructor deallocates this automatically allocated map,

    ///of course.

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

    Bfs &processedMap(ProcessedMap &m)

    {

      if(local_processed) {

        delete _processed;

        local_processed=false;

      }

      _processed = &m;

      return *this;

    }

    ///Sets the map that stores the distances of the nodes.

    ///Sets the map that stores the distances of the nodes calculated by

    ///the algorithm.

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

    ///or \ref init(), an instance will be allocated automatically.

    ///The destructor deallocates this automatically allocated map,

    ///of course.

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

    Bfs &distMap(DistMap &m)

    {

      if(local_dist) {

        delete _dist;

        local_dist=false;

      }

      _dist = &m;

      return *this;

    }

  public:

    ///\name Execution Control

    ///The simplest way to execute the BFS algorithm is to use one of the

    ///member functions called \ref run(Node) "run()".\n

    ///\ref addSource(). Finally the actual path computation can be

    ///performed with one of the \ref start() functions.

    ///@{

    ///\brief Initializes the internal data structures.

    ///

    ///Initializes the internal data structures.

    void init()

    {

      create_maps();

      _queue.resize(countNodes(*G));

      _queue_head=_queue_tail=0;

      _curr_dist=1;

      for ( NodeIt u(*G) ; u!=INVALID ; ++u ) {

        _pred->set(u,INVALID);

        _reached->set(u,false);

        _processed->set(u,false);

      }

    }

    ///Adds a new source node.

    ///Adds a new source node to the set of nodes to be processed.

    ///

    void addSource(Node s)

    {

      if(!(*_reached)[s])

        {

          _reached->set(s,true);

          _pred->set(s,INVALID);

          _dist->set(s,0);

          _queue[_queue_head++]=s;

          _queue_next_dist=_queue_head;

        }

    }

    ///Processes the next node.

    ///Processes the next node.

    ///

    ///\return The processed node.

    ///

    ///\pre The queue must not be empty.

    Node processNextNode()

    {

      if(_queue_tail==_queue_next_dist) {

        _curr_dist++;

    ///  b.start(t);

    ///\endcode

    bool run(Node s,Node t) {

      init();

      addSource(s);

      start(t);

      return reached(t);

    }

    ///Runs the algorithm to visit all nodes in the digraph.

    ///This method runs the %BFS algorithm in order to

    ///compute the shortest path to each node.

    ///

    ///The algorithm computes

    ///- the shortest path tree (forest),

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

    ///

    ///\note <tt>b.run(s)</tt> is just a shortcut of the following code.

    ///\code

    ///  b.init();

    ///  for (NodeIt n(gr); n != INVALID; ++n) {

    ///    if (!b.reached(n)) {

    ///      b.addSource(n);

    ///      b.start();

    ///    }

    ///  }

    ///\endcode

    void run() {

      init();

      for (NodeIt n(*G); n != INVALID; ++n) {

        if (!reached(n)) {

          addSource(n);

          start();

        }

      }

    }

    ///@}

    ///\name Query Functions

    ///The results of the BFS algorithm can be obtained using these

    ///functions.\n

    ///Either \ref run(Node) "run()" or \ref start() should be called

    ///before using them.

    ///@{

    ///

    ///\warning \c t should be reached from the root(s).

    ///

    ///\pre Either \ref run(Node) "run()" or \ref init()

    ///must be called before using this function.

    Path path(Node t) const { return Path(*G, *_pred, t); }

    ///

    ///\warning If node \c v is not reached from the root(s), then

    ///the return value of this function is undefined.

    ///

    ///\pre Either \ref run(Node) "run()" or \ref init()

    ///must be called before using this function.

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

    ///This function returns the 'previous arc' of the shortest path

    ///tree for the node \c v, i.e. it returns the last arc of a

    ///shortest path from a root to \c v. It is \c INVALID if \c v

    ///is not reached from the root(s) or if \c v is a root.

    ///

    ///The shortest path tree used here is equal to the shortest path

    ///

    ///\pre Either \ref run(Node) "run()" or \ref init()

    ///must be called before using this function.

    Arc predArc(Node v) const { return (*_pred)[v];}

    ///This function returns the 'previous node' of the shortest path

    ///tree for the node \c v, i.e. it returns the last but one node

    ///if \c v is not reached from the root(s) or if \c v is a root.

    ///

    ///The shortest path tree used here is equal to the shortest path

    ///

    ///\pre Either \ref run(Node) "run()" or \ref init()

    ///must be called before using this function.

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

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

    ///\brief Returns a const reference to the node map that stores the

    /// distances of the nodes.

    ///

    ///Returns a const reference to the node map that stores the distances

    ///of the nodes calculated by the algorithm.

    ///

    ///\pre Either \ref run(Node) "run()" or \ref init()

    ///must be called before using this function.

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

    ///\brief Returns a const reference to the node map that stores the

    ///predecessor arcs.

    ///

    ///Returns a const reference to the node map that stores the predecessor

    ///

    ///\pre Either \ref run(Node) "run()" or \ref init()

    ///must be called before using this function.

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

    ///Returns \c true if \c v is reached from the root(s).

    ///

    ///\pre Either \ref run(Node) "run()" or \ref init()

    ///must be called before using this function.

    bool reached(Node v) const { return (*_reached)[v]; }

    ///@}

  };

  ///Default traits class of bfs() function.

  ///Default traits class of bfs() function.

  ///\tparam GR Digraph type.

  template<class GR>

  struct BfsWizardDefaultTraits

  {

    ///The type of the digraph the algorithm runs on.

    typedef GR Digraph;

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

    ///arcs of the shortest paths.