/* -*- 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_CAPACITY_SCALING_H

 #define LEMON_CAPACITY_SCALING_H

 /// \ingroup min_cost_flow_algs

 ///

 /// \file

 /// \brief Capacity Scaling algorithm for finding a minimum cost flow.

 #include <vector>

 #include <limits>

 #include <lemon/core.h>

 #include <lemon/maps.h>

 #include <lemon/bin_heap.h>

 namespace lemon {

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

   ///

   /// Default traits class of CapacityScaling 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.

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

   struct CapacityScalingDefaultTraits

   {

     /// 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 type of the heap used for internal Dijkstra computations.

     ///

     /// The type of the heap used for internal Dijkstra computations.

     /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,

     /// its priority type must be \c Cost and its cross reference type

     /// must be \ref RangeMap "RangeMap<int>".

     typedef BinHeap<Cost, RangeMap<int> > Heap;

   };

   /// \addtogroup min_cost_flow_algs

   /// @{

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

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

   ///

   /// \ref CapacityScaling implements the capacity scaling version

   /// of the successive shortest path algorithm for finding a

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

   /// \ref edmondskarp72theoretical. It is an efficient dual

   /// solution method.

   ///

   /// 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 CapacityScalingDefaultTraits

   /// "CapacityScalingDefaultTraits<GR, V, C>".

   /// In most cases, this parameter should not be set directly,

   /// consider to use the named template parameters instead.

   ///

   /// \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.

 #ifdef DOXYGEN

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

 #else

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

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

 #endif

   class CapacityScaling

   {

   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;

     /// The type of the heap used for internal Dijkstra computations

     typedef typename TR::Heap Heap;

     /// The \ref CapacityScalingDefaultTraits "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

     };

   private:

     TEMPLATE_DIGRAPH_TYPEDEFS(GR);

     typedef std::vector<int> IntVector;

     typedef std::vector<Value> ValueVector;

     typedef std::vector<Cost> CostVector;

     typedef std::vector<char> BoolVector;

     // Note: vector<char> is used instead of vector<bool> for efficiency reasons

   private:

     // Data related to the underlying digraph

     const GR &_graph;

     int _node_num;

     int _arc_num;

     int _res_arc_num;

     int _root;

     // Parameters of the problem

     bool _have_lower;

     Value _sum_supply;

     // Data structures for storing the digraph

     IntNodeMap _node_id;

     IntArcMap _arc_idf;

     IntArcMap _arc_idb;

     IntVector _first_out;

     BoolVector _forward;

     IntVector _source;

     IntVector _target;

     IntVector _reverse;

     // Node and arc data

     ValueVector _lower;

     ValueVector _upper;

     CostVector _cost;

     ValueVector _supply;

     ValueVector _res_cap;

     CostVector _pi;

     ValueVector _excess;

     IntVector _excess_nodes;

     IntVector _deficit_nodes;

     Value _delta;

     int _factor;

     IntVector _pred;

   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;

   private:

     // Special implementation of the Dijkstra algorithm for finding

     // shortest paths in the residual network of the digraph with

     // respect to the reduced arc costs and modifying the node

     // potentials according to the found distance labels.

     class ResidualDijkstra

     {

     private:

       int _node_num;

       bool _geq;

       const IntVector &_first_out;

       const IntVector &_target;

       const CostVector &_cost;

       const ValueVector &_res_cap;

       const ValueVector &_excess;

       CostVector &_pi;

       IntVector &_pred;

       IntVector _proc_nodes;

       CostVector _dist;

     public:

       ResidualDijkstra(CapacityScaling& cs) :

         _node_num(cs._node_num), _geq(cs._sum_supply < 0),

         _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),

         _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),

         _pred(cs._pred), _dist(cs._node_num)

       {}

       int run(int s, Value delta = 1) {

         RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);

         Heap heap(heap_cross_ref);

         heap.push(s, 0);

         _pred[s] = -1;

         _proc_nodes.clear();

         // Process nodes

         while (!heap.empty() && _excess[heap.top()] > -delta) {

           int u = heap.top(), v;

           Cost d = heap.prio() + _pi[u], dn;

           _dist[u] = heap.prio();

           _proc_nodes.push_back(u);

           heap.pop();

           // Traverse outgoing residual arcs

           int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;

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

             if (_res_cap[a] < delta) continue;

             v = _target[a];

             switch (heap.state(v)) {

               case Heap::PRE_HEAP:

                 heap.push(v, d + _cost[a] - _pi[v]);

                 _pred[v] = a;

                 break;

               case Heap::IN_HEAP:

                 dn = d + _cost[a] - _pi[v];

                 if (dn < heap[v]) {

                   heap.decrease(v, dn);

                   _pred[v] = a;

                 }

                 break;

               case Heap::POST_HEAP:

                 break;

             }

           }

         }

         if (heap.empty()) return -1;

         // Update potentials of processed nodes

         int t = heap.top();

         Cost dt = heap.prio();

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

           _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;

         }

         return t;

       }

     }; //class ResidualDijkstra

   public:

     /// \name Named Template Parameters

     /// @{

     template <typename T>

     struct SetHeapTraits : public Traits {

       typedef T Heap;

     };

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

     /// \c Heap type.

     ///

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

     /// type, which is used for internal Dijkstra computations.

     /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,

     /// its priority type must be \c Cost and its cross reference type

     /// must be \ref RangeMap "RangeMap<int>".

     template <typename T>

     struct SetHeap

       : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {

       typedef  CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;

     };

     /// @}

   protected:

     CapacityScaling() {}

   public:

     /// \brief Constructor.

     ///

     /// The constructor of the class.

     ///

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

     CapacityScaling(const GR& graph) :

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

       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 CapacityScaling must be signed");

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

         "The cost type of CapacityScaling 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>

     CapacityScaling& 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>

     CapacityScaling& 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>

     CapacityScaling& 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>

     CapacityScaling& 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>

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

       for (int i = 0; i != _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

     ///   CapacityScaling<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 factor The capacity scaling factor. It must be larger than

     /// one to use scaling. If it is less or equal to one, then scaling

     /// will be disabled.

     ///

     /// \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

     /// \see resetParams(), reset()

     ProblemType run(int factor = 4) {

       _factor = factor;

       ProblemType pt = init();

       if (pt != OPTIMAL) return pt;

       return start();

     }

     /// \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

     ///   CapacityScaling<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()

     CapacityScaling& resetParams() {

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

         _supply[i] = 0;

       }

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

         _lower[j] = 0;

         _upper[j] = INF;

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

       }

       _have_lower = false;

       return *this;

     }

     /// \brief Reset the internal data structures and all the parameters

     /// that have been given before.

     ///

     /// This function resets the internal data structures and 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.

     ///

     /// See \ref resetParams() for examples.

     ///

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

     ///

     /// \see resetParams(), run()

     CapacityScaling& reset() {

       // Resize vectors

       _node_num = countNodes(_graph);

       _arc_num = countArcs(_graph);

       _res_arc_num = 2 * (_arc_num + _node_num);

       _root = _node_num;

       ++_node_num;

       _first_out.resize(_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(_node_num);

       _res_cap.resize(_res_arc_num);

       _pi.resize(_node_num);

       _excess.resize(_node_num);

       _pred.resize(_node_num);

       // Copy the graph

       int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;

       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[_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>(_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 _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, _pi[_node_id[n]]);

       }

     }

     /// @}

   private:

     // Initialize the algorithm

     ProblemType init() {

       if (_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 != _root; ++i) {

         _pi[i] = 0;

         _excess[i] = _supply[i];

       }

       // Remove non-zero lower bounds

       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 = _lower[j];

               if (c >= 0) {

                 _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;

               } else {

                 _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;

               }

               _excess[i] -= c;

               _excess[_target[j]] += c;

             } else {

               _res_cap[j] = 0;

             }

           }

         }

       } else {

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

           _res_cap[j] = _forward[j] ? _upper[j] : 0;

         }

       }

       // Handle negative costs

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

         last_out = _first_out[i+1] - 1;

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

           Value rc = _res_cap[j];

           if (_cost[j] < 0 && rc > 0) {

             if (rc >= MAX) return UNBOUNDED;

             _excess[i] -= rc;

             _excess[_target[j]] += rc;

             _res_cap[j] = 0;

             _res_cap[_reverse[j]] += rc;

           }

         }

       }

       // Handle GEQ supply type

       if (_sum_supply < 0) {

         _pi[_root] = 0;

         _excess[_root] = -_sum_supply;

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

           int ra = _reverse[a];

           _res_cap[a] = -_sum_supply + 1;

           _res_cap[ra] = 0;

           _cost[a] = 0;

           _cost[ra] = 0;

         }

       } else {

         _pi[_root] = 0;

         _excess[_root] = 0;

         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;

         }

       }

       // Initialize delta value

       if (_factor > 1) {

         // With scaling

         Value max_sup = 0, max_dem = 0, max_cap = 0;

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

           Value ex = _excess[i];

           if ( ex > max_sup) max_sup =  ex;

           if (-ex > max_dem) max_dem = -ex;

           int last_out = _first_out[i+1] - 1;

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

             if (_res_cap[j] > max_cap) max_cap = _res_cap[j];

           }

         }

         max_sup = std::min(std::min(max_sup, max_dem), max_cap);

         for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;

       } else {

         // Without scaling

         _delta = 1;

       }

       return OPTIMAL;

     }

     ProblemType start() {

       // Execute the algorithm

       ProblemType pt;

       if (_delta > 1)

         pt = startWithScaling();

       else

         pt = startWithoutScaling();

       // 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];

         }

       }

       // Shift potentials if necessary

       Cost pr = _pi[_root];

       if (_sum_supply < 0 || pr > 0) {

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

           _pi[i] -= pr;

         }

       }

       return pt;

     }

     // Execute the capacity scaling algorithm

     ProblemType startWithScaling() {

       // Perform capacity scaling phases

       int s, t;

       ResidualDijkstra _dijkstra(*this);

       while (true) {

         // Saturate all arcs not satisfying the optimality condition

         int last_out;

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

           last_out = _sum_supply < 0 ?

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

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

             int v = _target[a];

             Cost c = _cost[a] + _pi[u] - _pi[v];

             Value rc = _res_cap[a];

             if (c < 0 && rc >= _delta) {

               _excess[u] -= rc;

               _excess[v] += rc;

               _res_cap[a] = 0;

               _res_cap[_reverse[a]] += rc;

             }

           }

         }

         // Find excess nodes and deficit nodes

         _excess_nodes.clear();

         _deficit_nodes.clear();

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

           Value ex = _excess[u];

           if (ex >=  _delta) _excess_nodes.push_back(u);

           if (ex <= -_delta) _deficit_nodes.push_back(u);

         }

         int next_node = 0, next_def_node = 0;

         // Find augmenting shortest paths

         while (next_node < int(_excess_nodes.size())) {

           // Check deficit nodes

           if (_delta > 1) {

             bool delta_deficit = false;

             for ( ; next_def_node < int(_deficit_nodes.size());

                     ++next_def_node ) {

               if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {

                 delta_deficit = true;

                 break;

               }

             }

             if (!delta_deficit) break;

           }

           // Run Dijkstra in the residual network

           s = _excess_nodes[next_node];

           if ((t = _dijkstra.run(s, _delta)) == -1) {

             if (_delta > 1) {

               ++next_node;

               continue;

             }

             return INFEASIBLE;

           }

           // Augment along a shortest path from s to t

           Value d = std::min(_excess[s], -_excess[t]);

           int u = t;

           int a;

           if (d > _delta) {

             while ((a = _pred[u]) != -1) {

               if (_res_cap[a] < d) d = _res_cap[a];

               u = _source[a];

             }

           }

           u = t;

           while ((a = _pred[u]) != -1) {

             _res_cap[a] -= d;

             _res_cap[_reverse[a]] += d;

             u = _source[a];

           }

           _excess[s] -= d;

           _excess[t] += d;

           if (_excess[s] < _delta) ++next_node;

         }

         if (_delta == 1) break;

         _delta = _delta <= _factor ? 1 : _delta / _factor;

       }

       return OPTIMAL;

     }

     // Execute the successive shortest path algorithm

     ProblemType startWithoutScaling() {

       // Find excess nodes

       _excess_nodes.clear();

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

         if (_excess[i] > 0) _excess_nodes.push_back(i);

       }

       if (_excess_nodes.size() == 0) return OPTIMAL;

       int next_node = 0;

       // Find shortest paths

       int s, t;

       ResidualDijkstra _dijkstra(*this);

       while ( _excess[_excess_nodes[next_node]] > 0 ||

               ++next_node < int(_excess_nodes.size()) )

       {

         // Run Dijkstra in the residual network

         s = _excess_nodes[next_node];

         if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;

         // Augment along a shortest path from s to t

         Value d = std::min(_excess[s], -_excess[t]);

         int u = t;

         int a;

         if (d > 1) {

           while ((a = _pred[u]) != -1) {

             if (_res_cap[a] < d) d = _res_cap[a];

             u = _source[a];

           }

         }

         u = t;

         while ((a = _pred[u]) != -1) {

           _res_cap[a] -= d;

           _res_cap[_reverse[a]] += d;

           u = _source[a];

         }

         _excess[s] -= d;

         _excess[t] += d;

       }

       return OPTIMAL;

     }

   }; //class CapacityScaling

   ///@}

 } //namespace lemon

 #endif //LEMON_CAPACITY_SCALING_H