/* -*- 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 The capacity scaling algorithm for finding a minimum cost flow.

#include <lemon/graph_adaptor.h>
#include <lemon/bin_heap.h>
#include <vector>

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 CapacityScaling implements the capacity scaling version
  /// of the successive shortest path 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 non-negative integers.
  ///   However \c CostMap::Value should be signed type.
  /// - Supply values should be signed 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
  {
    GRAPH_TYPEDEFS(typename Graph);

    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> CapacityEdgeMap;
    typedef typename Graph::template NodeMap<Supply> SupplyNodeMap;
    typedef typename Graph::template NodeMap<Edge> PredMap;

  public:

    /// Type to enable or disable capacity scaling.
    enum ScalingEnum {
      WITH_SCALING = 0,
      WITHOUT_SCALING = -1
    };

    /// 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;

  protected:

    /// \brief 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<Cost> CostNodeMap;
      typedef typename Graph::template NodeMap<Edge> PredMap;

      typedef typename Graph::template NodeMap<int> HeapCrossRef;
      typedef BinHeap<Cost, HeapCrossRef> Heap;

    protected:

      /// The directed graph the algorithm runs on.
      const Graph &graph;

      /// The flow map.
      const FlowMap &flow;
      /// The residual capacity map.
      const CapacityEdgeMap &res_cap;
      /// The cost map.
      const CostMap &cost;
      /// The excess map.
      const SupplyNodeMap &excess;

      /// The potential map.
      PotentialMap &potential;

      /// The distance map.
      CostNodeMap dist;
      /// The map of predecessors edges.
      PredMap &pred;
      /// The processed (i.e. permanently labeled) nodes.
      std::vector<Node> proc_nodes;

    public:

      /// The constructor of the class.
      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) {
        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 dt = heap.prio();
        for (int i = 0; i < proc_nodes.size(); ++i)
          potential[proc_nodes[i]] -= dist[proc_nodes[i]] - dt;

        return t;
      }

    }; //class ResidualDijkstra

  protected:

    /// 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 cost map.
    const CostMap &cost;
    /// The modified supply map.
    SupplyNodeMap supply;
    bool valid_supply;

    /// The edge map of the current flow.
    FlowMap flow;
    /// The potential node map.
    PotentialMap potential;

    /// The residual capacity map.
    CapacityEdgeMap res_cap;
    /// The excess map.
    SupplyNodeMap excess;
    /// The excess nodes (i.e. the nodes with positive excess).
    std::vector<Node> excess_nodes;
    /// The deficit nodes (i.e. the nodes with negative excess).
    std::vector<Node> deficit_nodes;

    /// The scaling status (enabled or disabled).
    ScalingEnum scaling;
    /// The \c delta parameter used for capacity scaling.
    Capacity delta;
    /// The maximum number of phases.
    int phase_num;

    /// \brief Implementation of the \ref Dijkstra algorithm for
    /// finding augmenting shortest paths in the residual network.
    ResidualDijkstra dijkstra;
    /// The map of predecessors edges.
    PredMap pred;

  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_cap(_graph), excess(_graph), pred(_graph),
      dijkstra(graph, flow, res_cap, cost, excess, potential, pred)
    {
      // 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 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_cap(_capacity), excess(_graph), pred(_graph),
      dijkstra(graph, flow, res_cap, cost, excess, potential)
    {
      // 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_cap(_graph), excess(_graph), pred(_graph),
      dijkstra(graph, flow, res_cap, cost, excess, potential)
    {
      // Removing non-zero lower bounds
      capacity = subMap(_capacity, _lower);
      res_cap = capacity;
      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] = 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_cap(_capacity), excess(_graph), pred(_graph),
      dijkstra(graph, flow, res_cap, cost, excess, potential)
    {
      supply[_s] =  _flow_value;
      supply[_t] = -_flow_value;
      valid_supply = true;
    }

    /// \brief Runs the algorithm.
    ///
    /// Runs the algorithm.
    ///
    /// \param scaling_mode The scaling mode. In case of WITH_SCALING
    /// capacity scaling is enabled in the algorithm (this is the
    /// default) otherwise it is disabled.
    /// If the maximum edge capacity and/or the amount of total supply
    /// is small, the algorithm could be slightly faster without
    /// scaling.
    ///
    /// \return \c true if a feasible flow can be found.
    bool run(int scaling_mode = WITH_SCALING) {
      return init(scaling_mode) && start();
    }

    /// \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;
    }

  protected:

    /// Initializes the algorithm.
    bool init(int scaling_mode) {
      if (!valid_supply) return false;
      excess = supply;

      // Initilaizing delta value
      if (scaling_mode == WITH_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];
        }
        if (max_dem < max_sup) max_sup = max_dem;
        phase_num = 0;
        for (delta = 1; 2 * delta <= max_sup; delta *= 2)
          ++phase_num;
      } else {
        // Without scaling
        delta = 1;
      }
      return true;
    }

    /// Executes the algorithm.
    bool start() {
      if (delta > 1)
        return startWithScaling();
      else
        return startWithoutScaling();
    }

    /// \brief Executes the capacity scaling version of the successive
    /// shortest path 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 < excess_nodes.size()) {
          // Checking deficit nodes
          if (delta > 1) {
            bool delta_deficit = false;
            for (int i = 0; i < 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 = excess[s] < -excess[t] ? 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;
    }

    /// \brief Executes the successive shortest path algorithm without
    /// capacity scaling.
    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 < excess_nodes.size() )
      {
        // Running Dijkstra
        s = excess_nodes[next_node];
        if ((t = dijkstra.run(s, 1)) == INVALID)
          return false;

        // Augmenting along a shortest path from s to t
        Capacity d = excess[s] < -excess[t] ? excess[s] : -excess[t];
        Node u = t;
        Edge e;
        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