lemon/howard.h
changeset 784 1a7fe3bef514
parent 769 e746fb14e680
child 825 75e6020b19b1
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/lemon/howard.h	Thu Nov 05 15:50:01 2009 +0100
     1.3 @@ -0,0 +1,597 @@
     1.4 +/* -*- C++ -*-
     1.5 + *
     1.6 + * This file is a part of LEMON, a generic C++ optimization library
     1.7 + *
     1.8 + * Copyright (C) 2003-2008
     1.9 + * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
    1.10 + * (Egervary Research Group on Combinatorial Optimization, EGRES).
    1.11 + *
    1.12 + * Permission to use, modify and distribute this software is granted
    1.13 + * provided that this copyright notice appears in all copies. For
    1.14 + * precise terms see the accompanying LICENSE file.
    1.15 + *
    1.16 + * This software is provided "AS IS" with no warranty of any kind,
    1.17 + * express or implied, and with no claim as to its suitability for any
    1.18 + * purpose.
    1.19 + *
    1.20 + */
    1.21 +
    1.22 +#ifndef LEMON_HOWARD_H
    1.23 +#define LEMON_HOWARD_H
    1.24 +
    1.25 +/// \ingroup min_mean_cycle
    1.26 +///
    1.27 +/// \file
    1.28 +/// \brief Howard's algorithm for finding a minimum mean cycle.
    1.29 +
    1.30 +#include <vector>
    1.31 +#include <limits>
    1.32 +#include <lemon/core.h>
    1.33 +#include <lemon/path.h>
    1.34 +#include <lemon/tolerance.h>
    1.35 +#include <lemon/connectivity.h>
    1.36 +
    1.37 +namespace lemon {
    1.38 +
    1.39 +  /// \brief Default traits class of Howard class.
    1.40 +  ///
    1.41 +  /// Default traits class of Howard class.
    1.42 +  /// \tparam GR The type of the digraph.
    1.43 +  /// \tparam LEN The type of the length map.
    1.44 +  /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.
    1.45 +#ifdef DOXYGEN
    1.46 +  template <typename GR, typename LEN>
    1.47 +#else
    1.48 +  template <typename GR, typename LEN,
    1.49 +    bool integer = std::numeric_limits<typename LEN::Value>::is_integer>
    1.50 +#endif
    1.51 +  struct HowardDefaultTraits
    1.52 +  {
    1.53 +    /// The type of the digraph
    1.54 +    typedef GR Digraph;
    1.55 +    /// The type of the length map
    1.56 +    typedef LEN LengthMap;
    1.57 +    /// The type of the arc lengths
    1.58 +    typedef typename LengthMap::Value Value;
    1.59 +
    1.60 +    /// \brief The large value type used for internal computations
    1.61 +    ///
    1.62 +    /// The large value type used for internal computations.
    1.63 +    /// It is \c long \c long if the \c Value type is integer,
    1.64 +    /// otherwise it is \c double.
    1.65 +    /// \c Value must be convertible to \c LargeValue.
    1.66 +    typedef double LargeValue;
    1.67 +
    1.68 +    /// The tolerance type used for internal computations
    1.69 +    typedef lemon::Tolerance<LargeValue> Tolerance;
    1.70 +
    1.71 +    /// \brief The path type of the found cycles
    1.72 +    ///
    1.73 +    /// The path type of the found cycles.
    1.74 +    /// It must conform to the \ref lemon::concepts::Path "Path" concept
    1.75 +    /// and it must have an \c addBack() function.
    1.76 +    typedef lemon::Path<Digraph> Path;
    1.77 +  };
    1.78 +
    1.79 +  // Default traits class for integer value types
    1.80 +  template <typename GR, typename LEN>
    1.81 +  struct HowardDefaultTraits<GR, LEN, true>
    1.82 +  {
    1.83 +    typedef GR Digraph;
    1.84 +    typedef LEN LengthMap;
    1.85 +    typedef typename LengthMap::Value Value;
    1.86 +#ifdef LEMON_HAVE_LONG_LONG
    1.87 +    typedef long long LargeValue;
    1.88 +#else
    1.89 +    typedef long LargeValue;
    1.90 +#endif
    1.91 +    typedef lemon::Tolerance<LargeValue> Tolerance;
    1.92 +    typedef lemon::Path<Digraph> Path;
    1.93 +  };
    1.94 +
    1.95 +
    1.96 +  /// \addtogroup min_mean_cycle
    1.97 +  /// @{
    1.98 +
    1.99 +  /// \brief Implementation of Howard's algorithm for finding a minimum
   1.100 +  /// mean cycle.
   1.101 +  ///
   1.102 +  /// This class implements Howard's policy iteration algorithm for finding
   1.103 +  /// a directed cycle of minimum mean length (cost) in a digraph
   1.104 +  /// \ref amo93networkflows, \ref dasdan98minmeancycle.
   1.105 +  /// This class provides the most efficient algorithm for the
   1.106 +  /// minimum mean cycle problem, though the best known theoretical
   1.107 +  /// bound on its running time is exponential.
   1.108 +  ///
   1.109 +  /// \tparam GR The type of the digraph the algorithm runs on.
   1.110 +  /// \tparam LEN The type of the length map. The default
   1.111 +  /// map type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".
   1.112 +#ifdef DOXYGEN
   1.113 +  template <typename GR, typename LEN, typename TR>
   1.114 +#else
   1.115 +  template < typename GR,
   1.116 +             typename LEN = typename GR::template ArcMap<int>,
   1.117 +             typename TR = HowardDefaultTraits<GR, LEN> >
   1.118 +#endif
   1.119 +  class Howard
   1.120 +  {
   1.121 +  public:
   1.122 +  
   1.123 +    /// The type of the digraph
   1.124 +    typedef typename TR::Digraph Digraph;
   1.125 +    /// The type of the length map
   1.126 +    typedef typename TR::LengthMap LengthMap;
   1.127 +    /// The type of the arc lengths
   1.128 +    typedef typename TR::Value Value;
   1.129 +
   1.130 +    /// \brief The large value type
   1.131 +    ///
   1.132 +    /// The large value type used for internal computations.
   1.133 +    /// Using the \ref HowardDefaultTraits "default traits class",
   1.134 +    /// it is \c long \c long if the \c Value type is integer,
   1.135 +    /// otherwise it is \c double.
   1.136 +    typedef typename TR::LargeValue LargeValue;
   1.137 +
   1.138 +    /// The tolerance type
   1.139 +    typedef typename TR::Tolerance Tolerance;
   1.140 +
   1.141 +    /// \brief The path type of the found cycles
   1.142 +    ///
   1.143 +    /// The path type of the found cycles.
   1.144 +    /// Using the \ref HowardDefaultTraits "default traits class",
   1.145 +    /// it is \ref lemon::Path "Path<Digraph>".
   1.146 +    typedef typename TR::Path Path;
   1.147 +
   1.148 +    /// The \ref HowardDefaultTraits "traits class" of the algorithm
   1.149 +    typedef TR Traits;
   1.150 +
   1.151 +  private:
   1.152 +
   1.153 +    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);
   1.154 +  
   1.155 +    // The digraph the algorithm runs on
   1.156 +    const Digraph &_gr;
   1.157 +    // The length of the arcs
   1.158 +    const LengthMap &_length;
   1.159 +
   1.160 +    // Data for the found cycles
   1.161 +    bool _curr_found, _best_found;
   1.162 +    LargeValue _curr_length, _best_length;
   1.163 +    int _curr_size, _best_size;
   1.164 +    Node _curr_node, _best_node;
   1.165 +
   1.166 +    Path *_cycle_path;
   1.167 +    bool _local_path;
   1.168 +
   1.169 +    // Internal data used by the algorithm
   1.170 +    typename Digraph::template NodeMap<Arc> _policy;
   1.171 +    typename Digraph::template NodeMap<bool> _reached;
   1.172 +    typename Digraph::template NodeMap<int> _level;
   1.173 +    typename Digraph::template NodeMap<LargeValue> _dist;
   1.174 +
   1.175 +    // Data for storing the strongly connected components
   1.176 +    int _comp_num;
   1.177 +    typename Digraph::template NodeMap<int> _comp;
   1.178 +    std::vector<std::vector<Node> > _comp_nodes;
   1.179 +    std::vector<Node>* _nodes;
   1.180 +    typename Digraph::template NodeMap<std::vector<Arc> > _in_arcs;
   1.181 +    
   1.182 +    // Queue used for BFS search
   1.183 +    std::vector<Node> _queue;
   1.184 +    int _qfront, _qback;
   1.185 +
   1.186 +    Tolerance _tolerance;
   1.187 +  
   1.188 +    // Infinite constant
   1.189 +    const LargeValue INF;
   1.190 +
   1.191 +  public:
   1.192 +  
   1.193 +    /// \name Named Template Parameters
   1.194 +    /// @{
   1.195 +
   1.196 +    template <typename T>
   1.197 +    struct SetLargeValueTraits : public Traits {
   1.198 +      typedef T LargeValue;
   1.199 +      typedef lemon::Tolerance<T> Tolerance;
   1.200 +    };
   1.201 +
   1.202 +    /// \brief \ref named-templ-param "Named parameter" for setting
   1.203 +    /// \c LargeValue type.
   1.204 +    ///
   1.205 +    /// \ref named-templ-param "Named parameter" for setting \c LargeValue
   1.206 +    /// type. It is used for internal computations in the algorithm.
   1.207 +    template <typename T>
   1.208 +    struct SetLargeValue
   1.209 +      : public Howard<GR, LEN, SetLargeValueTraits<T> > {
   1.210 +      typedef Howard<GR, LEN, SetLargeValueTraits<T> > Create;
   1.211 +    };
   1.212 +
   1.213 +    template <typename T>
   1.214 +    struct SetPathTraits : public Traits {
   1.215 +      typedef T Path;
   1.216 +    };
   1.217 +
   1.218 +    /// \brief \ref named-templ-param "Named parameter" for setting
   1.219 +    /// \c %Path type.
   1.220 +    ///
   1.221 +    /// \ref named-templ-param "Named parameter" for setting the \c %Path
   1.222 +    /// type of the found cycles.
   1.223 +    /// It must conform to the \ref lemon::concepts::Path "Path" concept
   1.224 +    /// and it must have an \c addBack() function.
   1.225 +    template <typename T>
   1.226 +    struct SetPath
   1.227 +      : public Howard<GR, LEN, SetPathTraits<T> > {
   1.228 +      typedef Howard<GR, LEN, SetPathTraits<T> > Create;
   1.229 +    };
   1.230 +    
   1.231 +    /// @}
   1.232 +
   1.233 +  public:
   1.234 +
   1.235 +    /// \brief Constructor.
   1.236 +    ///
   1.237 +    /// The constructor of the class.
   1.238 +    ///
   1.239 +    /// \param digraph The digraph the algorithm runs on.
   1.240 +    /// \param length The lengths (costs) of the arcs.
   1.241 +    Howard( const Digraph &digraph,
   1.242 +            const LengthMap &length ) :
   1.243 +      _gr(digraph), _length(length), _best_found(false),
   1.244 +      _best_length(0), _best_size(1), _cycle_path(NULL), _local_path(false),
   1.245 +      _policy(digraph), _reached(digraph), _level(digraph), _dist(digraph),
   1.246 +      _comp(digraph), _in_arcs(digraph),
   1.247 +      INF(std::numeric_limits<LargeValue>::has_infinity ?
   1.248 +          std::numeric_limits<LargeValue>::infinity() :
   1.249 +          std::numeric_limits<LargeValue>::max())
   1.250 +    {}
   1.251 +
   1.252 +    /// Destructor.
   1.253 +    ~Howard() {
   1.254 +      if (_local_path) delete _cycle_path;
   1.255 +    }
   1.256 +
   1.257 +    /// \brief Set the path structure for storing the found cycle.
   1.258 +    ///
   1.259 +    /// This function sets an external path structure for storing the
   1.260 +    /// found cycle.
   1.261 +    ///
   1.262 +    /// If you don't call this function before calling \ref run() or
   1.263 +    /// \ref findMinMean(), it will allocate a local \ref Path "path"
   1.264 +    /// structure. The destuctor deallocates this automatically
   1.265 +    /// allocated object, of course.
   1.266 +    ///
   1.267 +    /// \note The algorithm calls only the \ref lemon::Path::addBack()
   1.268 +    /// "addBack()" function of the given path structure.
   1.269 +    ///
   1.270 +    /// \return <tt>(*this)</tt>
   1.271 +    Howard& cycle(Path &path) {
   1.272 +      if (_local_path) {
   1.273 +        delete _cycle_path;
   1.274 +        _local_path = false;
   1.275 +      }
   1.276 +      _cycle_path = &path;
   1.277 +      return *this;
   1.278 +    }
   1.279 +
   1.280 +    /// \brief Set the tolerance used by the algorithm.
   1.281 +    ///
   1.282 +    /// This function sets the tolerance object used by the algorithm.
   1.283 +    ///
   1.284 +    /// \return <tt>(*this)</tt>
   1.285 +    Howard& tolerance(const Tolerance& tolerance) {
   1.286 +      _tolerance = tolerance;
   1.287 +      return *this;
   1.288 +    }
   1.289 +
   1.290 +    /// \brief Return a const reference to the tolerance.
   1.291 +    ///
   1.292 +    /// This function returns a const reference to the tolerance object
   1.293 +    /// used by the algorithm.
   1.294 +    const Tolerance& tolerance() const {
   1.295 +      return _tolerance;
   1.296 +    }
   1.297 +
   1.298 +    /// \name Execution control
   1.299 +    /// The simplest way to execute the algorithm is to call the \ref run()
   1.300 +    /// function.\n
   1.301 +    /// If you only need the minimum mean length, you may call
   1.302 +    /// \ref findMinMean().
   1.303 +
   1.304 +    /// @{
   1.305 +
   1.306 +    /// \brief Run the algorithm.
   1.307 +    ///
   1.308 +    /// This function runs the algorithm.
   1.309 +    /// It can be called more than once (e.g. if the underlying digraph
   1.310 +    /// and/or the arc lengths have been modified).
   1.311 +    ///
   1.312 +    /// \return \c true if a directed cycle exists in the digraph.
   1.313 +    ///
   1.314 +    /// \note <tt>mmc.run()</tt> is just a shortcut of the following code.
   1.315 +    /// \code
   1.316 +    ///   return mmc.findMinMean() && mmc.findCycle();
   1.317 +    /// \endcode
   1.318 +    bool run() {
   1.319 +      return findMinMean() && findCycle();
   1.320 +    }
   1.321 +
   1.322 +    /// \brief Find the minimum cycle mean.
   1.323 +    ///
   1.324 +    /// This function finds the minimum mean length of the directed
   1.325 +    /// cycles in the digraph.
   1.326 +    ///
   1.327 +    /// \return \c true if a directed cycle exists in the digraph.
   1.328 +    bool findMinMean() {
   1.329 +      // Initialize and find strongly connected components
   1.330 +      init();
   1.331 +      findComponents();
   1.332 +      
   1.333 +      // Find the minimum cycle mean in the components
   1.334 +      for (int comp = 0; comp < _comp_num; ++comp) {
   1.335 +        // Find the minimum mean cycle in the current component
   1.336 +        if (!buildPolicyGraph(comp)) continue;
   1.337 +        while (true) {
   1.338 +          findPolicyCycle();
   1.339 +          if (!computeNodeDistances()) break;
   1.340 +        }
   1.341 +        // Update the best cycle (global minimum mean cycle)
   1.342 +        if ( _curr_found && (!_best_found ||
   1.343 +             _curr_length * _best_size < _best_length * _curr_size) ) {
   1.344 +          _best_found = true;
   1.345 +          _best_length = _curr_length;
   1.346 +          _best_size = _curr_size;
   1.347 +          _best_node = _curr_node;
   1.348 +        }
   1.349 +      }
   1.350 +      return _best_found;
   1.351 +    }
   1.352 +
   1.353 +    /// \brief Find a minimum mean directed cycle.
   1.354 +    ///
   1.355 +    /// This function finds a directed cycle of minimum mean length
   1.356 +    /// in the digraph using the data computed by findMinMean().
   1.357 +    ///
   1.358 +    /// \return \c true if a directed cycle exists in the digraph.
   1.359 +    ///
   1.360 +    /// \pre \ref findMinMean() must be called before using this function.
   1.361 +    bool findCycle() {
   1.362 +      if (!_best_found) return false;
   1.363 +      _cycle_path->addBack(_policy[_best_node]);
   1.364 +      for ( Node v = _best_node;
   1.365 +            (v = _gr.target(_policy[v])) != _best_node; ) {
   1.366 +        _cycle_path->addBack(_policy[v]);
   1.367 +      }
   1.368 +      return true;
   1.369 +    }
   1.370 +
   1.371 +    /// @}
   1.372 +
   1.373 +    /// \name Query Functions
   1.374 +    /// The results of the algorithm can be obtained using these
   1.375 +    /// functions.\n
   1.376 +    /// The algorithm should be executed before using them.
   1.377 +
   1.378 +    /// @{
   1.379 +
   1.380 +    /// \brief Return the total length of the found cycle.
   1.381 +    ///
   1.382 +    /// This function returns the total length of the found cycle.
   1.383 +    ///
   1.384 +    /// \pre \ref run() or \ref findMinMean() must be called before
   1.385 +    /// using this function.
   1.386 +    LargeValue cycleLength() const {
   1.387 +      return _best_length;
   1.388 +    }
   1.389 +
   1.390 +    /// \brief Return the number of arcs on the found cycle.
   1.391 +    ///
   1.392 +    /// This function returns the number of arcs on the found cycle.
   1.393 +    ///
   1.394 +    /// \pre \ref run() or \ref findMinMean() must be called before
   1.395 +    /// using this function.
   1.396 +    int cycleArcNum() const {
   1.397 +      return _best_size;
   1.398 +    }
   1.399 +
   1.400 +    /// \brief Return the mean length of the found cycle.
   1.401 +    ///
   1.402 +    /// This function returns the mean length of the found cycle.
   1.403 +    ///
   1.404 +    /// \note <tt>alg.cycleMean()</tt> is just a shortcut of the
   1.405 +    /// following code.
   1.406 +    /// \code
   1.407 +    ///   return static_cast<double>(alg.cycleLength()) / alg.cycleArcNum();
   1.408 +    /// \endcode
   1.409 +    ///
   1.410 +    /// \pre \ref run() or \ref findMinMean() must be called before
   1.411 +    /// using this function.
   1.412 +    double cycleMean() const {
   1.413 +      return static_cast<double>(_best_length) / _best_size;
   1.414 +    }
   1.415 +
   1.416 +    /// \brief Return the found cycle.
   1.417 +    ///
   1.418 +    /// This function returns a const reference to the path structure
   1.419 +    /// storing the found cycle.
   1.420 +    ///
   1.421 +    /// \pre \ref run() or \ref findCycle() must be called before using
   1.422 +    /// this function.
   1.423 +    const Path& cycle() const {
   1.424 +      return *_cycle_path;
   1.425 +    }
   1.426 +
   1.427 +    ///@}
   1.428 +
   1.429 +  private:
   1.430 +
   1.431 +    // Initialize
   1.432 +    void init() {
   1.433 +      if (!_cycle_path) {
   1.434 +        _local_path = true;
   1.435 +        _cycle_path = new Path;
   1.436 +      }
   1.437 +      _queue.resize(countNodes(_gr));
   1.438 +      _best_found = false;
   1.439 +      _best_length = 0;
   1.440 +      _best_size = 1;
   1.441 +      _cycle_path->clear();
   1.442 +    }
   1.443 +    
   1.444 +    // Find strongly connected components and initialize _comp_nodes
   1.445 +    // and _in_arcs
   1.446 +    void findComponents() {
   1.447 +      _comp_num = stronglyConnectedComponents(_gr, _comp);
   1.448 +      _comp_nodes.resize(_comp_num);
   1.449 +      if (_comp_num == 1) {
   1.450 +        _comp_nodes[0].clear();
   1.451 +        for (NodeIt n(_gr); n != INVALID; ++n) {
   1.452 +          _comp_nodes[0].push_back(n);
   1.453 +          _in_arcs[n].clear();
   1.454 +          for (InArcIt a(_gr, n); a != INVALID; ++a) {
   1.455 +            _in_arcs[n].push_back(a);
   1.456 +          }
   1.457 +        }
   1.458 +      } else {
   1.459 +        for (int i = 0; i < _comp_num; ++i)
   1.460 +          _comp_nodes[i].clear();
   1.461 +        for (NodeIt n(_gr); n != INVALID; ++n) {
   1.462 +          int k = _comp[n];
   1.463 +          _comp_nodes[k].push_back(n);
   1.464 +          _in_arcs[n].clear();
   1.465 +          for (InArcIt a(_gr, n); a != INVALID; ++a) {
   1.466 +            if (_comp[_gr.source(a)] == k) _in_arcs[n].push_back(a);
   1.467 +          }
   1.468 +        }
   1.469 +      }
   1.470 +    }
   1.471 +
   1.472 +    // Build the policy graph in the given strongly connected component
   1.473 +    // (the out-degree of every node is 1)
   1.474 +    bool buildPolicyGraph(int comp) {
   1.475 +      _nodes = &(_comp_nodes[comp]);
   1.476 +      if (_nodes->size() < 1 ||
   1.477 +          (_nodes->size() == 1 && _in_arcs[(*_nodes)[0]].size() == 0)) {
   1.478 +        return false;
   1.479 +      }
   1.480 +      for (int i = 0; i < int(_nodes->size()); ++i) {
   1.481 +        _dist[(*_nodes)[i]] = INF;
   1.482 +      }
   1.483 +      Node u, v;
   1.484 +      Arc e;
   1.485 +      for (int i = 0; i < int(_nodes->size()); ++i) {
   1.486 +        v = (*_nodes)[i];
   1.487 +        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
   1.488 +          e = _in_arcs[v][j];
   1.489 +          u = _gr.source(e);
   1.490 +          if (_length[e] < _dist[u]) {
   1.491 +            _dist[u] = _length[e];
   1.492 +            _policy[u] = e;
   1.493 +          }
   1.494 +        }
   1.495 +      }
   1.496 +      return true;
   1.497 +    }
   1.498 +
   1.499 +    // Find the minimum mean cycle in the policy graph
   1.500 +    void findPolicyCycle() {
   1.501 +      for (int i = 0; i < int(_nodes->size()); ++i) {
   1.502 +        _level[(*_nodes)[i]] = -1;
   1.503 +      }
   1.504 +      LargeValue clength;
   1.505 +      int csize;
   1.506 +      Node u, v;
   1.507 +      _curr_found = false;
   1.508 +      for (int i = 0; i < int(_nodes->size()); ++i) {
   1.509 +        u = (*_nodes)[i];
   1.510 +        if (_level[u] >= 0) continue;
   1.511 +        for (; _level[u] < 0; u = _gr.target(_policy[u])) {
   1.512 +          _level[u] = i;
   1.513 +        }
   1.514 +        if (_level[u] == i) {
   1.515 +          // A cycle is found
   1.516 +          clength = _length[_policy[u]];
   1.517 +          csize = 1;
   1.518 +          for (v = u; (v = _gr.target(_policy[v])) != u; ) {
   1.519 +            clength += _length[_policy[v]];
   1.520 +            ++csize;
   1.521 +          }
   1.522 +          if ( !_curr_found ||
   1.523 +               (clength * _curr_size < _curr_length * csize) ) {
   1.524 +            _curr_found = true;
   1.525 +            _curr_length = clength;
   1.526 +            _curr_size = csize;
   1.527 +            _curr_node = u;
   1.528 +          }
   1.529 +        }
   1.530 +      }
   1.531 +    }
   1.532 +
   1.533 +    // Contract the policy graph and compute node distances
   1.534 +    bool computeNodeDistances() {
   1.535 +      // Find the component of the main cycle and compute node distances
   1.536 +      // using reverse BFS
   1.537 +      for (int i = 0; i < int(_nodes->size()); ++i) {
   1.538 +        _reached[(*_nodes)[i]] = false;
   1.539 +      }
   1.540 +      _qfront = _qback = 0;
   1.541 +      _queue[0] = _curr_node;
   1.542 +      _reached[_curr_node] = true;
   1.543 +      _dist[_curr_node] = 0;
   1.544 +      Node u, v;
   1.545 +      Arc e;
   1.546 +      while (_qfront <= _qback) {
   1.547 +        v = _queue[_qfront++];
   1.548 +        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
   1.549 +          e = _in_arcs[v][j];
   1.550 +          u = _gr.source(e);
   1.551 +          if (_policy[u] == e && !_reached[u]) {
   1.552 +            _reached[u] = true;
   1.553 +            _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;
   1.554 +            _queue[++_qback] = u;
   1.555 +          }
   1.556 +        }
   1.557 +      }
   1.558 +
   1.559 +      // Connect all other nodes to this component and compute node
   1.560 +      // distances using reverse BFS
   1.561 +      _qfront = 0;
   1.562 +      while (_qback < int(_nodes->size())-1) {
   1.563 +        v = _queue[_qfront++];
   1.564 +        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
   1.565 +          e = _in_arcs[v][j];
   1.566 +          u = _gr.source(e);
   1.567 +          if (!_reached[u]) {
   1.568 +            _reached[u] = true;
   1.569 +            _policy[u] = e;
   1.570 +            _dist[u] = _dist[v] + _length[e] * _curr_size - _curr_length;
   1.571 +            _queue[++_qback] = u;
   1.572 +          }
   1.573 +        }
   1.574 +      }
   1.575 +
   1.576 +      // Improve node distances
   1.577 +      bool improved = false;
   1.578 +      for (int i = 0; i < int(_nodes->size()); ++i) {
   1.579 +        v = (*_nodes)[i];
   1.580 +        for (int j = 0; j < int(_in_arcs[v].size()); ++j) {
   1.581 +          e = _in_arcs[v][j];
   1.582 +          u = _gr.source(e);
   1.583 +          LargeValue delta = _dist[v] + _length[e] * _curr_size - _curr_length;
   1.584 +          if (_tolerance.less(delta, _dist[u])) {
   1.585 +            _dist[u] = delta;
   1.586 +            _policy[u] = e;
   1.587 +            improved = true;
   1.588 +          }
   1.589 +        }
   1.590 +      }
   1.591 +      return improved;
   1.592 +    }
   1.593 +
   1.594 +  }; //class Howard
   1.595 +
   1.596 +  ///@}
   1.597 +
   1.598 +} //namespace lemon
   1.599 +
   1.600 +#endif //LEMON_HOWARD_H