/* -*- 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