/* -*- C++ -*-

  *

  * This file is a part of LEMON, a generic C++ optimization library

  *

  * Copyright (C) 2003-2008

  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

  * (Egervary Research Group on Combinatorial Optimization, EGRES).

  *

  * Permission to use, modify and distribute this software is granted

  * provided that this copyright notice appears in all copies. For

  * precise terms see the accompanying LICENSE file.

  *

  * This software is provided "AS IS" with no warranty of any kind,

  * express or implied, and with no claim as to its suitability for any

  * purpose.

  *

  */

 #ifndef LEMON_COST_SCALING_H

 #define LEMON_COST_SCALING_H

 /// \ingroup min_cost_flow

 ///

 /// \file

 /// \brief Cost scaling algorithm for finding a minimum cost flow.

 #include <deque>

 #include <lemon/graph_adaptor.h>

 #include <lemon/graph_utils.h>

 #include <lemon/maps.h>

 #include <lemon/math.h>

 #include <lemon/circulation.h>

 #include <lemon/bellman_ford.h>

 namespace lemon {

   /// \addtogroup min_cost_flow

   /// @{

   /// \brief Implementation of the cost scaling algorithm for finding a

   /// minimum cost flow.

   ///

   /// \ref CostScaling implements the cost scaling algorithm performing

   /// generalized push-relabel operations for finding a minimum cost

   /// flow.

   ///

   /// \tparam Graph The directed graph type the algorithm runs on.

   /// \tparam LowerMap The type of the lower bound map.

   /// \tparam CapacityMap The type of the capacity (upper bound) map.

   /// \tparam CostMap The type of the cost (length) map.

   /// \tparam SupplyMap The type of the supply map.

   ///

   /// \warning

   /// - Edge capacities and costs should be \e non-negative \e integers.

   /// - Supply values should be \e signed \e integers.

   /// - The value types of the maps should be convertible to each other.

   /// - \c CostMap::Value must be signed type.

   ///

   /// \note Edge costs are multiplied with the number of nodes during

   /// the algorithm so overflow problems may arise more easily than with

   /// other minimum cost flow algorithms.

   /// If it is available, <tt>long long int</tt> type is used instead of

   /// <tt>long int</tt> in the inside computations.

   ///

   /// \author Peter Kovacs

   template < typename Graph,

              typename LowerMap = typename Graph::template EdgeMap<int>,

              typename CapacityMap = typename Graph::template EdgeMap<int>,

              typename CostMap = typename Graph::template EdgeMap<int>,

              typename SupplyMap = typename Graph::template NodeMap<int> >

   class CostScaling

   {

     GRAPH_TYPEDEFS(typename Graph);

     typedef typename CapacityMap::Value Capacity;

     typedef typename CostMap::Value Cost;

     typedef typename SupplyMap::Value Supply;

     typedef typename Graph::template EdgeMap<Capacity> CapacityEdgeMap;

     typedef typename Graph::template NodeMap<Supply> SupplyNodeMap;

     typedef ResGraphAdaptor< const Graph, Capacity,

                              CapacityEdgeMap, CapacityEdgeMap > ResGraph;

     typedef typename ResGraph::Edge ResEdge;

 #if defined __GNUC__ && !defined __STRICT_ANSI__

     typedef long long int LCost;

 #else

     typedef long int LCost;

 #endif

     typedef typename Graph::template EdgeMap<LCost> LargeCostMap;

   public:

     /// The type of the flow map.

     typedef typename Graph::template EdgeMap<Capacity> FlowMap;

     /// The type of the potential map.

     typedef typename Graph::template NodeMap<LCost> PotentialMap;

   private:

     /// \brief Map adaptor class for handling residual edge costs.

     ///

     /// Map adaptor class for handling residual edge costs.

     template <typename Map>

     class ResidualCostMap : public MapBase<ResEdge, typename Map::Value>

     {

     private:

       const Map &_cost_map;

     public:

       ///\e

       ResidualCostMap(const Map &cost_map) :

         _cost_map(cost_map) {}

       ///\e

       typename Map::Value operator[](const ResEdge &e) const {

         return ResGraph::forward(e) ?  _cost_map[e] : -_cost_map[e];

       }

     }; //class ResidualCostMap

     /// \brief Map adaptor class for handling reduced edge costs.

     ///

     /// Map adaptor class for handling reduced edge costs.

     class ReducedCostMap : public MapBase<Edge, LCost>

     {

     private:

       const Graph &_gr;

       const LargeCostMap &_cost_map;

       const PotentialMap &_pot_map;

     public:

       ///\e

       ReducedCostMap( const Graph &gr,

                       const LargeCostMap &cost_map,

                       const PotentialMap &pot_map ) :

         _gr(gr), _cost_map(cost_map), _pot_map(pot_map) {}

       ///\e

       LCost operator[](const Edge &e) const {

         return _cost_map[e] + _pot_map[_gr.source(e)]

                             - _pot_map[_gr.target(e)];

       }

     }; //class ReducedCostMap

   private:

     // Scaling factor

     static const int ALPHA = 4;

     // Paramters for heuristics

     static const int BF_HEURISTIC_EPSILON_BOUND = 5000;

     static const int BF_HEURISTIC_BOUND_FACTOR  = 3;

   private:

     // The directed graph the algorithm runs on

     const Graph &_graph;

     // The original lower bound map

     const LowerMap *_lower;

     // The modified capacity map

     CapacityEdgeMap _capacity;

     // The original cost map

     const CostMap &_orig_cost;

     // The scaled cost map

     LargeCostMap _cost;

     // The modified supply map

     SupplyNodeMap _supply;

     bool _valid_supply;

     // Edge map of the current flow

     FlowMap *_flow;

     bool _local_flow;

     // Node map of the current potentials

     PotentialMap *_potential;

     bool _local_potential;

     // The residual cost map

     ResidualCostMap<LargeCostMap> _res_cost;

     // The residual graph

     ResGraph *_res_graph;

     // The reduced cost map

     ReducedCostMap *_red_cost;

     // The excess map

     SupplyNodeMap _excess;

     // The epsilon parameter used for cost scaling

     LCost _epsilon;

   public:

     /// \brief General constructor (with lower bounds).

     ///

     /// General constructor (with lower bounds).

     ///

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

     /// \param lower The lower bounds of the edges.

     /// \param capacity The capacities (upper bounds) of the edges.

     /// \param cost The cost (length) values of the edges.

     /// \param supply The supply values of the nodes (signed).

     CostScaling( const Graph &graph,

                  const LowerMap &lower,

                  const CapacityMap &capacity,

                  const CostMap &cost,

                  const SupplyMap &supply ) :

       _graph(graph), _lower(&lower), _capacity(graph), _orig_cost(cost),

       _cost(graph), _supply(graph), _flow(NULL), _local_flow(false),

       _potential(NULL), _local_potential(false), _res_cost(_cost),

       _res_graph(NULL), _red_cost(NULL), _excess(graph, 0)

     {

       // Removing non-zero lower bounds

       _capacity = subMap(capacity, lower);

       Supply sum = 0;

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

         Supply s = supply[n];

         for (InEdgeIt e(_graph, n); e != INVALID; ++e)

           s += lower[e];

         for (OutEdgeIt e(_graph, n); e != INVALID; ++e)

           s -= lower[e];

         _supply[n] = s;

         sum += s;

       }

       _valid_supply = sum == 0;

     }

     /// \brief General constructor (without lower bounds).

     ///

     /// General constructor (without lower bounds).

     ///

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

     /// \param capacity The capacities (upper bounds) of the edges.

     /// \param cost The cost (length) values of the edges.

     /// \param supply The supply values of the nodes (signed).

     CostScaling( const Graph &graph,

                  const CapacityMap &capacity,

                  const CostMap &cost,

                  const SupplyMap &supply ) :

       _graph(graph), _lower(NULL), _capacity(capacity), _orig_cost(cost),

       _cost(graph), _supply(supply), _flow(NULL), _local_flow(false),

       _potential(NULL), _local_potential(false), _res_cost(_cost),

       _res_graph(NULL), _red_cost(NULL), _excess(graph, 0)

     {

       // Checking the sum of supply values

       Supply sum = 0;

       for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n];

       _valid_supply = sum == 0;

     }

     /// \brief Simple constructor (with lower bounds).

     ///

     /// Simple constructor (with lower bounds).

     ///

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

     /// \param lower The lower bounds of the edges.

     /// \param capacity The capacities (upper bounds) of the edges.

     /// \param cost The cost (length) values of the edges.

     /// \param s The source node.

     /// \param t The target node.

     /// \param flow_value 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).

     CostScaling( const Graph &graph,

                  const LowerMap &lower,

                  const CapacityMap &capacity,

                  const CostMap &cost,

                  Node s, Node t,

                  Supply flow_value ) :

       _graph(graph), _lower(&lower), _capacity(graph), _orig_cost(cost),

       _cost(graph), _supply(graph), _flow(NULL), _local_flow(false),

       _potential(NULL), _local_potential(false), _res_cost(_cost),

       _res_graph(NULL), _red_cost(NULL), _excess(graph, 0)

     {

       // Removing nonzero lower bounds

       _capacity = subMap(capacity, lower);

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

         Supply sum = 0;

         if (n == s) sum =  flow_value;

         if (n == t) sum = -flow_value;

         for (InEdgeIt e(_graph, n); e != INVALID; ++e)

           sum += lower[e];

         for (OutEdgeIt e(_graph, n); e != INVALID; ++e)

           sum -= lower[e];

         _supply[n] = sum;

       }

       _valid_supply = true;

     }

     /// \brief Simple constructor (without lower bounds).

     ///

     /// Simple constructor (without lower bounds).

     ///

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

     /// \param capacity The capacities (upper bounds) of the edges.

     /// \param cost The cost (length) values of the edges.

     /// \param s The source node.

     /// \param t The target node.

     /// \param flow_value 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).

     CostScaling( const Graph &graph,

                  const CapacityMap &capacity,

                  const CostMap &cost,

                  Node s, Node t,

                  Supply flow_value ) :

       _graph(graph), _lower(NULL), _capacity(capacity), _orig_cost(cost),

       _cost(graph), _supply(graph, 0), _flow(NULL), _local_flow(false),

       _potential(NULL), _local_potential(false), _res_cost(_cost),

       _res_graph(NULL), _red_cost(NULL), _excess(graph, 0)

     {

       _supply[s] =  flow_value;

       _supply[t] = -flow_value;

       _valid_supply = true;

     }

     /// Destructor.

     ~CostScaling() {

       if (_local_flow) delete _flow;

       if (_local_potential) delete _potential;

       delete _res_graph;

       delete _red_cost;

     }

     /// \brief Set the flow map.

     ///

     /// Set the flow map.

     ///

     /// \return \c (*this)

     CostScaling& flowMap(FlowMap &map) {

       if (_local_flow) {

         delete _flow;

         _local_flow = false;

       }

       _flow = ↦

       return *this;

     }

     /// \brief Set the potential map.

     ///

     /// Set the potential map.

     ///

     /// \return \c (*this)

     CostScaling& potentialMap(PotentialMap &map) {

       if (_local_potential) {

         delete _potential;

         _local_potential = false;

       }

       _potential = ↦

       return *this;

     }

     /// \name Execution control

     /// @{

     /// \brief Run the algorithm.

     ///

     /// Run the algorithm.

     ///

     /// \return \c true if a feasible flow can be found.

     bool run() {

       return init() && start();

     }

     /// @}

     /// \name Query Functions

     /// The result of the algorithm can be obtained using these

     /// functions.\n

     /// \ref lemon::CostScaling::run() "run()" must be called before

     /// using them.

     /// @{

     /// \brief Return a const reference to the edge map storing the

     /// found flow.

     ///

     /// Return a const reference to the edge map storing the found flow.

     ///

     /// \pre \ref run() must be called before using this function.

     const FlowMap& flowMap() const {

       return *_flow;

     }

     /// \brief Return a const reference to the node map storing the

     /// found potentials (the dual solution).

     ///

     /// Return a const reference to the node map storing the found

     /// potentials (the dual solution).

     ///

     /// \pre \ref run() must be called before using this function.

     const PotentialMap& potentialMap() const {

       return *_potential;

     }

     /// \brief Return the flow on the given edge.

     ///

     /// Return the flow on the given edge.

     ///

     /// \pre \ref run() must be called before using this function.

     Capacity flow(const Edge& edge) const {

       return (*_flow)[edge];

     }

     /// \brief Return the potential of the given node.

     ///

     /// Return the potential of the given node.

     ///

     /// \pre \ref run() must be called before using this function.

     Cost potential(const Node& node) const {

       return (*_potential)[node];

     }

     /// \brief Return the total cost of the found flow.

     ///

     /// Return the total cost of the found flow. The complexity of the

     /// function is \f$ O(e) \f$.

     ///

     /// \pre \ref run() must be called before using this function.

     Cost totalCost() const {

       Cost c = 0;

       for (EdgeIt e(_graph); e != INVALID; ++e)

         c += (*_flow)[e] * _orig_cost[e];

       return c;

     }

     /// @}

   private:

     /// Initialize the algorithm.

     bool init() {

       if (!_valid_supply) return false;

       // Initializing flow and potential maps

       if (!_flow) {

         _flow = new FlowMap(_graph);

         _local_flow = true;

       }

       if (!_potential) {

         _potential = new PotentialMap(_graph);

         _local_potential = true;

       }

       _red_cost = new ReducedCostMap(_graph, _cost, *_potential);

       _res_graph = new ResGraph(_graph, _capacity, *_flow);

       // Initializing the scaled cost map and the epsilon parameter

       Cost max_cost = 0;

       int node_num = countNodes(_graph);

       for (EdgeIt e(_graph); e != INVALID; ++e) {

         _cost[e] = LCost(_orig_cost[e]) * node_num * ALPHA;

         if (_orig_cost[e] > max_cost) max_cost = _orig_cost[e];

       }

       _epsilon = max_cost * node_num;

       // Finding a feasible flow using Circulation

       Circulation< Graph, ConstMap<Edge, Capacity>, CapacityEdgeMap,

                    SupplyMap >

         circulation( _graph, constMap<Edge>(Capacity(0)), _capacity,

                      _supply );

       return circulation.flowMap(*_flow).run();

     }

     /// Execute the algorithm.

     bool start() {

       std::deque<Node> active_nodes;

       typename Graph::template NodeMap<bool> hyper(_graph, false);

       int node_num = countNodes(_graph);

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

                                         1 : _epsilon / ALPHA )

       {

         // Performing price refinement heuristic using Bellman-Ford

         // algorithm

         if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {

           typedef ShiftMap< ResidualCostMap<LargeCostMap> > ShiftCostMap;

           ShiftCostMap shift_cost(_res_cost, _epsilon);

           BellmanFord<ResGraph, ShiftCostMap> bf(*_res_graph, shift_cost);

           bf.init(0);

           bool done = false;

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

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

             done = bf.processNextWeakRound();

           if (done) {

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

               (*_potential)[n] = bf.dist(n);

             continue;

           }

         }

         // Saturating edges not satisfying the optimality condition

         Capacity delta;

         for (EdgeIt e(_graph); e != INVALID; ++e) {

           if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {

             delta = _capacity[e] - (*_flow)[e];

             _excess[_graph.source(e)] -= delta;

             _excess[_graph.target(e)] += delta;

             (*_flow)[e] = _capacity[e];

           }

           if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {

             _excess[_graph.target(e)] -= (*_flow)[e];

             _excess[_graph.source(e)] += (*_flow)[e];

             (*_flow)[e] = 0;

           }

         }

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

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

           if (_excess[n] > 0) active_nodes.push_back(n);

         // Performing push and relabel operations

         while (active_nodes.size() > 0) {

           Node n = active_nodes[0], t;

           bool relabel_enabled = true;

           // Performing push operations if there are admissible edges

           if (_excess[n] > 0) {

             for (OutEdgeIt e(_graph, n); e != INVALID; ++e) {

               if (_capacity[e] - (*_flow)[e] > 0 && (*_red_cost)[e] < 0) {

                 delta = _capacity[e] - (*_flow)[e] <= _excess[n] ?

                         _capacity[e] - (*_flow)[e] : _excess[n];

                 t = _graph.target(e);

                 // Push-look-ahead heuristic

                 Capacity ahead = -_excess[t];

                 for (OutEdgeIt oe(_graph, t); oe != INVALID; ++oe) {

                   if (_capacity[oe] - (*_flow)[oe] > 0 && (*_red_cost)[oe] < 0)

                     ahead += _capacity[oe] - (*_flow)[oe];

                 }

                 for (InEdgeIt ie(_graph, t); ie != INVALID; ++ie) {

                   if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < 0)

                     ahead += (*_flow)[ie];

                 }

                 if (ahead < 0) ahead = 0;

                 // Pushing flow along the edge

                 if (ahead < delta) {

                   (*_flow)[e] += ahead;

                   _excess[n] -= ahead;

                   _excess[t] += ahead;

                   active_nodes.push_front(t);

                   hyper[t] = true;

                   relabel_enabled = false;

                   break;

                 } else {

                   (*_flow)[e] += delta;

                   _excess[n] -= delta;

                   _excess[t] += delta;

                   if (_excess[t] > 0 && _excess[t] <= delta)

                     active_nodes.push_back(t);

                 }

                 if (_excess[n] == 0) break;

               }

             }

           }

           if (_excess[n] > 0) {

             for (InEdgeIt e(_graph, n); e != INVALID; ++e) {

               if ((*_flow)[e] > 0 && -(*_red_cost)[e] < 0) {

                 delta = (*_flow)[e] <= _excess[n] ? (*_flow)[e] : _excess[n];

                 t = _graph.source(e);

                 // Push-look-ahead heuristic

                 Capacity ahead = -_excess[t];

                 for (OutEdgeIt oe(_graph, t); oe != INVALID; ++oe) {

                   if (_capacity[oe] - (*_flow)[oe] > 0 && (*_red_cost)[oe] < 0)

                     ahead += _capacity[oe] - (*_flow)[oe];

                 }

                 for (InEdgeIt ie(_graph, t); ie != INVALID; ++ie) {

                   if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < 0)

                     ahead += (*_flow)[ie];

                 }

                 if (ahead < 0) ahead = 0;

                 // Pushing flow along the edge

                 if (ahead < delta) {

                   (*_flow)[e] -= ahead;

                   _excess[n] -= ahead;

                   _excess[t] += ahead;

                   active_nodes.push_front(t);

                   hyper[t] = true;

                   relabel_enabled = false;

                   break;

                 } else {

                   (*_flow)[e] -= delta;

                   _excess[n] -= delta;

                   _excess[t] += delta;

                   if (_excess[t] > 0 && _excess[t] <= delta)

                     active_nodes.push_back(t);

                 }

                 if (_excess[n] == 0) break;

               }

             }

           }

           if (relabel_enabled && (_excess[n] > 0 || hyper[n])) {

             // Performing relabel operation if the node is still active

             LCost min_red_cost = std::numeric_limits<LCost>::max();

             for (OutEdgeIt oe(_graph, n); oe != INVALID; ++oe) {

               if ( _capacity[oe] - (*_flow)[oe] > 0 &&

                    (*_red_cost)[oe] < min_red_cost )

                 min_red_cost = (*_red_cost)[oe];

             }

             for (InEdgeIt ie(_graph, n); ie != INVALID; ++ie) {

               if ((*_flow)[ie] > 0 && -(*_red_cost)[ie] < min_red_cost)

                 min_red_cost = -(*_red_cost)[ie];

             }

             (*_potential)[n] -= min_red_cost + _epsilon;

             hyper[n] = false;

           }

           // Removing active nodes with non-positive excess

           while ( active_nodes.size() > 0 &&

                   _excess[active_nodes[0]] <= 0 &&

                   !hyper[active_nodes[0]] ) {

             active_nodes.pop_front();

           }

         }

       }

       // Computing node potentials for the original costs

       ResidualCostMap<CostMap> res_cost(_orig_cost);

       BellmanFord< ResGraph, ResidualCostMap<CostMap> >

         bf(*_res_graph, res_cost);

       bf.init(0); bf.start();

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

         (*_potential)[n] = bf.dist(n);

       // Handling non-zero lower bounds

       if (_lower) {

         for (EdgeIt e(_graph); e != INVALID; ++e)

           (*_flow)[e] += (*_lower)[e];

       }

       return true;

     }

   }; //class CostScaling

   ///@}

 } //namespace lemon

 #endif //LEMON_COST_SCALING_H