RhodeCode

\code

ALL_UPPER_CASE_WITH_UNDERSCORES

\endcode

\subsection cs-loc-var Class and instance member variables, auto variables

The names of class and instance member variables and auto variables

(=variables used locally in methods) should look like the following.

\code

all_lower_case_with_underscores

\endcode

\subsection pri-loc-var Private member variables

\code

\endcode

\subsection cs-excep Exceptions

When writing exceptions please comply the following naming conventions.

\code

ClassNameEndsWithException

\endcode

or

\code

ClassNameEndsWithError

\endcode

This group contains the algorithms for finding minimum cost flows and

circulations \ref amo93networkflows. 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 dantzig63linearprog, \ref kellyoneill91netsimplex.

 - \ref CostScaling Cost Scaling algorithm based on push/augment and

   relabel operations \ref goldberg90approximation, \ref goldberg97efficient,

   \ref bunnagel98efficient.

 - \ref CapacityScaling Capacity Scaling algorithm based on the successive

   shortest path method \ref edmondskarp72theoretical.

 - \ref CycleCanceling Cycle-Canceling algorithms, two of which are

   strongly polynomial \ref klein67primal, \ref goldberg89cyclecanceling.

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

*/

/**

@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:

This problem has an important connection to \e conservative \e length

\e functions, too. A length function on the arcs of a digraph is called

conservative if and only if there is no directed cycle of negative total

length. For an arbitrary length function, the negative of the minimum

cycle mean is the smallest \f$\epsilon\f$ value so that increasing the

arc lengths uniformly by \f$\epsilon\f$ results in a conservative length

function.

LEMON contains three algorithms for solving the minimum mean cycle problem:

- \ref KarpMmc Karp's original algorithm \ref amo93networkflows,

  \ref dasdan98minmeancycle.

- \ref HartmannOrlinMmc Hartmann-Orlin's algorithm, which is an improved

  version of Karp's algorithm \ref dasdan98minmeancycle.

- \ref HowardMmc Howard's policy iteration algorithm

  \ref dasdan98minmeancycle.

most efficient one, though the best known theoretical bound on its running

time is exponential.

Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" algorithms

run in time O(ne) and use space O(n<sup>2</sup>+e), but the latter one is

typically faster due to the applied early termination scheme.

*/

/**

@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.

\image latex matching.eps "Min Cost Perfect Matching" width=\textwidth

*/

/**

@defgroup graph_properties Connectivity and Other Graph Properties

@ingroup algs

\brief Algorithms for discovering the graph properties

This group contains the algorithms for discovering the graph properties

like connectivity, bipartiteness, euler property, simplicity etc.

\image html connected_components.png

\image latex connected_components.eps "Connected components" width=\textwidth

*/

/**

@ingroup algs

\brief Algorithms for planarity checking, embedding and drawing

This group contains the algorithms for planarity checking,

embedding and drawing.

\image html planar.png

\image latex planar.eps "Plane graph" width=\textwidth

*/

/**

@defgroup approx_algs Approximation Algorithms

@ingroup algs

\brief Approximation algorithms.

This group contains the approximation and heuristic algorithms

  /// specified, then default values will be used.

  ///

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

  /// \tparam V The number type used for flow amounts, capacity bounds

  /// and supply values in the algorithm. By default, it is \c int.

  /// \tparam C The number type used for costs and potentials in the

  /// algorithm. By default, it is the same as \c V.

  /// \tparam TR The traits class that defines various types used by the

  /// algorithm. By default, it is \ref CapacityScalingDefaultTraits

  /// "CapacityScalingDefaultTraits<GR, V, C>".

  /// In most cases, this parameter should not be set directly,

  /// consider to use the named template parameters instead.

  ///

  /// \warning Both \c V and \c C must be signed number types.

  /// \warning All input data (capacities, supply values, and costs) must

  /// be integer.

#ifdef DOXYGEN

  template <typename GR, typename V, typename C, typename TR>

#else

  template < typename GR, typename V = int, typename C = V,

             typename TR = CapacityScalingDefaultTraits<GR, V, C> >

#endif

  class CapacityScaling

  {

  public:

    /// The type of the digraph

    typedef typename TR::Digraph Digraph;

    /// The type of the flow amounts, capacity bounds and supply values

    typedef typename TR::Value Value;

    /// The type of the arc costs

    typedef typename TR::Cost Cost;

    template<typename SupplyMap>

    CapacityScaling& supplyMap(const SupplyMap& map) {

      for (NodeIt n(_graph); n != INVALID; ++n) {

        _supply[_node_id[n]] = map[n];

      }

      return *this;

    }

    /// \brief Set single source and target nodes and a supply value.

    ///

    /// This function sets a single source node and a single target node

    /// and the required flow value.

    /// If neither this function nor \ref supplyMap() is used before

    /// calling \ref run(), the supply of each node will be set to zero.

    ///

    /// Using this function has the same effect as using \ref supplyMap()

    /// assigned to \c t and all other nodes have zero supply value.

    ///

    /// \param s The source node.

    /// \param t The target node.

    /// \param k The required amount of flow from node \c s to node \c t

    /// (i.e. the supply of \c s and the demand of \c t).

    ///

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

    CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {

      for (int i = 0; i != _node_num; ++i) {

        _supply[i] = 0;

      }

      _supply[_node_id[s]] =  k;

      _supply[_node_id[t]] = -k;

      return *this;

    }

    };

    template <typename Graph>

    struct GraphCopySelector<

      Graph,

      typename enable_if<typename Graph::BuildTag, void>::type>

    {

      template <typename From, typename NodeRefMap, typename EdgeRefMap>

      static void copy(const From& from, Graph &to,

                       NodeRefMap& nodeRefMap, EdgeRefMap& edgeRefMap) {

        to.build(from, nodeRefMap, edgeRefMap);

      }

    };

  }

  ///

  /// This function returns \c true if the given graph is undirected.

#ifdef DOXYGEN

  template <typename GR>

  bool undirected(const GR& g) { return false; }

#else

  template <typename GR>

  typename enable_if<UndirectedTagIndicator<GR>, bool>::type

  undirected(const GR&) {

    return true;

  }

  template <typename GR>

  typename disable_if<UndirectedTagIndicator<GR>, bool>::type

  undirected(const GR&) {

    return false;

  }

  /// \addtogroup min_cost_flow_algs

  /// @{

  /// \brief Implementation of the Cost Scaling algorithm for

  /// finding a \ref min_cost_flow "minimum cost flow".

  ///

  /// \ref CostScaling implements a cost scaling algorithm that performs

  /// push/augment and relabel operations for finding a \ref min_cost_flow

  /// "minimum cost flow" \ref amo93networkflows, \ref goldberg90approximation,

  /// \ref goldberg97efficient, \ref bunnagel98efficient.

  /// It is a highly efficient primal-dual solution method, which

  /// can be viewed as the generalization of the \ref Preflow

  /// "preflow push-relabel" algorithm for the maximum flow problem.

  ///

  /// Most of the parameters of the problem (except for the digraph)

  /// can be given using separate functions, and the algorithm can be

  /// executed using the \ref run() function. If some parameters are not

  /// specified, then default values will be used.

  ///

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

  /// \tparam V The number type used for flow amounts, capacity bounds

  /// and supply values in the algorithm. By default, it is \c int.

  /// \tparam C The number type used for costs and potentials in the

  /// algorithm. By default, it is the same as \c V.

  /// \tparam TR The traits class that defines various types used by the

  /// algorithm. By default, it is \ref CostScalingDefaultTraits

  /// "CostScalingDefaultTraits<GR, V, C>".

  /// In most cases, this parameter should not be set directly,

  /// consider to use the named template parameters instead.

  ///

  /// \warning Both \c V and \c C must be signed number types.

  /// \warning All input data (capacities, supply values, and costs) must

  /// be integer.

  ///

  /// \note %CostScaling provides three different internal methods,

  /// from which the most efficient one is used by default.

  /// For more information, see \ref Method.

#ifdef DOXYGEN

  template <typename GR, typename V, typename C, typename TR>

#else

  template < typename GR, typename V = int, typename C = V,

             typename TR = CostScalingDefaultTraits<GR, V, C> >

#endif

  class CostScaling

  {

  public:

    /// The type of the digraph

    typedef typename TR::Digraph Digraph;

      /// upper bound. It means that the objective function is unbounded

      /// on that arc, however, note that it could actually be bounded

      /// over the feasible flows, but this algroithm cannot handle

      /// these cases.

      UNBOUNDED

    };

    /// \brief Constants for selecting the internal method.

    ///

    /// Enum type containing constants for selecting the internal method

    /// for the \ref run() function.

    ///

    /// \ref CostScaling provides three internal methods that differ mainly

    /// in their base operations, which are used in conjunction with the

    /// relabel operation.

    /// By default, the so called \ref PARTIAL_AUGMENT

    /// the most efficient and the most robust on various test inputs.

    /// However, the other methods can be selected using the \ref run()

    /// function with the proper parameter.

    enum Method {

      /// Local push operations are used, i.e. flow is moved only on one

      /// admissible arc at once.

      PUSH,

      /// Augment operations are used, i.e. flow is moved on admissible

      /// paths from a node with excess to a node with deficit.

      AUGMENT,

      /// Partial augment operations are used, i.e. flow is moved on

      /// admissible paths started from a node with excess, but the

      /// lengths of these paths are limited. This method can be viewed

      /// as a combined version of the previous two operations.

      PARTIAL_AUGMENT

    };

    template<typename SupplyMap>

    CostScaling& supplyMap(const SupplyMap& map) {

      for (NodeIt n(_graph); n != INVALID; ++n) {

        _supply[_node_id[n]] = map[n];

      }

      return *this;

    }

    /// \brief Set single source and target nodes and a supply value.

    ///

    /// This function sets a single source node and a single target node

    /// and the required flow value.

    /// If neither this function nor \ref supplyMap() is used before

    /// calling \ref run(), the supply of each node will be set to zero.

    ///

    /// Using this function has the same effect as using \ref supplyMap()

    /// assigned to \c t and all other nodes have zero supply value.

    ///

    /// \param s The source node.

    /// \param t The target node.

    /// \param k The required amount of flow from node \c s to node \c t

    /// (i.e. the supply of \c s and the demand of \c t).

    ///

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

    CostScaling& stSupply(const Node& s, const Node& t, Value k) {

      for (int i = 0; i != _res_node_num; ++i) {

        _supply[i] = 0;

      }

      _supply[_node_id[s]] =  k;

      _supply[_node_id[t]] = -k;

      return *this;

    }

  /// slower than the scaling algorithms and NetworkSimplex.

  ///

  /// Most of the parameters of the problem (except for the digraph)

  /// can be given using separate functions, and the algorithm can be

  /// executed using the \ref run() function. If some parameters are not

  /// specified, then default values will be used.

  ///

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

  /// \tparam V The number type used for flow amounts, capacity bounds

  /// and supply values in the algorithm. By default, it is \c int.

  /// \tparam C The number type used for costs and potentials in the

  /// algorithm. By default, it is the same as \c V.

  ///

  /// \warning Both \c V and \c C must be signed number types.

  /// \warning All input data (capacities, supply values, and costs) must

  /// be integer.

  ///

  /// \note For more information about the three available methods,

  /// see \ref Method.

#ifdef DOXYGEN

  template <typename GR, typename V, typename C>

#else

  template <typename GR, typename V = int, typename C = V>

#endif

  class CycleCanceling

  {

  public:

    /// The type of the digraph

    typedef GR Digraph;

    /// The type of the flow amounts, capacity bounds and supply values

    typedef V Value;

      OPTIMAL,

      /// The digraph contains an arc of negative cost and infinite

      /// upper bound. It means that the objective function is unbounded

      /// on that arc, however, note that it could actually be bounded

      /// over the feasible flows, but this algroithm cannot handle

      /// these cases.

      UNBOUNDED

    };

    /// \brief Constants for selecting the used method.

    ///

    /// Enum type containing constants for selecting the used method

    /// for the \ref run() function.

    ///

    /// \ref CycleCanceling provides three different cycle-canceling

    /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel and Tighten"

    /// However, the other methods can be selected using the \ref run()

    /// function with the proper parameter.

    enum Method {

      /// A simple cycle-canceling method, which uses the

      /// \ref BellmanFord "Bellman-Ford" algorithm with limited iteration

      /// number for detecting negative cycles in the residual network.

      SIMPLE_CYCLE_CANCELING,

      /// The "Minimum Mean Cycle-Canceling" algorithm, which is a

      /// well-known strongly polynomial method

      /// \ref goldberg89cyclecanceling. It improves along a

      /// \ref min_mean_cycle "minimum mean cycle" in each iteration.

      /// Its running time complexity is O(n<sup>2</sup>m<sup>3</sup>log(n)).

      MINIMUM_MEAN_CYCLE_CANCELING,

      /// The "Cancel And Tighten" algorithm, which can be viewed as an

      /// improved version of the previous method

      /// \ref goldberg89cyclecanceling.

    template<typename SupplyMap>

    CycleCanceling& supplyMap(const SupplyMap& map) {

      for (NodeIt n(_graph); n != INVALID; ++n) {

        _supply[_node_id[n]] = map[n];

      }

      return *this;

    }

    /// \brief Set single source and target nodes and a supply value.

    ///

    /// This function sets a single source node and a single target node

    /// and the required flow value.

    /// If neither this function nor \ref supplyMap() is used before

    /// calling \ref run(), the supply of each node will be set to zero.

    ///

    /// Using this function has the same effect as using \ref supplyMap()

    /// assigned to \c t and all other nodes have zero supply value.

    ///

    /// \param s The source node.

    /// \param t The target node.

    /// \param k The required amount of flow from node \c s to node \c t

    /// (i.e. the supply of \c s and the demand of \c t).

    ///

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

    CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {

      for (int i = 0; i != _res_node_num; ++i) {

        _supply[i] = 0;

      }

      _supply[_node_id[s]] =  k;

      _supply[_node_id[t]] = -k;

      return *this;

    }

#include<lemon/adaptors.h>

#include<lemon/connectivity.h>

#include <list>

/// \ingroup graph_properties

/// \file

/// \brief Euler tour iterators and a function for checking the \e Eulerian

/// property.

///

///This file provides Euler tour iterators and a function to check

///if a (di)graph is \e Eulerian.

namespace lemon {

  ///Euler tour iterator for digraphs.

  ///This iterator provides an Euler tour (Eulerian circuit) of a \e directed

  ///graph (if there exists) and it converts to the \c Arc type of the digraph.

  ///

  ///For example, if the given digraph has an Euler tour (i.e it has only one

  ///non-trivial component and the in-degree is equal to the out-degree

  ///for all nodes), then the following code will put the arcs of \c g

  ///to the vector \c et according to an Euler tour of \c g.

  ///\code

  ///  std::vector<ListDigraph::Arc> et;

  ///  for(DiEulerIt<ListDigraph> e(g); e!=INVALID; ++e)

  ///    et.push_back(e);

  ///\endcode

  ///If \c g has no Euler tour, then the resulted walk will not be closed

  ///or not contain all arcs.

  ///\sa EulerIt

  template<typename GR>

namespace lemon {

  /// \addtogroup approx_algs

  /// @{

  /// \brief Implementation of the iterated local search algorithm of Grosso,

  /// Locatelli, and Pullan for the maximum clique problem

  ///

  /// \ref GrossoLocatelliPullanMc implements the iterated local search

  /// algorithm of Grosso, Locatelli, and Pullan for solving the \e maximum

  /// \e clique \e problem \ref grosso08maxclique.

  /// It is to find the largest complete subgraph (\e clique) in an

  /// undirected graph, i.e., the largest set of nodes where each

  /// pair of nodes is connected.

  ///

  /// This class provides a simple but highly efficient and robust heuristic

  /// largest one.

  ///

  /// \tparam GR The undirected graph type the algorithm runs on.

  ///

  /// \note %GrossoLocatelliPullanMc provides three different node selection

  /// rules, from which the most powerful one is used by default.

  /// For more information, see \ref SelectionRule.

  template <typename GR>

  class GrossoLocatelliPullanMc

  {

  public:

    /// \brief Constants for specifying the node selection rule.

    ///

    /// Enum type containing constants for specifying the node selection rule

    /// for the \ref run() function.

    ///

    /// However, another selection rule can be specified using the \ref run()

    /// function with the proper parameter.

    enum SelectionRule {

      /// A node is selected randomly without any evaluation at each step.

      RANDOM,

      /// A node of maximum degree is selected randomly at each step.

      DEGREE_BASED,

      /// A node of minimum penalty is selected randomly at each step.

      /// The node penalties are updated adaptively after each stage of the

      /// search process.

      PENALTY_BASED

    };

  private:

    TEMPLATE_GRAPH_TYPEDEFS(GR);

    typedef std::vector<int> IntVector;

    typedef std::vector<char> BoolVector;

    typedef std::vector<BoolVector> BoolMatrix;

    // Note: vector<char> is used instead of vector<bool> for efficiency reasons

    const GR &_graph;

    IntNodeMap _id;

    // Internal matrix representation of the graph

    BoolMatrix _gr;

    int _n;

    // The current clique

    BoolVector _clique;

    int _size;

    // The best clique found so far

    BoolVector _best_clique;

    int _best_size;

    // The "distances" of the nodes from the current clique.

    // _delta[u] is the number of nodes in the clique that are

    // not connected with u.

    IntVector _delta;

    // The current tabu set

    BoolVector _tabu;

      // Update internal data structures between stages (if necessary)

      void update() {}

    }; //class PenaltyBasedSelectionRule

  public:

    /// \brief Constructor.

    ///

    /// Constructor.

    /// The global \ref rnd "random number generator instance" is used

    /// during the algorithm.

    ///

    /// \param graph The undirected graph the algorithm runs on.

    GrossoLocatelliPullanMc(const GR& graph) :

      _graph(graph), _id(_graph), _rnd(rnd)

    /// \brief Constructor with random seed.

    ///

    /// Constructor with random seed.

    ///

    /// \param graph The undirected graph the algorithm runs on.

    /// \param seed Seed value for the internal random number generator

    /// that is used during the algorithm.

    GrossoLocatelliPullanMc(const GR& graph, int seed) :

      _graph(graph), _id(_graph), _rnd(seed)

    /// \brief Constructor with random number generator.

    ///

    /// Constructor with random number generator.

    ///

    /// \param graph The undirected graph the algorithm runs on.

    /// \param random A random number generator that is used during the

    /// algorithm.

    GrossoLocatelliPullanMc(const GR& graph, const Random& random) :

      _graph(graph), _id(_graph), _rnd(random)

    /// \name Execution Control

    /// @{

    /// \brief Runs the algorithm.

    ///

    ///

    /// \param rule The node selection rule. For more information, see

    /// \ref SelectionRule.

    ///

    {

      init();

      switch (rule) {

        case RANDOM:

        case DEGREE_BASED:

      }

    }

    /// @}

    /// \name Query Functions

    /// @{

    /// \brief The size of the found clique

    ///

    /// This function returns the size of the found clique.

    ///

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

    int cliqueSize() const {

      return _best_size;

    }

    /// \brief Gives back the found clique in a \c bool node map

    ///

    /// This function gives back the characteristic vector of the found

    /// clique in the given node map.

    /// It must be a \ref concepts::WriteMap "writable" node map with

      /// Postfix incrementation.

      ///

      /// \warning This incrementation returns a \c Node, not a

      /// \c CliqueNodeIt as one may expect.

      typename GR::Node operator++(int) {

        Node n=*this;

        ++(*this);

        return n;

      }

    };

    /// @}

  private:

    // Adds a node to the current clique

    void addCliqueNode(int u) {

      if (_clique[u]) return;

      _clique[u] = true;

      _size++;

      BoolVector &row = _gr[u];

      for (int i = 0; i != _n; i++) {

        if (!row[i]) _delta[i]++;

      }

    }

    // Removes a node from the current clique

    void delCliqueNode(int u) {

      if (!_clique[u]) return;

      _clique[u] = false;

      }

      _clique.clear();

      _clique.resize(_n, false);

      _size = 0;

      _best_clique.clear();

      _best_clique.resize(_n, false);

      _best_size = 0;

      _delta.clear();

      _delta.resize(_n, 0);

      _tabu.clear();

      _tabu.resize(_n, false);

    }

    // Executes the algorithm

    template <typename SelectionRuleImpl>

      if (_n == 1) {

        _best_clique[0] = true;

        _best_size = 1;

      }

      SelectionRuleImpl sel_method(*this);

      IntVector restart_nodes;

        // Perturbation/restart

          restart_nodes.clear();

          for (int i = 0; i != _n; i++) {

              restart_nodes.push_back(i);

          }

        }

        int rs_node = -1;

        if (restart_nodes.size() > 0) {

          rs_node = restart_nodes[_rnd[restart_nodes.size()]];

        } else {

          rs_node = _rnd[_n];

        }

        BoolVector &row = _gr[rs_node];

        for (int i = 0; i != _n; i++) {

          if (_clique[i] && !row[i]) delCliqueNode(i);

        }

        addCliqueNode(rs_node);

        // Local search

            }

            addCliqueNode(u);

            delCliqueNode(v);

            _tabu[v] = true;

            tabu_empty = false;

            if (--max_swap <= 0) break;

          }

          else if ((u = sel_method.nextAddNode()) != -1) {

            // Non-feasible add move

            addCliqueNode(u);

          }

          else break;

        }

        if (_size > _best_size) {

          _best_clique = _clique;

          _best_size = _size;

        }

        sel_method.update();

      }

    }

  }; //class GrossoLocatelliPullanMc

  ///@}

} //namespace lemon

#endif //LEMON_GROSSO_LOCATELLI_PULLAN_MC_H

namespace lemon {

  /// \addtogroup min_cost_flow_algs

  /// @{

  /// \brief Implementation of the primal Network Simplex algorithm

  /// for finding a \ref min_cost_flow "minimum cost flow".

  ///

  /// \ref NetworkSimplex implements the primal Network Simplex algorithm

  /// for finding a \ref min_cost_flow "minimum cost flow"

  /// \ref amo93networkflows, \ref dantzig63linearprog,

  /// \ref kellyoneill91netsimplex.

  /// This algorithm is a highly efficient specialized version of the

  /// linear programming simplex method directly for the minimum cost

  /// flow problem.

  ///

  ///

  /// Most of the parameters of the problem (except for the digraph)

  /// can be given using separate functions, and the algorithm can be

  /// executed using the \ref run() function. If some parameters are not

  /// specified, then default values will be used.

  ///

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

  /// \tparam V The number type used for flow amounts, capacity bounds

  /// and supply values in the algorithm. By default, it is \c int.

  /// \tparam C The number type used for costs and potentials in the

  /// algorithm. By default, it is the same as \c V.

  ///

  /// \warning Both \c V and \c C must be signed number types.

  /// \warning All input data (capacities, supply values, and costs) must

  /// be integer.

  ///

      /// supply/demand constraints in the definition of the problem.

      GEQ,

      /// This option means that there are <em>"less or equal"</em>

      /// supply/demand constraints in the definition of the problem.

      LEQ

    };

    /// \brief Constants for selecting the pivot rule.

    ///

    /// Enum type containing constants for selecting the pivot rule for

    /// the \ref run() function.

    ///