kpeter@808: /* -*- C++ -*-
kpeter@808:  *
kpeter@808:  * This file is a part of LEMON, a generic C++ optimization library
kpeter@808:  *
kpeter@808:  * Copyright (C) 2003-2008
kpeter@808:  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
kpeter@808:  * (Egervary Research Group on Combinatorial Optimization, EGRES).
kpeter@808:  *
kpeter@808:  * Permission to use, modify and distribute this software is granted
kpeter@808:  * provided that this copyright notice appears in all copies. For
kpeter@808:  * precise terms see the accompanying LICENSE file.
kpeter@808:  *
kpeter@808:  * This software is provided "AS IS" with no warranty of any kind,
kpeter@808:  * express or implied, and with no claim as to its suitability for any
kpeter@808:  * purpose.
kpeter@808:  *
kpeter@808:  */
kpeter@808: 
kpeter@808: #ifndef LEMON_COST_SCALING_H
kpeter@808: #define LEMON_COST_SCALING_H
kpeter@808: 
kpeter@808: /// \ingroup min_cost_flow_algs
kpeter@808: /// \file
kpeter@808: /// \brief Cost scaling algorithm for finding a minimum cost flow.
kpeter@808: 
kpeter@808: #include <vector>
kpeter@808: #include <deque>
kpeter@808: #include <limits>
kpeter@808: 
kpeter@808: #include <lemon/core.h>
kpeter@808: #include <lemon/maps.h>
kpeter@808: #include <lemon/math.h>
kpeter@809: #include <lemon/static_graph.h>
kpeter@808: #include <lemon/circulation.h>
kpeter@808: #include <lemon/bellman_ford.h>
kpeter@808: 
kpeter@808: namespace lemon {
kpeter@808: 
kpeter@809:   /// \brief Default traits class of CostScaling algorithm.
kpeter@809:   ///
kpeter@809:   /// Default traits class of CostScaling algorithm.
kpeter@809:   /// \tparam GR Digraph type.
kpeter@812:   /// \tparam V The number type used for flow amounts, capacity bounds
kpeter@809:   /// and supply values. By default it is \c int.
kpeter@812:   /// \tparam C The number type used for costs and potentials.
kpeter@809:   /// By default it is the same as \c V.
kpeter@809: #ifdef DOXYGEN
kpeter@809:   template <typename GR, typename V = int, typename C = V>
kpeter@809: #else
kpeter@809:   template < typename GR, typename V = int, typename C = V,
kpeter@809:              bool integer = std::numeric_limits<C>::is_integer >
kpeter@809: #endif
kpeter@809:   struct CostScalingDefaultTraits
kpeter@809:   {
kpeter@809:     /// The type of the digraph
kpeter@809:     typedef GR Digraph;
kpeter@809:     /// The type of the flow amounts, capacity bounds and supply values
kpeter@809:     typedef V Value;
kpeter@809:     /// The type of the arc costs
kpeter@809:     typedef C Cost;
kpeter@809: 
kpeter@809:     /// \brief The large cost type used for internal computations
kpeter@809:     ///
kpeter@809:     /// The large cost type used for internal computations.
kpeter@809:     /// It is \c long \c long if the \c Cost type is integer,
kpeter@809:     /// otherwise it is \c double.
kpeter@809:     /// \c Cost must be convertible to \c LargeCost.
kpeter@809:     typedef double LargeCost;
kpeter@809:   };
kpeter@809: 
kpeter@809:   // Default traits class for integer cost types
kpeter@809:   template <typename GR, typename V, typename C>
kpeter@809:   struct CostScalingDefaultTraits<GR, V, C, true>
kpeter@809:   {
kpeter@809:     typedef GR Digraph;
kpeter@809:     typedef V Value;
kpeter@809:     typedef C Cost;
kpeter@809: #ifdef LEMON_HAVE_LONG_LONG
kpeter@809:     typedef long long LargeCost;
kpeter@809: #else
kpeter@809:     typedef long LargeCost;
kpeter@809: #endif
kpeter@809:   };
kpeter@809: 
kpeter@809: 
kpeter@808:   /// \addtogroup min_cost_flow_algs
kpeter@808:   /// @{
kpeter@808: 
kpeter@809:   /// \brief Implementation of the Cost Scaling algorithm for
kpeter@809:   /// finding a \ref min_cost_flow "minimum cost flow".
kpeter@808:   ///
kpeter@809:   /// \ref CostScaling implements a cost scaling algorithm that performs
kpeter@813:   /// push/augment and relabel operations for finding a \ref min_cost_flow
kpeter@813:   /// "minimum cost flow" \ref amo93networkflows, \ref goldberg90approximation,
kpeter@813:   /// \ref goldberg97efficient, \ref bunnagel98efficient. 
kpeter@813:   /// It is a highly efficient primal-dual solution method, which
kpeter@809:   /// can be viewed as the generalization of the \ref Preflow
kpeter@809:   /// "preflow push-relabel" algorithm for the maximum flow problem.
kpeter@808:   ///
kpeter@809:   /// Most of the parameters of the problem (except for the digraph)
kpeter@809:   /// can be given using separate functions, and the algorithm can be
kpeter@809:   /// executed using the \ref run() function. If some parameters are not
kpeter@809:   /// specified, then default values will be used.
kpeter@808:   ///
kpeter@809:   /// \tparam GR The digraph type the algorithm runs on.
kpeter@812:   /// \tparam V The number type used for flow amounts, capacity bounds
kpeter@809:   /// and supply values in the algorithm. By default it is \c int.
kpeter@812:   /// \tparam C The number type used for costs and potentials in the
kpeter@809:   /// algorithm. By default it is the same as \c V.
kpeter@808:   ///
kpeter@812:   /// \warning Both number types must be signed and all input data must
kpeter@809:   /// be integer.
kpeter@809:   /// \warning This algorithm does not support negative costs for such
kpeter@809:   /// arcs that have infinite upper bound.
kpeter@810:   ///
kpeter@810:   /// \note %CostScaling provides three different internal methods,
kpeter@810:   /// from which the most efficient one is used by default.
kpeter@810:   /// For more information, see \ref Method.
kpeter@809: #ifdef DOXYGEN
kpeter@809:   template <typename GR, typename V, typename C, typename TR>
kpeter@809: #else
kpeter@809:   template < typename GR, typename V = int, typename C = V,
kpeter@809:              typename TR = CostScalingDefaultTraits<GR, V, C> >
kpeter@809: #endif
kpeter@808:   class CostScaling
kpeter@808:   {
kpeter@809:   public:
kpeter@808: 
kpeter@809:     /// The type of the digraph
kpeter@809:     typedef typename TR::Digraph Digraph;
kpeter@809:     /// The type of the flow amounts, capacity bounds and supply values
kpeter@809:     typedef typename TR::Value Value;
kpeter@809:     /// The type of the arc costs
kpeter@809:     typedef typename TR::Cost Cost;
kpeter@808: 
kpeter@809:     /// \brief The large cost type
kpeter@809:     ///
kpeter@809:     /// The large cost type used for internal computations.
kpeter@809:     /// Using the \ref CostScalingDefaultTraits "default traits class",
kpeter@809:     /// it is \c long \c long if the \c Cost type is integer,
kpeter@809:     /// otherwise it is \c double.
kpeter@809:     typedef typename TR::LargeCost LargeCost;
kpeter@808: 
kpeter@809:     /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
kpeter@809:     typedef TR Traits;
kpeter@808: 
kpeter@808:   public:
kpeter@808: 
kpeter@809:     /// \brief Problem type constants for the \c run() function.
kpeter@809:     ///
kpeter@809:     /// Enum type containing the problem type constants that can be
kpeter@809:     /// returned by the \ref run() function of the algorithm.
kpeter@809:     enum ProblemType {
kpeter@809:       /// The problem has no feasible solution (flow).
kpeter@809:       INFEASIBLE,
kpeter@809:       /// The problem has optimal solution (i.e. it is feasible and
kpeter@809:       /// bounded), and the algorithm has found optimal flow and node
kpeter@809:       /// potentials (primal and dual solutions).
kpeter@809:       OPTIMAL,
kpeter@809:       /// The digraph contains an arc of negative cost and infinite
kpeter@809:       /// upper bound. It means that the objective function is unbounded
kpeter@812:       /// on that arc, however, note that it could actually be bounded
kpeter@809:       /// over the feasible flows, but this algroithm cannot handle
kpeter@809:       /// these cases.
kpeter@809:       UNBOUNDED
kpeter@809:     };
kpeter@808: 
kpeter@810:     /// \brief Constants for selecting the internal method.
kpeter@810:     ///
kpeter@810:     /// Enum type containing constants for selecting the internal method
kpeter@810:     /// for the \ref run() function.
kpeter@810:     ///
kpeter@810:     /// \ref CostScaling provides three internal methods that differ mainly
kpeter@810:     /// in their base operations, which are used in conjunction with the
kpeter@810:     /// relabel operation.
kpeter@810:     /// By default, the so called \ref PARTIAL_AUGMENT
kpeter@810:     /// "Partial Augment-Relabel" method is used, which proved to be
kpeter@810:     /// the most efficient and the most robust on various test inputs.
kpeter@810:     /// However, the other methods can be selected using the \ref run()
kpeter@810:     /// function with the proper parameter.
kpeter@810:     enum Method {
kpeter@810:       /// Local push operations are used, i.e. flow is moved only on one
kpeter@810:       /// admissible arc at once.
kpeter@810:       PUSH,
kpeter@810:       /// Augment operations are used, i.e. flow is moved on admissible
kpeter@810:       /// paths from a node with excess to a node with deficit.
kpeter@810:       AUGMENT,
kpeter@810:       /// Partial augment operations are used, i.e. flow is moved on 
kpeter@810:       /// admissible paths started from a node with excess, but the
kpeter@810:       /// lengths of these paths are limited. This method can be viewed
kpeter@810:       /// as a combined version of the previous two operations.
kpeter@810:       PARTIAL_AUGMENT
kpeter@810:     };
kpeter@810: 
kpeter@808:   private:
kpeter@808: 
kpeter@809:     TEMPLATE_DIGRAPH_TYPEDEFS(GR);
kpeter@808: 
kpeter@809:     typedef std::vector<int> IntVector;
kpeter@809:     typedef std::vector<char> BoolVector;
kpeter@809:     typedef std::vector<Value> ValueVector;
kpeter@809:     typedef std::vector<Cost> CostVector;
kpeter@809:     typedef std::vector<LargeCost> LargeCostVector;
kpeter@808: 
kpeter@809:   private:
kpeter@809:   
kpeter@809:     template <typename KT, typename VT>
kpeter@820:     class StaticVectorMap {
kpeter@808:     public:
kpeter@809:       typedef KT Key;
kpeter@809:       typedef VT Value;
kpeter@809:       
kpeter@820:       StaticVectorMap(std::vector<Value>& v) : _v(v) {}
kpeter@809:       
kpeter@809:       const Value& operator[](const Key& key) const {
kpeter@809:         return _v[StaticDigraph::id(key)];
kpeter@808:       }
kpeter@808: 
kpeter@809:       Value& operator[](const Key& key) {
kpeter@809:         return _v[StaticDigraph::id(key)];
kpeter@809:       }
kpeter@809:       
kpeter@809:       void set(const Key& key, const Value& val) {
kpeter@809:         _v[StaticDigraph::id(key)] = val;
kpeter@808:       }
kpeter@808: 
kpeter@809:     private:
kpeter@809:       std::vector<Value>& _v;
kpeter@809:     };
kpeter@809: 
kpeter@820:     typedef StaticVectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;
kpeter@820:     typedef StaticVectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
kpeter@808: 
kpeter@808:   private:
kpeter@808: 
kpeter@809:     // Data related to the underlying digraph
kpeter@809:     const GR &_graph;
kpeter@809:     int _node_num;
kpeter@809:     int _arc_num;
kpeter@809:     int _res_node_num;
kpeter@809:     int _res_arc_num;
kpeter@809:     int _root;
kpeter@808: 
kpeter@809:     // Parameters of the problem
kpeter@809:     bool _have_lower;
kpeter@809:     Value _sum_supply;
kpeter@808: 
kpeter@809:     // Data structures for storing the digraph
kpeter@809:     IntNodeMap _node_id;
kpeter@809:     IntArcMap _arc_idf;
kpeter@809:     IntArcMap _arc_idb;
kpeter@809:     IntVector _first_out;
kpeter@809:     BoolVector _forward;
kpeter@809:     IntVector _source;
kpeter@809:     IntVector _target;
kpeter@809:     IntVector _reverse;
kpeter@809: 
kpeter@809:     // Node and arc data
kpeter@809:     ValueVector _lower;
kpeter@809:     ValueVector _upper;
kpeter@809:     CostVector _scost;
kpeter@809:     ValueVector _supply;
kpeter@809: 
kpeter@809:     ValueVector _res_cap;
kpeter@809:     LargeCostVector _cost;
kpeter@809:     LargeCostVector _pi;
kpeter@809:     ValueVector _excess;
kpeter@809:     IntVector _next_out;
kpeter@809:     std::deque<int> _active_nodes;
kpeter@809: 
kpeter@809:     // Data for scaling
kpeter@809:     LargeCost _epsilon;
kpeter@808:     int _alpha;
kpeter@808: 
kpeter@809:     // Data for a StaticDigraph structure
kpeter@809:     typedef std::pair<int, int> IntPair;
kpeter@809:     StaticDigraph _sgr;
kpeter@809:     std::vector<IntPair> _arc_vec;
kpeter@809:     std::vector<LargeCost> _cost_vec;
kpeter@809:     LargeCostArcMap _cost_map;
kpeter@809:     LargeCostNodeMap _pi_map;
kpeter@809:   
kpeter@809:   public:
kpeter@809:   
kpeter@809:     /// \brief Constant for infinite upper bounds (capacities).
kpeter@809:     ///
kpeter@809:     /// Constant for infinite upper bounds (capacities).
kpeter@809:     /// It is \c std::numeric_limits<Value>::infinity() if available,
kpeter@809:     /// \c std::numeric_limits<Value>::max() otherwise.
kpeter@809:     const Value INF;
kpeter@809: 
kpeter@808:   public:
kpeter@808: 
kpeter@809:     /// \name Named Template Parameters
kpeter@809:     /// @{
kpeter@809: 
kpeter@809:     template <typename T>
kpeter@809:     struct SetLargeCostTraits : public Traits {
kpeter@809:       typedef T LargeCost;
kpeter@809:     };
kpeter@809: 
kpeter@809:     /// \brief \ref named-templ-param "Named parameter" for setting
kpeter@809:     /// \c LargeCost type.
kpeter@808:     ///
kpeter@809:     /// \ref named-templ-param "Named parameter" for setting \c LargeCost
kpeter@809:     /// type, which is used for internal computations in the algorithm.
kpeter@809:     /// \c Cost must be convertible to \c LargeCost.
kpeter@809:     template <typename T>
kpeter@809:     struct SetLargeCost
kpeter@809:       : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
kpeter@809:       typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
kpeter@809:     };
kpeter@809: 
kpeter@809:     /// @}
kpeter@809: 
kpeter@809:   public:
kpeter@809: 
kpeter@809:     /// \brief Constructor.
kpeter@808:     ///
kpeter@809:     /// The constructor of the class.
kpeter@809:     ///
kpeter@809:     /// \param graph The digraph the algorithm runs on.
kpeter@809:     CostScaling(const GR& graph) :
kpeter@809:       _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
kpeter@809:       _cost_map(_cost_vec), _pi_map(_pi),
kpeter@809:       INF(std::numeric_limits<Value>::has_infinity ?
kpeter@809:           std::numeric_limits<Value>::infinity() :
kpeter@809:           std::numeric_limits<Value>::max())
kpeter@808:     {
kpeter@812:       // Check the number types
kpeter@809:       LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
kpeter@809:         "The flow type of CostScaling must be signed");
kpeter@809:       LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
kpeter@809:         "The cost type of CostScaling must be signed");
kpeter@809: 
kpeter@809:       // Resize vectors
kpeter@809:       _node_num = countNodes(_graph);
kpeter@809:       _arc_num = countArcs(_graph);
kpeter@809:       _res_node_num = _node_num + 1;
kpeter@809:       _res_arc_num = 2 * (_arc_num + _node_num);
kpeter@809:       _root = _node_num;
kpeter@809: 
kpeter@809:       _first_out.resize(_res_node_num + 1);
kpeter@809:       _forward.resize(_res_arc_num);
kpeter@809:       _source.resize(_res_arc_num);
kpeter@809:       _target.resize(_res_arc_num);
kpeter@809:       _reverse.resize(_res_arc_num);
kpeter@809: 
kpeter@809:       _lower.resize(_res_arc_num);
kpeter@809:       _upper.resize(_res_arc_num);
kpeter@809:       _scost.resize(_res_arc_num);
kpeter@809:       _supply.resize(_res_node_num);
kpeter@808:       
kpeter@809:       _res_cap.resize(_res_arc_num);
kpeter@809:       _cost.resize(_res_arc_num);
kpeter@809:       _pi.resize(_res_node_num);
kpeter@809:       _excess.resize(_res_node_num);
kpeter@809:       _next_out.resize(_res_node_num);
kpeter@808: 
kpeter@809:       _arc_vec.reserve(_res_arc_num);
kpeter@809:       _cost_vec.reserve(_res_arc_num);
kpeter@809: 
kpeter@809:       // Copy the graph
kpeter@809:       int i = 0, j = 0, k = 2 * _arc_num + _node_num;
kpeter@809:       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
kpeter@809:         _node_id[n] = i;
kpeter@809:       }
kpeter@809:       i = 0;
kpeter@809:       for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
kpeter@809:         _first_out[i] = j;
kpeter@809:         for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
kpeter@809:           _arc_idf[a] = j;
kpeter@809:           _forward[j] = true;
kpeter@809:           _source[j] = i;
kpeter@809:           _target[j] = _node_id[_graph.runningNode(a)];
kpeter@808:         }
kpeter@809:         for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
kpeter@809:           _arc_idb[a] = j;
kpeter@809:           _forward[j] = false;
kpeter@809:           _source[j] = i;
kpeter@809:           _target[j] = _node_id[_graph.runningNode(a)];
kpeter@809:         }
kpeter@809:         _forward[j] = false;
kpeter@809:         _source[j] = i;
kpeter@809:         _target[j] = _root;
kpeter@809:         _reverse[j] = k;
kpeter@809:         _forward[k] = true;
kpeter@809:         _source[k] = _root;
kpeter@809:         _target[k] = i;
kpeter@809:         _reverse[k] = j;
kpeter@809:         ++j; ++k;
kpeter@808:       }
kpeter@809:       _first_out[i] = j;
kpeter@809:       _first_out[_res_node_num] = k;
kpeter@809:       for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:         int fi = _arc_idf[a];
kpeter@809:         int bi = _arc_idb[a];
kpeter@809:         _reverse[fi] = bi;
kpeter@809:         _reverse[bi] = fi;
kpeter@809:       }
kpeter@809:       
kpeter@809:       // Reset parameters
kpeter@809:       reset();
kpeter@808:     }
kpeter@808: 
kpeter@809:     /// \name Parameters
kpeter@809:     /// The parameters of the algorithm can be specified using these
kpeter@809:     /// functions.
kpeter@809: 
kpeter@809:     /// @{
kpeter@809: 
kpeter@809:     /// \brief Set the lower bounds on the arcs.
kpeter@808:     ///
kpeter@809:     /// This function sets the lower bounds on the arcs.
kpeter@809:     /// If it is not used before calling \ref run(), the lower bounds
kpeter@809:     /// will be set to zero on all arcs.
kpeter@808:     ///
kpeter@809:     /// \param map An arc map storing the lower bounds.
kpeter@809:     /// Its \c Value type must be convertible to the \c Value type
kpeter@809:     /// of the algorithm.
kpeter@809:     ///
kpeter@809:     /// \return <tt>(*this)</tt>
kpeter@809:     template <typename LowerMap>
kpeter@809:     CostScaling& lowerMap(const LowerMap& map) {
kpeter@809:       _have_lower = true;
kpeter@809:       for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:         _lower[_arc_idf[a]] = map[a];
kpeter@809:         _lower[_arc_idb[a]] = map[a];
kpeter@808:       }
kpeter@808:       return *this;
kpeter@808:     }
kpeter@808: 
kpeter@809:     /// \brief Set the upper bounds (capacities) on the arcs.
kpeter@808:     ///
kpeter@809:     /// This function sets the upper bounds (capacities) on the arcs.
kpeter@809:     /// If it is not used before calling \ref run(), the upper bounds
kpeter@809:     /// will be set to \ref INF on all arcs (i.e. the flow value will be
kpeter@812:     /// unbounded from above).
kpeter@808:     ///
kpeter@809:     /// \param map An arc map storing the upper bounds.
kpeter@809:     /// Its \c Value type must be convertible to the \c Value type
kpeter@809:     /// of the algorithm.
kpeter@809:     ///
kpeter@809:     /// \return <tt>(*this)</tt>
kpeter@809:     template<typename UpperMap>
kpeter@809:     CostScaling& upperMap(const UpperMap& map) {
kpeter@809:       for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:         _upper[_arc_idf[a]] = map[a];
kpeter@808:       }
kpeter@808:       return *this;
kpeter@808:     }
kpeter@808: 
kpeter@809:     /// \brief Set the costs of the arcs.
kpeter@809:     ///
kpeter@809:     /// This function sets the costs of the arcs.
kpeter@809:     /// If it is not used before calling \ref run(), the costs
kpeter@809:     /// will be set to \c 1 on all arcs.
kpeter@809:     ///
kpeter@809:     /// \param map An arc map storing the costs.
kpeter@809:     /// Its \c Value type must be convertible to the \c Cost type
kpeter@809:     /// of the algorithm.
kpeter@809:     ///
kpeter@809:     /// \return <tt>(*this)</tt>
kpeter@809:     template<typename CostMap>
kpeter@809:     CostScaling& costMap(const CostMap& map) {
kpeter@809:       for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:         _scost[_arc_idf[a]] =  map[a];
kpeter@809:         _scost[_arc_idb[a]] = -map[a];
kpeter@809:       }
kpeter@809:       return *this;
kpeter@809:     }
kpeter@809: 
kpeter@809:     /// \brief Set the supply values of the nodes.
kpeter@809:     ///
kpeter@809:     /// This function sets the supply values of the nodes.
kpeter@809:     /// If neither this function nor \ref stSupply() is used before
kpeter@809:     /// calling \ref run(), the supply of each node will be set to zero.
kpeter@809:     ///
kpeter@809:     /// \param map A node map storing the supply values.
kpeter@809:     /// Its \c Value type must be convertible to the \c Value type
kpeter@809:     /// of the algorithm.
kpeter@809:     ///
kpeter@809:     /// \return <tt>(*this)</tt>
kpeter@809:     template<typename SupplyMap>
kpeter@809:     CostScaling& supplyMap(const SupplyMap& map) {
kpeter@809:       for (NodeIt n(_graph); n != INVALID; ++n) {
kpeter@809:         _supply[_node_id[n]] = map[n];
kpeter@809:       }
kpeter@809:       return *this;
kpeter@809:     }
kpeter@809: 
kpeter@809:     /// \brief Set single source and target nodes and a supply value.
kpeter@809:     ///
kpeter@809:     /// This function sets a single source node and a single target node
kpeter@809:     /// and the required flow value.
kpeter@809:     /// If neither this function nor \ref supplyMap() is used before
kpeter@809:     /// calling \ref run(), the supply of each node will be set to zero.
kpeter@809:     ///
kpeter@809:     /// Using this function has the same effect as using \ref supplyMap()
kpeter@809:     /// with such a map in which \c k is assigned to \c s, \c -k is
kpeter@809:     /// assigned to \c t and all other nodes have zero supply value.
kpeter@809:     ///
kpeter@809:     /// \param s The source node.
kpeter@809:     /// \param t The target node.
kpeter@809:     /// \param k The required amount of flow from node \c s to node \c t
kpeter@809:     /// (i.e. the supply of \c s and the demand of \c t).
kpeter@809:     ///
kpeter@809:     /// \return <tt>(*this)</tt>
kpeter@809:     CostScaling& stSupply(const Node& s, const Node& t, Value k) {
kpeter@809:       for (int i = 0; i != _res_node_num; ++i) {
kpeter@809:         _supply[i] = 0;
kpeter@809:       }
kpeter@809:       _supply[_node_id[s]] =  k;
kpeter@809:       _supply[_node_id[t]] = -k;
kpeter@809:       return *this;
kpeter@809:     }
kpeter@809:     
kpeter@809:     /// @}
kpeter@809: 
kpeter@808:     /// \name Execution control
kpeter@809:     /// The algorithm can be executed using \ref run().
kpeter@808: 
kpeter@808:     /// @{
kpeter@808: 
kpeter@808:     /// \brief Run the algorithm.
kpeter@808:     ///
kpeter@809:     /// This function runs the algorithm.
kpeter@809:     /// The paramters can be specified using functions \ref lowerMap(),
kpeter@809:     /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
kpeter@809:     /// For example,
kpeter@809:     /// \code
kpeter@809:     ///   CostScaling<ListDigraph> cs(graph);
kpeter@809:     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
kpeter@809:     ///     .supplyMap(sup).run();
kpeter@809:     /// \endcode
kpeter@809:     ///
kpeter@809:     /// This function can be called more than once. All the parameters
kpeter@809:     /// that have been given are kept for the next call, unless
kpeter@809:     /// \ref reset() is called, thus only the modified parameters
kpeter@809:     /// have to be set again. See \ref reset() for examples.
kpeter@810:     /// However, the underlying digraph must not be modified after this
kpeter@810:     /// class have been constructed, since it copies and extends the graph.
kpeter@808:     ///
kpeter@810:     /// \param method The internal method that will be used in the
kpeter@810:     /// algorithm. For more information, see \ref Method.
kpeter@810:     /// \param factor The cost scaling factor. It must be larger than one.
kpeter@808:     ///
kpeter@809:     /// \return \c INFEASIBLE if no feasible flow exists,
kpeter@809:     /// \n \c OPTIMAL if the problem has optimal solution
kpeter@809:     /// (i.e. it is feasible and bounded), and the algorithm has found
kpeter@809:     /// optimal flow and node potentials (primal and dual solutions),
kpeter@809:     /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
kpeter@809:     /// and infinite upper bound. It means that the objective function
kpeter@812:     /// is unbounded on that arc, however, note that it could actually be
kpeter@809:     /// bounded over the feasible flows, but this algroithm cannot handle
kpeter@809:     /// these cases.
kpeter@809:     ///
kpeter@810:     /// \see ProblemType, Method
kpeter@810:     ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
kpeter@810:       _alpha = factor;
kpeter@809:       ProblemType pt = init();
kpeter@809:       if (pt != OPTIMAL) return pt;
kpeter@810:       start(method);
kpeter@809:       return OPTIMAL;
kpeter@809:     }
kpeter@809: 
kpeter@809:     /// \brief Reset all the parameters that have been given before.
kpeter@809:     ///
kpeter@809:     /// This function resets all the paramaters that have been given
kpeter@809:     /// before using functions \ref lowerMap(), \ref upperMap(),
kpeter@809:     /// \ref costMap(), \ref supplyMap(), \ref stSupply().
kpeter@809:     ///
kpeter@809:     /// It is useful for multiple run() calls. If this function is not
kpeter@809:     /// used, all the parameters given before are kept for the next
kpeter@809:     /// \ref run() call.
kpeter@812:     /// However, the underlying digraph must not be modified after this
kpeter@809:     /// class have been constructed, since it copies and extends the graph.
kpeter@809:     ///
kpeter@809:     /// For example,
kpeter@809:     /// \code
kpeter@809:     ///   CostScaling<ListDigraph> cs(graph);
kpeter@809:     ///
kpeter@809:     ///   // First run
kpeter@809:     ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
kpeter@809:     ///     .supplyMap(sup).run();
kpeter@809:     ///
kpeter@809:     ///   // Run again with modified cost map (reset() is not called,
kpeter@809:     ///   // so only the cost map have to be set again)
kpeter@809:     ///   cost[e] += 100;
kpeter@809:     ///   cs.costMap(cost).run();
kpeter@809:     ///
kpeter@809:     ///   // Run again from scratch using reset()
kpeter@809:     ///   // (the lower bounds will be set to zero on all arcs)
kpeter@809:     ///   cs.reset();
kpeter@809:     ///   cs.upperMap(capacity).costMap(cost)
kpeter@809:     ///     .supplyMap(sup).run();
kpeter@809:     /// \endcode
kpeter@809:     ///
kpeter@809:     /// \return <tt>(*this)</tt>
kpeter@809:     CostScaling& reset() {
kpeter@809:       for (int i = 0; i != _res_node_num; ++i) {
kpeter@809:         _supply[i] = 0;
kpeter@808:       }
kpeter@809:       int limit = _first_out[_root];
kpeter@809:       for (int j = 0; j != limit; ++j) {
kpeter@809:         _lower[j] = 0;
kpeter@809:         _upper[j] = INF;
kpeter@809:         _scost[j] = _forward[j] ? 1 : -1;
kpeter@809:       }
kpeter@809:       for (int j = limit; j != _res_arc_num; ++j) {
kpeter@809:         _lower[j] = 0;
kpeter@809:         _upper[j] = INF;
kpeter@809:         _scost[j] = 0;
kpeter@809:         _scost[_reverse[j]] = 0;
kpeter@809:       }      
kpeter@809:       _have_lower = false;
kpeter@809:       return *this;
kpeter@808:     }
kpeter@808: 
kpeter@808:     /// @}
kpeter@808: 
kpeter@808:     /// \name Query Functions
kpeter@809:     /// The results of the algorithm can be obtained using these
kpeter@808:     /// functions.\n
kpeter@809:     /// The \ref run() function must be called before using them.
kpeter@808: 
kpeter@808:     /// @{
kpeter@808: 
kpeter@809:     /// \brief Return the total cost of the found flow.
kpeter@808:     ///
kpeter@809:     /// This function returns the total cost of the found flow.
kpeter@809:     /// Its complexity is O(e).
kpeter@809:     ///
kpeter@809:     /// \note The return type of the function can be specified as a
kpeter@809:     /// template parameter. For example,
kpeter@809:     /// \code
kpeter@809:     ///   cs.totalCost<double>();
kpeter@809:     /// \endcode
kpeter@809:     /// It is useful if the total cost cannot be stored in the \c Cost
kpeter@809:     /// type of the algorithm, which is the default return type of the
kpeter@809:     /// function.
kpeter@808:     ///
kpeter@808:     /// \pre \ref run() must be called before using this function.
kpeter@809:     template <typename Number>
kpeter@809:     Number totalCost() const {
kpeter@809:       Number c = 0;
kpeter@809:       for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:         int i = _arc_idb[a];
kpeter@809:         c += static_cast<Number>(_res_cap[i]) *
kpeter@809:              (-static_cast<Number>(_scost[i]));
kpeter@809:       }
kpeter@809:       return c;
kpeter@808:     }
kpeter@808: 
kpeter@809: #ifndef DOXYGEN
kpeter@809:     Cost totalCost() const {
kpeter@809:       return totalCost<Cost>();
kpeter@808:     }
kpeter@809: #endif
kpeter@808: 
kpeter@808:     /// \brief Return the flow on the given arc.
kpeter@808:     ///
kpeter@809:     /// This function returns the flow on the given arc.
kpeter@808:     ///
kpeter@808:     /// \pre \ref run() must be called before using this function.
kpeter@809:     Value flow(const Arc& a) const {
kpeter@809:       return _res_cap[_arc_idb[a]];
kpeter@808:     }
kpeter@808: 
kpeter@809:     /// \brief Return the flow map (the primal solution).
kpeter@808:     ///
kpeter@809:     /// This function copies the flow value on each arc into the given
kpeter@809:     /// map. The \c Value type of the algorithm must be convertible to
kpeter@809:     /// the \c Value type of the map.
kpeter@808:     ///
kpeter@808:     /// \pre \ref run() must be called before using this function.
kpeter@809:     template <typename FlowMap>
kpeter@809:     void flowMap(FlowMap &map) const {
kpeter@809:       for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:         map.set(a, _res_cap[_arc_idb[a]]);
kpeter@809:       }
kpeter@808:     }
kpeter@808: 
kpeter@809:     /// \brief Return the potential (dual value) of the given node.
kpeter@808:     ///
kpeter@809:     /// This function returns the potential (dual value) of the
kpeter@809:     /// given node.
kpeter@808:     ///
kpeter@808:     /// \pre \ref run() must be called before using this function.
kpeter@809:     Cost potential(const Node& n) const {
kpeter@809:       return static_cast<Cost>(_pi[_node_id[n]]);
kpeter@809:     }
kpeter@809: 
kpeter@809:     /// \brief Return the potential map (the dual solution).
kpeter@809:     ///
kpeter@809:     /// This function copies the potential (dual value) of each node
kpeter@809:     /// into the given map.
kpeter@809:     /// The \c Cost type of the algorithm must be convertible to the
kpeter@809:     /// \c Value type of the map.
kpeter@809:     ///
kpeter@809:     /// \pre \ref run() must be called before using this function.
kpeter@809:     template <typename PotentialMap>
kpeter@809:     void potentialMap(PotentialMap &map) const {
kpeter@809:       for (NodeIt n(_graph); n != INVALID; ++n) {
kpeter@809:         map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
kpeter@809:       }
kpeter@808:     }
kpeter@808: 
kpeter@808:     /// @}
kpeter@808: 
kpeter@808:   private:
kpeter@808: 
kpeter@809:     // Initialize the algorithm
kpeter@809:     ProblemType init() {
kpeter@809:       if (_res_node_num == 0) return INFEASIBLE;
kpeter@809: 
kpeter@809:       // Check the sum of supply values
kpeter@809:       _sum_supply = 0;
kpeter@809:       for (int i = 0; i != _root; ++i) {
kpeter@809:         _sum_supply += _supply[i];
kpeter@808:       }
kpeter@809:       if (_sum_supply > 0) return INFEASIBLE;
kpeter@809:       
kpeter@809: 
kpeter@809:       // Initialize vectors
kpeter@809:       for (int i = 0; i != _res_node_num; ++i) {
kpeter@809:         _pi[i] = 0;
kpeter@809:         _excess[i] = _supply[i];
kpeter@809:       }
kpeter@809:       
kpeter@809:       // Remove infinite upper bounds and check negative arcs
kpeter@809:       const Value MAX = std::numeric_limits<Value>::max();
kpeter@809:       int last_out;
kpeter@809:       if (_have_lower) {
kpeter@809:         for (int i = 0; i != _root; ++i) {
kpeter@809:           last_out = _first_out[i+1];
kpeter@809:           for (int j = _first_out[i]; j != last_out; ++j) {
kpeter@809:             if (_forward[j]) {
kpeter@809:               Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
kpeter@809:               if (c >= MAX) return UNBOUNDED;
kpeter@809:               _excess[i] -= c;
kpeter@809:               _excess[_target[j]] += c;
kpeter@809:             }
kpeter@809:           }
kpeter@809:         }
kpeter@809:       } else {
kpeter@809:         for (int i = 0; i != _root; ++i) {
kpeter@809:           last_out = _first_out[i+1];
kpeter@809:           for (int j = _first_out[i]; j != last_out; ++j) {
kpeter@809:             if (_forward[j] && _scost[j] < 0) {
kpeter@809:               Value c = _upper[j];
kpeter@809:               if (c >= MAX) return UNBOUNDED;
kpeter@809:               _excess[i] -= c;
kpeter@809:               _excess[_target[j]] += c;
kpeter@809:             }
kpeter@809:           }
kpeter@809:         }
kpeter@809:       }
kpeter@809:       Value ex, max_cap = 0;
kpeter@809:       for (int i = 0; i != _res_node_num; ++i) {
kpeter@809:         ex = _excess[i];
kpeter@809:         _excess[i] = 0;
kpeter@809:         if (ex < 0) max_cap -= ex;
kpeter@809:       }
kpeter@809:       for (int j = 0; j != _res_arc_num; ++j) {
kpeter@809:         if (_upper[j] >= MAX) _upper[j] = max_cap;
kpeter@808:       }
kpeter@808: 
kpeter@809:       // Initialize the large cost vector and the epsilon parameter
kpeter@809:       _epsilon = 0;
kpeter@809:       LargeCost lc;
kpeter@809:       for (int i = 0; i != _root; ++i) {
kpeter@809:         last_out = _first_out[i+1];
kpeter@809:         for (int j = _first_out[i]; j != last_out; ++j) {
kpeter@809:           lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
kpeter@809:           _cost[j] = lc;
kpeter@809:           if (lc > _epsilon) _epsilon = lc;
kpeter@809:         }
kpeter@809:       }
kpeter@809:       _epsilon /= _alpha;
kpeter@808: 
kpeter@809:       // Initialize maps for Circulation and remove non-zero lower bounds
kpeter@809:       ConstMap<Arc, Value> low(0);
kpeter@809:       typedef typename Digraph::template ArcMap<Value> ValueArcMap;
kpeter@809:       typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
kpeter@809:       ValueArcMap cap(_graph), flow(_graph);
kpeter@809:       ValueNodeMap sup(_graph);
kpeter@809:       for (NodeIt n(_graph); n != INVALID; ++n) {
kpeter@809:         sup[n] = _supply[_node_id[n]];
kpeter@808:       }
kpeter@809:       if (_have_lower) {
kpeter@809:         for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:           int j = _arc_idf[a];
kpeter@809:           Value c = _lower[j];
kpeter@809:           cap[a] = _upper[j] - c;
kpeter@809:           sup[_graph.source(a)] -= c;
kpeter@809:           sup[_graph.target(a)] += c;
kpeter@809:         }
kpeter@809:       } else {
kpeter@809:         for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:           cap[a] = _upper[_arc_idf[a]];
kpeter@809:         }
kpeter@809:       }
kpeter@808: 
kpeter@808:       // Find a feasible flow using Circulation
kpeter@809:       Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
kpeter@809:         circ(_graph, low, cap, sup);
kpeter@809:       if (!circ.flowMap(flow).run()) return INFEASIBLE;
kpeter@809: 
kpeter@809:       // Set residual capacities and handle GEQ supply type
kpeter@809:       if (_sum_supply < 0) {
kpeter@809:         for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:           Value fa = flow[a];
kpeter@809:           _res_cap[_arc_idf[a]] = cap[a] - fa;
kpeter@809:           _res_cap[_arc_idb[a]] = fa;
kpeter@809:           sup[_graph.source(a)] -= fa;
kpeter@809:           sup[_graph.target(a)] += fa;
kpeter@809:         }
kpeter@809:         for (NodeIt n(_graph); n != INVALID; ++n) {
kpeter@809:           _excess[_node_id[n]] = sup[n];
kpeter@809:         }
kpeter@809:         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
kpeter@809:           int u = _target[a];
kpeter@809:           int ra = _reverse[a];
kpeter@809:           _res_cap[a] = -_sum_supply + 1;
kpeter@809:           _res_cap[ra] = -_excess[u];
kpeter@809:           _cost[a] = 0;
kpeter@809:           _cost[ra] = 0;
kpeter@809:           _excess[u] = 0;
kpeter@809:         }
kpeter@809:       } else {
kpeter@809:         for (ArcIt a(_graph); a != INVALID; ++a) {
kpeter@809:           Value fa = flow[a];
kpeter@809:           _res_cap[_arc_idf[a]] = cap[a] - fa;
kpeter@809:           _res_cap[_arc_idb[a]] = fa;
kpeter@809:         }
kpeter@809:         for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
kpeter@809:           int ra = _reverse[a];
kpeter@809:           _res_cap[a] = 1;
kpeter@809:           _res_cap[ra] = 0;
kpeter@809:           _cost[a] = 0;
kpeter@809:           _cost[ra] = 0;
kpeter@809:         }
kpeter@809:       }
kpeter@809:       
kpeter@809:       return OPTIMAL;
kpeter@809:     }
kpeter@809: 
kpeter@809:     // Execute the algorithm and transform the results
kpeter@810:     void start(Method method) {
kpeter@810:       // Maximum path length for partial augment
kpeter@810:       const int MAX_PATH_LENGTH = 4;
kpeter@810:       
kpeter@809:       // Execute the algorithm
kpeter@810:       switch (method) {
kpeter@810:         case PUSH:
kpeter@810:           startPush();
kpeter@810:           break;
kpeter@810:         case AUGMENT:
kpeter@810:           startAugment();
kpeter@810:           break;
kpeter@810:         case PARTIAL_AUGMENT:
kpeter@810:           startAugment(MAX_PATH_LENGTH);
kpeter@810:           break;
kpeter@809:       }
kpeter@809: 
kpeter@809:       // Compute node potentials for the original costs
kpeter@809:       _arc_vec.clear();
kpeter@809:       _cost_vec.clear();
kpeter@809:       for (int j = 0; j != _res_arc_num; ++j) {
kpeter@809:         if (_res_cap[j] > 0) {
kpeter@809:           _arc_vec.push_back(IntPair(_source[j], _target[j]));
kpeter@809:           _cost_vec.push_back(_scost[j]);
kpeter@809:         }
kpeter@809:       }
kpeter@809:       _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
kpeter@809: 
kpeter@809:       typename BellmanFord<StaticDigraph, LargeCostArcMap>
kpeter@809:         ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
kpeter@809:       bf.distMap(_pi_map);
kpeter@809:       bf.init(0);
kpeter@809:       bf.start();
kpeter@809: 
kpeter@809:       // Handle non-zero lower bounds
kpeter@809:       if (_have_lower) {
kpeter@809:         int limit = _first_out[_root];
kpeter@809:         for (int j = 0; j != limit; ++j) {
kpeter@809:           if (!_forward[j]) _res_cap[j] += _lower[j];
kpeter@809:         }
kpeter@809:       }
kpeter@808:     }
kpeter@808: 
kpeter@810:     /// Execute the algorithm performing augment and relabel operations
kpeter@810:     void startAugment(int max_length = std::numeric_limits<int>::max()) {
kpeter@808:       // Paramters for heuristics
kpeter@809:       const int BF_HEURISTIC_EPSILON_BOUND = 1000;
kpeter@809:       const int BF_HEURISTIC_BOUND_FACTOR  = 3;
kpeter@808: 
kpeter@809:       // Perform cost scaling phases
kpeter@809:       IntVector pred_arc(_res_node_num);
kpeter@809:       std::vector<int> path_nodes;
kpeter@808:       for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
kpeter@808:                                         1 : _epsilon / _alpha )
kpeter@808:       {
kpeter@808:         // "Early Termination" heuristic: use Bellman-Ford algorithm
kpeter@808:         // to check if the current flow is optimal
kpeter@808:         if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
kpeter@809:           _arc_vec.clear();
kpeter@809:           _cost_vec.clear();
kpeter@809:           for (int j = 0; j != _res_arc_num; ++j) {
kpeter@809:             if (_res_cap[j] > 0) {
kpeter@809:               _arc_vec.push_back(IntPair(_source[j], _target[j]));
kpeter@809:               _cost_vec.push_back(_cost[j] + 1);
kpeter@809:             }
kpeter@809:           }
kpeter@809:           _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
kpeter@809: 
kpeter@809:           BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
kpeter@808:           bf.init(0);
kpeter@808:           bool done = false;
kpeter@809:           int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
kpeter@808:           for (int i = 0; i < K && !done; ++i)
kpeter@808:             done = bf.processNextWeakRound();
kpeter@808:           if (done) break;
kpeter@808:         }
kpeter@809: 
kpeter@808:         // Saturate arcs not satisfying the optimality condition
kpeter@809:         for (int a = 0; a != _res_arc_num; ++a) {
kpeter@809:           if (_res_cap[a] > 0 &&
kpeter@809:               _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
kpeter@809:             Value delta = _res_cap[a];
kpeter@809:             _excess[_source[a]] -= delta;
kpeter@809:             _excess[_target[a]] += delta;
kpeter@809:             _res_cap[a] = 0;
kpeter@809:             _res_cap[_reverse[a]] += delta;
kpeter@808:           }
kpeter@808:         }
kpeter@809:         
kpeter@808:         // Find active nodes (i.e. nodes with positive excess)
kpeter@809:         for (int u = 0; u != _res_node_num; ++u) {
kpeter@809:           if (_excess[u] > 0) _active_nodes.push_back(u);
kpeter@808:         }
kpeter@808: 
kpeter@809:         // Initialize the next arcs
kpeter@809:         for (int u = 0; u != _res_node_num; ++u) {
kpeter@809:           _next_out[u] = _first_out[u];
kpeter@808:         }
kpeter@808: 
kpeter@808:         // Perform partial augment and relabel operations
kpeter@809:         while (true) {
kpeter@808:           // Select an active node (FIFO selection)
kpeter@809:           while (_active_nodes.size() > 0 &&
kpeter@809:                  _excess[_active_nodes.front()] <= 0) {
kpeter@809:             _active_nodes.pop_front();
kpeter@808:           }
kpeter@809:           if (_active_nodes.size() == 0) break;
kpeter@809:           int start = _active_nodes.front();
kpeter@808:           path_nodes.clear();
kpeter@808:           path_nodes.push_back(start);
kpeter@808: 
kpeter@808:           // Find an augmenting path from the start node
kpeter@809:           int tip = start;
kpeter@809:           while (_excess[tip] >= 0 &&
kpeter@810:                  int(path_nodes.size()) <= max_length) {
kpeter@809:             int u;
kpeter@809:             LargeCost min_red_cost, rc;
kpeter@809:             int last_out = _sum_supply < 0 ?
kpeter@809:               _first_out[tip+1] : _first_out[tip+1] - 1;
kpeter@809:             for (int a = _next_out[tip]; a != last_out; ++a) {
kpeter@809:               if (_res_cap[a] > 0 &&
kpeter@809:                   _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
kpeter@809:                 u = _target[a];
kpeter@809:                 pred_arc[u] = a;
kpeter@809:                 _next_out[tip] = a;
kpeter@808:                 tip = u;
kpeter@808:                 path_nodes.push_back(tip);
kpeter@808:                 goto next_step;
kpeter@808:               }
kpeter@808:             }
kpeter@808: 
kpeter@808:             // Relabel tip node
kpeter@809:             min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
kpeter@809:             for (int a = _first_out[tip]; a != last_out; ++a) {
kpeter@809:               rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
kpeter@809:               if (_res_cap[a] > 0 && rc < min_red_cost) {
kpeter@809:                 min_red_cost = rc;
kpeter@809:               }
kpeter@808:             }
kpeter@809:             _pi[tip] -= min_red_cost + _epsilon;
kpeter@808: 
kpeter@809:             // Reset the next arc of tip
kpeter@809:             _next_out[tip] = _first_out[tip];
kpeter@808: 
kpeter@808:             // Step back
kpeter@808:             if (tip != start) {
kpeter@808:               path_nodes.pop_back();
kpeter@809:               tip = path_nodes.back();
kpeter@808:             }
kpeter@808: 
kpeter@809:           next_step: ;
kpeter@808:           }
kpeter@808: 
kpeter@808:           // Augment along the found path (as much flow as possible)
kpeter@809:           Value delta;
kpeter@809:           int u, v = path_nodes.front(), pa;
kpeter@808:           for (int i = 1; i < int(path_nodes.size()); ++i) {
kpeter@809:             u = v;
kpeter@809:             v = path_nodes[i];
kpeter@809:             pa = pred_arc[v];
kpeter@809:             delta = std::min(_res_cap[pa], _excess[u]);
kpeter@809:             _res_cap[pa] -= delta;
kpeter@809:             _res_cap[_reverse[pa]] += delta;
kpeter@809:             _excess[u] -= delta;
kpeter@809:             _excess[v] += delta;
kpeter@809:             if (_excess[v] > 0 && _excess[v] <= delta)
kpeter@809:               _active_nodes.push_back(v);
kpeter@808:           }
kpeter@808:         }
kpeter@808:       }
kpeter@808:     }
kpeter@808: 
kpeter@809:     /// Execute the algorithm performing push and relabel operations
kpeter@810:     void startPush() {
kpeter@808:       // Paramters for heuristics
kpeter@809:       const int BF_HEURISTIC_EPSILON_BOUND = 1000;
kpeter@809:       const int BF_HEURISTIC_BOUND_FACTOR  = 3;
kpeter@808: 
kpeter@809:       // Perform cost scaling phases
kpeter@809:       BoolVector hyper(_res_node_num, false);
kpeter@808:       for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
kpeter@808:                                         1 : _epsilon / _alpha )
kpeter@808:       {
kpeter@808:         // "Early Termination" heuristic: use Bellman-Ford algorithm
kpeter@808:         // to check if the current flow is optimal
kpeter@808:         if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
kpeter@809:           _arc_vec.clear();
kpeter@809:           _cost_vec.clear();
kpeter@809:           for (int j = 0; j != _res_arc_num; ++j) {
kpeter@809:             if (_res_cap[j] > 0) {
kpeter@809:               _arc_vec.push_back(IntPair(_source[j], _target[j]));
kpeter@809:               _cost_vec.push_back(_cost[j] + 1);
kpeter@809:             }
kpeter@809:           }
kpeter@809:           _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
kpeter@809: 
kpeter@809:           BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
kpeter@808:           bf.init(0);
kpeter@808:           bool done = false;
kpeter@809:           int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
kpeter@808:           for (int i = 0; i < K && !done; ++i)
kpeter@808:             done = bf.processNextWeakRound();
kpeter@808:           if (done) break;
kpeter@808:         }
kpeter@808: 
kpeter@808:         // Saturate arcs not satisfying the optimality condition
kpeter@809:         for (int a = 0; a != _res_arc_num; ++a) {
kpeter@809:           if (_res_cap[a] > 0 &&
kpeter@809:               _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
kpeter@809:             Value delta = _res_cap[a];
kpeter@809:             _excess[_source[a]] -= delta;
kpeter@809:             _excess[_target[a]] += delta;
kpeter@809:             _res_cap[a] = 0;
kpeter@809:             _res_cap[_reverse[a]] += delta;
kpeter@808:           }
kpeter@808:         }
kpeter@808: 
kpeter@808:         // Find active nodes (i.e. nodes with positive excess)
kpeter@809:         for (int u = 0; u != _res_node_num; ++u) {
kpeter@809:           if (_excess[u] > 0) _active_nodes.push_back(u);
kpeter@808:         }
kpeter@808: 
kpeter@809:         // Initialize the next arcs
kpeter@809:         for (int u = 0; u != _res_node_num; ++u) {
kpeter@809:           _next_out[u] = _first_out[u];
kpeter@808:         }
kpeter@808: 
kpeter@808:         // Perform push and relabel operations
kpeter@809:         while (_active_nodes.size() > 0) {
kpeter@809:           LargeCost min_red_cost, rc;
kpeter@809:           Value delta;
kpeter@809:           int n, t, a, last_out = _res_arc_num;
kpeter@809: 
kpeter@808:           // Select an active node (FIFO selection)
kpeter@809:         next_node:
kpeter@809:           n = _active_nodes.front();
kpeter@809:           last_out = _sum_supply < 0 ?
kpeter@809:             _first_out[n+1] : _first_out[n+1] - 1;
kpeter@808: 
kpeter@808:           // Perform push operations if there are admissible arcs
kpeter@809:           if (_excess[n] > 0) {
kpeter@809:             for (a = _next_out[n]; a != last_out; ++a) {
kpeter@809:               if (_res_cap[a] > 0 &&
kpeter@809:                   _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
kpeter@809:                 delta = std::min(_res_cap[a], _excess[n]);
kpeter@809:                 t = _target[a];
kpeter@808: 
kpeter@808:                 // Push-look-ahead heuristic
kpeter@809:                 Value ahead = -_excess[t];
kpeter@809:                 int last_out_t = _sum_supply < 0 ?
kpeter@809:                   _first_out[t+1] : _first_out[t+1] - 1;
kpeter@809:                 for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
kpeter@809:                   if (_res_cap[ta] > 0 && 
kpeter@809:                       _cost[ta] + _pi[_source[ta]] - _pi[_target[ta]] < 0)
kpeter@809:                     ahead += _res_cap[ta];
kpeter@809:                   if (ahead >= delta) break;
kpeter@808:                 }
kpeter@808:                 if (ahead < 0) ahead = 0;
kpeter@808: 
kpeter@808:                 // Push flow along the arc
kpeter@808:                 if (ahead < delta) {
kpeter@809:                   _res_cap[a] -= ahead;
kpeter@809:                   _res_cap[_reverse[a]] += ahead;
kpeter@808:                   _excess[n] -= ahead;
kpeter@808:                   _excess[t] += ahead;
kpeter@809:                   _active_nodes.push_front(t);
kpeter@808:                   hyper[t] = true;
kpeter@809:                   _next_out[n] = a;
kpeter@809:                   goto next_node;
kpeter@808:                 } else {
kpeter@809:                   _res_cap[a] -= delta;
kpeter@809:                   _res_cap[_reverse[a]] += delta;
kpeter@808:                   _excess[n] -= delta;
kpeter@808:                   _excess[t] += delta;
kpeter@808:                   if (_excess[t] > 0 && _excess[t] <= delta)
kpeter@809:                     _active_nodes.push_back(t);
kpeter@808:                 }
kpeter@808: 
kpeter@809:                 if (_excess[n] == 0) {
kpeter@809:                   _next_out[n] = a;
kpeter@809:                   goto remove_nodes;
kpeter@809:                 }
kpeter@808:               }
kpeter@808:             }
kpeter@809:             _next_out[n] = a;
kpeter@808:           }
kpeter@808: 
kpeter@808:           // Relabel the node if it is still active (or hyper)
kpeter@809:           if (_excess[n] > 0 || hyper[n]) {
kpeter@809:             min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
kpeter@809:             for (int a = _first_out[n]; a != last_out; ++a) {
kpeter@809:               rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
kpeter@809:               if (_res_cap[a] > 0 && rc < min_red_cost) {
kpeter@809:                 min_red_cost = rc;
kpeter@809:               }
kpeter@808:             }
kpeter@809:             _pi[n] -= min_red_cost + _epsilon;
kpeter@808:             hyper[n] = false;
kpeter@808: 
kpeter@809:             // Reset the next arc
kpeter@809:             _next_out[n] = _first_out[n];
kpeter@808:           }
kpeter@809:         
kpeter@808:           // Remove nodes that are not active nor hyper
kpeter@809:         remove_nodes:
kpeter@809:           while ( _active_nodes.size() > 0 &&
kpeter@809:                   _excess[_active_nodes.front()] <= 0 &&
kpeter@809:                   !hyper[_active_nodes.front()] ) {
kpeter@809:             _active_nodes.pop_front();
kpeter@808:           }
kpeter@808:         }
kpeter@808:       }
kpeter@808:     }
kpeter@808: 
kpeter@808:   }; //class CostScaling
kpeter@808: 
kpeter@808:   ///@}
kpeter@808: 
kpeter@808: } //namespace lemon
kpeter@808: 
kpeter@808: #endif //LEMON_COST_SCALING_H