/* -*- C++ -*-

 *

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

 *

 * Copyright (C) 2003-2007

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

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

 *

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

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

 * precise terms see the accompanying LICENSE file.

 *

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

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

 * purpose.

 *

 */

#ifndef LEMON_CAPACITY_SCALING_H

#define LEMON_CAPACITY_SCALING_H

/// \ingroup min_cost_flow

///

/// \file

/// \brief The capacity scaling algorithm for finding a minimum cost

/// flow.

#include <vector>

#include <lemon/dijkstra.h>

#include <lemon/graph_adaptor.h>

#define WITH_SCALING

//#define _DEBUG_ITER_

namespace lemon {

  /// \addtogroup min_cost_flow

  /// @{

  /// \brief Implementation of the capacity scaling version of the

  /// successive shortest path algorithm for finding a minimum cost flow.

  ///

  /// \ref lemon::CapacityScaling "CapacityScaling" implements a

  /// capacity scaling algorithm for finding a minimum cost flow.

  ///

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

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

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

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

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

  ///

  /// \warning

  /// - Edge capacities and costs should be nonnegative integers.

  ///	However \c CostMap::Value should be signed type.

  /// - Supply values should be integers.

  /// - \c LowerMap::Value must be convertible to

  ///	\c CapacityMap::Value and \c CapacityMap::Value must be

  ///	convertible to \c SupplyMap::Value.

  ///

  /// \author Peter Kovacs

template < typename Graph,

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

	   typename CapacityMap = LowerMap,

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

	   typename SupplyMap = typename Graph::template NodeMap

				<typename CapacityMap::Value> >

  class CapacityScaling

  {

    typedef typename Graph::Node Node;

    typedef typename Graph::NodeIt NodeIt;

    typedef typename Graph::Edge Edge;

    typedef typename Graph::EdgeIt EdgeIt;

    typedef typename Graph::InEdgeIt InEdgeIt;

    typedef typename Graph::OutEdgeIt OutEdgeIt;

    typedef typename LowerMap::Value Lower;

    typedef typename CapacityMap::Value Capacity;

    typedef typename CostMap::Value Cost;

    typedef typename SupplyMap::Value Supply;

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

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

    typedef ResGraphAdaptor< const Graph, Capacity,

			     CapacityRefMap, CapacityRefMap > ResGraph;

    typedef typename ResGraph::Node ResNode;

    typedef typename ResGraph::NodeIt ResNodeIt;

    typedef typename ResGraph::Edge ResEdge;

    typedef typename ResGraph::EdgeIt ResEdgeIt;

  public:

    /// \brief The type of the flow map.

    typedef CapacityRefMap FlowMap;

    /// \brief The type of the potential map.

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

  protected:

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

    class ReducedCostMap : public MapBase<ResEdge, Cost>

    {

    private:

      const ResGraph &gr;

      const CostMap &cost_map;

      const PotentialMap &pot_map;

    public:

      typedef typename MapBase<ResEdge, Cost>::Value Value;

      typedef typename MapBase<ResEdge, Cost>::Key Key;

      ReducedCostMap( const ResGraph &_gr,

		      const CostMap &_cost,

		      const PotentialMap &_pot ) :

	gr(_gr), cost_map(_cost), pot_map(_pot) {}

      Value operator[](const Key &e) const {

	return ResGraph::forward(e) ?

	   cost_map[e] - pot_map[gr.source(e)] + pot_map[gr.target(e)] :

	  -cost_map[e] - pot_map[gr.source(e)] + pot_map[gr.target(e)];

      }

    }; //class ReducedCostMap

    /// \brief Map adaptor for \ref lemon::Dijkstra "Dijkstra" class to

    /// update node potentials.

    class PotentialUpdateMap : public MapBase<ResNode, Cost>

    {

    private:

      PotentialMap *pot;

      typedef std::pair<ResNode, Cost> Pair;

      std::vector<Pair> data;

    public:

      typedef typename MapBase<ResNode, Cost>::Value Value;

      typedef typename MapBase<ResNode, Cost>::Key Key;

      void potentialMap(PotentialMap &_pot) {

	pot = &_pot;

      }

      void init() {

	data.clear();

      }

      void set(const Key &n, const Value &v) {

	data.push_back(Pair(n, v));

      }

      void update() {

	Cost val = data[data.size()-1].second;

	for (int i = 0; i < data.size()-1; ++i)

	  (*pot)[data[i].first] += val - data[i].second;

      }

    }; //class PotentialUpdateMap

#ifdef WITH_SCALING

    /// \brief Map adaptor class for identifing deficit nodes.

    class DeficitBoolMap : public MapBase<ResNode, bool>

    {

    private:

      const SupplyRefMap &imb;

      const Capacity δ

    public:

      DeficitBoolMap(const SupplyRefMap &_imb, const Capacity &_delta) :

	imb(_imb), delta(_delta) {}

      bool operator[](const ResNode &n) const {

	return imb[n] <= -delta;

      }

    }; //class DeficitBoolMap

    /// \brief Map adaptor class for filtering edges with at least

    /// \c delta residual capacity

    class DeltaFilterMap : public MapBase<ResEdge, bool>

    {

    private:

      const ResGraph &gr;

      const Capacity δ

    public:

      typedef typename MapBase<ResEdge, Cost>::Value Value;

      typedef typename MapBase<ResEdge, Cost>::Key Key;

      DeltaFilterMap(const ResGraph &_gr, const Capacity &_delta) :

	gr(_gr), delta(_delta) {}

      Value operator[](const Key &e) const {

	return gr.rescap(e) >= delta;

      }

    }; //class DeltaFilterMap

    typedef EdgeSubGraphAdaptor<const ResGraph, const DeltaFilterMap>

      DeltaResGraph;

    /// \brief Traits class for \ref lemon::Dijkstra "Dijkstra" class.

    class ResDijkstraTraits :

      public DijkstraDefaultTraits<DeltaResGraph, ReducedCostMap>

    {

    public:

      typedef PotentialUpdateMap DistMap;

      static DistMap *createDistMap(const DeltaResGraph&) {

	return new DistMap();

      }

    }; //class ResDijkstraTraits

#else //WITHOUT_CAPACITY_SCALING

    /// \brief Map adaptor class for identifing deficit nodes.

    class DeficitBoolMap : public MapBase<ResNode, bool>

    {

    private:

      const SupplyRefMap &imb;

    public:

      DeficitBoolMap(const SupplyRefMap &_imb) : imb(_imb) {}

      bool operator[](const ResNode &n) const {

	return imb[n] < 0;

      }

    }; //class DeficitBoolMap

    /// \brief Traits class for \ref lemon::Dijkstra "Dijkstra" class.

    class ResDijkstraTraits :

      public DijkstraDefaultTraits<ResGraph, ReducedCostMap>

    {

    public:

      typedef PotentialUpdateMap DistMap;

      static DistMap *createDistMap(const ResGraph&) {

	return new DistMap();

      }

    }; //class ResDijkstraTraits

#endif

  protected:

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

    const Graph &graph;

    /// \brief The original lower bound map.

    const LowerMap *lower;

    /// \brief The modified capacity map.

    CapacityRefMap capacity;

    /// \brief The cost map.

    const CostMap &cost;

    /// \brief The modified supply map.

    SupplyRefMap supply;

    /// \brief The sum of supply values equals zero.

    bool valid_supply;

    /// \brief The edge map of the current flow.

    FlowMap flow;

    /// \brief The potential node map.

    PotentialMap potential;

    /// \brief The residual graph.

    ResGraph res_graph;

    /// \brief The reduced cost map.

    ReducedCostMap red_cost;

    /// \brief The imbalance map.

    SupplyRefMap imbalance;

    /// \brief The excess nodes.

    std::vector<ResNode> excess_nodes;

    /// \brief The index of the next excess node.

    int next_node;

#ifdef WITH_SCALING

    typedef Dijkstra<DeltaResGraph, ReducedCostMap, ResDijkstraTraits>

      ResDijkstra;

    /// \brief \ref lemon::Dijkstra "Dijkstra" class for finding

    /// shortest paths in the residual graph with respect to the

    /// reduced edge costs.

    ResDijkstra dijkstra;

    /// \brief The delta parameter used for capacity scaling.

    Capacity delta;

    /// \brief Edge filter map.

    DeltaFilterMap delta_filter;

    /// \brief The delta residual graph.

    DeltaResGraph dres_graph;

    /// \brief Map for identifing deficit nodes.

    DeficitBoolMap delta_deficit;

    /// \brief The deficit nodes.

    std::vector<ResNode> deficit_nodes;

#else //WITHOUT_CAPACITY_SCALING

    typedef Dijkstra<ResGraph, ReducedCostMap, ResDijkstraTraits>

      ResDijkstra;

    /// \brief \ref lemon::Dijkstra "Dijkstra" class for finding

    /// shortest paths in the residual graph with respect to the

    /// reduced edge costs.

    ResDijkstra dijkstra;

    /// \brief Map for identifing deficit nodes.

    DeficitBoolMap has_deficit;

#endif

    /// \brief Pred map for the \ref lemon::Dijkstra "Dijkstra" class.

    typename ResDijkstra::PredMap pred;

    /// \brief Dist map for the \ref lemon::Dijkstra "Dijkstra" class to

    /// update node potentials.

    PotentialUpdateMap updater;

  public :

    /// \brief General constructor of the class (with lower bounds).

    ///

    /// General constructor of the class (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).

    CapacityScaling( const Graph &_graph,

		     const LowerMap &_lower,

		     const CapacityMap &_capacity,

		     const CostMap &_cost,

		     const SupplyMap &_supply ) :

      graph(_graph), lower(&_lower), capacity(_graph), cost(_cost),

      supply(_graph), flow(_graph, 0), potential(_graph, 0),

      res_graph(_graph, capacity, flow),

      red_cost(res_graph, cost, potential), imbalance(_graph),

#ifdef WITH_SCALING

      delta(0), delta_filter(res_graph, delta),

      dres_graph(res_graph, delta_filter),

      dijkstra(dres_graph, red_cost), pred(dres_graph),

      delta_deficit(imbalance, delta)

#else //WITHOUT_CAPACITY_SCALING

      dijkstra(res_graph, red_cost), pred(res_graph),

      has_deficit(imbalance)

#endif

    {

      // Removing nonzero 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] = imbalance[n] = s;

	sum += s;

      }

      valid_supply = sum == 0;

    }

    /// \brief General constructor of the class (without lower bounds).

    ///

    /// General constructor of the class (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).

    CapacityScaling( const Graph &_graph,

		     const CapacityMap &_capacity,

		     const CostMap &_cost,

		     const SupplyMap &_supply ) :

      graph(_graph), lower(NULL), capacity(_capacity), cost(_cost),

      supply(_supply), flow(_graph, 0), potential(_graph, 0),

      res_graph(_graph, capacity, flow),

      red_cost(res_graph, cost, potential), imbalance(_graph),

#ifdef WITH_SCALING

      delta(0), delta_filter(res_graph, delta),

      dres_graph(res_graph, delta_filter),

      dijkstra(dres_graph, red_cost), pred(dres_graph),

      delta_deficit(imbalance, delta)

#else //WITHOUT_CAPACITY_SCALING

      dijkstra(res_graph, red_cost), pred(res_graph),

      has_deficit(imbalance)

#endif

    {

      // 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 of the class (with lower bounds).

    ///

    /// Simple constructor of the class (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).

    CapacityScaling( 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), cost(_cost),

      supply(_graph), flow(_graph, 0), potential(_graph, 0),

      res_graph(_graph, capacity, flow),

      red_cost(res_graph, cost, potential), imbalance(_graph),

#ifdef WITH_SCALING

      delta(0), delta_filter(res_graph, delta),

      dres_graph(res_graph, delta_filter),

      dijkstra(dres_graph, red_cost), pred(dres_graph),

      delta_deficit(imbalance, delta)

#else //WITHOUT_CAPACITY_SCALING

      dijkstra(res_graph, red_cost), pred(res_graph),

      has_deficit(imbalance)

#endif

    {

      // Removing nonzero lower bounds

      capacity = subMap(_capacity, _lower);

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

	Supply s = 0;

	if (n == _s) s =  _flow_value;

	if (n == _t) s = -_flow_value;

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

	  s += _lower[e];

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

	  s -= _lower[e];

	supply[n] = imbalance[n] = s;

      }

      valid_supply = true;

    }

    /// \brief Simple constructor of the class (without lower bounds).

    ///

    /// Simple constructor of the class (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).

    CapacityScaling( const Graph &_graph,

		     const CapacityMap &_capacity,

		     const CostMap &_cost,

		     Node _s, Node _t,

		     Supply _flow_value ) :

      graph(_graph), lower(NULL), capacity(_capacity), cost(_cost),

      supply(_graph, 0), flow(_graph, 0), potential(_graph, 0),

      res_graph(_graph, capacity, flow),

      red_cost(res_graph, cost, potential), imbalance(_graph),

#ifdef WITH_SCALING

      delta(0), delta_filter(res_graph, delta),

      dres_graph(res_graph, delta_filter),

      dijkstra(dres_graph, red_cost), pred(dres_graph),

      delta_deficit(imbalance, delta)

#else //WITHOUT_CAPACITY_SCALING

      dijkstra(res_graph, red_cost), pred(res_graph),

      has_deficit(imbalance)

#endif

    {

      supply[_s] =  _flow_value;

      supply[_t] = -_flow_value;

      valid_supply = true;

    }

    /// \brief Returns a const reference to the flow map.

    ///

    /// Returns a const reference to the flow map.

    ///

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

    const FlowMap& flowMap() const {

      return flow;

    }

    /// \brief Returns a const reference to the potential map (the dual

    /// solution).

    ///

    /// Returns a const reference to the potential map (the dual

    /// solution).

    ///

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

    const PotentialMap& potentialMap() const {

      return potential;

    }

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

    ///

    /// Returns 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] * cost[e];

      return c;

    }

    /// \brief Runs the algorithm.

    ///

    /// Runs the algorithm.

    ///

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

    bool run() {

      return init() && start();

    }

  protected:

    /// \brief Initializes the algorithm.

    bool init() {

      if (!valid_supply) return false;

      // Initalizing Dijkstra class

      updater.potentialMap(potential);

      dijkstra.distMap(updater).predMap(pred);

#ifdef WITH_SCALING

      // Initilaizing delta value

      Supply max_sup = 0, max_dem = 0;

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

	if (supply[n] >  max_sup) max_sup =  supply[n];

	if (supply[n] < -max_dem) max_dem = -supply[n];

      }

      if (max_dem < max_sup) max_sup = max_dem;

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

#endif

      return true;

    }

#ifdef WITH_SCALING

    /// \brief Executes the capacity scaling version of the successive

    /// shortest path algorithm.

    bool start() {

      typedef typename DeltaResGraph::EdgeIt DeltaResEdgeIt;

      typedef typename DeltaResGraph::Edge DeltaResEdge;

#ifdef _DEBUG_ITER_

      int dijk_num = 0;

#endif

      // Processing capacity scaling phases

      ResNode s, t;

      for ( ; delta >= 1; delta = delta < 4 && delta > 1 ?

				  1 : delta / 4 )

      {

	// Saturating edges not satisfying the optimality condition

	Capacity r;

	for (DeltaResEdgeIt e(dres_graph); e != INVALID; ++e) {

	  if (red_cost[e] < 0) {

	    res_graph.augment(e, r = res_graph.rescap(e));

	    imbalance[dres_graph.target(e)] += r;

	    imbalance[dres_graph.source(e)] -= r;

	  }

	}

	// Finding excess nodes

	excess_nodes.clear();

	deficit_nodes.clear();

	for (ResNodeIt n(res_graph); n != INVALID; ++n) {

	  if (imbalance[n] >=  delta) excess_nodes.push_back(n);

	  if (imbalance[n] <= -delta) deficit_nodes.push_back(n);

	}

	next_node = 0;

	// Finding shortest paths

	while (next_node < excess_nodes.size()) {

	  // Checking deficit nodes

	  if (delta > 1) {

	    bool delta_def = false;

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

	      if (imbalance[deficit_nodes[i]] <= -delta) {

		delta_def = true;

		break;

	      }

	    }

	    if (!delta_def) break;

	  }

	  // Running Dijkstra

	  s = excess_nodes[next_node];

	  updater.init();

	  dijkstra.init();

	  dijkstra.addSource(s);

#ifdef _DEBUG_ITER_

	  ++dijk_num;

#endif

	  if ((t = dijkstra.start(delta_deficit)) == INVALID) {

	    if (delta > 1) {

	      ++next_node;

	      continue;

	    }

	    return false;

	  }

	  // Updating node potentials

	  updater.update();

	  // Augment along a shortest path from s to t

	  Capacity d = imbalance[s] < -imbalance[t] ?

	    imbalance[s] : -imbalance[t];

	  ResNode u = t;

	  ResEdge e;

	  if (d > delta) {

	    while ((e = pred[u]) != INVALID) {

	      if (res_graph.rescap(e) < d) d = res_graph.rescap(e);

	      u = dres_graph.source(e);

	    }

	  }

	  u = t;

	  while ((e = pred[u]) != INVALID) {

	    res_graph.augment(e, d);

	    u = dres_graph.source(e);

	  }

	  imbalance[s] -= d;

	  imbalance[t] += d;

	  if (imbalance[s] < delta) ++next_node;

	}

      }

#ifdef _DEBUG_ITER_

      std::cout << "Cost Scaling algorithm finished with running Dijkstra algorithm "

	<< dijk_num << " times." << std::endl;

#endif

      // Handling nonzero lower bounds

      if (lower) {

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

	  flow[e] += (*lower)[e];

      }

      return true;

    }

#else //WITHOUT_CAPACITY_SCALING

    /// \brief Executes the successive shortest path algorithm without

    /// capacity scaling.

    bool start() {

      // Finding excess nodes

      for (ResNodeIt n(res_graph); n != INVALID; ++n) {

	if (imbalance[n] > 0) excess_nodes.push_back(n);

      }

      if (excess_nodes.size() == 0) return true;

      next_node = 0;

      // Finding shortest paths

      ResNode s, t;

      while ( imbalance[excess_nodes[next_node]] > 0 ||

	      ++next_node < excess_nodes.size() )

      {

	// Running Dijkstra

	s = excess_nodes[next_node];

	updater.init();

	dijkstra.init();

	dijkstra.addSource(s);

	if ((t = dijkstra.start(has_deficit)) == INVALID)

	  return false;

	// Updating node potentials

	updater.update();

	// Augmenting along a shortest path from s to t

	Capacity delta = imbalance[s] < -imbalance[t] ?

	  imbalance[s] : -imbalance[t];

	ResNode u = t;

	ResEdge e;

	while ((e = pred[u]) != INVALID) {

	  if (res_graph.rescap(e) < delta) delta = res_graph.rescap(e);

	  u = res_graph.source(e);

	}

	u = t;

	while ((e = pred[u]) != INVALID) {

	  res_graph.augment(e, delta);

	  u = res_graph.source(e);

	}

	imbalance[s] -= delta;

	imbalance[t] += delta;

      }

      // Handling nonzero lower bounds

      if (lower) {

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

	  flow[e] += (*lower)[e];

      }

      return true;

    }

#endif

  }; //class CapacityScaling

  ///@}

} //namespace lemon

#endif //LEMON_CAPACITY_SCALING_H