source: lemon/lemon/cost_scaling.h @ 1025:140c953ad5d1

/* -*- mode: C++; indent-tabs-mode: nil; -*-
 *
 * This file is a part of LEMON, a generic C++ optimization library.
 *
 * Copyright (C) 2003-2010
 * 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.
  /// \tparam TR The traits class that defines various types used by the
  /// algorithm. By default, it is \ref CostScalingDefaultTraits
  /// "CostScalingDefaultTraits<GR, V, C>".
  /// In most cases, this parameter should not be set directly,
  /// consider to use the named template parameters instead.
  ///
  /// \warning Both \c V and \c C must be signed number types.
  /// \warning All input data (capacities, supply values, and costs) 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.
    /// By default, 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<Value> ValueVector;
    typedef std::vector<Cost> CostVector;
    typedef std::vector<LargeCost> LargeCostVector;
    typedef std::vector<char> BoolVector;
    // Note: vector<char> is used instead of vector<bool> for efficiency reasons

  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;
    int _sup_node_num;

    // 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;

    IntVector _buckets;
    IntVector _bucket_next;
    IntVector _bucket_prev;
    IntVector _rank;
    int _max_rank;

    // 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;
    };

    /// @}

  protected:

    CostScaling() {}

  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");

      // Reset data structures
      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 given parameters
    /// are kept for the next call, unless \ref resetParams() or \ref reset()
    /// is used, thus only the modified parameters have to be set again.
    /// If the underlying digraph was also modified after the construction
    /// of the class (or the last \ref reset() call), then the \ref reset()
    /// function must be called.
    ///
    /// \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
    /// \see resetParams(), reset()
    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 \ref run() calls. Basically, all the given
    /// parameters are kept for the next \ref run() call, unless
    /// \ref resetParams() or \ref reset() is used.
    /// If the underlying digraph was also modified after the construction
    /// of the class or the last \ref reset() call, then the \ref reset()
    /// function must be used, otherwise \ref resetParams() is sufficient.
    ///
    /// 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 (resetParams() 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 resetParams()
    ///   // (the lower bounds will be set to zero on all arcs)
    ///   cs.resetParams();
    ///   cs.upperMap(capacity).costMap(cost)
    ///     .supplyMap(sup).run();
    /// \endcode
    ///
    /// \return <tt>(*this)</tt>
    ///
    /// \see reset(), run()
    CostScaling& resetParams() {
      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;
    }

    /// \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.
    /// \return <tt>(*this)</tt>
    CostScaling& reset() {
      // 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
      resetParams();
      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]];
        }
      }

      _sup_node_num = 0;
      for (NodeIt n(_graph); n != INVALID; ++n) {
        if (sup[n] > 0) ++_sup_node_num;
      }

      // 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] = 0;
          _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;

      // Initialize data structures for buckets
      _max_rank = _alpha * _res_node_num;
      _buckets.resize(_max_rank);
      _bucket_next.resize(_res_node_num + 1);
      _bucket_prev.resize(_res_node_num + 1);
      _rank.resize(_res_node_num + 1);

      // 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];
        }
      }
    }

    // Initialize a cost scaling phase
    void initPhase() {
      // Saturate arcs not satisfying the optimality condition
      for (int u = 0; u != _res_node_num; ++u) {
        int last_out = _first_out[u+1];
        LargeCost pi_u = _pi[u];
        for (int a = _first_out[u]; a != last_out; ++a) {
          int v = _target[a];
          if (_res_cap[a] > 0 && _cost[a] + pi_u - _pi[v] < 0) {
            Value delta = _res_cap[a];
            _excess[u] -= delta;
            _excess[v] += 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];
      }
    }

    // Early termination heuristic
    bool earlyTermination() {
      const double EARLY_TERM_FACTOR = 3.0;

      // Build a static residual graph
      _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());

      // Run Bellman-Ford algorithm to check if the current flow is optimal
      BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
      bf.init(0);
      bool done = false;
      int K = int(EARLY_TERM_FACTOR * std::sqrt(double(_res_node_num)));
      for (int i = 0; i < K && !done; ++i) {
        done = bf.processNextWeakRound();
      }
      return done;
    }

    // Global potential update heuristic
    void globalUpdate() {
      int bucket_end = _root + 1;

      // Initialize buckets
      for (int r = 0; r != _max_rank; ++r) {
        _buckets[r] = bucket_end;
      }
      Value total_excess = 0;
      for (int i = 0; i != _res_node_num; ++i) {
        if (_excess[i] < 0) {
          _rank[i] = 0;
          _bucket_next[i] = _buckets[0];
          _bucket_prev[_buckets[0]] = i;
          _buckets[0] = i;
        } else {
          total_excess += _excess[i];
          _rank[i] = _max_rank;
        }
      }
      if (total_excess == 0) return;

      // Search the buckets
      int r = 0;
      for ( ; r != _max_rank; ++r) {
        while (_buckets[r] != bucket_end) {
          // Remove the first node from the current bucket
          int u = _buckets[r];
          _buckets[r] = _bucket_next[u];

          // Search the incomming arcs of u
          LargeCost pi_u = _pi[u];
          int last_out = _first_out[u+1];
          for (int a = _first_out[u]; a != last_out; ++a) {
            int ra = _reverse[a];
            if (_res_cap[ra] > 0) {
              int v = _source[ra];
              int old_rank_v = _rank[v];
              if (r < old_rank_v) {
                // Compute the new rank of v
                LargeCost nrc = (_cost[ra] + _pi[v] - pi_u) / _epsilon;
                int new_rank_v = old_rank_v;
                if (nrc < LargeCost(_max_rank))
                  new_rank_v = r + 1 + int(nrc);

                // Change the rank of v
                if (new_rank_v < old_rank_v) {
                  _rank[v] = new_rank_v;
                  _next_out[v] = _first_out[v];

                  // Remove v from its old bucket
                  if (old_rank_v < _max_rank) {
                    if (_buckets[old_rank_v] == v) {
                      _buckets[old_rank_v] = _bucket_next[v];
                    } else {
                      _bucket_next[_bucket_prev[v]] = _bucket_next[v];
                      _bucket_prev[_bucket_next[v]] = _bucket_prev[v];
                    }
                  }

                  // Insert v to its new bucket
                  _bucket_next[v] = _buckets[new_rank_v];
                  _bucket_prev[_buckets[new_rank_v]] = v;
                  _buckets[new_rank_v] = v;
                }
              }
            }
          }

          // Finish search if there are no more active nodes
          if (_excess[u] > 0) {
            total_excess -= _excess[u];
            if (total_excess <= 0) break;
          }
        }
        if (total_excess <= 0) break;
      }

      // Relabel nodes
      for (int u = 0; u != _res_node_num; ++u) {
        int k = std::min(_rank[u], r);
        if (k > 0) {
          _pi[u] -= _epsilon * k;
          _next_out[u] = _first_out[u];
        }
      }
    }

    /// Execute the algorithm performing augment and relabel operations
    void startAugment(int max_length = std::numeric_limits<int>::max()) {
      // Paramters for heuristics
      const int EARLY_TERM_EPSILON_LIMIT = 1000;
      const double GLOBAL_UPDATE_FACTOR = 3.0;

      const int global_update_freq = int(GLOBAL_UPDATE_FACTOR *
        (_res_node_num + _sup_node_num * _sup_node_num));
      int next_update_limit = global_update_freq;

      int relabel_cnt = 0;

      // Perform cost scaling phases
      std::vector<int> path;
      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
                                        1 : _epsilon / _alpha )
      {
        // Early termination heuristic
        if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
          if (earlyTermination()) break;
        }

        // Initialize current phase
        initPhase();

        // 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();

          // Find an augmenting path from the start node
          path.clear();
          int tip = start;
          while (_excess[tip] >= 0 && int(path.size()) < max_length) {
            int u;
            LargeCost min_red_cost, rc, pi_tip = _pi[tip];
            int last_out = _first_out[tip+1];
            for (int a = _next_out[tip]; a != last_out; ++a) {
              u = _target[a];
              if (_res_cap[a] > 0 && _cost[a] + pi_tip - _pi[u] < 0) {
                path.push_back(a);
                _next_out[tip] = a;
                tip = u;
                goto next_step;
              }
            }

            // Relabel tip node
            min_red_cost = std::numeric_limits<LargeCost>::max();
            if (tip != start) {
              int ra = _reverse[path.back()];
              min_red_cost = _cost[ra] + pi_tip - _pi[_target[ra]];
            }
            for (int a = _first_out[tip]; a != last_out; ++a) {
              rc = _cost[a] + pi_tip - _pi[_target[a]];
              if (_res_cap[a] > 0 && rc < min_red_cost) {
                min_red_cost = rc;
              }
            }
            _pi[tip] -= min_red_cost + _epsilon;
            _next_out[tip] = _first_out[tip];
            ++relabel_cnt;

            // Step back
            if (tip != start) {
              tip = _source[path.back()];
              path.pop_back();
            }

          next_step: ;
          }

          // Augment along the found path (as much flow as possible)
          Value delta;
          int pa, u, v = start;
          for (int i = 0; i != int(path.size()); ++i) {
            pa = path[i];
            u = v;
            v = _target[pa];
            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);
          }

          // Global update heuristic
          if (relabel_cnt >= next_update_limit) {
            globalUpdate();
            next_update_limit += global_update_freq;
          }
        }
      }
    }

    /// Execute the algorithm performing push and relabel operations
    void startPush() {
      // Paramters for heuristics
      const int EARLY_TERM_EPSILON_LIMIT = 1000;
      const double GLOBAL_UPDATE_FACTOR = 2.0;

      const int global_update_freq = int(GLOBAL_UPDATE_FACTOR *
        (_res_node_num + _sup_node_num * _sup_node_num));
      int next_update_limit = global_update_freq;

      int relabel_cnt = 0;

      // Perform cost scaling phases
      BoolVector hyper(_res_node_num, false);
      LargeCostVector hyper_cost(_res_node_num);
      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
                                        1 : _epsilon / _alpha )
      {
        // Early termination heuristic
        if (_epsilon <= EARLY_TERM_EPSILON_LIMIT) {
          if (earlyTermination()) break;
        }

        // Initialize current phase
        initPhase();

        // Perform push and relabel operations
        while (_active_nodes.size() > 0) {
          LargeCost min_red_cost, rc, pi_n;
          Value delta;
          int n, t, a, last_out = _res_arc_num;

        next_node:
          // Select an active node (FIFO selection)
          n = _active_nodes.front();
          last_out = _first_out[n+1];
          pi_n = _pi[n];

          // 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_n - _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 = _first_out[t+1];
                LargeCost pi_t = _pi[t];
                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
                  if (_res_cap[ta] > 0 &&
                      _cost[ta] + pi_t - _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 && !hyper[t]) {
                  _res_cap[a] -= ahead;
                  _res_cap[_reverse[a]] += ahead;
                  _excess[n] -= ahead;
                  _excess[t] += ahead;
                  _active_nodes.push_front(t);
                  hyper[t] = true;
                  hyper_cost[t] = _cost[a] + pi_n - pi_t;
                  _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 = hyper[n] ? -hyper_cost[n] :
               std::numeric_limits<LargeCost>::max();
            for (int a = _first_out[n]; a != last_out; ++a) {
              rc = _cost[a] + pi_n - _pi[_target[a]];
              if (_res_cap[a] > 0 && rc < min_red_cost) {
                min_red_cost = rc;
              }
            }
            _pi[n] -= min_red_cost + _epsilon;
            _next_out[n] = _first_out[n];
            hyper[n] = false;
            ++relabel_cnt;
          }

          // 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();
          }

          // Global update heuristic
          if (relabel_cnt >= next_update_limit) {
            globalUpdate();
            for (int u = 0; u != _res_node_num; ++u)
              hyper[u] = false;
            next_update_limit += global_update_freq;
          }
        }
      }
    }

  }; //class CostScaling

  ///@}

} //namespace lemon

#endif //LEMON_COST_SCALING_H