RhodeCode

      UNBOUNDED

    };

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(GR);

    typedef std::vector<int> IntVector;

    typedef std::vector<bool> BoolVector;

    typedef std::vector<Value> ValueVector;

    typedef std::vector<Cost> CostVector;

  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;

    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;

      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), _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();

    /// 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 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

    ///

    ///

    /// \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

      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 run() calls. If this function is not

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

    /// \ref run() call.

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

    ///

    /// 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 (reset() is not called,

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

    ///   cost[e] += 100;

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

    ///

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

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

    ///   cs.reset();

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

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

    /// \endcode

    ///

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

    CapacityScaling& reset() {

      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;

    }

    /// @}

    /// \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.

    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

      if (_node_num == 0) 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 maps

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

        _pi[i] = 0;

        _excess[i] = _supply[i];

      }

      // Remove non-zero lower bounds

      if (_have_lower) {

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

          for (int j = _first_out[i]; j != _first_out[i+1]; ++j) {

            if (_forward[j]) {

              Value c = _lower[j];

              if (c >= 0) {

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

              } else {

                _res_cap[j] = _upper[j] < INF + 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 u = 0; u != _root; ++u) {

        for (int a = _first_out[u]; a != _first_out[u+1]; ++a) {

          Value rc = _res_cap[a];

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

            if (rc == INF) return UNBOUNDED;

            _excess[u] -= rc;

            _excess[_target[a]] += rc;

            _res_cap[a] = 0;

            _res_cap[_reverse[a]] += 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 u = _target[a];

          if (_excess[u] < 0) {

            _res_cap[a] = -_excess[u] + 1;

          } else {

            _res_cap[a] = 1;

          }

          _res_cap[_reverse[a]] = 0;

          _cost[a] = 0;

          _cost[_reverse[a]] = 0;

        }

      } else {

        _pi[_root] = 0;

        _excess[_root] = 0;

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

          _res_cap[a] = 1;

          _res_cap[_reverse[a]] = 0;

          _cost[a] = 0;

          _cost[_reverse[a]] = 0;

        }

      }

      // Initialize delta value

        // With scaling

        Value max_sup = 0, max_dem = 0;

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

          if ( _excess[i] > max_sup) max_sup =  _excess[i];

          if (-_excess[i] > max_dem) max_dem = -_excess[i];

        }

        Value max_cap = 0;

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

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

        }

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

      } 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) {

        for (int j = 0; j != _res_arc_num - _node_num + 1; ++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

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

          for (int a = _first_out[u]; a != _first_out[u+1]; ++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) {

          if (_excess[u] >=  _delta) _excess_nodes.push_back(u);

          if (_excess[u] <= -_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;

      }

      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;

    }

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

    /// otherwise it is \c double.

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

    typedef double LargeCost;

  };

  // Default traits class for integer cost types

  template <typename GR, typename V, typename C>

  struct CostScalingDefaultTraits<GR, V, C, true>

  {

    typedef GR Digraph;

    typedef V Value;

    typedef C Cost;

#ifdef LEMON_HAVE_LONG_LONG

    typedef long long LargeCost;

#else

    typedef long LargeCost;

#endif

  };

  /// \addtogroup min_cost_flow_algs

  /// @{

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

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

  ///

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

  /// push/augment and relabel operations for finding a minimum cost

  /// flow. It is an efficient primal-dual solution method, which

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

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

  ///

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

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

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

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

  ///

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

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

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

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

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

  ///

  /// \warning Both value 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 = CostScalingDefaultTraits<GR, V, C> >

#endif

  class CostScaling

  {

  public:

    /// The type of the digraph

    typedef typename TR::Digraph Digraph;

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

    typedef typename TR::Value Value;

    /// The type of the arc costs

    typedef typename TR::Cost Cost;

    /// \brief The large cost type

    ///

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

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

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

    /// otherwise it is \c double.

    typedef typename TR::LargeCost LargeCost;

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

    typedef TR Traits;

  public:

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

    ///

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

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

    enum ProblemType {

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

      INFEASIBLE,

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

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

      /// potentials (primal and dual solutions).

      OPTIMAL,

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

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

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

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

      /// these cases.

      UNBOUNDED

    };

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(GR);

    typedef std::vector<int> IntVector;

    typedef std::vector<char> BoolVector;

    typedef std::vector<Value> ValueVector;

    typedef std::vector<Cost> CostVector;

    typedef std::vector<LargeCost> LargeCostVector;

  private:

    template <typename KT, typename VT>

    class 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, LargeCost> LargeCostNodeMap;

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

  private:

    // Data related to the underlying digraph

    const GR &_graph;

    int _node_num;

    int _arc_num;

    int _res_node_num;

    int _res_arc_num;

    int _root;

    ///

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

    /// and the required flow value.

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

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

    ///

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

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

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

    ///

    /// \param s The source node.

    /// \param t The target node.

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

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

    ///

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

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

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

        _supply[i] = 0;

      }

      _supply[_node_id[s]] =  k;

      _supply[_node_id[t]] = -k;

      return *this;

    }

    /// @}

    /// \name Execution control

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

    /// @{

    /// \brief Run the algorithm.

    ///

    /// This function runs the algorithm.

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

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

    /// For example,

    /// \code

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

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

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

    /// \endcode

    ///

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

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

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

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

    ///

    ///

    /// \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.

    ///

      ProblemType pt = init();

      if (pt != OPTIMAL) return pt;

      return OPTIMAL;

    }

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

    ///

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

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

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

    ///

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

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

    /// \ref run() call.

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

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

    ///

    /// For example,

    /// \code

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

    ///

    ///   // First run

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

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

    ///

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

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

    ///   cost[e] += 100;

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

    ///

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

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

    ///   cs.reset();

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

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

    /// \endcode

    ///

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

    CostScaling& reset() {

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

        _supply[i] = 0;

      }

      int limit = _first_out[_root];

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

        _lower[j] = 0;

        _upper[j] = INF;

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

      }

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

        _lower[j] = 0;

    /// \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 == 0) return INFEASIBLE;

      // Check the sum of supply values

      _sum_supply = 0;

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

        _sum_supply += _supply[i];

      }

      if (_sum_supply > 0) return INFEASIBLE;

      // Initialize vectors

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

        _pi[i] = 0;

        _excess[i] = _supply[i];

      }

      // Remove infinite upper bounds and check negative arcs

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

      int last_out;

      if (_have_lower) {

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

          last_out = _first_out[i+1];

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

            if (_forward[j]) {

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

              if (c >= MAX) return UNBOUNDED;

              _excess[i] -= c;

              _excess[_target[j]] += c;

            }

          }

        }

      } else {

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

          last_out = _first_out[i+1];

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

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

              Value c = _upper[j];

              if (c >= MAX) return UNBOUNDED;

              _excess[i] -= c;

              _excess[_target[j]] += c;

            }

          }

        }

      }

      Value ex, max_cap = 0;

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

        ex = _excess[i];

        _excess[i] = 0;

        if (ex < 0) max_cap -= ex;

      }

        }

      }

      // Find a feasible flow using Circulation

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

        circ(_graph, low, cap, sup);

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

      // Set residual capacities and handle GEQ supply type

      if (_sum_supply < 0) {

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

          Value fa = flow[a];

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

          _res_cap[_arc_idb[a]] = fa;

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

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

        }

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

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

        }

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

          int u = _target[a];

          int ra = _reverse[a];

          _res_cap[a] = -_sum_supply + 1;

          _res_cap[ra] = -_excess[u];

          _cost[a] = 0;

          _cost[ra] = 0;

          _excess[u] = 0;

        }

      } else {

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

          Value fa = flow[a];

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

          _res_cap[_arc_idb[a]] = fa;

        }

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

          int ra = _reverse[a];

          _res_cap[a] = 1;

          _res_cap[ra] = 0;

          _cost[a] = 0;

          _cost[ra] = 0;

        }

      }

      return OPTIMAL;

    }

    // Execute the algorithm and transform the results

      // Execute the algorithm

      }

      // Compute node potentials for the original costs

      _arc_vec.clear();

      _cost_vec.clear();

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

        if (_res_cap[j] > 0) {

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

          _cost_vec.push_back(_scost[j]);

        }

      }

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

      typename BellmanFord<StaticDigraph, LargeCostArcMap>

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

      bf.distMap(_pi_map);

      bf.init(0);

      bf.start();

      // Handle non-zero lower bounds

      if (_have_lower) {

        int limit = _first_out[_root];

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

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

        }

      }

    }

      // Paramters for heuristics

      const int BF_HEURISTIC_EPSILON_BOUND = 1000;

      const int BF_HEURISTIC_BOUND_FACTOR  = 3;

      // Perform cost scaling phases

      IntVector pred_arc(_res_node_num);

      std::vector<int> path_nodes;

      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?

                                        1 : _epsilon / _alpha )

      {

        // "Early Termination" heuristic: use Bellman-Ford algorithm

        // to check if the current flow is optimal

        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {

          _arc_vec.clear();

          _cost_vec.clear();

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

            if (_res_cap[j] > 0) {

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

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

            }

          }

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

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

          bf.init(0);

          bool done = false;

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

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

            done = bf.processNextWeakRound();

          if (done) break;

        }

        // Saturate arcs not satisfying the optimality condition

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

          if (_res_cap[a] > 0 &&

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

            Value delta = _res_cap[a];

            _excess[_source[a]] -= delta;

            _excess[_target[a]] += delta;

            _res_cap[a] = 0;

            _res_cap[_reverse[a]] += delta;

          }

        }

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

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

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

        }

        // Initialize the next arcs