/* -*- 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_NETWORK_SIMPLEX_H

 #define LEMON_NETWORK_SIMPLEX_H

 /// \ingroup min_cost_flow

 ///

 /// \file

 /// \brief The network simplex algorithm for finding a minimum cost

 /// flow.

 #include <limits>

 #include <lemon/smart_graph.h>

 #include <lemon/graph_utils.h>

 /// \brief The pivot rule used in the algorithm.

 #define EDGE_BLOCK_PIVOT

 //#define FIRST_ELIGIBLE_PIVOT

 //#define BEST_ELIGIBLE_PIVOT

 //#define CANDIDATE_LIST_PIVOT

 //#define SORTED_LIST_PIVOT

 /// \brief State constant for edges at their lower bounds.

 #define LOWER	1

 /// \brief State constant for edges in the spanning tree.

 #define TREE	0

 /// \brief State constant for edges at their upper bounds.

 #define UPPER	-1

 #ifdef EDGE_BLOCK_PIVOT

   /// \brief Number of blocks for the "Edge Block" pivot rule.

   #define BLOCK_NUM	  100

   /// \brief Lower bound for the number of edges to use "Edge Block"

   // pivot rule instead of "First Eligible" pivot rule.

   #define MIN_BOUND	  1000

 #endif

 #ifdef CANDIDATE_LIST_PIVOT

   #include <list>

   /// \brief The maximum length of the edge list for the

   /// "Candidate List" pivot rule.

   #define LIST_LENGTH	  100

   /// \brief The maximum number of minor iterations between two major

   /// itarations.

   #define MINOR_LIMIT	  50

 #endif

 #ifdef SORTED_LIST_PIVOT

   #include <deque>

   #include <algorithm>

   /// \brief The maximum length of the edge list for the

   /// "Sorted List" pivot rule.

   #define LIST_LENGTH	  500

   #define LOWER_DIV	  4

 #endif

 //#define _DEBUG_ITER_

 namespace lemon {

   /// \addtogroup min_cost_flow

   /// @{

   /// \brief Implementation of the network simplex algorithm for

   /// finding a minimum cost flow.

   ///

   /// \ref lemon::NetworkSimplex "NetworkSimplex" implements the

   /// network simplex 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 NetworkSimplex

   {

     typedef typename LowerMap::Value Lower;

     typedef typename CapacityMap::Value Capacity;

     typedef typename CostMap::Value Cost;

     typedef typename SupplyMap::Value Supply;

     typedef SmartGraph SGraph;

     typedef typename SGraph::Node Node;

     typedef typename SGraph::NodeIt NodeIt;

     typedef typename SGraph::Edge Edge;

     typedef typename SGraph::EdgeIt EdgeIt;

     typedef typename SGraph::InEdgeIt InEdgeIt;

     typedef typename SGraph::OutEdgeIt OutEdgeIt;

     typedef typename SGraph::template EdgeMap<Lower> SLowerMap;

     typedef typename SGraph::template EdgeMap<Capacity> SCapacityMap;

     typedef typename SGraph::template EdgeMap<Cost> SCostMap;

     typedef typename SGraph::template NodeMap<Supply> SSupplyMap;

     typedef typename SGraph::template NodeMap<Cost> SPotentialMap;

     typedef typename SGraph::template NodeMap<int> IntNodeMap;

     typedef typename SGraph::template NodeMap<bool> BoolNodeMap;

     typedef typename SGraph::template NodeMap<Node> NodeNodeMap;

     typedef typename SGraph::template NodeMap<Edge> EdgeNodeMap;

     typedef typename SGraph::template EdgeMap<int> IntEdgeMap;

     typedef typename Graph::template NodeMap<Node> NodeRefMap;

     typedef typename Graph::template EdgeMap<Edge> EdgeRefMap;

   public:

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

     typedef typename Graph::template EdgeMap<Capacity> 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<Edge, Cost>

     {

     private:

       const SGraph &gr;

       const SCostMap &cost_map;

       const SPotentialMap &pot_map;

     public:

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

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

       ReducedCostMap( const SGraph &_gr,

 		      const SCostMap &_cm,

 		      const SPotentialMap &_pm ) :

 	gr(_gr), cost_map(_cm), pot_map(_pm) {}

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

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

       }

     }; //class ReducedCostMap

   protected:

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

     SGraph graph;

     /// \brief The original graph.

     const Graph &graph_ref;

     /// \brief The original lower bound map.

     const LowerMap *lower;

     /// \brief The capacity map.

     SCapacityMap capacity;

     /// \brief The cost map.

     SCostMap cost;

     /// \brief The supply map.

     SSupplyMap supply;

     /// \brief The reduced cost map.

     ReducedCostMap red_cost;

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

     bool valid_supply;

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

     SCapacityMap flow;

     /// \brief The edge map of the found flow on the original graph.

     FlowMap flow_result;

     /// \brief The potential node map.

     SPotentialMap potential;

     /// \brief The potential node map on the original graph.

     PotentialMap potential_result;

     /// \brief Node reference for the original graph.

     NodeRefMap node_ref;

     /// \brief Edge reference for the original graph.

     EdgeRefMap edge_ref;

     /// \brief The depth node map of the spanning tree structure.

     IntNodeMap depth;

     /// \brief The parent node map of the spanning tree structure.

     NodeNodeMap parent;

     /// \brief The pred_edge node map of the spanning tree structure.

     EdgeNodeMap pred_edge;

     /// \brief The thread node map of the spanning tree structure.

     NodeNodeMap thread;

     /// \brief The forward node map of the spanning tree structure.

     BoolNodeMap forward;

     /// \brief The state edge map.

     IntEdgeMap state;

 #ifdef EDGE_BLOCK_PIVOT

     /// \brief The size of blocks for the "Edge Block" pivot rule.

     int block_size;

 #endif

 #ifdef CANDIDATE_LIST_PIVOT

     /// \brief The list of candidate edges for the "Candidate List"

     /// pivot rule.

     std::list<Edge> candidates;

     /// \brief The number of minor iterations.

     int minor_count;

 #endif

 #ifdef SORTED_LIST_PIVOT

     /// \brief The list of candidate edges for the "Sorted List"

     /// pivot rule.

     std::deque<Edge> candidates;

 #endif

     // Root node of the starting spanning tree.

     Node root;

     // The entering edge of the current pivot iteration.

     Edge in_edge;

     // Temporary nodes used in the current pivot iteration.

     Node join, u_in, v_in, u_out, v_out;

     Node right, first, second, last;

     Node stem, par_stem, new_stem;

     // The maximum augment amount along the cycle in the current pivot

     // iteration.

     Capacity delta;

   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).

     NetworkSimplex( const Graph &_graph,

 		    const LowerMap &_lower,

 		    const CapacityMap &_capacity,

 		    const CostMap &_cost,

 		    const SupplyMap &_supply ) :

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

       supply(graph), flow(graph), flow_result(_graph), potential(graph),

       potential_result(_graph), depth(graph), parent(graph),

       pred_edge(graph), thread(graph), forward(graph), state(graph),

       node_ref(graph_ref), edge_ref(graph_ref),

       red_cost(graph, cost, potential)

     {

       // Checking the sum of supply values

       Supply sum = 0;

       for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n)

 	sum += _supply[n];

       if (!(valid_supply = sum == 0)) return;

       // Copying graph_ref to graph

       copyGraph(graph, graph_ref)

 	.edgeMap(cost, _cost)

 	.nodeRef(node_ref)

 	.edgeRef(edge_ref)

 	.run();

       // Removing nonzero lower bounds

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

 	capacity[edge_ref[e]] = _capacity[e] - _lower[e];

       }

       for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n) {

 	Supply s = _supply[n];

 	for (typename Graph::InEdgeIt e(graph_ref, n); e != INVALID; ++e)

 	  s += _lower[e];

 	for (typename Graph::OutEdgeIt e(graph_ref, n); e != INVALID; ++e)

 	  s -= _lower[e];

 	supply[node_ref[n]] = s;

       }

     }

     /// \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).

     NetworkSimplex( const Graph &_graph,

 		    const CapacityMap &_capacity,

 		    const CostMap &_cost,

 		    const SupplyMap &_supply ) :

       graph_ref(_graph), lower(NULL), capacity(graph), cost(graph),

       supply(graph), flow(graph), flow_result(_graph), potential(graph),

       potential_result(_graph), depth(graph), parent(graph),

       pred_edge(graph), thread(graph), forward(graph), state(graph),

       node_ref(graph_ref), edge_ref(graph_ref),

       red_cost(graph, cost, potential)

     {

       // Checking the sum of supply values

       Supply sum = 0;

       for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n)

 	sum += _supply[n];

       if (!(valid_supply = sum == 0)) return;

       // Copying graph_ref to graph

       copyGraph(graph, graph_ref)

 	.edgeMap(capacity, _capacity)

 	.edgeMap(cost, _cost)

 	.nodeMap(supply, _supply)

 	.nodeRef(node_ref)

 	.edgeRef(edge_ref)

 	.run();

     }

     /// \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).

     NetworkSimplex( const Graph &_graph,

 		    const LowerMap &_lower,

 		    const CapacityMap &_capacity,

 		    const CostMap &_cost,

 		    typename Graph::Node _s,

 		    typename Graph::Node _t,

 		    typename SupplyMap::Value _flow_value ) :

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

       supply(graph), flow(graph), flow_result(_graph), potential(graph),

       potential_result(_graph), depth(graph), parent(graph),

       pred_edge(graph), thread(graph), forward(graph), state(graph),

       node_ref(graph_ref), edge_ref(graph_ref),

       red_cost(graph, cost, potential)

     {

       // Copying graph_ref to graph

       copyGraph(graph, graph_ref)

 	.edgeMap(cost, _cost)

 	.nodeRef(node_ref)

 	.edgeRef(edge_ref)

 	.run();

       // Removing nonzero lower bounds

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

 	capacity[edge_ref[e]] = _capacity[e] - _lower[e];

       }

       for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n) {

 	Supply s = 0;

 	if (n == _s) s =  _flow_value;

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

 	for (typename Graph::InEdgeIt e(graph_ref, n); e != INVALID; ++e)

 	  s += _lower[e];

 	for (typename Graph::OutEdgeIt e(graph_ref, n); e != INVALID; ++e)

 	  s -= _lower[e];

 	supply[node_ref[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).

     NetworkSimplex( const Graph &_graph,

 		    const CapacityMap &_capacity,

 		    const CostMap &_cost,

 		    typename Graph::Node _s,

 		    typename Graph::Node _t,

 		    typename SupplyMap::Value _flow_value ) :

       graph_ref(_graph), lower(NULL), capacity(graph), cost(graph),

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

       potential_result(_graph), depth(graph), parent(graph),

       pred_edge(graph), thread(graph), forward(graph), state(graph),

       node_ref(graph_ref), edge_ref(graph_ref),

       red_cost(graph, cost, potential)

     {

       // Copying graph_ref to graph

       copyGraph(graph, graph_ref)

 	.edgeMap(capacity, _capacity)

 	.edgeMap(cost, _cost)

 	.nodeRef(node_ref)

 	.edgeRef(edge_ref)

 	.run();

       supply[node_ref[_s]] =  _flow_value;

       supply[node_ref[_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_result;

     }

     /// \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_result;

     }

     /// \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 (typename Graph::EdgeIt e(graph_ref); e != INVALID; ++e)

 	c += flow_result[e] * cost[edge_ref[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 Extends the underlaying graph and initializes all the

     /// node and edge maps.

     bool init() {

       if (!valid_supply) return false;

       // Initializing state and flow maps

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

 	flow[e] = 0;

 	state[e] = LOWER;

       }

       // Adding an artificial root node to the graph

       root = graph.addNode();

       parent[root] = INVALID;

       pred_edge[root] = INVALID;

       depth[root] = supply[root] = potential[root] = 0;

       // Adding artificial edges to the graph and initializing the node

       // maps of the spanning tree data structure

       Supply sum = 0;

       Node last = root;

       Edge e;

       Cost max_cost = std::numeric_limits<Cost>::max() / 4;

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

 	if (u == root) continue;

 	thread[last] = u;

 	last = u;

 	parent[u] = root;

 	depth[u] = 1;

 	sum += supply[u];

 	if (supply[u] >= 0) {

 	  e = graph.addEdge(u, root);

 	  flow[e] = supply[u];

 	  forward[u] = true;

 	  potential[u] = max_cost;

 	} else {

 	  e = graph.addEdge(root, u);

 	  flow[e] = -supply[u];

 	  forward[u] = false;

 	  potential[u] = -max_cost;

 	}

 	cost[e] = max_cost;

 	capacity[e] = std::numeric_limits<Capacity>::max();

 	state[e] = TREE;

 	pred_edge[u] = e;

       }

       thread[last] = root;

 #ifdef EDGE_BLOCK_PIVOT

       // Initializing block_size for the edge block pivot rule

       int edge_num = countEdges(graph);

       block_size = edge_num > MIN_BOUND ? edge_num / BLOCK_NUM + 1 : 1;

 #endif

 #ifdef CANDIDATE_LIST_PIVOT

       minor_count = 0;

 #endif

       return sum == 0;

     }

 #ifdef FIRST_ELIGIBLE_PIVOT

     /// \brief Finds entering edge according to the "First Eligible"

     /// pivot rule.

     bool findEnteringEdge(EdgeIt &next_edge) {

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

 	if (state[e] * red_cost[e] < 0) {

 	  in_edge = e;

 	  next_edge = ++e;

 	  return true;

 	}

       }

       for (EdgeIt e(graph); e != next_edge; ++e) {

 	if (state[e] * red_cost[e] < 0) {

 	  in_edge = e;

 	  next_edge = ++e;

 	  return true;

 	}

       }

       return false;

     }

 #endif

 #ifdef BEST_ELIGIBLE_PIVOT

     /// \brief Finds entering edge according to the "Best Eligible"

     /// pivot rule.

     bool findEnteringEdge() {

       Cost min = 0;

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

 	if (state[e] * red_cost[e] < min) {

 	  min = state[e] * red_cost[e];

 	  in_edge = e;

 	}

       }

       return min < 0;

     }

 #endif

 #ifdef EDGE_BLOCK_PIVOT

     /// \brief Finds entering edge according to the "Edge Block"

     /// pivot rule.

     bool findEnteringEdge(EdgeIt &next_edge) {

       if (block_size == 1) {

 	// Performing first eligible selection

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

 	  if (state[e] * red_cost[e] < 0) {

 	    in_edge = e;

 	    next_edge = ++e;

 	    return true;

 	  }

 	}

 	for (EdgeIt e(graph); e != next_edge; ++e) {

 	  if (state[e] * red_cost[e] < 0) {

 	    in_edge = e;

 	    next_edge = ++e;

 	    return true;

 	  }

 	}

 	return false;

       } else {

 	// Performing edge block selection

 	Cost curr, min = 0;

 	EdgeIt min_edge(graph);

 	int cnt = 0;

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

 	  if ((curr = state[e] * red_cost[e]) < min) {

 	    min = curr;

 	    min_edge = e;

 	  }

 	  if (++cnt == block_size) {

 	    if (min < 0) break;

 	    cnt = 0;

 	  }

 	}

 	if (!(min < 0)) {

 	  for (EdgeIt e(graph); e != next_edge; ++e) {

 	    if ((curr = state[e] * red_cost[e]) < min) {

 	      min = curr;

 	      min_edge = e;

 	    }

 	    if (++cnt == block_size) {

 	      if (min < 0) break;

 	      cnt = 0;

 	    }

 	  }

 	}

 	in_edge = min_edge;

 	if ((next_edge = ++min_edge) == INVALID)

 	  next_edge = EdgeIt(graph);

 	return min < 0;

       }

     }

 #endif

 #ifdef CANDIDATE_LIST_PIVOT

     /// \brief Functor class for removing non-eligible edges from the

     /// candidate list.

     class RemoveFunc

     {

     private:

       const IntEdgeMap &st;

       const ReducedCostMap &rc;

     public:

       RemoveFunc(const IntEdgeMap &_st, const ReducedCostMap &_rc) :

 	st(_st), rc(_rc) {}

       bool operator()(const Edge &e) {

 	return st[e] * rc[e] >= 0;

       }

     };

     /// \brief Finds entering edge according to the "Candidate List"

     /// pivot rule.

     bool findEnteringEdge() {

       static RemoveFunc remove_func(state, red_cost);

       typedef typename std::list<Edge>::iterator ListIt;

       candidates.remove_if(remove_func);

       if (minor_count >= MINOR_LIMIT || candidates.size() == 0) {

 	// Major iteration

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

 	  if (state[e] * red_cost[e] < 0) {

 	    candidates.push_back(e);

 	    if (candidates.size() == LIST_LENGTH) break;

 	  }

 	}

 	if (candidates.size() == 0) return false;

       }

       // Minor iteration

       ++minor_count;

       Cost min = 0;

       for (ListIt it = candidates.begin(); it != candidates.end(); ++it) {

 	if (state[*it] * red_cost[*it] < min) {

 	  min = state[*it] * red_cost[*it];

 	  in_edge = *it;

 	}

       }

       return true;

     }

 #endif

 #ifdef SORTED_LIST_PIVOT

     /// \brief Functor class to compare edges during sort of the

     /// candidate list.

     class SortFunc

     {

     private:

       const IntEdgeMap &st;

       const ReducedCostMap &rc;

     public:

       SortFunc(const IntEdgeMap &_st, const ReducedCostMap &_rc) :

 	st(_st), rc(_rc) {}

       bool operator()(const Edge &e1, const Edge &e2) {

 	return st[e1] * rc[e1] < st[e2] * rc[e2];

       }

     };

     /// \brief Finds entering edge according to the "Sorted List"

     /// pivot rule.

     bool findEnteringEdge() {

       static SortFunc sort_func(state, red_cost);

       // Minor iteration

       while (candidates.size() > 0) {

 	in_edge = candidates.front();

 	candidates.pop_front();

 	if (state[in_edge] * red_cost[in_edge] < 0) return true;

       }

       // Major iteration

       Cost curr, min = 0;

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

 	if ((curr = state[e] * red_cost[e]) < min / LOWER_DIV) {

 	  candidates.push_back(e);

 	  if (curr < min) min = curr;

 	  if (candidates.size() >= LIST_LENGTH) break;

 	}

       }

       if (candidates.size() == 0) return false;

       sort(candidates.begin(), candidates.end(), sort_func);

       return true;

     }

 #endif

     /// \brief Finds the join node.

     Node findJoinNode() {

       // Finding the join node

       Node u = graph.source(in_edge);

       Node v = graph.target(in_edge);

       while (u != v) {

 	if (depth[u] == depth[v]) {

 	  u = parent[u];

 	  v = parent[v];

 	}

 	else if (depth[u] > depth[v]) u = parent[u];

 	else v = parent[v];

       }

       return u;

     }

     /// \brief Finds the leaving edge of the cycle. Returns \c true if

     /// the leaving edge is not the same as the entering edge.

     bool findLeavingEdge() {

       // Initializing first and second nodes according to the direction

       // of the cycle

       if (state[in_edge] == LOWER) {

 	first = graph.source(in_edge);

 	second	= graph.target(in_edge);

       } else {

 	first = graph.target(in_edge);

 	second	= graph.source(in_edge);

       }

       delta = capacity[in_edge];

       bool result = false;

       Capacity d;

       Edge e;

       // Searching the cycle along the path form the first node to the

       // root node

       for (Node u = first; u != join; u = parent[u]) {

 	e = pred_edge[u];

 	d = forward[u] ? flow[e] : capacity[e] - flow[e];

 	if (d < delta) {

 	  delta = d;

 	  u_out = u;

 	  u_in = first;

 	  v_in = second;

 	  result = true;

 	}

       }

       // Searching the cycle along the path form the second node to the

       // root node

       for (Node u = second; u != join; u = parent[u]) {

 	e = pred_edge[u];

 	d = forward[u] ? capacity[e] - flow[e] : flow[e];

 	if (d <= delta) {

 	  delta = d;

 	  u_out = u;

 	  u_in = second;

 	  v_in = first;

 	  result = true;

 	}

       }

       return result;

     }

     /// \brief Changes flow and state edge maps.

     void changeFlows(bool change) {

       // Augmenting along the cycle

       if (delta > 0) {

 	Capacity val = state[in_edge] * delta;

 	flow[in_edge] += val;

 	for (Node u = graph.source(in_edge); u != join; u = parent[u]) {

 	  flow[pred_edge[u]] += forward[u] ? -val : val;

 	}

 	for (Node u = graph.target(in_edge); u != join; u = parent[u]) {

 	  flow[pred_edge[u]] += forward[u] ? val : -val;

 	}

       }

       // Updating the state of the entering and leaving edges

       if (change) {

 	state[in_edge] = TREE;

 	state[pred_edge[u_out]] =

 	  (flow[pred_edge[u_out]] == 0) ? LOWER : UPPER;

       } else {

 	state[in_edge] = -state[in_edge];

       }

     }

     /// \brief Updates thread and parent node maps.

     void updateThreadParent() {

       Node u;

       v_out = parent[u_out];

       // Handling the case when join and v_out coincide

       bool par_first = false;

       if (join == v_out) {

 	for (u = join; u != u_in && u != v_in; u = thread[u]) ;

 	if (u == v_in) {

 	  par_first = true;

 	  while (thread[u] != u_out) u = thread[u];

 	  first = u;

 	}

       }

       // Finding the last successor of u_in (u) and the node after it

       // (right) according to the thread index

       for (u = u_in; depth[thread[u]] > depth[u_in]; u = thread[u]) ;

       right = thread[u];

       if (thread[v_in] == u_out) {

 	for (last = u; depth[last] > depth[u_out]; last = thread[last]) ;

 	if (last == u_out) last = thread[last];

       }

       else last = thread[v_in];

       // Updating stem nodes

       thread[v_in] = stem = u_in;

       par_stem = v_in;

       while (stem != u_out) {

 	thread[u] = new_stem = parent[stem];

 	// Finding the node just before the stem node (u) according to

 	// the original thread index

 	for (u = new_stem; thread[u] != stem; u = thread[u]) ;

 	thread[u] = right;

 	// Changing the parent node of stem and shifting stem and

 	// par_stem nodes

 	parent[stem] = par_stem;

 	par_stem = stem;

 	stem = new_stem;

 	// Finding the last successor of stem (u) and the node after it

 	// (right) according to the thread index

 	for (u = stem; depth[thread[u]] > depth[stem]; u = thread[u]) ;

 	right = thread[u];

       }

       parent[u_out] = par_stem;

       thread[u] = last;

       if (join == v_out && par_first) {

 	if (first != v_in) thread[first] = right;

       } else {

 	for (u = v_out; thread[u] != u_out; u = thread[u]) ;

 	thread[u] = right;

       }

     }

     /// \brief Updates pred_edge and forward node maps.

     void updatePredEdge() {

       Node u = u_out, v;

       while (u != u_in) {

 	v = parent[u];

 	pred_edge[u] = pred_edge[v];

 	forward[u] = !forward[v];

 	u = v;

       }

       pred_edge[u_in] = in_edge;

       forward[u_in] = (u_in == graph.source(in_edge));

     }

     /// \brief Updates depth and potential node maps.

     void updateDepthPotential() {

       depth[u_in] = depth[v_in] + 1;

       potential[u_in] = forward[u_in] ?

 	potential[v_in] + cost[pred_edge[u_in]] :

 	potential[v_in] - cost[pred_edge[u_in]];

       Node u = thread[u_in], v;

       while (true) {

 	v = parent[u];

 	if (v == INVALID) break;

 	depth[u] = depth[v] + 1;

 	potential[u] = forward[u] ?

 	  potential[v] + cost[pred_edge[u]] :

 	  potential[v] - cost[pred_edge[u]];

 	if (depth[u] <= depth[v_in]) break;

 	u = thread[u];

       }

     }

     /// \brief Executes the algorithm.

     bool start() {

       // Processing pivots

 #ifdef _DEBUG_ITER_

       int iter_num = 0;

 #endif

 #if defined(FIRST_ELIGIBLE_PIVOT) || defined(EDGE_BLOCK_PIVOT)

       EdgeIt next_edge(graph);

       while (findEnteringEdge(next_edge))

 #else

       while (findEnteringEdge())

 #endif

       {

 	join = findJoinNode();

 	bool change = findLeavingEdge();

 	changeFlows(change);

 	if (change) {

 	  updateThreadParent();

 	  updatePredEdge();

 	  updateDepthPotential();

 	}

 #ifdef _DEBUG_ITER_

 	++iter_num;

 #endif

       }

 #ifdef _DEBUG_ITER_

       t_iter.stop();

       std::cout << "Network Simplex algorithm finished. " << iter_num

 		<< " pivot iterations performed.";

 #endif

       // Checking if the flow amount equals zero on all the

       // artificial edges

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

 	if (flow[e] > 0) return false;

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

 	if (flow[e] > 0) return false;

       // Copying flow values to flow_result

       if (lower) {

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

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

       } else {

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

 	  flow_result[e] = flow[edge_ref[e]];

       }

       // Copying potential values to potential_result

       for (typename Graph::NodeIt n(graph_ref); n != INVALID; ++n)

 	potential_result[n] = potential[node_ref[n]];

       return true;

     }

   }; //class NetworkSimplex

   ///@}

 } //namespace lemon

 #endif //LEMON_NETWORK_SIMPLEX_H