/* -*- 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/static_graph.h>

 #include <lemon/circulation.h>

 #include <lemon/bellman_ford.h>

 namespace lemon {

   /// \brief Default traits class of CostScaling algorithm.

   ///

   /// Default traits class of CostScaling algorithm.

   /// \tparam GR Digraph type.

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

   /// and supply values. By default it is \c int.

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

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

 #ifdef DOXYGEN

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

 #else

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

              bool integer = std::numeric_limits<C>::is_integer >

 #endif

   struct CostScalingDefaultTraits

   {

     /// The type of the digraph

     typedef GR Digraph;

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

     typedef V Value;

     /// The type of the arc costs

     typedef C Cost;

     /// \brief The large cost type used for internal computations

     ///

     /// The large cost type used for internal computations.

     /// It is \c long \c long if the \c Cost type is integer,

     /// otherwise it is \c double.

     /// \c Cost must be convertible to \c LargeCost.

     typedef double LargeCost;

   };

   // Default traits class for integer cost types

   template <typename GR, typename V, typename C>

   struct CostScalingDefaultTraits<GR, V, C, true>

   {

     typedef GR Digraph;

     typedef V Value;

     typedef C Cost;

 #ifdef LEMON_HAVE_LONG_LONG

     typedef long long LargeCost;

 #else

     typedef long LargeCost;

 #endif

   };

   /// \addtogroup min_cost_flow_algs

   /// @{

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

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

   ///

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

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

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

   /// \ref goldberg97efficient, \ref bunnagel98efficient. 

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

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

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

   ///

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

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

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

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

   ///

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

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

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

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

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

   ///

   /// \warning Both number types must be signed and all input data must

   /// be integer.

   /// \warning This algorithm does not support negative costs for such

   /// arcs that have infinite upper bound.

   ///

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

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

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

 #ifdef DOXYGEN

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

 #else

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

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

 #endif

   class CostScaling

   {

   public:

     /// The type of the digraph

     typedef typename TR::Digraph Digraph;

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

     typedef typename TR::Value Value;

     /// The type of the arc costs

     typedef typename TR::Cost Cost;

     /// \brief The large cost type

     ///

     /// The large cost type used for internal computations.

     /// Using the \ref CostScalingDefaultTraits "default traits class",

     /// it is \c long \c long if the \c Cost type is integer,

     /// otherwise it is \c double.

     typedef typename TR::LargeCost LargeCost;

     /// The \ref CostScalingDefaultTraits "traits class" of the algorithm

     typedef TR Traits;

   public:

     /// \brief Problem type constants for the \c run() function.

     ///

     /// Enum type containing the problem type constants that can be

     /// returned by the \ref run() function of the algorithm.

     enum ProblemType {

       /// The problem has no feasible solution (flow).

       INFEASIBLE,

       /// The problem has optimal solution (i.e. it is feasible and

       /// bounded), and the algorithm has found optimal flow and node

       /// potentials (primal and dual solutions).

       OPTIMAL,

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

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

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

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

       /// these cases.

       UNBOUNDED

     };

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

     ///

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

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

     ///

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

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

     /// relabel operation.

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

     /// "Partial Augment-Relabel" method is used, which proved to be

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

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

     /// function with the proper parameter.

     enum Method {

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

       /// admissible arc at once.

       PUSH,

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

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

       AUGMENT,

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

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

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

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

       PARTIAL_AUGMENT

     };

   private:

     TEMPLATE_DIGRAPH_TYPEDEFS(GR);

     typedef std::vector<int> IntVector;

     typedef std::vector<char> BoolVector;

     typedef std::vector<Value> ValueVector;

     typedef std::vector<Cost> CostVector;

     typedef std::vector<LargeCost> LargeCostVector;

   private:

     template <typename KT, typename VT>

     class StaticVectorMap {

     public:

       typedef KT Key;

       typedef VT Value;

       StaticVectorMap(std::vector<Value>& v) : _v(v) {}

       const Value& operator[](const Key& key) const {

         return _v[StaticDigraph::id(key)];

       }

       Value& operator[](const Key& key) {

         return _v[StaticDigraph::id(key)];

       }

       void set(const Key& key, const Value& val) {

         _v[StaticDigraph::id(key)] = val;

       }

     private:

       std::vector<Value>& _v;

     };

     typedef StaticVectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;

     typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;

   private:

     // Data related to the underlying digraph

     const GR &_graph;

     int _node_num;

     int _arc_num;

     int _res_node_num;

     int _res_arc_num;

     int _root;

     // Parameters of the problem

     bool _have_lower;

     Value _sum_supply;

     // Data structures for storing the digraph

     IntNodeMap _node_id;

     IntArcMap _arc_idf;

     IntArcMap _arc_idb;

     IntVector _first_out;

     BoolVector _forward;

     IntVector _source;

     IntVector _target;

     IntVector _reverse;

     // Node and arc data

     ValueVector _lower;

     ValueVector _upper;

     CostVector _scost;

     ValueVector _supply;

     ValueVector _res_cap;

     LargeCostVector _cost;

     LargeCostVector _pi;

     ValueVector _excess;

     IntVector _next_out;

     std::deque<int> _active_nodes;

     // Data for scaling

     LargeCost _epsilon;

     int _alpha;

     // Data for a StaticDigraph structure

     typedef std::pair<int, int> IntPair;

     StaticDigraph _sgr;

     std::vector<IntPair> _arc_vec;

     std::vector<LargeCost> _cost_vec;

     LargeCostArcMap _cost_map;

     LargeCostNodeMap _pi_map;

   public:

     /// \brief Constant for infinite upper bounds (capacities).

     ///

     /// Constant for infinite upper bounds (capacities).

     /// It is \c std::numeric_limits<Value>::infinity() if available,

     /// \c std::numeric_limits<Value>::max() otherwise.

     const Value INF;

   public:

     /// \name Named Template Parameters

     /// @{

     template <typename T>

     struct SetLargeCostTraits : public Traits {

       typedef T LargeCost;

     };

     /// \brief \ref named-templ-param "Named parameter" for setting

     /// \c LargeCost type.

     ///

     /// \ref named-templ-param "Named parameter" for setting \c LargeCost

     /// type, which is used for internal computations in the algorithm.

     /// \c Cost must be convertible to \c LargeCost.

     template <typename T>

     struct SetLargeCost

       : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {

       typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;

     };

     /// @}

   public:

     /// \brief Constructor.

     ///

     /// The constructor of the class.

     ///

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

     CostScaling(const GR& graph) :

       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),

       _cost_map(_cost_vec), _pi_map(_pi),

       INF(std::numeric_limits<Value>::has_infinity ?

           std::numeric_limits<Value>::infinity() :

           std::numeric_limits<Value>::max())

     {

       // Check the number types

       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,

         "The flow type of CostScaling must be signed");

       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,

         "The cost type of CostScaling must be signed");

       // Resize vectors

       _node_num = countNodes(_graph);

       _arc_num = countArcs(_graph);

       _res_node_num = _node_num + 1;

       _res_arc_num = 2 * (_arc_num + _node_num);

       _root = _node_num;

       _first_out.resize(_res_node_num + 1);

       _forward.resize(_res_arc_num);

       _source.resize(_res_arc_num);

       _target.resize(_res_arc_num);

       _reverse.resize(_res_arc_num);

       _lower.resize(_res_arc_num);

       _upper.resize(_res_arc_num);

       _scost.resize(_res_arc_num);

       _supply.resize(_res_node_num);

       _res_cap.resize(_res_arc_num);

       _cost.resize(_res_arc_num);

       _pi.resize(_res_node_num);

       _excess.resize(_res_node_num);

       _next_out.resize(_res_node_num);

       _arc_vec.reserve(_res_arc_num);

       _cost_vec.reserve(_res_arc_num);

       // Copy the graph

       int i = 0, j = 0, k = 2 * _arc_num + _node_num;

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

         _node_id[n] = i;

       }

       i = 0;

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

         _first_out[i] = j;

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

           _arc_idf[a] = j;

           _forward[j] = true;

           _source[j] = i;

           _target[j] = _node_id[_graph.runningNode(a)];

         }

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

           _arc_idb[a] = j;

           _forward[j] = false;

           _source[j] = i;

           _target[j] = _node_id[_graph.runningNode(a)];

         }

         _forward[j] = false;

         _source[j] = i;

         _target[j] = _root;

         _reverse[j] = k;

         _forward[k] = true;

         _source[k] = _root;

         _target[k] = i;

         _reverse[k] = j;

         ++j; ++k;

       }

       _first_out[i] = j;

       _first_out[_res_node_num] = k;

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

         int fi = _arc_idf[a];

         int bi = _arc_idb[a];

         _reverse[fi] = bi;

         _reverse[bi] = fi;

       }

       // Reset parameters

       reset();

     }

     /// \name Parameters

     /// The parameters of the algorithm can be specified using these

     /// functions.

     /// @{

     /// \brief Set the lower bounds on the arcs.

     ///

     /// This function sets the lower bounds on the arcs.

     /// If it is not used before calling \ref run(), the lower bounds

     /// will be set to zero on all arcs.

     ///

     /// \param map An arc map storing the lower bounds.

     /// Its \c Value type must be convertible to the \c Value type

     /// of the algorithm.

     ///

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

     template <typename LowerMap>

     CostScaling& lowerMap(const LowerMap& map) {

       _have_lower = true;

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

         _lower[_arc_idf[a]] = map[a];

         _lower[_arc_idb[a]] = map[a];

       }

       return *this;

     }

     /// \brief Set the upper bounds (capacities) on the arcs.

     ///

     /// This function sets the upper bounds (capacities) on the arcs.

     /// If it is not used before calling \ref run(), the upper bounds

     /// will be set to \ref INF on all arcs (i.e. the flow value will be

     /// unbounded from above).

     ///

     /// \param map An arc map storing the upper bounds.

     /// Its \c Value type must be convertible to the \c Value type

     /// of the algorithm.

     ///

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

     template<typename UpperMap>

     CostScaling& upperMap(const UpperMap& map) {

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

         _upper[_arc_idf[a]] = map[a];

       }

       return *this;

     }

     /// \brief Set the costs of the arcs.

     ///

     /// This function sets the costs of the arcs.

     /// If it is not used before calling \ref run(), the costs

     /// will be set to \c 1 on all arcs.

     ///

     /// \param map An arc map storing the costs.

     /// Its \c Value type must be convertible to the \c Cost type

     /// of the algorithm.

     ///

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

     template<typename CostMap>

     CostScaling& costMap(const CostMap& map) {

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

         _scost[_arc_idf[a]] =  map[a];

         _scost[_arc_idb[a]] = -map[a];

       }

       return *this;

     }

     /// \brief Set the supply values of the nodes.

     ///

     /// This function sets the supply values of the nodes.

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

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

     ///

     /// \param map A node map storing the supply values.

     /// Its \c Value type must be convertible to the \c Value type

     /// of the algorithm.

     ///

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

     template<typename SupplyMap>

     CostScaling& supplyMap(const SupplyMap& map) {

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

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

       }

       return *this;

     }

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

     ///

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

     /// and the required flow value.

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

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

     ///

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

     /// with such a map in which \c k is assigned to \c s, \c -k is

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

     ///

     /// \param s The source node.

     /// \param t The target node.

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

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

     ///

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

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

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

         _supply[i] = 0;

       }

       _supply[_node_id[s]] =  k;

       _supply[_node_id[t]] = -k;

       return *this;

     }

     /// @}

     /// \name Execution control

     /// The algorithm can be executed using \ref run().

     /// @{

     /// \brief Run the algorithm.

     ///

     /// This function runs the algorithm.

     /// The paramters can be specified using functions \ref lowerMap(),

     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().

     /// For example,

     /// \code

     ///   CostScaling<ListDigraph> cs(graph);

     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)

     ///     .supplyMap(sup).run();

     /// \endcode

     ///

     /// This function can be called more than once. All the parameters

     /// that have been given are kept for the next call, unless

     /// \ref reset() is called, thus only the modified parameters

     /// have to be set again. See \ref reset() for examples.

     /// However, the underlying digraph must not be modified after this

     /// class have been constructed, since it copies and extends the graph.

     ///

     /// \param method The internal method that will be used in the

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

     /// \param factor The cost scaling factor. It must be larger than one.

     ///

     /// \return \c INFEASIBLE if no feasible flow exists,

     /// \n \c OPTIMAL if the problem has optimal solution

     /// (i.e. it is feasible and bounded), and the algorithm has found

     /// optimal flow and node potentials (primal and dual solutions),

     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost

     /// and infinite upper bound. It means that the objective function

     /// is unbounded on that arc, however, note that it could actually be

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

     /// these cases.

     ///

     /// \see ProblemType, Method

     ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {

       _alpha = factor;

       ProblemType pt = init();

       if (pt != OPTIMAL) return pt;

       start(method);

       return OPTIMAL;

     }

     /// \brief Reset all the parameters that have been given before.

     ///

     /// This function resets all the paramaters that have been given

     /// before using functions \ref lowerMap(), \ref upperMap(),

     /// \ref costMap(), \ref supplyMap(), \ref stSupply().

     ///

     /// It is useful for multiple run() calls. If this function is not

     /// used, all the parameters given before are kept for the next

     /// \ref run() call.

     /// However, the underlying digraph must not be modified after this

     /// class have been constructed, since it copies and extends the graph.

     ///

     /// For example,

     /// \code

     ///   CostScaling<ListDigraph> cs(graph);

     ///

     ///   // First run

     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)

     ///     .supplyMap(sup).run();

     ///

     ///   // Run again with modified cost map (reset() is not called,

     ///   // so only the cost map have to be set again)

     ///   cost[e] += 100;

     ///   cs.costMap(cost).run();

     ///

     ///   // Run again from scratch using reset()

     ///   // (the lower bounds will be set to zero on all arcs)

     ///   cs.reset();

     ///   cs.upperMap(capacity).costMap(cost)

     ///     .supplyMap(sup).run();

     /// \endcode

     ///

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

     CostScaling& reset() {

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

         _supply[i] = 0;

       }

       int limit = _first_out[_root];

       for (int j = 0; j != limit; ++j) {

         _lower[j] = 0;

         _upper[j] = INF;

         _scost[j] = _forward[j] ? 1 : -1;

       }

       for (int j = limit; j != _res_arc_num; ++j) {

         _lower[j] = 0;

         _upper[j] = INF;

         _scost[j] = 0;

         _scost[_reverse[j]] = 0;

       }      

       _have_lower = false;

       return *this;

     }

     /// @}

     /// \name Query Functions

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

     /// functions.\n

     /// The \ref run() function must be called before using them.

     /// @{

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

     ///

     /// This function returns the total cost of the found flow.

     /// Its complexity is O(e).

     ///

     /// \note The return type of the function can be specified as a

     /// template parameter. For example,

     /// \code

     ///   cs.totalCost<double>();

     /// \endcode

     /// It is useful if the total cost cannot be stored in the \c Cost

     /// type of the algorithm, which is the default return type of the

     /// function.

     ///

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

     template <typename Number>

     Number totalCost() const {

       Number c = 0;

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

         int i = _arc_idb[a];

         c += static_cast<Number>(_res_cap[i]) *

              (-static_cast<Number>(_scost[i]));

       }

       return c;

     }

 #ifndef DOXYGEN

     Cost totalCost() const {

       return totalCost<Cost>();

     }

 #endif

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

     ///

     /// This function returns the flow on the given arc.

     ///

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

     Value flow(const Arc& a) const {

       return _res_cap[_arc_idb[a]];

     }

     /// \brief Return the flow map (the primal solution).

     ///

     /// This function copies the flow value on each arc into the given

     /// map. The \c Value type of the algorithm must be convertible to

     /// the \c Value type of the map.

     ///

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

     template <typename FlowMap>

     void flowMap(FlowMap &map) const {

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

         map.set(a, _res_cap[_arc_idb[a]]);

       }

     }

     /// \brief Return the potential (dual value) of the given node.

     ///

     /// This function returns the potential (dual value) of the

     /// given node.

     ///

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

     Cost potential(const Node& n) const {

       return static_cast<Cost>(_pi[_node_id[n]]);

     }

     /// \brief Return the potential map (the dual solution).

     ///

     /// This function copies the potential (dual value) of each node

     /// into the given map.

     /// The \c Cost type of the algorithm must be convertible to the

     /// \c Value type of the map.

     ///

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

     template <typename PotentialMap>

     void potentialMap(PotentialMap &map) const {

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

         map.set(n, static_cast<Cost>(_pi[_node_id[n]]));

       }

     }

     /// @}

   private:

     // Initialize the algorithm

     ProblemType init() {

       if (_res_node_num <= 1) return INFEASIBLE;

       // Check the sum of supply values

       _sum_supply = 0;

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

         _sum_supply += _supply[i];

       }

       if (_sum_supply > 0) return INFEASIBLE;

       // Initialize vectors

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

         _pi[i] = 0;

         _excess[i] = _supply[i];

       }

       // Remove infinite upper bounds and check negative arcs

       const Value MAX = std::numeric_limits<Value>::max();

       int last_out;

       if (_have_lower) {

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

           last_out = _first_out[i+1];

           for (int j = _first_out[i]; j != last_out; ++j) {

             if (_forward[j]) {

               Value c = _scost[j] < 0 ? _upper[j] : _lower[j];

               if (c >= MAX) return UNBOUNDED;

               _excess[i] -= c;

               _excess[_target[j]] += c;

             }

           }

         }

       } else {

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

           last_out = _first_out[i+1];

           for (int j = _first_out[i]; j != last_out; ++j) {

             if (_forward[j] && _scost[j] < 0) {

               Value c = _upper[j];

               if (c >= MAX) return UNBOUNDED;

               _excess[i] -= c;

               _excess[_target[j]] += c;

             }

           }

         }

       }

       Value ex, max_cap = 0;

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

         ex = _excess[i];

         _excess[i] = 0;

         if (ex < 0) max_cap -= ex;

       }

       for (int j = 0; j != _res_arc_num; ++j) {

         if (_upper[j] >= MAX) _upper[j] = max_cap;

       }

       // Initialize the large cost vector and the epsilon parameter

       _epsilon = 0;

       LargeCost lc;

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

         last_out = _first_out[i+1];

         for (int j = _first_out[i]; j != last_out; ++j) {

           lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;

           _cost[j] = lc;

           if (lc > _epsilon) _epsilon = lc;

         }

       }

       _epsilon /= _alpha;

       // Initialize maps for Circulation and remove non-zero lower bounds

       ConstMap<Arc, Value> low(0);

       typedef typename Digraph::template ArcMap<Value> ValueArcMap;

       typedef typename Digraph::template NodeMap<Value> ValueNodeMap;

       ValueArcMap cap(_graph), flow(_graph);

       ValueNodeMap sup(_graph);

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

         sup[n] = _supply[_node_id[n]];

       }

       if (_have_lower) {

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

           int j = _arc_idf[a];

           Value c = _lower[j];

           cap[a] = _upper[j] - c;

           sup[_graph.source(a)] -= c;

           sup[_graph.target(a)] += c;

         }

       } else {

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

           cap[a] = _upper[_arc_idf[a]];

         }

       }

       // Find a feasible flow using Circulation

       Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>

         circ(_graph, low, cap, sup);

       if (!circ.flowMap(flow).run()) return INFEASIBLE;

       // Set residual capacities and handle GEQ supply type

       if (_sum_supply < 0) {

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

           Value fa = flow[a];

           _res_cap[_arc_idf[a]] = cap[a] - fa;

           _res_cap[_arc_idb[a]] = fa;

           sup[_graph.source(a)] -= fa;

           sup[_graph.target(a)] += fa;

         }

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

           _excess[_node_id[n]] = sup[n];

         }

         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {

           int u = _target[a];

           int ra = _reverse[a];

           _res_cap[a] = -_sum_supply + 1;

           _res_cap[ra] = -_excess[u];

           _cost[a] = 0;

           _cost[ra] = 0;

           _excess[u] = 0;

         }

       } else {

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

           Value fa = flow[a];

           _res_cap[_arc_idf[a]] = cap[a] - fa;

           _res_cap[_arc_idb[a]] = fa;

         }

         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {

           int ra = _reverse[a];

           _res_cap[a] = 1;

           _res_cap[ra] = 0;

           _cost[a] = 0;

           _cost[ra] = 0;

         }

       }

       return OPTIMAL;

     }

     // Execute the algorithm and transform the results

     void start(Method method) {

       // Maximum path length for partial augment

       const int MAX_PATH_LENGTH = 4;

       // Execute the algorithm

       switch (method) {

         case PUSH:

           startPush();

           break;

         case AUGMENT:

           startAugment();

           break;

         case PARTIAL_AUGMENT:

           startAugment(MAX_PATH_LENGTH);

           break;

       }

       // Compute node potentials for the original costs

       _arc_vec.clear();

       _cost_vec.clear();

       for (int j = 0; j != _res_arc_num; ++j) {

         if (_res_cap[j] > 0) {

           _arc_vec.push_back(IntPair(_source[j], _target[j]));

           _cost_vec.push_back(_scost[j]);

         }

       }

       _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());

       typename BellmanFord<StaticDigraph, LargeCostArcMap>

         ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);

       bf.distMap(_pi_map);

       bf.init(0);

       bf.start();

       // Handle non-zero lower bounds

       if (_have_lower) {

         int limit = _first_out[_root];

         for (int j = 0; j != limit; ++j) {

           if (!_forward[j]) _res_cap[j] += _lower[j];

         }

       }

     }

     /// Execute the algorithm performing augment and relabel operations

     void startAugment(int max_length = std::numeric_limits<int>::max()) {

       // Paramters for heuristics

       const int BF_HEURISTIC_EPSILON_BOUND = 1000;

       const int BF_HEURISTIC_BOUND_FACTOR  = 3;

       // Perform cost scaling phases

       IntVector pred_arc(_res_node_num);

       std::vector<int> path_nodes;

       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) {

           _arc_vec.clear();

           _cost_vec.clear();

           for (int j = 0; j != _res_arc_num; ++j) {

             if (_res_cap[j] > 0) {

               _arc_vec.push_back(IntPair(_source[j], _target[j]));

               _cost_vec.push_back(_cost[j] + 1);

             }

           }

           _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());

           BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);

           bf.init(0);

           bool done = false;

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

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

             done = bf.processNextWeakRound();

           if (done) break;

         }

         // Saturate arcs not satisfying the optimality condition

         for (int a = 0; a != _res_arc_num; ++a) {

           if (_res_cap[a] > 0 &&

               _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {

             Value delta = _res_cap[a];

             _excess[_source[a]] -= delta;

             _excess[_target[a]] += delta;

             _res_cap[a] = 0;

             _res_cap[_reverse[a]] += delta;

           }

         }

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

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

           if (_excess[u] > 0) _active_nodes.push_back(u);

         }

         // Initialize the next arcs

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

           _next_out[u] = _first_out[u];

         }

         // Perform partial augment and relabel operations

         while (true) {

           // Select an active node (FIFO selection)

           while (_active_nodes.size() > 0 &&

                  _excess[_active_nodes.front()] <= 0) {

             _active_nodes.pop_front();

           }

           if (_active_nodes.size() == 0) break;

           int start = _active_nodes.front();

           path_nodes.clear();

           path_nodes.push_back(start);

           // Find an augmenting path from the start node

           int tip = start;

           while (_excess[tip] >= 0 &&

                  int(path_nodes.size()) <= max_length) {

             int u;

             LargeCost min_red_cost, rc;

             int last_out = _sum_supply < 0 ?

               _first_out[tip+1] : _first_out[tip+1] - 1;

             for (int a = _next_out[tip]; a != last_out; ++a) {

               if (_res_cap[a] > 0 &&

                   _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {

                 u = _target[a];

                 pred_arc[u] = a;

                 _next_out[tip] = a;

                 tip = u;

                 path_nodes.push_back(tip);

                 goto next_step;

               }

             }

             // Relabel tip node

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

             for (int a = _first_out[tip]; a != last_out; ++a) {

               rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];

               if (_res_cap[a] > 0 && rc < min_red_cost) {

                 min_red_cost = rc;

               }

             }

             _pi[tip] -= min_red_cost + _epsilon;

             // Reset the next arc of tip

             _next_out[tip] = _first_out[tip];

             // Step back

             if (tip != start) {

               path_nodes.pop_back();

               tip = path_nodes.back();

             }

           next_step: ;

           }

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

           Value delta;

           int u, v = path_nodes.front(), pa;

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

             u = v;

             v = path_nodes[i];

             pa = pred_arc[v];

             delta = std::min(_res_cap[pa], _excess[u]);

             _res_cap[pa] -= delta;

             _res_cap[_reverse[pa]] += delta;

             _excess[u] -= delta;

             _excess[v] += delta;

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

               _active_nodes.push_back(v);

           }

         }

       }

     }

     /// Execute the algorithm performing push and relabel operations

     void startPush() {

       // Paramters for heuristics

       const int BF_HEURISTIC_EPSILON_BOUND = 1000;

       const int BF_HEURISTIC_BOUND_FACTOR  = 3;

       // Perform cost scaling phases

       BoolVector hyper(_res_node_num, false);

       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) {

           _arc_vec.clear();

           _cost_vec.clear();

           for (int j = 0; j != _res_arc_num; ++j) {

             if (_res_cap[j] > 0) {

               _arc_vec.push_back(IntPair(_source[j], _target[j]));

               _cost_vec.push_back(_cost[j] + 1);

             }

           }

           _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());

           BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);

           bf.init(0);

           bool done = false;

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

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

             done = bf.processNextWeakRound();

           if (done) break;

         }

         // Saturate arcs not satisfying the optimality condition

         for (int a = 0; a != _res_arc_num; ++a) {

           if (_res_cap[a] > 0 &&

               _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {

             Value delta = _res_cap[a];

             _excess[_source[a]] -= delta;

             _excess[_target[a]] += delta;

             _res_cap[a] = 0;

             _res_cap[_reverse[a]] += delta;

           }

         }

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

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

           if (_excess[u] > 0) _active_nodes.push_back(u);

         }

         // Initialize the next arcs

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

           _next_out[u] = _first_out[u];

         }

         // Perform push and relabel operations

         while (_active_nodes.size() > 0) {

           LargeCost min_red_cost, rc;

           Value delta;

           int n, t, a, last_out = _res_arc_num;

           // Select an active node (FIFO selection)

         next_node:

           n = _active_nodes.front();

           last_out = _sum_supply < 0 ?

             _first_out[n+1] : _first_out[n+1] - 1;

           // Perform push operations if there are admissible arcs

           if (_excess[n] > 0) {

             for (a = _next_out[n]; a != last_out; ++a) {

               if (_res_cap[a] > 0 &&

                   _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {

                 delta = std::min(_res_cap[a], _excess[n]);

                 t = _target[a];

                 // Push-look-ahead heuristic

                 Value ahead = -_excess[t];

                 int last_out_t = _sum_supply < 0 ?

                   _first_out[t+1] : _first_out[t+1] - 1;

                 for (int ta = _next_out[t]; ta != last_out_t; ++ta) {

                   if (_res_cap[ta] > 0 && 

                       _cost[ta] + _pi[_source[ta]] - _pi[_target[ta]] < 0)

                     ahead += _res_cap[ta];

                   if (ahead >= delta) break;

                 }

                 if (ahead < 0) ahead = 0;

                 // Push flow along the arc

                 if (ahead < delta) {

                   _res_cap[a] -= ahead;

                   _res_cap[_reverse[a]] += ahead;

                   _excess[n] -= ahead;

                   _excess[t] += ahead;

                   _active_nodes.push_front(t);

                   hyper[t] = true;

                   _next_out[n] = a;

                   goto next_node;

                 } else {

                   _res_cap[a] -= delta;

                   _res_cap[_reverse[a]] += delta;

                   _excess[n] -= delta;

                   _excess[t] += delta;

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

                     _active_nodes.push_back(t);

                 }

                 if (_excess[n] == 0) {

                   _next_out[n] = a;

                   goto remove_nodes;

                 }

               }

             }

             _next_out[n] = a;

           }

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

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

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

             for (int a = _first_out[n]; a != last_out; ++a) {

               rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];

               if (_res_cap[a] > 0 && rc < min_red_cost) {

                 min_red_cost = rc;

               }

             }

             _pi[n] -= min_red_cost + _epsilon;

             hyper[n] = false;

             // Reset the next arc

             _next_out[n] = _first_out[n];

           }

           // Remove nodes that are not active nor hyper

         remove_nodes:

           while ( _active_nodes.size() > 0 &&

                   _excess[_active_nodes.front()] <= 0 &&

                   !hyper[_active_nodes.front()] ) {

             _active_nodes.pop_front();

           }

         }

       }

     }

   }; //class CostScaling

   ///@}

 } //namespace lemon

 #endif //LEMON_COST_SCALING_H