/* -*- 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_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/graph_adaptor.h>

 #include <lemon/graph_utils.h>

 #include <lemon/smart_graph.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_

 // State constant for edges at their lower bounds.

 #define LOWER   1

 // State constant for edges in the spanning tree.

 #define TREE    0

 // State constant for edges at their upper bounds.

 #define UPPER   -1

 #ifdef EDGE_BLOCK_PIVOT

   #include <cmath>

   #define MIN_BLOCK_SIZE        10

 #endif

 #ifdef CANDIDATE_LIST_PIVOT

   #include <vector>

   #define LIST_LENGTH_DIV       50

   #define MINOR_LIMIT_DIV       200

 #endif

 #ifdef SORTED_LIST_PIVOT

   #include <vector>

   #include <algorithm>

   #define LIST_LENGTH_DIV       100

   #define LOWER_DIV             4

 #endif

 namespace lemon {

   /// \addtogroup min_cost_flow

   /// @{

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

   /// finding a minimum cost flow.

   ///

   /// \ref 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 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 NetworkSimplex

   {

     typedef typename LowerMap::Value Lower;

     typedef typename CapacityMap::Value Capacity;

     typedef typename CostMap::Value Cost;

     typedef typename SupplyMap::Value Supply;

     typedef SmartGraph SGraph;

     GRAPH_TYPEDEFS(typename SGraph);

     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:

     /// 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:

     /// 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:

       ReducedCostMap( const SGraph &_gr,

                       const SCostMap &_cm,

                       const SPotentialMap &_pm ) :

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

       Cost operator[](const Edge &e) const {

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

       }

     }; //class ReducedCostMap

   protected:

     /// The directed graph the algorithm runs on.

     SGraph graph;

     /// The original graph.

     const Graph &graph_ref;

     /// The original lower bound map.

     const LowerMap *lower;

     /// The capacity map.

     SCapacityMap capacity;

     /// The cost map.

     SCostMap cost;

     /// The supply map.

     SSupplyMap supply;

     /// The reduced cost map.

     ReducedCostMap red_cost;

     bool valid_supply;

     /// The edge map of the current flow.

     SCapacityMap flow;

     /// The potential node map.

     SPotentialMap potential;

     FlowMap flow_result;

     PotentialMap potential_result;

     /// Node reference for the original graph.

     NodeRefMap node_ref;

     /// Edge reference for the original graph.

     EdgeRefMap edge_ref;

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

     IntNodeMap depth;

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

     NodeNodeMap parent;

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

     EdgeNodeMap pred_edge;

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

     NodeNodeMap thread;

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

     BoolNodeMap forward;

     /// The state edge map.

     IntEdgeMap state;

     /// The 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 found cycle in the current

     // pivot iteration.

     Capacity delta;

 #ifdef EDGE_BLOCK_PIVOT

     /// 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::vector<Edge> candidates;

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

     /// "Candidate List" pivot rule.

     int list_length;

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

     /// itarations.

     int minor_limit;

     /// \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::vector<Edge> candidates;

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

     /// "Sorted List" pivot rule.

     int list_length;

     int list_index;

 #endif

   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

       graph.reserveNode(countNodes(graph_ref) + 1);

       graph.reserveEdge(countEdges(graph_ref) + countNodes(graph_ref));

       copyGraph(graph, graph_ref)

         .edgeMap(cost, _cost)

         .nodeRef(node_ref)

         .edgeRef(edge_ref)

         .run();

       // Removing non-zero 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 non-zero 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 Runs the algorithm.

     ///

     /// Runs the algorithm.

     ///

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

     bool run() {

       return init() && 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_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;

     }

   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] = 0;

       supply[root] = 0;

       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 = 2 * int(sqrt(countEdges(graph)));

       if (block_size < MIN_BLOCK_SIZE) block_size = MIN_BLOCK_SIZE;

 #endif

 #ifdef CANDIDATE_LIST_PIVOT

       int edge_num = countEdges(graph);

       minor_count = 0;

       list_length = edge_num / LIST_LENGTH_DIV;

       minor_limit = edge_num / MINOR_LIMIT_DIV;

 #endif

 #ifdef SORTED_LIST_PIVOT

       int edge_num = countEdges(graph);

       list_index = 0;

       list_length = edge_num / LIST_LENGTH_DIV;

 #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 Finds entering edge according to the "Candidate List"

     /// pivot rule.

     bool findEnteringEdge() {

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

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

         // Major iteration

         candidates.clear();

         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;

       Edge e;

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

         e = candidates[i];

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

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

           in_edge = e;

         }

       }

       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 (list_index < candidates.size()) {

         in_edge = candidates[list_index++];

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

       }

       // Major iteration

       candidates.clear();

       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[0];

       list_index = 1;

       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 \c flow and \c 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 \c thread and \c 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 \c pred_edge and \c 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 \c depth and \c 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