lemon/howard.h
author Peter Kovacs <kpeter@inf.elte.hu>
Fri, 19 Feb 2010 14:08:32 +0100
changeset 844 a6eb9698c321
parent 771 8452ca46e29a
child 841 aa8c9008b3de
permissions -rw-r--r--
Support tolerance technique for BellmanFord (#51)

A new operation traits class BellmanFordToleranceOperationTraits
is introduced, which uses the tolerance technique in its less()
function. This class can be used with the SetOperationTraits
named template parameter.
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/* -*- C++ -*-
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 *
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 * This file is a part of LEMON, a generic C++ optimization library
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 *
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 * Copyright (C) 2003-2008
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 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
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 * (Egervary Research Group on Combinatorial Optimization, EGRES).
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 *
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 * Permission to use, modify and distribute this software is granted
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 * provided that this copyright notice appears in all copies. For
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 * precise terms see the accompanying LICENSE file.
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 *
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 * This software is provided "AS IS" with no warranty of any kind,
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 * express or implied, and with no claim as to its suitability for any
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 * purpose.
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 *
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 */
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#ifndef LEMON_HOWARD_H
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#define LEMON_HOWARD_H
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/// \ingroup min_mean_cycle
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///
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/// \file
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/// \brief Howard's algorithm for finding a minimum mean cycle.
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#include <vector>
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#include <limits>
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#include <lemon/core.h>
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#include <lemon/path.h>
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#include <lemon/tolerance.h>
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#include <lemon/connectivity.h>
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namespace lemon {
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  /// \brief Default traits class of Howard class.
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  ///
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  /// Default traits class of Howard class.
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  /// \tparam GR The type of the digraph.
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  /// \tparam LEN The type of the length map.
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  /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.
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#ifdef DOXYGEN
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  template <typename GR, typename LEN>
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#else
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  template <typename GR, typename LEN,
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    bool integer = std::numeric_limits<typename LEN::Value>::is_integer>
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#endif
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  struct HowardDefaultTraits
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  {
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    /// The type of the digraph
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    typedef GR Digraph;
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    /// The type of the length map
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    typedef LEN LengthMap;
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    /// The type of the arc lengths
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    typedef typename LengthMap::Value Value;
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    /// \brief The large value type used for internal computations
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    ///
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    /// The large value type used for internal computations.
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    /// It is \c long \c long if the \c Value type is integer,
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    /// otherwise it is \c double.
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    /// \c Value must be convertible to \c LargeValue.
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    typedef double LargeValue;
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    /// The tolerance type used for internal computations
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    typedef lemon::Tolerance<LargeValue> Tolerance;
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    /// \brief The path type of the found cycles
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    ///
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    /// The path type of the found cycles.
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    /// It must conform to the \ref lemon::concepts::Path "Path" concept
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    /// and it must have an \c addBack() function.
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    typedef lemon::Path<Digraph> Path;
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  };
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  // Default traits class for integer value types
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  template <typename GR, typename LEN>
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  struct HowardDefaultTraits<GR, LEN, true>
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  {
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    typedef GR Digraph;
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    typedef LEN LengthMap;
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    typedef typename LengthMap::Value Value;
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#ifdef LEMON_HAVE_LONG_LONG
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    typedef long long LargeValue;
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#else
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    typedef long LargeValue;
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#endif
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    typedef lemon::Tolerance<LargeValue> Tolerance;
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    typedef lemon::Path<Digraph> Path;
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  };
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  /// \addtogroup min_mean_cycle
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  /// @{
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  /// \brief Implementation of Howard's algorithm for finding a minimum
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  /// mean cycle.
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  ///
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  /// This class implements Howard's policy iteration algorithm for finding
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  /// a directed cycle of minimum mean length (cost) in a digraph
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  /// \ref amo93networkflows, \ref dasdan98minmeancycle.
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  /// This class provides the most efficient algorithm for the
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  /// minimum mean cycle problem, though the best known theoretical
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  /// bound on its running time is exponential.
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  ///
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  /// \tparam GR The type of the digraph the algorithm runs on.
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  /// \tparam LEN The type of the length map. The default
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  /// map type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".
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  /// \tparam TR The traits class that defines various types used by the
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  /// algorithm. By default, it is \ref HowardDefaultTraits
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  /// "HowardDefaultTraits<GR, LEN>".
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  /// In most cases, this parameter should not be set directly,
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  /// consider to use the named template parameters instead.
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#ifdef DOXYGEN
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  template <typename GR, typename LEN, typename TR>
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#else
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  template < typename GR,
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             typename LEN = typename GR::template ArcMap<int>,
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             typename TR = HowardDefaultTraits<GR, LEN> >
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#endif
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  class Howard
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  {
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  public:
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    /// The type of the digraph
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    typedef typename TR::Digraph Digraph;
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    /// The type of the length map
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    typedef typename TR::LengthMap LengthMap;
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    /// The type of the arc lengths
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    typedef typename TR::Value Value;
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    /// \brief The large value type
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    ///
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    /// The large value type used for internal computations.
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    /// By default, it is \c long \c long if the \c Value type is integer,
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    /// otherwise it is \c double.
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    typedef typename TR::LargeValue LargeValue;
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    /// The tolerance type
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    typedef typename TR::Tolerance Tolerance;
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    /// \brief The path type of the found cycles
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    ///
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    /// The path type of the found cycles.
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    /// Using the \ref HowardDefaultTraits "default traits class",
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    /// it is \ref lemon::Path "Path<Digraph>".
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    typedef typename TR::Path Path;
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    /// The \ref HowardDefaultTraits "traits class" of the algorithm
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    typedef TR Traits;
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  private:
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    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
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    // The digraph the algorithm runs on
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    const Digraph &_gr;
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    // The length of the arcs
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    const LengthMap &_length;
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    // Data for the found cycles
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    bool _curr_found, _best_found;
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    LargeValue _curr_length, _best_length;
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    int _curr_size, _best_size;
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    Node _curr_node, _best_node;
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    Path *_cycle_path;
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    bool _local_path;
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    // Internal data used by the algorithm
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    typename Digraph::template NodeMap<Arc> _policy;
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    typename Digraph::template NodeMap<bool> _reached;
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    typename Digraph::template NodeMap<int> _level;
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    typename Digraph::template NodeMap<LargeValue> _dist;
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    // Data for storing the strongly connected components
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    int _comp_num;
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    typename Digraph::template NodeMap<int> _comp;
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    std::vector<std::vector<Node> > _comp_nodes;
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    std::vector<Node>* _nodes;
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    typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs;
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    // Queue used for BFS search
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    std::vector<Node> _queue;
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    int _qfront, _qback;
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    Tolerance _tolerance;
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    // Infinite constant
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    const LargeValue INF;
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  public:
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    /// \name Named Template Parameters
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    /// @{
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    template <typename T>
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    struct SetLargeValueTraits : public Traits {
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      typedef T LargeValue;
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      typedef lemon::Tolerance<T> Tolerance;
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    };
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    /// \brief \ref named-templ-param "Named parameter" for setting
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    /// \c LargeValue type.
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    ///
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    /// \ref named-templ-param "Named parameter" for setting \c LargeValue
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    /// type. It is used for internal computations in the algorithm.
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    template <typename T>
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    struct SetLargeValue
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      : public Howard<GR, LEN, SetLargeValueTraits<T> > {
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      typedef Howard<GR, LEN, SetLargeValueTraits<T> > Create;
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    };
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    template <typename T>
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    struct SetPathTraits : public Traits {
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      typedef T Path;
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    };
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    /// \brief \ref named-templ-param "Named parameter" for setting
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    /// \c %Path type.
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    ///
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    /// \ref named-templ-param "Named parameter" for setting the \c %Path
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    /// type of the found cycles.
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    /// It must conform to the \ref lemon::concepts::Path "Path" concept
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    /// and it must have an \c addBack() function.
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    template <typename T>
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    struct SetPath
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      : public Howard<GR, LEN, SetPathTraits<T> > {
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      typedef Howard<GR, LEN, SetPathTraits<T> > Create;
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    };
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    /// @}
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  public:
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    /// \brief Constructor.
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    ///
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    /// The constructor of the class.
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    ///
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    /// \param digraph The digraph the algorithm runs on.
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    /// \param length The lengths (costs) of the arcs.
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    Howard( const Digraph &digraph,
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            const LengthMap &length ) :
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      _gr(digraph), _length(length), _best_found(false),
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      _best_length(0), _best_size(1), _cycle_path(NULL), _local_path(false),
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      _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph),
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      _comp(digraph), _in_arcs(digraph),
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      INF(std::numeric_limits<LargeValue>::has_infinity ?
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          std::numeric_limits<LargeValue>::infinity() :
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          std::numeric_limits<LargeValue>::max())
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    {}
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    /// Destructor.
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    ~Howard() {
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      if (_local_path) delete _cycle_path;
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    }
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    /// \brief Set the path structure for storing the found cycle.
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    ///
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    /// This function sets an external path structure for storing the
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    /// found cycle.
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    ///
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    /// If you don't call this function before calling \ref run() or
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    /// \ref findMinMean(), it will allocate a local \ref Path "path"
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    /// structure. The destuctor deallocates this automatically
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    /// allocated object, of course.
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    ///
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    /// \note The algorithm calls only the \ref lemon::Path::addBack()
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    /// "addBack()" function of the given path structure.
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    ///
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    /// \return <tt>(*this)</tt>
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    Howard& cycle(Path &path) {
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      if (_local_path) {
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        delete _cycle_path;
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        _local_path = false;
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      }
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      _cycle_path = &path;
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      return *this;
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    }
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    /// \brief Set the tolerance used by the algorithm.
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    ///
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    /// This function sets the tolerance object used by the algorithm.
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    ///
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    /// \return <tt>(*this)</tt>
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    Howard& tolerance(const Tolerance& tolerance) {
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      _tolerance = tolerance;
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      return *this;
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    }
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    /// \brief Return a const reference to the tolerance.
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    ///
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    /// This function returns a const reference to the tolerance object
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    /// used by the algorithm.
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    const Tolerance& tolerance() const {
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      return _tolerance;
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    }
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    /// \name Execution control
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    /// The simplest way to execute the algorithm is to call the \ref run()
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    /// function.\n
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    /// If you only need the minimum mean length, you may call
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    /// \ref findMinMean().
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    /// @{
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    /// \brief Run the algorithm.
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    ///
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    /// This function runs the algorithm.
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    /// It can be called more than once (e.g. if the underlying digraph
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    /// and/or the arc lengths have been modified).
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    ///
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    /// \return \c true if a directed cycle exists in the digraph.
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    ///
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    /// \note <tt>mmc.run()</tt> is just a shortcut of the following code.
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    /// \code
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    ///   return mmc.findMinMean() && mmc.findCycle();
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    /// \endcode
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    bool run() {
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      return findMinMean() && findCycle();
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    }
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    /// \brief Find the minimum cycle mean.
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    ///
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    /// This function finds the minimum mean length of the directed
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    /// cycles in the digraph.
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    ///
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    /// \return \c true if a directed cycle exists in the digraph.
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    bool findMinMean() {
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      // Initialize and find strongly connected components
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      init();
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      findComponents();
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      // Find the minimum cycle mean in the components
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      for (int comp = 0; comp < _comp_num; ++comp) {
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        // Find the minimum mean cycle in the current component
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        if (!buildPolicyGraph(comp)) continue;
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        while (true) {
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          findPolicyCycle();
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          if (!computeNodeDistances()) break;
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        }
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        // Update the best cycle (global minimum mean cycle)
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        if ( _curr_found && (!_best_found ||
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             _curr_length * _best_size < _best_length * _curr_size) ) {
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          _best_found = true;
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          _best_length = _curr_length;
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          _best_size = _curr_size;
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          _best_node = _curr_node;
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        }
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      }
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      return _best_found;
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    }
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    /// \brief Find a minimum mean directed cycle.
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    ///
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    /// This function finds a directed cycle of minimum mean length
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    /// in the digraph using the data computed by findMinMean().
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    ///
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    /// \return \c true if a directed cycle exists in the digraph.
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    ///
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    /// \pre \ref findMinMean() must be called before using this function.
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    bool findCycle() {
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      if (!_best_found) return false;
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      _cycle_path->addBack(_policy[_best_node]);
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      for ( Node v = _best_node;
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            (v = _gr.target(_policy[v])) != _best_node; ) {
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        _cycle_path->addBack(_policy[v]);
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      }
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      return true;
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    }
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    /// @}
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    /// \name Query Functions
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    /// The results of the algorithm can be obtained using these
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    /// functions.\n
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    /// The algorithm should be executed before using them.
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    /// @{
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    /// \brief Return the total length of the found cycle.
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    ///
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    /// This function returns the total length of the found cycle.
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    ///
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    /// \pre \ref run() or \ref findMinMean() must be called before
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    /// using this function.
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    LargeValue cycleLength() const {
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      return _best_length;
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    }
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    /// \brief Return the number of arcs on the found cycle.
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    ///
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    /// This function returns the number of arcs on the found cycle.
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    ///
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    /// \pre \ref run() or \ref findMinMean() must be called before
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    /// using this function.
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    int cycleArcNum() const {
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      return _best_size;
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    }
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    /// \brief Return the mean length of the found cycle.
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    ///
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    /// This function returns the mean length of the found cycle.
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    ///
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   405
    /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the
kpeter@758
   406
    /// following code.
kpeter@758
   407
    /// \code
kpeter@760
   408
    ///   return static_cast<double>(alg.cycleLength()) / alg.cycleArcNum();
kpeter@758
   409
    /// \endcode
kpeter@758
   410
    ///
kpeter@758
   411
    /// \pre \ref run() or \ref findMinMean() must be called before
kpeter@758
   412
    /// using this function.
kpeter@758
   413
    double cycleMean() const {
kpeter@760
   414
      return static_cast<double>(_best_length) / _best_size;
kpeter@758
   415
    }
kpeter@758
   416
kpeter@758
   417
    /// \brief Return the found cycle.
kpeter@758
   418
    ///
kpeter@758
   419
    /// This function returns a const reference to the path structure
kpeter@758
   420
    /// storing the found cycle.
kpeter@758
   421
    ///
kpeter@758
   422
    /// \pre \ref run() or \ref findCycle() must be called before using
kpeter@758
   423
    /// this function.
kpeter@758
   424
    const Path& cycle() const {
kpeter@758
   425
      return *_cycle_path;
kpeter@758
   426
    }
kpeter@758
   427
kpeter@758
   428
    ///@}
kpeter@758
   429
kpeter@758
   430
  private:
kpeter@758
   431
kpeter@760
   432
    // Initialize
kpeter@760
   433
    void init() {
kpeter@760
   434
      if (!_cycle_path) {
kpeter@760
   435
        _local_path = true;
kpeter@760
   436
        _cycle_path = new Path;
kpeter@758
   437
      }
kpeter@760
   438
      _queue.resize(countNodes(_gr));
kpeter@760
   439
      _best_found = false;
kpeter@760
   440
      _best_length = 0;
kpeter@760
   441
      _best_size = 1;
kpeter@760
   442
      _cycle_path->clear();
kpeter@760
   443
    }
kpeter@760
   444
    
kpeter@760
   445
    // Find strongly connected components and initialize _comp_nodes
kpeter@760
   446
    // and _in_arcs
kpeter@760
   447
    void findComponents() {
kpeter@760
   448
      _comp_num = stronglyConnectedComponents(_gr, _comp);
kpeter@760
   449
      _comp_nodes.resize(_comp_num);
kpeter@760
   450
      if (_comp_num == 1) {
kpeter@760
   451
        _comp_nodes[0].clear();
kpeter@760
   452
        for (NodeIt n(_gr); n != INVALID; ++n) {
kpeter@760
   453
          _comp_nodes[0].push_back(n);
kpeter@760
   454
          _in_arcs[n].clear();
kpeter@760
   455
          for (InArcIt a(_gr, n); a != INVALID; ++a) {
kpeter@760
   456
            _in_arcs[n].push_back(a);
kpeter@760
   457
          }
kpeter@760
   458
        }
kpeter@760
   459
      } else {
kpeter@760
   460
        for (int i = 0; i < _comp_num; ++i)
kpeter@760
   461
          _comp_nodes[i].clear();
kpeter@760
   462
        for (NodeIt n(_gr); n != INVALID; ++n) {
kpeter@760
   463
          int k = _comp[n];
kpeter@760
   464
          _comp_nodes[k].push_back(n);
kpeter@760
   465
          _in_arcs[n].clear();
kpeter@760
   466
          for (InArcIt a(_gr, n); a != INVALID; ++a) {
kpeter@760
   467
            if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a);
kpeter@760
   468
          }
kpeter@760
   469
        }
kpeter@758
   470
      }
kpeter@760
   471
    }
kpeter@760
   472
kpeter@760
   473
    // Build the policy graph in the given strongly connected component
kpeter@760
   474
    // (the out-degree of every node is 1)
kpeter@760
   475
    bool buildPolicyGraph(int comp) {
kpeter@760
   476
      _nodes = &(_comp_nodes[comp]);
kpeter@760
   477
      if (_nodes->size() < 1 ||
kpeter@760
   478
          (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) {
kpeter@760
   479
        return false;
kpeter@758
   480
      }
kpeter@760
   481
      for (int i = 0; i < int(_nodes->size()); ++i) {
kpeter@767
   482
        _dist[(*_nodes)[i]] = INF;
kpeter@760
   483
      }
kpeter@760
   484
      Node u, v;
kpeter@760
   485
      Arc e;
kpeter@760
   486
      for (int i = 0; i < int(_nodes->size()); ++i) {
kpeter@760
   487
        v = (*_nodes)[i];
kpeter@760
   488
        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
kpeter@760
   489
          e = _in_arcs[v][j];
kpeter@760
   490
          u = _gr.source(e);
kpeter@760
   491
          if (_length[e] < _dist[u]) {
kpeter@760
   492
            _dist[u] = _length[e];
kpeter@760
   493
            _policy[u] = e;
kpeter@760
   494
          }
kpeter@758
   495
        }
kpeter@758
   496
      }
kpeter@758
   497
      return true;
kpeter@758
   498
    }
kpeter@758
   499
kpeter@760
   500
    // Find the minimum mean cycle in the policy graph
kpeter@760
   501
    void findPolicyCycle() {
kpeter@760
   502
      for (int i = 0; i < int(_nodes->size()); ++i) {
kpeter@760
   503
        _level[(*_nodes)[i]] = -1;
kpeter@760
   504
      }
kpeter@761
   505
      LargeValue clength;
kpeter@758
   506
      int csize;
kpeter@758
   507
      Node u, v;
kpeter@760
   508
      _curr_found = false;
kpeter@760
   509
      for (int i = 0; i < int(_nodes->size()); ++i) {
kpeter@760
   510
        u = (*_nodes)[i];
kpeter@760
   511
        if (_level[u] >= 0) continue;
kpeter@760
   512
        for (; _level[u] < 0; u = _gr.target(_policy[u])) {
kpeter@760
   513
          _level[u] = i;
kpeter@760
   514
        }
kpeter@760
   515
        if (_level[u] == i) {
kpeter@760
   516
          // A cycle is found
kpeter@760
   517
          clength = _length[_policy[u]];
kpeter@760
   518
          csize = 1;
kpeter@760
   519
          for (v = u; (v = _gr.target(_policy[v])) != u; ) {
kpeter@760
   520
            clength += _length[_policy[v]];
kpeter@760
   521
            ++csize;
kpeter@758
   522
          }
kpeter@760
   523
          if ( !_curr_found ||
kpeter@760
   524
               (clength * _curr_size < _curr_length * csize) ) {
kpeter@760
   525
            _curr_found = true;
kpeter@760
   526
            _curr_length = clength;
kpeter@760
   527
            _curr_size = csize;
kpeter@760
   528
            _curr_node = u;
kpeter@758
   529
          }
kpeter@758
   530
        }
kpeter@758
   531
      }
kpeter@758
   532
    }
kpeter@758
   533
kpeter@760
   534
    // Contract the policy graph and compute node distances
kpeter@758
   535
    bool computeNodeDistances() {
kpeter@760
   536
      // Find the component of the main cycle and compute node distances
kpeter@760
   537
      // using reverse BFS
kpeter@760
   538
      for (int i = 0; i < int(_nodes->size()); ++i) {
kpeter@760
   539
        _reached[(*_nodes)[i]] = false;
kpeter@760
   540
      }
kpeter@760
   541
      _qfront = _qback = 0;
kpeter@760
   542
      _queue[0] = _curr_node;
kpeter@760
   543
      _reached[_curr_node] = true;
kpeter@760
   544
      _dist[_curr_node] = 0;
kpeter@758
   545
      Node u, v;
kpeter@760
   546
      Arc e;
kpeter@760
   547
      while (_qfront <= _qback) {
kpeter@760
   548
        v = _queue[_qfront++];
kpeter@760
   549
        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
kpeter@760
   550
          e = _in_arcs[v][j];
kpeter@758
   551
          u = _gr.source(e);
kpeter@760
   552
          if (_policy[u] == e && !_reached[u]) {
kpeter@760
   553
            _reached[u] = true;
kpeter@761
   554
            _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;
kpeter@760
   555
            _queue[++_qback] = u;
kpeter@758
   556
          }
kpeter@758
   557
        }
kpeter@758
   558
      }
kpeter@760
   559
kpeter@760
   560
      // Connect all other nodes to this component and compute node
kpeter@760
   561
      // distances using reverse BFS
kpeter@760
   562
      _qfront = 0;
kpeter@760
   563
      while (_qback < int(_nodes->size())-1) {
kpeter@760
   564
        v = _queue[_qfront++];
kpeter@760
   565
        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
kpeter@760
   566
          e = _in_arcs[v][j];
kpeter@760
   567
          u = _gr.source(e);
kpeter@760
   568
          if (!_reached[u]) {
kpeter@760
   569
            _reached[u] = true;
kpeter@760
   570
            _policy[u] = e;
kpeter@761
   571
            _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;
kpeter@760
   572
            _queue[++_qback] = u;
kpeter@760
   573
          }
kpeter@760
   574
        }
kpeter@760
   575
      }
kpeter@760
   576
kpeter@760
   577
      // Improve node distances
kpeter@758
   578
      bool improved = false;
kpeter@760
   579
      for (int i = 0; i < int(_nodes->size()); ++i) {
kpeter@760
   580
        v = (*_nodes)[i];
kpeter@760
   581
        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
kpeter@760
   582
          e = _in_arcs[v][j];
kpeter@760
   583
          u = _gr.source(e);
kpeter@761
   584
          LargeValue delta = _dist[v] + _length[e] * _curr_size - _curr_length;
kpeter@761
   585
          if (_tolerance.less(delta, _dist[u])) {
kpeter@760
   586
            _dist[u] = delta;
kpeter@760
   587
            _policy[u] = e;
kpeter@760
   588
            improved = true;
kpeter@760
   589
          }
kpeter@758
   590
        }
kpeter@758
   591
      }
kpeter@758
   592
      return improved;
kpeter@758
   593
    }
kpeter@758
   594
kpeter@764
   595
  }; //class Howard
kpeter@758
   596
kpeter@758
   597
  ///@}
kpeter@758
   598
kpeter@758
   599
} //namespace lemon
kpeter@758
   600
kpeter@764
   601
#endif //LEMON_HOWARD_H