/* -*- 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_CYCLE_CANCELING_H

#define LEMON_CYCLE_CANCELING_H

/// \ingroup min_cost_flow_algs

/// \file

/// \brief Cycle-canceling algorithms for finding a minimum cost flow.

#include <vector>

#include <limits>

#include <lemon/core.h>

#include <lemon/maps.h>

#include <lemon/path.h>

#include <lemon/math.h>

#include <lemon/static_graph.h>

#include <lemon/adaptors.h>

#include <lemon/circulation.h>

#include <lemon/bellman_ford.h>

#include <lemon/howard.h>

namespace lemon {

  /// \addtogroup min_cost_flow_algs

  /// @{

  /// \brief Implementation of cycle-canceling algorithms for

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

  ///

  /// \ref CycleCanceling implements three different cycle-canceling

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

  /// The most efficent one (both theoretically and practically)

  /// is the \ref CANCEL_AND_TIGHTEN "Cancel and Tighten" algorithm,

  /// thus it is the default method.

  /// It is strongly polynomial, but in practice, it is typically much

  /// slower than the scaling algorithms and NetworkSimplex.

  ///

  /// 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 For more information about the three available methods,

  /// see \ref Method.

#ifdef DOXYGEN

  template <typename GR, typename V, typename C>

#else

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

#endif

  class CycleCanceling

  {

  public:

    /// 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;

  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 used method.

    ///

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

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

    ///

    /// \ref CycleCanceling provides three different cycle-canceling

    /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel and Tighten"

    /// 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 {

      /// A simple cycle-canceling method, which uses the

      /// \ref BellmanFord "Bellman-Ford" algorithm with limited iteration

      /// number for detecting negative cycles in the residual network.

      SIMPLE_CYCLE_CANCELING,

      /// The "Minimum Mean Cycle-Canceling" algorithm, which is a

      /// well-known strongly polynomial method. It improves along a

      /// \ref min_mean_cycle "minimum mean cycle" in each iteration.

      /// Its running time complexity is O(n<sup>2</sup>m<sup>3</sup>log(n)).

      MINIMUM_MEAN_CYCLE_CANCELING,

      /// The "Cancel And Tighten" algorithm, which can be viewed as an

      /// improved version of the previous method.

      /// It is faster both in theory and in practice, its running time

      /// complexity is O(n<sup>2</sup>m<sup>2</sup>log(n)).

      CANCEL_AND_TIGHTEN

    };

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(GR);

    typedef std::vector<int> IntVector;

    typedef std::vector<char> CharVector;

    typedef std::vector<double> DoubleVector;

    typedef std::vector<Value> ValueVector;

    typedef std::vector<Cost> CostVector;

  private:

    template <typename KT, typename VT>

    class VectorMap {

    public:

      typedef KT Key;

      typedef VT Value;

      VectorMap(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 VectorMap<StaticDigraph::Node, Cost> CostNodeMap;

    typedef VectorMap<StaticDigraph::Arc, Cost> CostArcMap;

  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;

    CharVector _forward;

    IntVector _source;

    IntVector _target;

    IntVector _reverse;

    // Node and arc data

    ValueVector _lower;

    ValueVector _upper;

    CostVector _cost;

    ValueVector _supply;

    ValueVector _res_cap;

    CostVector _pi;

    // Data for a StaticDigraph structure

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

    StaticDigraph _sgr;

    std::vector<IntPair> _arc_vec;

    std::vector<Cost> _cost_vec;

    IntVector _id_vec;

    CostArcMap _cost_map;

    CostNodeMap _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:

    /// \brief Constructor.

    ///

    /// The constructor of the class.

    ///

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

    CycleCanceling(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 CycleCanceling must be signed");

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

        "The cost type of CycleCanceling 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);

      _cost.resize(_res_arc_num);

      _supply.resize(_res_node_num);

      _res_cap.resize(_res_arc_num);

      _pi.resize(_res_node_num);

      _arc_vec.reserve(_res_arc_num);

      _cost_vec.reserve(_res_arc_num);

      _id_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>

    CycleCanceling& 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>

    CycleCanceling& 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>

    CycleCanceling& costMap(const CostMap& map) {

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

        _cost[_arc_idf[a]] =  map[a];

        _cost[_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>

    CycleCanceling& 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>

    CycleCanceling& 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

    ///   CycleCanceling<ListDigraph> cc(graph);

    ///   cc.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 cycle-canceling method that will be used.

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

    ///

    /// \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 = CANCEL_AND_TIGHTEN) {

      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

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

    ///

    ///   // First run

    ///   cc.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;

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

    ///

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

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

    ///   cc.reset();

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

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

    /// \endcode

    ///

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

    CycleCanceling& 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;

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

      }

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

        _lower[j] = 0;

        _upper[j] = INF;

        _cost[j] = 0;

        _cost[_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

    ///   cc.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>(_cost[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;

      }

      ValueVector excess(_supply);

      // 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 = _cost[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] && _cost[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];

        if (ex < 0) max_cap -= ex;

      }

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

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

      }

      // 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;

        }

      } 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;

    }

    // Build a StaticDigraph structure containing the current

    // residual network

    void buildResidualNetwork() {

      _arc_vec.clear();

      _cost_vec.clear();

      _id_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]);

          _id_vec.push_back(j);

        }

      }

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

    }

    // Execute the algorithm and transform the results

    void start(Method method) {

      // Execute the algorithm

      switch (method) {

        case SIMPLE_CYCLE_CANCELING:

          startSimpleCycleCanceling();

          break;

        case MINIMUM_MEAN_CYCLE_CANCELING:

          startMinMeanCycleCanceling();

          break;

        case CANCEL_AND_TIGHTEN:

          startCancelAndTighten();

          break;

      }

      // Compute node potentials

      if (method != SIMPLE_CYCLE_CANCELING) {

        buildResidualNetwork();

        typename BellmanFord<StaticDigraph, CostArcMap>

          ::template SetDistMap<CostNodeMap>::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 "Simple Cycle Canceling" method

    void startSimpleCycleCanceling() {

      // Constants for computing the iteration limits

      const int BF_FIRST_LIMIT  = 2;

      const double BF_LIMIT_FACTOR = 1.5;

      typedef VectorMap<StaticDigraph::Arc, Value> FilterMap;

      typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;

      typedef VectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;

      typedef typename BellmanFord<ResDigraph, CostArcMap>

        ::template SetDistMap<CostNodeMap>

        ::template SetPredMap<PredMap>::Create BF;

      // Build the residual network

      _arc_vec.clear();

      _cost_vec.clear();

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

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

        _cost_vec.push_back(_cost[j]);

      }

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

      FilterMap filter_map(_res_cap);

      ResDigraph rgr(_sgr, filter_map);

      std::vector<int> cycle;

      std::vector<StaticDigraph::Arc> pred(_res_arc_num);

      PredMap pred_map(pred);

      BF bf(rgr, _cost_map);

      bf.distMap(_pi_map).predMap(pred_map);

      int length_bound = BF_FIRST_LIMIT;

      bool optimal = false;

      while (!optimal) {

        bf.init(0);

        int iter_num = 0;

        bool cycle_found = false;

        while (!cycle_found) {

          // Perform some iterations of the Bellman-Ford algorithm

          int curr_iter_num = iter_num + length_bound <= _node_num ?

            length_bound : _node_num - iter_num;

          iter_num += curr_iter_num;

          int real_iter_num = curr_iter_num;

          for (int i = 0; i < curr_iter_num; ++i) {

            if (bf.processNextWeakRound()) {

              real_iter_num = i;

              break;

            }

          }

          if (real_iter_num < curr_iter_num) {

            // Optimal flow is found

            optimal = true;

            break;

          } else {

            // Search for node disjoint negative cycles

            std::vector<int> state(_res_node_num, 0);

            int id = 0;

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

              if (state[u] != 0) continue;

              ++id;

              int v = u;

              for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?

                   -1 : rgr.id(rgr.source(pred[v]))) {

                state[v] = id;

              }

              if (v != -1 && state[v] == id) {

                // A negative cycle is found

                cycle_found = true;

                cycle.clear();

                StaticDigraph::Arc a = pred[v];

                Value d, delta = _res_cap[rgr.id(a)];

                cycle.push_back(rgr.id(a));

                while (rgr.id(rgr.source(a)) != v) {

                  a = pred_map[rgr.source(a)];

                  d = _res_cap[rgr.id(a)];

                  if (d < delta) delta = d;

                  cycle.push_back(rgr.id(a));

                }

                // Augment along the cycle

                for (int i = 0; i < int(cycle.size()); ++i) {

                  int j = cycle[i];

                  _res_cap[j] -= delta;

                  _res_cap[_reverse[j]] += delta;

                }

              }

            }

          }

          // Increase iteration limit if no cycle is found

          if (!cycle_found) {

            length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);

          }

        }

      }

    }

    // Execute the "Minimum Mean Cycle Canceling" method

    void startMinMeanCycleCanceling() {

      typedef SimplePath<StaticDigraph> SPath;

      typedef typename SPath::ArcIt SPathArcIt;

      typedef typename Howard<StaticDigraph, CostArcMap>

        ::template SetPath<SPath>::Create MMC;

      SPath cycle;

      MMC mmc(_sgr, _cost_map);

      mmc.cycle(cycle);

      buildResidualNetwork();

      while (mmc.findMinMean() && mmc.cycleLength() < 0) {

        // Find the cycle

        mmc.findCycle();

        // Compute delta value

        Value delta = INF;

        for (SPathArcIt a(cycle); a != INVALID; ++a) {

          Value d = _res_cap[_id_vec[_sgr.id(a)]];

          if (d < delta) delta = d;

        }

        // Augment along the cycle

        for (SPathArcIt a(cycle); a != INVALID; ++a) {

          int j = _id_vec[_sgr.id(a)];

          _res_cap[j] -= delta;

          _res_cap[_reverse[j]] += delta;

        }

        // Rebuild the residual network        

        buildResidualNetwork();

      }

    }

    // Execute the "Cancel And Tighten" method

    void startCancelAndTighten() {

      // Constants for the min mean cycle computations

      const double LIMIT_FACTOR = 1.0;

      const int MIN_LIMIT = 5;

      // Contruct auxiliary data vectors

      DoubleVector pi(_res_node_num, 0.0);

      IntVector level(_res_node_num);

      CharVector reached(_res_node_num);

      CharVector processed(_res_node_num);

      IntVector pred_node(_res_node_num);

      IntVector pred_arc(_res_node_num);

      std::vector<int> stack(_res_node_num);

      std::vector<int> proc_vector(_res_node_num);

      // Initialize epsilon

      double epsilon = 0;

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

        if (_res_cap[a] > 0 && -_cost[a] > epsilon)

          epsilon = -_cost[a];

      }

      // Start phases

      Tolerance<double> tol;

      tol.epsilon(1e-6);

      int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));

      if (limit < MIN_LIMIT) limit = MIN_LIMIT;

      int iter = limit;

      while (epsilon * _res_node_num >= 1) {

        // Find and cancel cycles in the admissible network using DFS

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

          reached[u] = false;

          processed[u] = false;

        }

        int stack_head = -1;

        int proc_head = -1;

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

          if (reached[start]) continue;

          // New start node

          reached[start] = true;

          pred_arc[start] = -1;

          pred_node[start] = -1;

          // Find the first admissible outgoing arc

          double p = pi[start];

          int a = _first_out[start];

          int last_out = _first_out[start+1];

          for (; a != last_out && (_res_cap[a] == 0 ||

               !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;

          if (a == last_out) {

            processed[start] = true;

            proc_vector[++proc_head] = start;

            continue;

          }

          stack[++stack_head] = a;

          while (stack_head >= 0) {

            int sa = stack[stack_head];

            int u = _source[sa];

            int v = _target[sa];

            if (!reached[v]) {

              // A new node is reached

              reached[v] = true;

              pred_node[v] = u;

              pred_arc[v] = sa;

              p = pi[v];

              a = _first_out[v];