/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

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

 *

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

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

 * precise terms see the accompanying LICENSE file.

 *

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

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

 * purpose.

 *

 */

#ifndef LEMON_COST_SCALING_H

#define LEMON_COST_SCALING_H

/// \ingroup min_cost_flow_algs

/// \file

/// \brief Cost scaling algorithm for finding a minimum cost flow.

#include <vector>

#include <deque>

#include <limits>

#include <lemon/core.h>

#include <lemon/maps.h>

#include <lemon/math.h>

#include <lemon/adaptors.h>

#include <lemon/circulation.h>

#include <lemon/bellman_ford.h>

namespace lemon {

  /// \addtogroup min_cost_flow_algs

  /// @{

  /// \brief Implementation of the cost scaling algorithm for finding a

  /// minimum cost flow.

  ///

  /// \ref CostScaling implements the cost scaling algorithm performing

  /// augment/push and relabel operations for finding a minimum cost

  /// flow.

  ///

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

  /// \tparam LowerMap The type of the lower bound map.

  /// \tparam CapacityMap The type of the capacity (upper bound) map.

  /// \tparam CostMap The type of the cost (length) map.

  /// \tparam SupplyMap The type of the supply map.

  ///

  /// \warning

  /// - Arc capacities and costs should be \e non-negative \e integers.

  /// - Supply values should be \e signed \e integers.

  /// - The value types of the maps should be convertible to each other.

  /// - \c CostMap::Value must be signed type.

  ///

  /// \note Arc costs are multiplied with the number of nodes during

  /// the algorithm so overflow problems may arise more easily than with

  /// other minimum cost flow algorithms.

  /// If it is available, <tt>long long int</tt> type is used instead of

  /// <tt>long int</tt> in the inside computations.

  ///

  /// \author Peter Kovacs

  template < typename Digraph,

             typename LowerMap = typename Digraph::template ArcMap<int>,

             typename CapacityMap = typename Digraph::template ArcMap<int>,

             typename CostMap = typename Digraph::template ArcMap<int>,

             typename SupplyMap = typename Digraph::template NodeMap<int> >

  class CostScaling

  {

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    typedef typename CapacityMap::Value Capacity;

    typedef typename CostMap::Value Cost;

    typedef typename SupplyMap::Value Supply;

    typedef typename Digraph::template ArcMap<Capacity> CapacityArcMap;

    typedef typename Digraph::template NodeMap<Supply> SupplyNodeMap;

    typedef ResidualDigraph< const Digraph,

                             CapacityArcMap, CapacityArcMap > ResDigraph;

    typedef typename ResDigraph::Arc ResArc;

#if defined __GNUC__ && !defined __STRICT_ANSI__

    typedef long long int LCost;

#else

    typedef long int LCost;

#endif

    typedef typename Digraph::template ArcMap<LCost> LargeCostMap;

  public:

    /// The type of the flow map.

    typedef typename Digraph::template ArcMap<Capacity> FlowMap;

    /// The type of the potential map.

    typedef typename Digraph::template NodeMap<LCost> PotentialMap;

  private:

    /// \brief Map adaptor class for handling residual arc costs.

    ///

    /// Map adaptor class for handling residual arc costs.

    template <typename Map>

    class ResidualCostMap : public MapBase<ResArc, typename Map::Value>

    {

    private:

      const Map &_cost_map;

    public:

      ///\e

      ResidualCostMap(const Map &cost_map) :

        _cost_map(cost_map) {}

      ///\e

      inline typename Map::Value operator[](const ResArc &e) const {

        return ResDigraph::forward(e) ? _cost_map[e] : -_cost_map[e];

      }

    }; //class ResidualCostMap

    /// \brief Map adaptor class for handling reduced arc costs.

    ///

    /// Map adaptor class for handling reduced arc costs.

    class ReducedCostMap : public MapBase<Arc, LCost>

    {

    private:

      const Digraph &_gr;

      const LargeCostMap &_cost_map;

      const PotentialMap &_pot_map;

    public:

      ///\e

      ReducedCostMap( const Digraph &gr,

                      const LargeCostMap &cost_map,

                      const PotentialMap &pot_map ) :

        _gr(gr), _cost_map(cost_map), _pot_map(pot_map) {}

      ///\e

      inline LCost operator[](const Arc &e) const {

        return _cost_map[e] + _pot_map[_gr.source(e)]

                            - _pot_map[_gr.target(e)];

      }

    }; //class ReducedCostMap

  private:

    // The digraph the algorithm runs on

    const Digraph &_graph;

    // The original lower bound map

    const LowerMap *_lower;

    // The modified capacity map

    CapacityArcMap _capacity;

    // The original cost map

    const CostMap &_orig_cost;

    // The scaled cost map

    LargeCostMap _cost;

    // The modified supply map

    SupplyNodeMap _supply;

    bool _valid_supply;

    // Arc map of the current flow

    FlowMap *_flow;

    bool _local_flow;

    // Node map of the current potentials

    PotentialMap *_potential;

    bool _local_potential;

    // The residual cost map

    ResidualCostMap<LargeCostMap> _res_cost;

    // The residual digraph

    ResDigraph *_res_graph;

    // The reduced cost map

    ReducedCostMap *_red_cost;

    // The excess map

    SupplyNodeMap _excess;

    // The epsilon parameter used for cost scaling

    LCost _epsilon;

    // The scaling factor

    int _alpha;

  public:

    /// \brief General constructor (with lower bounds).

    ///

    /// General constructor (with lower bounds).

    ///

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

    /// \param lower The lower bounds of the arcs.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param supply The supply values of the nodes (signed).

    CostScaling( const Digraph &digraph,

                 const LowerMap &lower,

                 const CapacityMap &capacity,

                 const CostMap &cost,

                 const SupplyMap &supply ) :

      _graph(digraph), _lower(&lower), _capacity(digraph), _orig_cost(cost),

      _cost(digraph), _supply(digraph), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false), _res_cost(_cost),

      _res_graph(NULL), _red_cost(NULL), _excess(digraph, 0)

    {

      // Check the sum of supply values

      Supply sum = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n];

      _valid_supply = sum == 0;

      for (ArcIt e(_graph); e != INVALID; ++e) _capacity[e] = capacity[e];

      for (NodeIt n(_graph); n != INVALID; ++n) _supply[n] = supply[n];

      // Remove non-zero lower bounds

      for (ArcIt e(_graph); e != INVALID; ++e) {

        if (lower[e] != 0) {

          _capacity[e] -= lower[e];

          _supply[_graph.source(e)] -= lower[e];

          _supply[_graph.target(e)] += lower[e];

        }

      }

    }

/*

    /// \brief General constructor (without lower bounds).

    ///

    /// General constructor (without lower bounds).

    ///

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

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param supply The supply values of the nodes (signed).

    CostScaling( const Digraph &digraph,

                 const CapacityMap &capacity,

                 const CostMap &cost,

                 const SupplyMap &supply ) :

      _graph(digraph), _lower(NULL), _capacity(capacity), _orig_cost(cost),

      _cost(digraph), _supply(supply), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false), _res_cost(_cost),

      _res_graph(NULL), _red_cost(NULL), _excess(digraph, 0)

    {

      // Check the sum of supply values

      Supply sum = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n];

      _valid_supply = sum == 0;

    }

    /// \brief Simple constructor (with lower bounds).

    ///

    /// Simple constructor (with lower bounds).

    ///

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

    /// \param lower The lower bounds of the arcs.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param s The source node.

    /// \param t The target node.

    /// \param flow_value 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).

    CostScaling( const Digraph &digraph,

                 const LowerMap &lower,

                 const CapacityMap &capacity,

                 const CostMap &cost,

                 Node s, Node t,

                 Supply flow_value ) :

      _graph(digraph), _lower(&lower), _capacity(capacity), _orig_cost(cost),

      _cost(digraph), _supply(digraph, 0), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false), _res_cost(_cost),

      _res_graph(NULL), _red_cost(NULL), _excess(digraph, 0)

    {

      // Remove non-zero lower bounds

      _supply[s] =  flow_value;

      _supply[t] = -flow_value;

      for (ArcIt e(_graph); e != INVALID; ++e) {

        if (lower[e] != 0) {

          _capacity[e] -= lower[e];

          _supply[_graph.source(e)] -= lower[e];

          _supply[_graph.target(e)] += lower[e];

        }

      }

      _valid_supply = true;

    }

    /// \brief Simple constructor (without lower bounds).

    ///

    /// Simple constructor (without lower bounds).

    ///

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

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param s The source node.

    /// \param t The target node.

    /// \param flow_value 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).

    CostScaling( const Digraph &digraph,

                 const CapacityMap &capacity,

                 const CostMap &cost,

                 Node s, Node t,

                 Supply flow_value ) :

      _graph(digraph), _lower(NULL), _capacity(capacity), _orig_cost(cost),

      _cost(digraph), _supply(digraph, 0), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false), _res_cost(_cost),

      _res_graph(NULL), _red_cost(NULL), _excess(digraph, 0)

    {

      _supply[s] =  flow_value;

      _supply[t] = -flow_value;

      _valid_supply = true;

    }

*/

    /// Destructor.

    ~CostScaling() {

      if (_local_flow) delete _flow;

      if (_local_potential) delete _potential;

      delete _res_graph;

      delete _red_cost;

    }

    /// \brief Set the flow map.

    ///

    /// Set the flow map.

    ///

    /// \return \c (*this)

    CostScaling& flowMap(FlowMap &map) {

      if (_local_flow) {

        delete _flow;

        _local_flow = false;

      }

      _flow = ↦

      return *this;

    }

    /// \brief Set the potential map.

    ///

    /// Set the potential map.

    ///

    /// \return \c (*this)

    CostScaling& potentialMap(PotentialMap &map) {

      if (_local_potential) {

        delete _potential;

        _local_potential = false;

      }

      _potential = ↦

      return *this;

    }

    /// \name Execution control

    /// @{

    /// \brief Run the algorithm.

    ///

    /// Run the algorithm.

    ///

    /// \param partial_augment By default the algorithm performs

    /// partial augment and relabel operations in the cost scaling

    /// phases. Set this parameter to \c false for using local push and

    /// relabel operations instead.

    ///

    /// \return \c true if a feasible flow can be found.

    bool run(bool partial_augment = true) {

      if (partial_augment) {

        return init() && startPartialAugment();

      } else {

        return init() && startPushRelabel();

      }

    }

    /// @}

    /// \name Query Functions

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

    /// functions.\n

    /// \ref lemon::CostScaling::run() "run()" must be called before

    /// using them.

    /// @{

    /// \brief Return a const reference to the arc map storing the

    /// found flow.

    ///

    /// Return a const reference to the arc map storing the found flow.

    ///

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

    const FlowMap& flowMap() const {

      return *_flow;

    }

    /// \brief Return a const reference to the node map storing the

    /// found potentials (the dual solution).

    ///

    /// Return a const reference to the node map storing the found

    /// potentials (the dual solution).

    ///

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

    const PotentialMap& potentialMap() const {

      return *_potential;

    }

    /// \brief Return the flow on the given arc.

    ///

    /// Return the flow on the given arc.

    ///

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

    Capacity flow(const Arc& arc) const {

      return (*_flow)[arc];

    }

    /// \brief Return the potential of the given node.

    ///

    /// Return the potential of the given node.

    ///

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

    Cost potential(const Node& node) const {

      return (*_potential)[node];

    }

    /// \brief Return the total cost of the found flow.

    ///

    /// Return the total cost of the found flow. The complexity of the

    /// function is \f$ O(e) \f$.

    ///

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

    Cost totalCost() const {

      Cost c = 0;

      for (ArcIt e(_graph); e != INVALID; ++e)

        c += (*_flow)[e] * _orig_cost[e];

      return c;

    }

    /// @}

  private:

    /// Initialize the algorithm.

    bool init() {

      if (!_valid_supply) return false;

      // The scaling factor

      _alpha = 8;

      // Initialize flow and potential maps

      if (!_flow) {

        _flow = new FlowMap(_graph);

        _local_flow = true;

      }

      if (!_potential) {

        _potential = new PotentialMap(_graph);

        _local_potential = true;

      }

      _red_cost = new ReducedCostMap(_graph, _cost, *_potential);

      _res_graph = new ResDigraph(_graph, _capacity, *_flow);

      // Initialize the scaled cost map and the epsilon parameter

      Cost max_cost = 0;

      int node_num = countNodes(_graph);

      for (ArcIt e(_graph); e != INVALID; ++e) {

        _cost[e] = LCost(_orig_cost[e]) * node_num * _alpha;

        if (_orig_cost[e] > max_cost) max_cost = _orig_cost[e];

      }

      _epsilon = max_cost * node_num;

      // Find a feasible flow using Circulation

      Circulation< Digraph, ConstMap<Arc, Capacity>, CapacityArcMap,

                   SupplyMap >

        circulation( _graph, constMap<Arc>(Capacity(0)), _capacity,

                     _supply );

      return circulation.flowMap(*_flow).run();

    }

    /// Execute the algorithm performing partial augmentation and

    /// relabel operations.

    bool startPartialAugment() {

      // Paramters for heuristics

//      const int BF_HEURISTIC_EPSILON_BOUND = 1000;

//      const int BF_HEURISTIC_BOUND_FACTOR  = 3;

      // Maximum augment path length

      const int MAX_PATH_LENGTH = 4;

      // Variables

      typename Digraph::template NodeMap<Arc> pred_arc(_graph);

      typename Digraph::template NodeMap<bool> forward(_graph);

      typename Digraph::template NodeMap<OutArcIt> next_out(_graph);

      typename Digraph::template NodeMap<InArcIt> next_in(_graph);

      typename Digraph::template NodeMap<bool> next_dir(_graph);

      std::deque<Node> active_nodes;

      std::vector<Node> path_nodes;

//      int node_num = countNodes(_graph);

      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?

                                        1 : _epsilon / _alpha )

      {

/*

        // "Early Termination" heuristic: use Bellman-Ford algorithm

        // to check if the current flow is optimal

        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {

          typedef ShiftMap< ResidualCostMap<LargeCostMap> > ShiftCostMap;

          ShiftCostMap shift_cost(_res_cost, 1);

          BellmanFord<ResDigraph, ShiftCostMap> bf(*_res_graph, shift_cost);

          bf.init(0);

          bool done = false;

          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(node_num));

          for (int i = 0; i < K && !done; ++i)

            done = bf.processNextWeakRound();

          if (done) break;

        }

*/

        // Saturate arcs not satisfying the optimality condition

        Capacity delta;

        for (ArcIt e(_graph); e != INVALID; ++e) {

          if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {

            delta = _capacity[e] - (*_flow)[e];

            _excess[_graph.source(e)] -= delta;

            _excess[_graph.target(e)] += delta;

            (*_flow)[e] = _capacity[e];

          }

          if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {

            _excess[_graph.target(e)] -= (*_flow)[e];

            _excess[_graph.source(e)] += (*_flow)[e];

            (*_flow)[e] = 0;

          }

        }

        // Find active nodes (i.e. nodes with positive excess)

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

          if (_excess[n] > 0) active_nodes.push_back(n);

        }

        // Initialize the next arc maps

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

          next_out[n] = OutArcIt(_graph, n);

          next_in[n] = InArcIt(_graph, n);

          next_dir[n] = true;

        }

        // Perform partial augment and relabel operations

        while (active_nodes.size() > 0) {

          // Select an active node (FIFO selection)

          if (_excess[active_nodes[0]] <= 0) {

            active_nodes.pop_front();

            continue;

          }

          Node start = active_nodes[0];

          path_nodes.clear();

          path_nodes.push_back(start);

          // Find an augmenting path from the start node

          Node u, tip = start;

          LCost min_red_cost;

          while ( _excess[tip] >= 0 &&

                  int(path_nodes.size()) <= MAX_PATH_LENGTH )

          {

            if (next_dir[tip]) {

              for (OutArcIt e = next_out[tip]; e != INVALID; ++e) {

                if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {

                  u = _graph.target(e);

                  pred_arc[u] = e;

                  forward[u] = true;

                  next_out[tip] = e;

                  tip = u;

                  path_nodes.push_back(tip);

                  goto next_step;

                }

              }

              next_dir[tip] = false;

            }

            for (InArcIt e = next_in[tip]; e != INVALID; ++e) {

              if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {

                u = _graph.source(e);

                pred_arc[u] = e;

                forward[u] = false;

                next_in[tip] = e;

                tip = u;

                path_nodes.push_back(tip);

                goto next_step;

              }

            }

            // Relabel tip node

            min_red_cost = std::numeric_limits<LCost>::max() / 2;

            for (OutArcIt oe(_graph, tip); oe != INVALID; ++oe) {

              if ( _capacity[oe] - (*_flow)[oe] > 0 &&

                   (*_red_cost)[oe] < min_red_cost )

                min_red_cost = (*_red_cost)[oe];

            }

            for (InArcIt ie(_graph, tip); ie != INVALID; ++ie) {

              if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < min_red_cost)

                min_red_cost = -(*_red_cost)[ie];

            }

            (*_potential)[tip] -= min_red_cost + _epsilon;

            // Reset the next arc maps

            next_out[tip] = OutArcIt(_graph, tip);

            next_in[tip] = InArcIt(_graph, tip);

            next_dir[tip] = true;

            // Step back

            if (tip != start) {

              path_nodes.pop_back();

              tip = path_nodes[path_nodes.size()-1];

            }

          next_step:

            continue;

          }

          // Augment along the found path (as much flow as possible)

          Capacity delta;

          for (int i = 1; i < int(path_nodes.size()); ++i) {

            u = path_nodes[i];

            delta = forward[u] ?

              _capacity[pred_arc[u]] - (*_flow)[pred_arc[u]] :

              (*_flow)[pred_arc[u]];

            delta = std::min(delta, _excess[path_nodes[i-1]]);

            (*_flow)[pred_arc[u]] += forward[u] ? delta : -delta;

            _excess[path_nodes[i-1]] -= delta;

            _excess[u] += delta;

            if (_excess[u] > 0 && _excess[u] <= delta) active_nodes.push_back(u);

          }

        }

      }

      // Compute node potentials for the original costs

      ResidualCostMap<CostMap> res_cost(_orig_cost);

      BellmanFord< ResDigraph, ResidualCostMap<CostMap> >

        bf(*_res_graph, res_cost);

      bf.init(0); bf.start();

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

        (*_potential)[n] = bf.dist(n);

      // Handle non-zero lower bounds

      if (_lower) {

        for (ArcIt e(_graph); e != INVALID; ++e)

          (*_flow)[e] += (*_lower)[e];

      }

      return true;

    }

    /// Execute the algorithm performing push and relabel operations.

    bool startPushRelabel() {

      // Paramters for heuristics

//      const int BF_HEURISTIC_EPSILON_BOUND = 1000;

//      const int BF_HEURISTIC_BOUND_FACTOR  = 3;

      typename Digraph::template NodeMap<bool> hyper(_graph, false);

      typename Digraph::template NodeMap<Arc> pred_arc(_graph);

      typename Digraph::template NodeMap<bool> forward(_graph);

      typename Digraph::template NodeMap<OutArcIt> next_out(_graph);

      typename Digraph::template NodeMap<InArcIt> next_in(_graph);

      typename Digraph::template NodeMap<bool> next_dir(_graph);

      std::deque<Node> active_nodes;

//      int node_num = countNodes(_graph);

      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?

                                        1 : _epsilon / _alpha )

      {

/*

        // "Early Termination" heuristic: use Bellman-Ford algorithm

        // to check if the current flow is optimal

        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {

          typedef ShiftMap< ResidualCostMap<LargeCostMap> > ShiftCostMap;

          ShiftCostMap shift_cost(_res_cost, 1);

          BellmanFord<ResDigraph, ShiftCostMap> bf(*_res_graph, shift_cost);

          bf.init(0);

          bool done = false;

          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(node_num));

          for (int i = 0; i < K && !done; ++i)

            done = bf.processNextWeakRound();

          if (done) break;

        }

*/

        // Saturate arcs not satisfying the optimality condition

        Capacity delta;

        for (ArcIt e(_graph); e != INVALID; ++e) {

          if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {

            delta = _capacity[e] - (*_flow)[e];

            _excess[_graph.source(e)] -= delta;

            _excess[_graph.target(e)] += delta;

            (*_flow)[e] = _capacity[e];

          }

          if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {

            _excess[_graph.target(e)] -= (*_flow)[e];

            _excess[_graph.source(e)] += (*_flow)[e];

            (*_flow)[e] = 0;

          }

        }

        // Find active nodes (i.e. nodes with positive excess)

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

          if (_excess[n] > 0) active_nodes.push_back(n);

        }

        // Initialize the next arc maps

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

          next_out[n] = OutArcIt(_graph, n);

          next_in[n] = InArcIt(_graph, n);

          next_dir[n] = true;

        }

        // Perform push and relabel operations

        while (active_nodes.size() > 0) {

          // Select an active node (FIFO selection)

          Node n = active_nodes[0], t;

          bool relabel_enabled = true;

          // Perform push operations if there are admissible arcs

          if (_excess[n] > 0 && next_dir[n]) {

            OutArcIt e = next_out[n];

            for ( ; e != INVALID; ++e) {

              if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {

                delta = std::min(_capacity[e] - (*_flow)[e], _excess[n]);

                t = _graph.target(e);

                // Push-look-ahead heuristic

                Capacity ahead = -_excess[t];

                for (OutArcIt oe(_graph, t); oe != INVALID; ++oe) {

                  if (_capacity[oe] - (*_flow)[oe] > 0 && (*_red_cost)[oe] < 0)

                    ahead += _capacity[oe] - (*_flow)[oe];

                }

                for (InArcIt ie(_graph, t); ie != INVALID; ++ie) {

                  if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < 0)

                    ahead += (*_flow)[ie];

                }

                if (ahead < 0) ahead = 0;

                // Push flow along the arc

                if (ahead < delta) {

                  (*_flow)[e] += ahead;

                  _excess[n] -= ahead;

                  _excess[t] += ahead;

                  active_nodes.push_front(t);

                  hyper[t] = true;

                  relabel_enabled = false;

                  break;

                } else {

                  (*_flow)[e] += delta;

                  _excess[n] -= delta;

                  _excess[t] += delta;

                  if (_excess[t] > 0 && _excess[t] <= delta)

                    active_nodes.push_back(t);

                }

                if (_excess[n] == 0) break;

              }

            }

            if (e != INVALID) {

              next_out[n] = e;

            } else {

              next_dir[n] = false;

            }

          }

          if (_excess[n] > 0 && !next_dir[n]) {

            InArcIt e = next_in[n];

            for ( ; e != INVALID; ++e) {

              if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {

                delta = std::min((*_flow)[e], _excess[n]);

                t = _graph.source(e);

                // Push-look-ahead heuristic

                Capacity ahead = -_excess[t];

                for (OutArcIt oe(_graph, t); oe != INVALID; ++oe) {

                  if (_capacity[oe] - (*_flow)[oe] > 0 && (*_red_cost)[oe] < 0)

                    ahead += _capacity[oe] - (*_flow)[oe];

                }

                for (InArcIt ie(_graph, t); ie != INVALID; ++ie) {

                  if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < 0)

                    ahead += (*_flow)[ie];

                }

                if (ahead < 0) ahead = 0;

                // Push flow along the arc

                if (ahead < delta) {

                  (*_flow)[e] -= ahead;

                  _excess[n] -= ahead;

                  _excess[t] += ahead;

                  active_nodes.push_front(t);

                  hyper[t] = true;

                  relabel_enabled = false;

                  break;

                } else {

                  (*_flow)[e] -= delta;

                  _excess[n] -= delta;

                  _excess[t] += delta;

                  if (_excess[t] > 0 && _excess[t] <= delta)

                    active_nodes.push_back(t);

                }

                if (_excess[n] == 0) break;

              }

            }

            next_in[n] = e;

          }

          // Relabel the node if it is still active (or hyper)

          if (relabel_enabled && (_excess[n] > 0 || hyper[n])) {

            LCost min_red_cost = std::numeric_limits<LCost>::max() / 2;

            for (OutArcIt oe(_graph, n); oe != INVALID; ++oe) {

              if ( _capacity[oe] - (*_flow)[oe] > 0 &&

                   (*_red_cost)[oe] < min_red_cost )

                min_red_cost = (*_red_cost)[oe];

            }

            for (InArcIt ie(_graph, n); ie != INVALID; ++ie) {

              if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < min_red_cost)

                min_red_cost = -(*_red_cost)[ie];

            }

            (*_potential)[n] -= min_red_cost + _epsilon;

            hyper[n] = false;

            // Reset the next arc maps

            next_out[n] = OutArcIt(_graph, n);

            next_in[n] = InArcIt(_graph, n);

            next_dir[n] = true;

          }

          // Remove nodes that are not active nor hyper

          while ( active_nodes.size() > 0 &&

                  _excess[active_nodes[0]] <= 0 &&

                  !hyper[active_nodes[0]] ) {

            active_nodes.pop_front();

          }

        }

      }

      // Compute node potentials for the original costs

      ResidualCostMap<CostMap> res_cost(_orig_cost);

      BellmanFord< ResDigraph, ResidualCostMap<CostMap> >

        bf(*_res_graph, res_cost);

      bf.init(0); bf.start();

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

        (*_potential)[n] = bf.dist(n);

      // Handle non-zero lower bounds

      if (_lower) {

        for (ArcIt e(_graph); e != INVALID; ++e)

          (*_flow)[e] += (*_lower)[e];

      }

      return true;

    }

  }; //class CostScaling

  ///@}

} //namespace lemon

#endif //LEMON_COST_SCALING_H