RhodeCode

    NetworkSimplex& upperMap(const UpperMap& map) {

      for (ArcIt a(_graph); a != INVALID; ++a) {

        _upper[_arc_id[a]] = map[a];

      }

      return *this;

    }

    /// \brief Set the costs of the arcs.

    ///

    /// This function sets the costs of the arcs.

    /// If it is not used before calling \ref run(), the costs

    /// will be set to \c 1 on all arcs.

    ///

    /// \param map An arc map storing the costs.

    /// Its \c Value type must be convertible to the \c Cost type

    /// of the algorithm.

    ///

    /// \return <tt>(*this)</tt>

    template<typename CostMap>

    NetworkSimplex& costMap(const CostMap& map) {

      for (ArcIt a(_graph); a != INVALID; ++a) {

        _cost[_arc_id[a]] = map[a];

      }

      return *this;

    }

    /// \brief Set the supply values of the nodes.

    ///

    /// This function sets the supply values of the nodes.

    /// If neither this function nor \ref stSupply() is used before

    /// calling \ref run(), the supply of each node will be set to zero.

    ///

    /// \param map A node map storing the supply values.

    /// Its \c Value type must be convertible to the \c Value type

    /// of the algorithm.

    ///

    /// \return <tt>(*this)</tt>

    template<typename SupplyMap>

    NetworkSimplex& supplyMap(const SupplyMap& map) {

      for (NodeIt n(_graph); n != INVALID; ++n) {

        _supply[_node_id[n]] = map[n];

      }

      return *this;

    }

    /// \brief Set single source and target nodes and a supply value.

    ///

    /// This function sets a single source node and a single target node

    /// and the required flow value.

    /// If neither this function nor \ref supplyMap() is used before

    /// calling \ref run(), the supply of each node will be set to zero.

    ///

    /// Using this function has the same effect as using \ref supplyMap()

    /// with such a map in which \c k is assigned to \c s, \c -k is

    /// assigned to \c t and all other nodes have zero supply value.

    ///

    /// \param s The source node.

    /// \param t The target node.

    /// \param k 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).

    ///

    /// \return <tt>(*this)</tt>

    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {

      for (int i = 0; i != _node_num; ++i) {

        _supply[i] = 0;

      }

      _supply[_node_id[s]] =  k;

      _supply[_node_id[t]] = -k;

      return *this;

    }

    /// \brief Set the type of the supply constraints.

    ///

    /// This function sets the type of the supply/demand constraints.

    /// If it is not used before calling \ref run(), the \ref GEQ supply

    /// type will be used.

    ///

    /// For more information, see \ref SupplyType.

    ///

    /// \return <tt>(*this)</tt>

    NetworkSimplex& supplyType(SupplyType supply_type) {

      _stype = supply_type;

      return *this;

    }

    /// @}

    /// \name Execution Control

    /// The algorithm can be executed using \ref run().

    /// @{

    /// \brief Run the algorithm.

    ///

    /// This function runs the algorithm.

    /// The paramters can be specified using functions \ref lowerMap(),

    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),

    /// \ref supplyType().

    /// For example,

    /// \code

    ///   NetworkSimplex<ListDigraph> ns(graph);

    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)

    ///     .supplyMap(sup).run();

    /// \endcode

    ///

    /// This function can be called more than once. All the given parameters

    /// are kept for the next call, unless \ref resetParams() or \ref reset()

    /// is used, thus only the modified parameters have to be set again.

    /// If the underlying digraph was also modified after the construction

    /// of the class (or the last \ref reset() call), then the \ref reset()

    /// function must be called.

    ///

    /// \param pivot_rule The pivot rule that will be used during the

    /// algorithm. For more information, see \ref PivotRule.

    ///

    /// \return \c INFEASIBLE if no feasible flow exists,

    /// \n \c OPTIMAL if the problem has optimal solution

    /// (i.e. it is feasible and bounded), and the algorithm has found

    /// optimal flow and node potentials (primal and dual solutions),

    /// \n \c UNBOUNDED if the objective function of the problem is

    /// unbounded, i.e. there is a directed cycle having negative total

    /// cost and infinite upper bound.

    ///

    /// \see ProblemType, PivotRule

    /// \see resetParams(), reset()

    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {

      if (!init()) return INFEASIBLE;

      return start(pivot_rule);

    }

    /// \brief Reset all the parameters that have been given before.

    ///

    /// This function resets all the paramaters that have been given

    /// before using functions \ref lowerMap(), \ref upperMap(),

    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().

    ///

    /// It is useful for multiple \ref run() calls. Basically, all the given

    /// parameters are kept for the next \ref run() call, unless

    /// \ref resetParams() or \ref reset() is used.

    /// If the underlying digraph was also modified after the construction

    /// of the class or the last \ref reset() call, then the \ref reset()

    /// function must be used, otherwise \ref resetParams() is sufficient.

    ///

    /// For example,

    /// \code

    ///   NetworkSimplex<ListDigraph> ns(graph);

    ///

    ///   // First run

    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)

    ///     .supplyMap(sup).run();

    ///

    ///   // Run again with modified cost map (resetParams() is not called,

    ///   // so only the cost map have to be set again)

    ///   cost[e] += 100;

    ///   ns.costMap(cost).run();

    ///

    ///   // Run again from scratch using resetParams()

    ///   // (the lower bounds will be set to zero on all arcs)

    ///   ns.resetParams();

    ///   ns.upperMap(capacity).costMap(cost)

    ///     .supplyMap(sup).run();

    /// \endcode

    ///

    /// \return <tt>(*this)</tt>

    ///

    /// \see reset(), run()

    NetworkSimplex& resetParams() {

      for (int i = 0; i != _node_num; ++i) {

        _supply[i] = 0;

      }

      for (int i = 0; i != _arc_num; ++i) {

        _lower[i] = 0;

        _upper[i] = INF;

        _cost[i] = 1;

      }

      _have_lower = false;

      _stype = GEQ;

      return *this;

    }

    /// \brief Reset the internal data structures and all the parameters

    /// that have been given before.

    ///

    /// This function resets the internal data structures and all the

    /// paramaters that have been given before using functions \ref lowerMap(),

    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),

    /// \ref supplyType().

    ///

    /// It is useful for multiple \ref run() calls. Basically, all the given

    /// parameters are kept for the next \ref run() call, unless

    /// \ref resetParams() or \ref reset() is used.

    /// If the underlying digraph was also modified after the construction

    /// of the class or the last \ref reset() call, then the \ref reset()

    /// function must be used, otherwise \ref resetParams() is sufficient.

    ///

    /// See \ref resetParams() for examples.

    ///

    /// \return <tt>(*this)</tt>

    ///

    /// \see resetParams(), run()

    NetworkSimplex& reset() {

      // Resize vectors

      _node_num = countNodes(_graph);

      _arc_num = countArcs(_graph);

      int all_node_num = _node_num + 1;

      int max_arc_num = _arc_num + 2 * _node_num;

      _source.resize(max_arc_num);

      _target.resize(max_arc_num);

      _lower.resize(_arc_num);

      _upper.resize(_arc_num);

      _cap.resize(max_arc_num);

      _cost.resize(max_arc_num);

      _supply.resize(all_node_num);

      _flow.resize(max_arc_num);

      _pi.resize(all_node_num);

      _parent.resize(all_node_num);

      _pred.resize(all_node_num);

      _forward.resize(all_node_num);

      _thread.resize(all_node_num);

      _rev_thread.resize(all_node_num);

      _succ_num.resize(all_node_num);

      _last_succ.resize(all_node_num);

      _state.resize(max_arc_num);

      // Copy the graph

      int i = 0;

      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {

        _node_id[n] = i;

      }

      if (_arc_mixing) {

        // Store the arcs in a mixed order

        int k = std::max(int(std::sqrt(double(_arc_num))), 10);

        int i = 0, j = 0;

        for (ArcIt a(_graph); a != INVALID; ++a) {

          _arc_id[a] = i;

          _source[i] = _node_id[_graph.source(a)];

          _target[i] = _node_id[_graph.target(a)];

          if ((i += k) >= _arc_num) i = ++j;

        }

      } else {

        // Store the arcs in the original order

        int i = 0;

        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {

          _arc_id[a] = i;

          _source[i] = _node_id[_graph.source(a)];

          _target[i] = _node_id[_graph.target(a)];

        }

      }

      // Reset parameters

      resetParams();

      return *this;

    }

    /// @}

    /// \name Query Functions

    /// The results of the algorithm can be obtained using these

    /// functions.\n

    /// The \ref run() function must be called before using them.

    /// @{

    /// \brief Return the total cost of the found flow.

    ///

    /// This function returns the total cost of the found flow.

    /// Its complexity is O(e).

    ///

    /// \note The return type of the function can be specified as a

    /// template parameter. For example,

    /// \code

    ///   ns.totalCost<double>();

    /// \endcode

    /// It is useful if the total cost cannot be stored in the \c Cost

    /// type of the algorithm, which is the default return type of the

    /// function.

    ///

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

    template <typename Number>

    Number totalCost() const {

      Number c = 0;

      for (ArcIt a(_graph); a != INVALID; ++a) {

        int i = _arc_id[a];

        c += Number(_flow[i]) * Number(_cost[i]);

      }

      return c;

    }

#ifndef DOXYGEN

    Cost totalCost() const {

      return totalCost<Cost>();

    }

#endif

    /// \brief Return the flow on the given arc.

    ///

    /// This function returns the flow on the given arc.

    ///

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

    Value flow(const Arc& a) const {

      return _flow[_arc_id[a]];

    }

    /// \brief Return the flow map (the primal solution).

    ///

    /// This function copies the flow value on each arc into the given

    /// map. The \c Value type of the algorithm must be convertible to

    /// the \c Value type of the map.

    ///

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

    template <typename FlowMap>

    void flowMap(FlowMap &map) const {

      for (ArcIt a(_graph); a != INVALID; ++a) {

        map.set(a, _flow[_arc_id[a]]);

      }

    }

    /// \brief Return the potential (dual value) of the given node.

    ///

    /// This function returns the potential (dual value) of the

    /// given node.

    ///

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

    Cost potential(const Node& n) const {

      return _pi[_node_id[n]];

    }

    /// \brief Return the potential map (the dual solution).

    ///

    /// This function copies the potential (dual value) of each node

    /// into the given map.

    /// The \c Cost type of the algorithm must be convertible to the

    /// \c Value type of the map.

    ///

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

    template <typename PotentialMap>

    void potentialMap(PotentialMap &map) const {

      for (NodeIt n(_graph); n != INVALID; ++n) {

        map.set(n, _pi[_node_id[n]]);

      }

    }

    /// @}

  private:

    // Initialize internal data structures

    bool init() {

      if (_node_num == 0) return false;

      // Check the sum of supply values

      _sum_supply = 0;

      for (int i = 0; i != _node_num; ++i) {

        _sum_supply += _supply[i];

      }

      if ( !((_stype == GEQ && _sum_supply <= 0) ||

             (_stype == LEQ && _sum_supply >= 0)) ) return false;

      // Remove non-zero lower bounds

      if (_have_lower) {

        for (int i = 0; i != _arc_num; ++i) {

          Value c = _lower[i];

          if (c >= 0) {

            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;

          } else {

            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;

          }

          _supply[_source[i]] -= c;

          _supply[_target[i]] += c;

        }

      } else {

        for (int i = 0; i != _arc_num; ++i) {

          _cap[i] = _upper[i];

        }

      }

      // Initialize artifical cost

      Cost ART_COST;

      if (std::numeric_limits<Cost>::is_exact) {

        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;

      } else {

        for (int i = 0; i != _arc_num; ++i) {

          if (_cost[i] > ART_COST) ART_COST = _cost[i];

        }

        ART_COST = (ART_COST + 1) * _node_num;

      }

      // Initialize arc maps

      for (int i = 0; i != _arc_num; ++i) {

        _flow[i] = 0;

        _state[i] = STATE_LOWER;

      }

      // Set data for the artificial root node

      _root = _node_num;

      _parent[_root] = -1;

      _pred[_root] = -1;

      _thread[_root] = 0;

      _rev_thread[0] = _root;

      _succ_num[_root] = _node_num + 1;

      _last_succ[_root] = _root - 1;

      _supply[_root] = -_sum_supply;

      _pi[_root] = 0;

      // Add artificial arcs and initialize the spanning tree data structure

      if (_sum_supply == 0) {

        // EQ supply constraints

        _search_arc_num = _arc_num;

        _all_arc_num = _arc_num + _node_num;

        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {

          _parent[u] = _root;

          _pred[u] = e;

          _thread[u] = u + 1;

          _rev_thread[u + 1] = u;

          _succ_num[u] = 1;

          _last_succ[u] = u;

          _cap[e] = INF;

          _state[e] = STATE_TREE;

          if (_supply[u] >= 0) {

            _forward[u] = true;

            _pi[u] = 0;

            _source[e] = u;

            _target[e] = _root;

            _flow[e] = _supply[u];

            _cost[e] = 0;

          } else {

            _forward[u] = false;

            _pi[u] = ART_COST;

            _source[e] = _root;

            _target[e] = u;

            _flow[e] = -_supply[u];

            _cost[e] = ART_COST;

          }

        }

      }

      else if (_sum_supply > 0) {

        // LEQ supply constraints

        _search_arc_num = _arc_num + _node_num;

        int f = _arc_num + _node_num;

        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {

          _parent[u] = _root;

          _thread[u] = u + 1;

          _rev_thread[u + 1] = u;

          _succ_num[u] = 1;

          _last_succ[u] = u;

          if (_supply[u] >= 0) {

            _forward[u] = true;

            _pi[u] = 0;

            _pred[u] = e;

            _source[e] = u;

            _target[e] = _root;

            _cap[e] = INF;

            _flow[e] = _supply[u];

            _cost[e] = 0;

            _state[e] = STATE_TREE;

          } else {

            _forward[u] = false;

            _pi[u] = ART_COST;

            _pred[u] = f;

            _source[f] = _root;

            _target[f] = u;

            _cap[f] = INF;

            _flow[f] = -_supply[u];

            _cost[f] = ART_COST;

            _state[f] = STATE_TREE;

            _source[e] = u;

            _target[e] = _root;

            _cap[e] = INF;

            _flow[e] = 0;

            _cost[e] = 0;

            _state[e] = STATE_LOWER;

            ++f;

          }

        }

        _all_arc_num = f;

      }

      else {

        // GEQ supply constraints

        _search_arc_num = _arc_num + _node_num;

        int f = _arc_num + _node_num;

        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {

          _parent[u] = _root;

          _thread[u] = u + 1;

          _rev_thread[u + 1] = u;

          _succ_num[u] = 1;

          _last_succ[u] = u;

          if (_supply[u] <= 0) {

            _forward[u] = false;

            _pi[u] = 0;

            _pred[u] = e;

            _source[e] = _root;

            _target[e] = u;

            _cap[e] = INF;

            _flow[e] = -_supply[u];

            _cost[e] = 0;

            _state[e] = STATE_TREE;

          } else {

            _forward[u] = true;

            _pi[u] = -ART_COST;

            _pred[u] = f;

            _source[f] = u;

            _target[f] = _root;

            _cap[f] = INF;

            _flow[f] = _supply[u];

            _state[f] = STATE_TREE;

            _cost[f] = ART_COST;

            _source[e] = _root;

            _target[e] = u;

            _cap[e] = INF;

            _flow[e] = 0;

            _cost[e] = 0;

            _state[e] = STATE_LOWER;

            ++f;

          }

        }

        _all_arc_num = f;

      }

      return true;

    }

    // Find the join node

    void findJoinNode() {

      int u = _source[in_arc];

      int v = _target[in_arc];

      while (u != v) {

        if (_succ_num[u] < _succ_num[v]) {

          u = _parent[u];

        } else {

          v = _parent[v];

        }

      }

      join = u;

    }

    // Find the leaving arc of the cycle and returns true if the

    // leaving arc is not the same as the entering arc

    bool findLeavingArc() {

      // Initialize first and second nodes according to the direction

      // of the cycle

      if (_state[in_arc] == STATE_LOWER) {

        first  = _source[in_arc];

        second = _target[in_arc];

      } else {

        first  = _target[in_arc];

        second = _source[in_arc];

      }

      delta = _cap[in_arc];

      int result = 0;

      Value d;

      int e;

      // Search the cycle along the path form the first node to the root

      for (int u = first; u != join; u = _parent[u]) {

        e = _pred[u];

        d = _forward[u] ?

          _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);

        if (d < delta) {

          delta = d;

          u_out = u;

          result = 1;

        }

      }

      // Search the cycle along the path form the second node to the root

      for (int u = second; u != join; u = _parent[u]) {

        e = _pred[u];

        d = _forward[u] ?

          (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];

        if (d <= delta) {

          delta = d;

          u_out = u;

          result = 2;

        }

      }

      if (result == 1) {

        u_in = first;

        v_in = second;

      } else {

        u_in = second;

        v_in = first;

      }

      return result != 0;

    }

    // Change _flow and _state vectors

    void changeFlow(bool change) {

      // Augment along the cycle

      if (delta > 0) {

        Value val = _state[in_arc] * delta;

        _flow[in_arc] += val;

        for (int u = _source[in_arc]; u != join; u = _parent[u]) {

          _flow[_pred[u]] += _forward[u] ? -val : val;

        }

        for (int u = _target[in_arc]; u != join; u = _parent[u]) {

          _flow[_pred[u]] += _forward[u] ? val : -val;

        }

      }

      // Update the state of the entering and leaving arcs

      if (change) {

        _state[in_arc] = STATE_TREE;

        _state[_pred[u_out]] =

          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;

      } else {

        _state[in_arc] = -_state[in_arc];

      }

    }

    // Update the tree structure

    void updateTreeStructure() {

      int u, w;

      int old_rev_thread = _rev_thread[u_out];

      int old_succ_num = _succ_num[u_out];

      int old_last_succ = _last_succ[u_out];

      v_out = _parent[u_out];

      u = _last_succ[u_in];  // the last successor of u_in

      right = _thread[u];    // the node after it

      // Handle the case when old_rev_thread equals to v_in

      // (it also means that join and v_out coincide)

      if (old_rev_thread == v_in) {

        last = _thread[_last_succ[u_out]];

      } else {

        last = _thread[v_in];

      }

      // Update _thread and _parent along the stem nodes (i.e. the nodes

      // between u_in and u_out, whose parent have to be changed)

      _thread[v_in] = stem = u_in;

      _dirty_revs.clear();

      _dirty_revs.push_back(v_in);

      par_stem = v_in;

      while (stem != u_out) {

        // Insert the next stem node into the thread list

        new_stem = _parent[stem];

        _thread[u] = new_stem;

        _dirty_revs.push_back(u);

        // Remove the subtree of stem from the thread list

        w = _rev_thread[stem];

        _thread[w] = right;

        _rev_thread[right] = w;

        // Change the parent node and shift stem nodes

        _parent[stem] = par_stem;

        par_stem = stem;

        stem = new_stem;

        // Update u and right

        u = _last_succ[stem] == _last_succ[par_stem] ?

          _rev_thread[par_stem] : _last_succ[stem];

        right = _thread[u];

      }

      _parent[u_out] = par_stem;

      _thread[u] = last;

      _rev_thread[last] = u;

      _last_succ[u_out] = u;

      // Remove the subtree of u_out from the thread list except for

      // the case when old_rev_thread equals to v_in

      // (it also means that join and v_out coincide)

      if (old_rev_thread != v_in) {

        _thread[old_rev_thread] = right;

        _rev_thread[right] = old_rev_thread;

      }

      // Update _rev_thread using the new _thread values

      for (int i = 0; i != int(_dirty_revs.size()); ++i) {

        u = _dirty_revs[i];

        _rev_thread[_thread[u]] = u;

      }

      // Update _pred, _forward, _last_succ and _succ_num for the

      // stem nodes from u_out to u_in

      int tmp_sc = 0, tmp_ls = _last_succ[u_out];

      u = u_out;

      while (u != u_in) {

        w = _parent[u];

        _pred[u] = _pred[w];

        _forward[u] = !_forward[w];

        tmp_sc += _succ_num[u] - _succ_num[w];

        _succ_num[u] = tmp_sc;

        _last_succ[w] = tmp_ls;

        u = w;

      }

      _pred[u_in] = in_arc;

      _forward[u_in] = (u_in == _source[in_arc]);

      _succ_num[u_in] = old_succ_num;

      // Set limits for updating _last_succ form v_in and v_out

      // towards the root

      int up_limit_in = -1;

      int up_limit_out = -1;

      if (_last_succ[join] == v_in) {

        up_limit_out = join;

      } else {

        up_limit_in = join;

      }

      // Update _last_succ from v_in towards the root

      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;

           u = _parent[u]) {

        _last_succ[u] = _last_succ[u_out];

      }

      // Update _last_succ from v_out towards the root

      if (join != old_rev_thread && v_in != old_rev_thread) {

        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;

             u = _parent[u]) {

          _last_succ[u] = old_rev_thread;

        }

      } else {

        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;

             u = _parent[u]) {

          _last_succ[u] = _last_succ[u_out];

        }

      }

      // Update _succ_num from v_in to join

      for (u = v_in; u != join; u = _parent[u]) {

        _succ_num[u] += old_succ_num;

      }

      // Update _succ_num from v_out to join

      for (u = v_out; u != join; u = _parent[u]) {

        _succ_num[u] -= old_succ_num;

      }

    }

    // Update potentials

    void updatePotential() {

      Cost sigma = _forward[u_in] ?

        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :

        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];

      // Update potentials in the subtree, which has been moved

      int end = _thread[_last_succ[u_in]];

      for (int u = u_in; u != end; u = _thread[u]) {

        _pi[u] += sigma;

      }

    }

    // Heuristic initial pivots

    bool initialPivots() {

      Value curr, total = 0;

      std::vector<Node> supply_nodes, demand_nodes;

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

        curr = _supply[_node_id[u]];

        if (curr > 0) {

          total += curr;

          supply_nodes.push_back(u);

        }

        else if (curr < 0) {

          demand_nodes.push_back(u);

        }

      }

      if (_sum_supply > 0) total -= _sum_supply;

      if (total <= 0) return true;

      IntVector arc_vector;

      if (_sum_supply >= 0) {

        if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {

          // Perform a reverse graph search from the sink to the source

          typename GR::template NodeMap<bool> reached(_graph, false);

          Node s = supply_nodes[0], t = demand_nodes[0];

          std::vector<Node> stack;

          reached[t] = true;

          stack.push_back(t);

          while (!stack.empty()) {

            Node u, v = stack.back();

            stack.pop_back();

            if (v == s) break;

            for (InArcIt a(_graph, v); a != INVALID; ++a) {

              if (reached[u = _graph.source(a)]) continue;

              int j = _arc_id[a];

              if (_cap[j] >= total) {

                arc_vector.push_back(j);

                reached[u] = true;

                stack.push_back(u);

              }

            }

          }

        } else {

          // Find the min. cost incomming arc for each demand node

          for (int i = 0; i != int(demand_nodes.size()); ++i) {

            Node v = demand_nodes[i];

            Cost c, min_cost = std::numeric_limits<Cost>::max();

            Arc min_arc = INVALID;

            for (InArcIt a(_graph, v); a != INVALID; ++a) {

              c = _cost[_arc_id[a]];

              if (c < min_cost) {

                min_cost = c;

                min_arc = a;

              }

            }

            if (min_arc != INVALID) {

              arc_vector.push_back(_arc_id[min_arc]);

            }

          }

        }

      } else {

        // Find the min. cost outgoing arc for each supply node

        for (int i = 0; i != int(supply_nodes.size()); ++i) {

          Node u = supply_nodes[i];

          Cost c, min_cost = std::numeric_limits<Cost>::max();

          Arc min_arc = INVALID;

          for (OutArcIt a(_graph, u); a != INVALID; ++a) {

            c = _cost[_arc_id[a]];

            if (c < min_cost) {

              min_cost = c;

              min_arc = a;

            }

          }

          if (min_arc != INVALID) {

            arc_vector.push_back(_arc_id[min_arc]);

          }

        }

      }

      // Perform heuristic initial pivots

      for (int i = 0; i != int(arc_vector.size()); ++i) {

        in_arc = arc_vector[i];

        if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -

            _pi[_target[in_arc]]) >= 0) continue;

        findJoinNode();

        bool change = findLeavingArc();

        if (delta >= MAX) return false;

        changeFlow(change);

        if (change) {

          updateTreeStructure();

          updatePotential();

        }

      }

      return true;

    }

    // Execute the algorithm

    ProblemType start(PivotRule pivot_rule) {

      // Select the pivot rule implementation

      switch (pivot_rule) {

        case FIRST_ELIGIBLE:

          return start<FirstEligiblePivotRule>();

        case BEST_ELIGIBLE:

          return start<BestEligiblePivotRule>();

        case BLOCK_SEARCH:

          return start<BlockSearchPivotRule>();

        case CANDIDATE_LIST:

          return start<CandidateListPivotRule>();

        case ALTERING_LIST:

          return start<AlteringListPivotRule>();

      }

      return INFEASIBLE; // avoid warning

    }

    template <typename PivotRuleImpl>

    ProblemType start() {

      PivotRuleImpl pivot(*this);

      // Perform heuristic initial pivots

      if (!initialPivots()) return UNBOUNDED;

      // Execute the Network Simplex algorithm

      while (pivot.findEnteringArc()) {

        findJoinNode();

        bool change = findLeavingArc();

        if (delta >= MAX) return UNBOUNDED;

        changeFlow(change);

        if (change) {

          updateTreeStructure();

          updatePotential();

        }

      }

      // Check feasibility

      for (int e = _search_arc_num; e != _all_arc_num; ++e) {

        if (_flow[e] != 0) return INFEASIBLE;

      }

      // Transform the solution and the supply map to the original form

      if (_have_lower) {

        for (int i = 0; i != _arc_num; ++i) {

          Value c = _lower[i];

          if (c != 0) {

            _flow[i] += c;

            _supply[_source[i]] += c;

            _supply[_target[i]] -= c;

          }

        }

      }

      // Shift potentials to meet the requirements of the GEQ/LEQ type

      // optimality conditions

      if (_sum_supply == 0) {

        if (_stype == GEQ) {

          for (int i = 0; i != _node_num; ++i) {

            if (_pi[i] > max_pot) max_pot = _pi[i];

          }

          if (max_pot > 0) {

            for (int i = 0; i != _node_num; ++i)

              _pi[i] -= max_pot;

          }

        } else {

          Cost min_pot = std::numeric_limits<Cost>::max();

          for (int i = 0; i != _node_num; ++i) {

            if (_pi[i] < min_pot) min_pot = _pi[i];

          }

          if (min_pot < 0) {

            for (int i = 0; i != _node_num; ++i)

              _pi[i] -= min_pot;

          }

        }

      }

      return OPTIMAL;

    }

  }; //class NetworkSimplex

  ///@}

} //namespace lemon

#endif //LEMON_NETWORK_SIMPLEX_H