/* -*- 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_CAPACITY_SCALING_H

#define LEMON_CAPACITY_SCALING_H

/// \ingroup min_cost_flow

///

/// \file

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

#include <vector>

#include <lemon/bin_heap.h>

namespace lemon {

  /// \addtogroup min_cost_flow

  /// @{

  /// \brief Implementation of the capacity scaling algorithm for

  /// finding a minimum cost flow.

  ///

  /// \ref CapacityScaling implements the capacity scaling version

  /// of the successive shortest path algorithm 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.

  ///

  /// \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 CapacityScaling

  {

    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 typename Graph::template NodeMap<Edge> PredMap;

  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<Cost> PotentialMap;

  private:

    /// \brief Special implementation of the \ref Dijkstra algorithm

    /// for finding shortest paths in the residual network.

    ///

    /// \ref ResidualDijkstra is a special implementation of the

    /// \ref Dijkstra algorithm for finding shortest paths in the

    /// residual network of the graph with respect to the reduced edge

    /// costs and modifying the node potentials according to the

    /// distance of the nodes.

    class ResidualDijkstra

    {

      typedef typename Graph::template NodeMap<int> HeapCrossRef;

      typedef BinHeap<Cost, HeapCrossRef> Heap;

    private:

      // The directed graph the algorithm runs on

      const Graph &_graph;

      // The main maps

      const FlowMap &_flow;

      const CapacityEdgeMap &_res_cap;

      const CostMap &_cost;

      const SupplyNodeMap &_excess;

      PotentialMap &_potential;

      // The distance map

      PotentialMap _dist;

      // The pred edge map

      PredMap &_pred;

      // The processed (i.e. permanently labeled) nodes

      std::vector<Node> _proc_nodes;

    public:

      /// Constructor.

      ResidualDijkstra( const Graph &graph,

                        const FlowMap &flow,

                        const CapacityEdgeMap &res_cap,

                        const CostMap &cost,

                        const SupplyMap &excess,

                        PotentialMap &potential,

                        PredMap &pred ) :

        _graph(graph), _flow(flow), _res_cap(res_cap), _cost(cost),

        _excess(excess), _potential(potential), _dist(graph),

        _pred(pred)

      {}

      /// Runs the algorithm from the given source node.

      Node run(Node s, Capacity delta = 1) {

        HeapCrossRef heap_cross_ref(_graph, Heap::PRE_HEAP);

        Heap heap(heap_cross_ref);

        heap.push(s, 0);

        _pred[s] = INVALID;

        _proc_nodes.clear();

        // Processing nodes

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

          Node u = heap.top(), v;

          Cost d = heap.prio() + _potential[u], nd;

          _dist[u] = heap.prio();

          heap.pop();

          _proc_nodes.push_back(u);

          // Traversing outgoing edges

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

            if (_res_cap[e] >= delta) {

              v = _graph.target(e);

              switch(heap.state(v)) {

              case Heap::PRE_HEAP:

                heap.push(v, d + _cost[e] - _potential[v]);

                _pred[v] = e;

                break;

              case Heap::IN_HEAP:

                nd = d + _cost[e] - _potential[v];

                if (nd < heap[v]) {

                  heap.decrease(v, nd);

                  _pred[v] = e;

                }

                break;

              case Heap::POST_HEAP:

                break;

              }

            }

          }

          // Traversing incoming edges

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

            if (_flow[e] >= delta) {

              v = _graph.source(e);

              switch(heap.state(v)) {

              case Heap::PRE_HEAP:

                heap.push(v, d - _cost[e] - _potential[v]);

                _pred[v] = e;

                break;

              case Heap::IN_HEAP:

                nd = d - _cost[e] - _potential[v];

                if (nd < heap[v]) {

                  heap.decrease(v, nd);

                  _pred[v] = e;

                }

                break;

              case Heap::POST_HEAP:

                break;

              }

            }

          }

        }

        if (heap.empty()) return INVALID;

        // Updating potentials of processed nodes

        Node t = heap.top();

        Cost t_dist = heap.prio();

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

          _potential[_proc_nodes[i]] += _dist[_proc_nodes[i]] - t_dist;

        return t;

      }

    }; //class ResidualDijkstra

  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 &_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 capacity map

    CapacityEdgeMap _res_cap;

    // The excess map

    SupplyNodeMap _excess;

    // The excess nodes (i.e. nodes with positive excess)

    std::vector<Node> _excess_nodes;

    // The deficit nodes (i.e. nodes with negative excess)

    std::vector<Node> _deficit_nodes;

    // The delta parameter used for capacity scaling

    Capacity _delta;

    // The maximum number of phases

    int _phase_num;

    // The pred edge map

    PredMap _pred;

    // Implementation of the Dijkstra algorithm for finding augmenting

    // shortest paths in the residual network

    ResidualDijkstra *_dijkstra;

  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).

    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(0), _local_flow(false),

      _potential(0), _local_potential(false),

      _res_cap(graph), _excess(graph), _pred(graph)

    {

      // Removing non-zero lower bounds

      _capacity = subMap(capacity, lower);

      _res_cap = _capacity;

      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).

    CapacityScaling( const Graph &graph,

                     const CapacityMap &capacity,

                     const CostMap &cost,

                     const SupplyMap &supply ) :

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

      _supply(supply), _flow(0), _local_flow(false),

      _potential(0), _local_potential(false),

      _res_cap(capacity), _excess(graph), _pred(graph)

    {

      // 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).

    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(0), _local_flow(false),

      _potential(0), _local_potential(false),

      _res_cap(graph), _excess(graph), _pred(graph)

    {

      // Removing non-zero lower bounds

      _capacity = subMap(capacity, lower);

      _res_cap = _capacity;

      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).

    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(0), _local_flow(false),

      _potential(0), _local_potential(false),

      _res_cap(capacity), _excess(graph), _pred(graph)

    {

      _supply[s] =  flow_value;

      _supply[t] = -flow_value;

      _valid_supply = true;

    }

    /// Destructor.

    ~CapacityScaling() {

      if (_local_flow) delete _flow;

      if (_local_potential) delete _potential;

      delete _dijkstra;

    }

    /// \brief Sets the flow map.

    ///

    /// Sets the flow map.

    ///

    /// \return \c (*this)

    CapacityScaling& flowMap(FlowMap &map) {

      if (_local_flow) {

        delete _flow;

        _local_flow = false;

      }

      _flow = ↦

      return *this;

    }

    /// \brief Sets the potential map.

    ///

    /// Sets the potential map.

    ///

    /// \return \c (*this)

    CapacityScaling& potentialMap(PotentialMap &map) {

      if (_local_potential) {

        delete _potential;

        _local_potential = false;

      }

      _potential = ↦

      return *this;

    }

    /// \name Execution control

    /// The only way to execute the algorithm is to call the run()

    /// function.

    /// @{

    /// \brief Runs the algorithm.

    ///

    /// Runs the algorithm.

    ///

    /// \param scaling Enable or disable capacity scaling.

    /// If the maximum edge capacity and/or the amount of total supply

    /// is rather small, the algorithm could be slightly faster without

    /// scaling.

    ///

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

    bool run(bool scaling = true) {

      return init(scaling) && start();

    }

    /// @}

    /// \name Query Functions

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

    /// functions.

    /// \n run() must be called before using them.

    /// @{

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

    /// found flow.

    ///

    /// Returns 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 Returns a const reference to the node map storing the

    /// found potentials (the dual solution).

    ///

    /// Returns 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 Returns the flow on the given edge.

    ///

    /// Returns 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 Returns the potential of the given node.

    ///

    /// Returns 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 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;

    }

    /// @}

  private:

    /// Initializes the algorithm.

    bool init(bool scaling) {

      if (!_valid_supply) return false;

      // Initializing maps

      if (!_flow) {

        _flow = new FlowMap(_graph);

        _local_flow = true;

      }

      if (!_potential) {

        _potential = new PotentialMap(_graph);

        _local_potential = true;

      }

      for (EdgeIt e(_graph); e != INVALID; ++e) (*_flow)[e] = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) (*_potential)[n] = 0;

      _excess = _supply;

      _dijkstra = new ResidualDijkstra( _graph, *_flow, _res_cap, _cost,

                                        _excess, *_potential, _pred );

      // Initializing delta value

      if (scaling) {

        // With scaling

        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];

        }

        Capacity max_cap = 0;

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

          if (_capacity[e] > max_cap) max_cap = _capacity[e];

        }

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

        _phase_num = 0;

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

          ++_phase_num;

      } else {

        // Without scaling

        _delta = 1;

      }

      return true;

    }

    bool start() {

      if (_delta > 1)

        return startWithScaling();

      else

        return startWithoutScaling();

    }

    /// Executes the capacity scaling algorithm.

    bool startWithScaling() {

      // Processing capacity scaling phases

      Node s, t;

      int phase_cnt = 0;

      int factor = 4;

      while (true) {

        // Saturating all edges not satisfying the optimality condition

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

          Node u = _graph.source(e), v = _graph.target(e);

          Cost c = _cost[e] + (*_potential)[u] - (*_potential)[v];

          if (c < 0 && _res_cap[e] >= _delta) {

            _excess[u] -= _res_cap[e];

            _excess[v] += _res_cap[e];

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

            _res_cap[e] = 0;

          }

          else if (c > 0 && (*_flow)[e] >= _delta) {

            _excess[u] += (*_flow)[e];

            _excess[v] -= (*_flow)[e];

            (*_flow)[e] = 0;

            _res_cap[e] = _capacity[e];

          }

        }

        // Finding excess nodes and deficit nodes

        _excess_nodes.clear();

        _deficit_nodes.clear();

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

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

          if (_excess[n] <= -_delta) _deficit_nodes.push_back(n);

        }

        int next_node = 0;

        // Finding augmenting shortest paths

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

          // Checking deficit nodes

          if (_delta > 1) {

            bool delta_deficit = false;

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

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

                delta_deficit = true;

                break;

              }

            }

            if (!delta_deficit) break;

          }

          // Running Dijkstra

          s = _excess_nodes[next_node];

          if ((t = _dijkstra->run(s, _delta)) == INVALID) {

            if (_delta > 1) {

              ++next_node;

              continue;

            }

            return false;

          }

          // Augmenting along a shortest path from s to t.

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

          Node u = t;

          Edge e;

          if (d > _delta) {

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

              Capacity rc;

              if (u == _graph.target(e)) {

                rc = _res_cap[e];

                u = _graph.source(e);

              } else {

                rc = (*_flow)[e];

                u = _graph.target(e);

              }

              if (rc < d) d = rc;

            }

          }

          u = t;

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

            if (u == _graph.target(e)) {

              (*_flow)[e] += d;

              _res_cap[e] -= d;

              u = _graph.source(e);

            } else {

              (*_flow)[e] -= d;

              _res_cap[e] += d;

              u = _graph.target(e);

            }

          }

          _excess[s] -= d;

          _excess[t] += d;

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

        }

        if (_delta == 1) break;

        if (++phase_cnt > _phase_num / 4) factor = 2;

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

      }

      // Handling non-zero lower bounds

      if (_lower) {

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

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

      }

      return true;

    }

    /// Executes the successive shortest path algorithm.

    bool startWithoutScaling() {

      // Finding excess nodes

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

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

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

      int next_node = 0;

      // Finding shortest paths

      Node s, t;

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

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

      {

        // Running Dijkstra

        s = _excess_nodes[next_node];

        if ((t = _dijkstra->run(s)) == INVALID) return false;

        // Augmenting along a shortest path from s to t

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

        Node u = t;

        Edge e;

        if (d > 1) {

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

            Capacity rc;

            if (u == _graph.target(e)) {

              rc = _res_cap[e];

              u = _graph.source(e);

            } else {

              rc = (*_flow)[e];

              u = _graph.target(e);

            }

            if (rc < d) d = rc;

          }

        }

        u = t;

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

          if (u == _graph.target(e)) {

            (*_flow)[e] += d;

            _res_cap[e] -= d;

            u = _graph.source(e);

          } else {

            (*_flow)[e] -= d;

            _res_cap[e] += d;

            u = _graph.target(e);

          }

        }

        _excess[s] -= d;

        _excess[t] += d;

      }

      // Handling non-zero lower bounds

      if (_lower) {

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

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

      }

      return true;

    }

  }; //class CapacityScaling

  ///@}

} //namespace lemon

#endif //LEMON_CAPACITY_SCALING_H