// -*- c++ -*-

 #ifndef HUGO_MINCOSTFLOWS_H

 #define HUGO_MINCOSTFLOWS_H

 ///\ingroup flowalgs

 ///\file

 ///\brief An algorithm for finding a flow of value \c k (for small values of \c k) having minimal total cost 

 #include <hugo/dijkstra.h>

 #include <hugo/graph_wrapper.h>

 #include <hugo/maps.h>

 #include <vector>

 namespace hugo {

 /// \addtogroup flowalgs

 /// @{

   ///\brief Implementation of an algorithm for finding a flow of value \c k 

   ///(for small values of \c k) having minimal total cost between 2 nodes 

   /// 

   ///

   /// The class \ref hugo::MinCostFlows "MinCostFlows" implements

   /// an algorithm for finding a flow of value \c k 

   /// having minimal total cost 

   /// from a given source node to a given target node in an

   /// edge-weighted directed graph. To this end, 

   /// the edge-capacities and edge-weitghs have to be nonnegative. 

   /// The edge-capacities should be integers, but the edge-weights can be 

   /// integers, reals or of other comparable numeric type.

   /// This algorithm is intended to use only for small values of \c k, 

   /// since it is only polynomial in k, 

   /// not in the length of k (which is log k). 

   /// In order to find the minimum cost flow of value \c k it 

   /// finds the minimum cost flow of value \c i for every 

   /// \c i between 0 and \c k. 

   ///

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

   ///\param LengthMap The type of the length map.

   ///\param CapacityMap The capacity map type.

   ///

   ///\author Attila Bernath

   template <typename Graph, typename LengthMap, typename CapacityMap>

   class MinCostFlows {

     typedef typename LengthMap::ValueType Length;

     //Warning: this should be integer type

     typedef typename CapacityMap::ValueType Capacity;

     typedef typename Graph::Node Node;

     typedef typename Graph::NodeIt NodeIt;

     typedef typename Graph::Edge Edge;

     typedef typename Graph::OutEdgeIt OutEdgeIt;

     typedef typename Graph::template EdgeMap<int> EdgeIntMap;

     typedef ResGraphWrapper<const Graph,int,CapacityMap,EdgeIntMap> ResGraphType;

     typedef typename ResGraphType::Edge ResGraphEdge;

     class ModLengthMap {   

       typedef typename Graph::template NodeMap<Length> NodeMap;

       const ResGraphType& G;

       const LengthMap &ol;

       const NodeMap &pot;

     public :

       typedef typename LengthMap::KeyType KeyType;

       typedef typename LengthMap::ValueType ValueType;

       ValueType operator[](typename ResGraphType::Edge e) const {     

 	if (G.forward(e))

 	  return  ol[e]-(pot[G.head(e)]-pot[G.tail(e)]);   

 	else

 	  return -ol[e]-(pot[G.head(e)]-pot[G.tail(e)]);   

       }     

       ModLengthMap(const ResGraphType& _G,

 		   const LengthMap &o,  const NodeMap &p) : 

 	G(_G), /*rev(_rev),*/ ol(o), pot(p){}; 

     };//ModLengthMap

   protected:

     //Input

     const Graph& G;

     const LengthMap& length;

     const CapacityMap& capacity;

     //auxiliary variables

     //To store the flow

     EdgeIntMap flow; 

     //To store the potential (dual variables)

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

     PotentialMap potential;

     Length total_length;

   public :

     /// The constructor of the class.

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

     ///\param _length The length (weight or cost) of the edges. 

     ///\param _cap The capacity of the edges. 

     MinCostFlows(Graph& _G, LengthMap& _length, CapacityMap& _cap) : G(_G), 

       length(_length), capacity(_cap), flow(_G), potential(_G){ }

     ///Runs the algorithm.

     ///Runs the algorithm.

     ///Returns k if there is a flow of value at least k edge-disjoint 

     ///from s to t.

     ///Otherwise it returns the maximum value of a flow from s to t.

     ///

     ///\param s The source node.

     ///\param t The target node.

     ///\param k The value of the flow we are looking for.

     ///

     ///\todo May be it does make sense to be able to start with a nonzero 

     /// feasible primal-dual solution pair as well.

     int run(Node s, Node t, int k) {

       //Resetting variables from previous runs

       total_length = 0;

       for (typename Graph::EdgeIt e(G); e!=INVALID; ++e) flow.set(e, 0);

       //Initialize the potential to zero

       for (typename Graph::NodeIt n(G); n!=INVALID; ++n) potential.set(n, 0);

       //We need a residual graph

       ResGraphType res_graph(G, capacity, flow);

       ModLengthMap mod_length(res_graph, length, potential);

       Dijkstra<ResGraphType, ModLengthMap> dijkstra(res_graph, mod_length);

       int i;

       for (i=0; i<k; ++i){

 	dijkstra.run(s);

 	if (!dijkstra.reached(t)){

 	  //There are no flow of value k from s to t

 	  break;

 	};

 	//We have to change the potential

         for(typename ResGraphType::NodeIt n(res_graph); n!=INVALID; ++n)

 	  potential[n] += dijkstra.distMap()[n];

 	//Augmenting on the sortest path

 	Node n=t;

 	ResGraphEdge e;

 	while (n!=s){

 	  e = dijkstra.pred(n);

 	  n = dijkstra.predNode(n);

 	  res_graph.augment(e,1);

 	  //Let's update the total length

 	  if (res_graph.forward(e))

 	    total_length += length[e];

 	  else 

 	    total_length -= length[e];	    

 	}

       }

       return i;

     }

     /// Gives back the total weight of the found flow.

     ///This function gives back the total weight of the found flow.

     ///Assumes that \c run() has been run and nothing changed since then.

     Length totalLength(){

       return total_length;

     }

     ///Returns a const reference to the EdgeMap \c flow. 

     ///Returns a const reference to the EdgeMap \c flow. 

     ///\pre \ref run() must

     ///be called before using this function.

     const EdgeIntMap &getFlow() const { return flow;}

     ///Returns a const reference to the NodeMap \c potential (the dual solution).

     ///Returns a const reference to the NodeMap \c potential (the dual solution).

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

     const PotentialMap &getPotential() const { return potential;}

     /// Checking the complementary slackness optimality criteria

     ///This function checks, whether the given solution is optimal

     ///If executed after the call of \c run() then it should return with true.

     ///This function only checks optimality, doesn't bother with feasibility.

     ///It is meant for testing purposes.

     ///

     bool checkComplementarySlackness(){

       Length mod_pot;

       Length fl_e;

         for(typename Graph::EdgeIt e(G); e!=INVALID; ++e) {

 	//C^{\Pi}_{i,j}

 	mod_pot = length[e]-potential[G.head(e)]+potential[G.tail(e)];

 	fl_e = flow[e];

 	if (0<fl_e && fl_e<capacity[e]) {

 	  /// \todo better comparison is needed for real types, moreover, 

 	  /// this comparison here is superfluous.

 	  if (mod_pot != 0)

 	    return false;

 	} 

 	else {

 	  if (mod_pot > 0 && fl_e != 0)

 	    return false;

 	  if (mod_pot < 0 && fl_e != capacity[e])

 	    return false;

 	}

       }

       return true;

     }

   }; //class MinCostFlows

   ///@}

 } //namespace hugo

 #endif //HUGO_MINCOSTFLOWS_H