/* -*- 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 FIRST_ELIGIBLE_PIVOT

//#define BEST_ELIGIBLE_PIVOT

#define EDGE_BLOCK_PIVOT

//#define CANDIDATE_LIST_PIVOT

//#define SORTED_LIST_PIVOT

//#define _DEBUG_ITER_

/// \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 size of blocks.

  #define MIN_BLOCK_SIZE	10

#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		10

#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		3

#endif

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 >= BLOCK_NUM * MIN_BLOCK_SIZE ? 

		   edge_num / BLOCK_NUM : MIN_BLOCK_SIZE;

#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) {

      // 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);

      in_edge = candidates.front();

      candidates.pop_front();

      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_

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

		<< " pivot iterations performed." << std::endl;

#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