/* -*- 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];

               last_out = _first_out[v+1];

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

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

               stack[++stack_head] = a == last_out ? -1 : a;

             } else {

               if (!processed[v]) {

                 // A cycle is found

                 int n, w = u;

                 Value d, delta = _res_cap[sa];

                 for (n = u; n != v; n = pred_node[n]) {

                   d = _res_cap[pred_arc[n]];

                   if (d <= delta) {

                     delta = d;

                     w = pred_node[n];

                   }

                 }

                 // Augment along the cycle

                 _res_cap[sa] -= delta;

                 _res_cap[_reverse[sa]] += delta;

                 for (n = u; n != v; n = pred_node[n]) {

                   int pa = pred_arc[n];

                   _res_cap[pa] -= delta;

                   _res_cap[_reverse[pa]] += delta;

                 }

                 for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {

                   --stack_head;

                   reached[n] = false;

                 }

                 u = w;

               }

               v = u;

               // Find the next admissible outgoing arc

               p = pi[v];

               a = stack[stack_head] + 1;

               last_out = _first_out[v+1];

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

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

               stack[stack_head] = a == last_out ? -1 : a;

             }

             while (stack_head >= 0 && stack[stack_head] == -1) {

               processed[v] = true;

               proc_vector[++proc_head] = v;

               if (--stack_head >= 0) {

                 // Find the next admissible outgoing arc

                 v = _source[stack[stack_head]];

                 p = pi[v];

                 a = stack[stack_head] + 1;

                 last_out = _first_out[v+1];

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

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

                 stack[stack_head] = a == last_out ? -1 : a;

               }

             }

           }

         }

         // Tighten potentials and epsilon

         if (--iter > 0) {

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

             level[u] = 0;

           }

           for (int i = proc_head; i > 0; --i) {

             int u = proc_vector[i];

             double p = pi[u];

             int l = level[u] + 1;

             int last_out = _first_out[u+1];

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

               int v = _target[a];

               if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&

                   l > level[v]) level[v] = l;

             }

           }

           // Modify potentials

           double q = std::numeric_limits<double>::max();

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

             int lu = level[u];

             double p, pu = pi[u];

             int last_out = _first_out[u+1];

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

               if (_res_cap[a] == 0) continue;

               int v = _target[a];

               int ld = lu - level[v];

               if (ld > 0) {

                 p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);

                 if (p < q) q = p;

               }

             }

           }

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

             pi[u] -= q * level[u];

           }

           // Modify epsilon

           epsilon = 0;

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

             double curr, pu = pi[u];

             int last_out = _first_out[u+1];

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

               if (_res_cap[a] == 0) continue;

               curr = _cost[a] + pu - pi[_target[a]];

               if (-curr > epsilon) epsilon = -curr;

             }

           }

         } else {

           typedef Howard<StaticDigraph, CostArcMap> MMC;

           typedef typename BellmanFord<StaticDigraph, CostArcMap>

             ::template SetDistMap<CostNodeMap>::Create BF;

           // Set epsilon to the minimum cycle mean

           buildResidualNetwork();

           MMC mmc(_sgr, _cost_map);

           mmc.findMinMean();

           epsilon = -mmc.cycleMean();

           Cost cycle_cost = mmc.cycleLength();

           int cycle_size = mmc.cycleArcNum();

           // Compute feasible potentials for the current epsilon

           for (int i = 0; i != int(_cost_vec.size()); ++i) {

             _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;

           }

           BF bf(_sgr, _cost_map);

           bf.distMap(_pi_map);

           bf.init(0);

           bf.start();

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

             pi[u] = static_cast<double>(_pi[u]) / cycle_size;

           }

           iter = limit;

         }

       }

     }

   }; //class CycleCanceling

   ///@}

 } //namespace lemon

 #endif //LEMON_CYCLE_CANCELING_H