LEMON/LEMON-official Changeset - r878:4b1b378823dc

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Changeset - r878:4b1b378823dc

« 877:fe80a8

879:25804e »

r878:4b1b378823dc

Thu, 12 Nov 2009 23:49:05

[Not Reviewed]

0 comments (0 inline)

kpeter (Peter Kovacs)
kpeter@inf.elte.hu

Small doc improvements + unifications in MCF classes (#180)

0 3 0

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3 files changed with 32 insertions and 32 deletions:

lemon/capacity_scaling.h

lemon/cost_scaling.h

lemon/network_simplex.h

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lemon/capacity_scaling.h

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 		/* -*- 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_CAPACITY_SCALING_H
 		#define LEMON_CAPACITY_SCALING_H
 		/// \ingroup min_cost_flow_algs
 		///
 		/// \file
 		/// \brief Capacity Scaling algorithm for finding a minimum cost flow.
 		#include <vector>
 		#include <limits>
 		#include <lemon/core.h>
 		#include <lemon/bin_heap.h>
 		namespace lemon {
 		  /// \brief Default traits class of CapacityScaling algorithm.
 		  ///
 		  /// Default traits class of CapacityScaling algorithm.
 		  /// \tparam GR Digraph type.
-		  /// \tparam V The value type used for flow amounts, capacity bounds
+		  /// \tparam V The number type used for flow amounts, capacity bounds
 		  /// and supply values. By default it is \c int.
-		  /// \tparam C The value type used for costs and potentials.
+		  /// \tparam C The number type used for costs and potentials.
 		  /// By default it is the same as \c V.
 		  template <typename GR, typename V = int, typename C = V>
 		  struct CapacityScalingDefaultTraits
+		  {
 		    /// The type of the digraph
 		    typedef GR Digraph;
 		    /// The type of the flow amounts, capacity bounds and supply values
 		    typedef V Value;
 		    /// The type of the arc costs
 		    typedef C Cost;
 		    /// \brief The type of the heap used for internal Dijkstra computations.
 		    ///
 		    /// The type of the heap used for internal Dijkstra computations.
 		    /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
 		    /// its priority type must be \c Cost and its cross reference type
 		    /// must be \ref RangeMap "RangeMap<int>".
 		    typedef BinHeap<Cost, RangeMap<int> > Heap;
 		  };
 		  /// \addtogroup min_cost_flow_algs
 		  /// @{
 		  /// \brief Implementation of the Capacity Scaling algorithm for
 		  /// finding a \ref min_cost_flow "minimum cost flow".
 		  ///
 		  /// \ref CapacityScaling implements the capacity scaling version
 		  /// of the successive shortest path algorithm for finding a
 		  /// \ref min_cost_flow "minimum cost flow". It is an efficient dual
 		  /// solution method.
 		  ///
 		  /// Most of the parameters of the problem (except for the digraph)
 		  /// can be given using separate functions, and the algorithm can be
 		  /// executed using the \ref run() function. If some parameters are not
 		  /// specified, then default values will be used.
 		  ///
 		  /// \tparam GR The digraph type the algorithm runs on.
-		  /// \tparam V The value type used for flow amounts, capacity bounds
+		  /// \tparam V The number type used for flow amounts, capacity bounds
 		  /// and supply values in the algorithm. By default it is \c int.
-		  /// \tparam C The value type used for costs and potentials in the
+		  /// \tparam C The number type used for costs and potentials in the
 		  /// algorithm. By default it is the same as \c V.
 		  ///
-		  /// \warning Both value types must be signed and all input data must
+		  /// \warning Both number types must be signed and all input data must
 		  /// be integer.
 		  /// \warning This algorithm does not support negative costs for such
 		  /// arcs that have infinite upper bound.
 		#ifdef DOXYGEN
 		  template <typename GR, typename V, typename C, typename TR>
 		#else
 		  template < typename GR, typename V = int, typename C = V,
 		             typename TR = CapacityScalingDefaultTraits<GR, V, C> >
 		#endif
 		  class CapacityScaling
+		  {
 		  public:
 		    /// The type of the digraph
 		    typedef typename TR::Digraph Digraph;
 		    /// The type of the flow amounts, capacity bounds and supply values
 		    typedef typename TR::Value Value;
 		    /// The type of the arc costs
 		    typedef typename TR::Cost Cost;
 		    /// The type of the heap used for internal Dijkstra computations
 		    typedef typename TR::Heap Heap;
 		    /// The \ref CapacityScalingDefaultTraits "traits class" of the algorithm
 		    typedef TR Traits;
 		  public:
 		    /// \brief Problem type constants for the \c run() function.
 		    ///
 		    /// Enum type containing the problem type constants that can be
 		    /// returned by the \ref run() function of the algorithm.
 		    enum ProblemType {
 		      /// The problem has no feasible solution (flow).
 		      INFEASIBLE,
 		      /// 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).
 		      OPTIMAL,
 		      /// The digraph contains an arc of negative cost and infinite
 		      /// upper bound. It means that the objective function is unbounded
 		      /// on that arc, however note that it could actually be bounded
+		      /// on that arc, however, note that it could actually be bounded
 		      /// over the feasible flows, but this algroithm cannot handle
 		      /// these cases.
 		      UNBOUNDED
 		    };
 		  private:
 		    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
 		    typedef std::vector<int> IntVector;
 		    typedef std::vector<char> BoolVector;
 		    typedef std::vector<Value> ValueVector;
 		    typedef std::vector<Cost> CostVector;
 		  private:
 		    // Data related to the underlying digraph
 		    const GR &_graph;
 		    int _node_num;
 		    int _arc_num;
 		    int _res_arc_num;
 		    int _root;
 		    // Parameters of the problem
 		    bool _have_lower;
 		    Value _sum_supply;
 		    // Data structures for storing the digraph
 		    IntNodeMap _node_id;
 		    IntArcMap _arc_idf;
 		    IntArcMap _arc_idb;
 		    IntVector _first_out;
 		    BoolVector _forward;
 		    IntVector _source;
 		    IntVector _target;
 		    IntVector _reverse;
 		    // Node and arc data
 		    ValueVector _lower;
 		    ValueVector _upper;
 		    CostVector _cost;
 		    ValueVector _supply;
 		    ValueVector _res_cap;
 		    CostVector _pi;
 		    ValueVector _excess;
 		    IntVector _excess_nodes;
 		    IntVector _deficit_nodes;
 		    Value _delta;
 		    int _factor;
 		    IntVector _pred;
 		  public:
 		    /// \brief Constant for infinite upper bounds (capacities).
 		    ///
 		    /// Constant for infinite upper bounds (capacities).
 		    /// It is \c std::numeric_limits<Value>::infinity() if available,
 		    /// \c std::numeric_limits<Value>::max() otherwise.
 		    const Value INF;
 		  private:
 		    // Special implementation of the Dijkstra algorithm for finding
 		    // shortest paths in the residual network of the digraph with
 		    // respect to the reduced arc costs and modifying the node
 		    // potentials according to the found distance labels.
 		    class ResidualDijkstra
+		    {
 		    private:
 		      int _node_num;
 		      bool _geq;
 		      const IntVector &_first_out;
 		      const IntVector &_target;
 		      const CostVector &_cost;
 		      const ValueVector &_res_cap;
 		      const ValueVector &_excess;
 		      CostVector &_pi;
 		      IntVector &_pred;
 		      IntVector _proc_nodes;
 		      CostVector _dist;
 		    public:
 		      ResidualDijkstra(CapacityScaling& cs) :
 		        _node_num(cs._node_num), _geq(cs._sum_supply < 0),
 		        _first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
 		        _res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
 		        _pred(cs._pred), _dist(cs._node_num)
 		      {}
 		      int run(int s, Value delta = 1) {
 		        RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
 		        Heap heap(heap_cross_ref);
 		        heap.push(s, 0);
 		        _pred[s] = -1;
 		        _proc_nodes.clear();
 		        // Process nodes
 		        while (!heap.empty() && _excess[heap.top()] > -delta) {
 		          int u = heap.top(), v;
 		          Cost d = heap.prio() + _pi[u], dn;
 		          _dist[u] = heap.prio();
 		          _proc_nodes.push_back(u);
 		          heap.pop();
 		          // Traverse outgoing residual arcs
 		          int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
 		          for (int a = _first_out[u]; a != last_out; ++a) {
 		            if (_res_cap[a] < delta) continue;
 		            v = _target[a];
 		            switch (heap.state(v)) {
 		              case Heap::PRE_HEAP:
 		                heap.push(v, d + _cost[a] - _pi[v]);
 		                _pred[v] = a;
 		                break;
 		              case Heap::IN_HEAP:
 		                dn = d + _cost[a] - _pi[v];
 		                if (dn < heap[v]) {
 		                  heap.decrease(v, dn);
 		                  _pred[v] = a;
+		                }
 		                break;
 		              case Heap::POST_HEAP:
 		                break;
+		            }
+		          }
+		        }
 		        if (heap.empty()) return -1;
 		        // Update potentials of processed nodes
 		        int t = heap.top();
 		        Cost dt = heap.prio();
 		        for (int i = 0; i < int(_proc_nodes.size()); ++i) {
 		          _pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
+		        }
 		        return t;
+		      }
 		    }; //class ResidualDijkstra
 		  public:
 		    /// \name Named Template Parameters
 		    /// @{
 		    template <typename T>
 		    struct SetHeapTraits : public Traits {
 		      typedef T Heap;
 		    };
 		    /// \brief \ref named-templ-param "Named parameter" for setting
 		    /// \c Heap type.
 		    ///
 		    /// \ref named-templ-param "Named parameter" for setting \c Heap
 		    /// type, which is used for internal Dijkstra computations.
 		    /// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
 		    /// its priority type must be \c Cost and its cross reference type
 		    /// must be \ref RangeMap "RangeMap<int>".
 		    template <typename T>
 		    struct SetHeap
 		      : public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
 		      typedef  CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
 		    };
 		    /// @}
 		  public:
 		    /// \brief Constructor.
 		    ///
 		    /// The constructor of the class.
 		    ///
 		    /// \param graph The digraph the algorithm runs on.
 		    CapacityScaling(const GR& graph) :
 		      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
 		      INF(std::numeric_limits<Value>::has_infinity ?
 		          std::numeric_limits<Value>::infinity() :
 		          std::numeric_limits<Value>::max())
+		    {
-		      // Check the value types
+		      // Check the number types
 		      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
 		        "The flow type of CapacityScaling must be signed");
 		      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
 		        "The cost type of CapacityScaling must be signed");
 		      // Resize vectors
 		      _node_num = countNodes(_graph);
 		      _arc_num = countArcs(_graph);
 		      _res_arc_num = 2 * (_arc_num + _node_num);
 		      _root = _node_num;
 		      ++_node_num;
 		      _first_out.resize(_node_num + 1);
 		      _forward.resize(_res_arc_num);
 		      _source.resize(_res_arc_num);
 		      _target.resize(_res_arc_num);
 		      _reverse.resize(_res_arc_num);
 		      _lower.resize(_res_arc_num);
 		      _upper.resize(_res_arc_num);
 		      _cost.resize(_res_arc_num);
 		      _supply.resize(_node_num);
 		      _res_cap.resize(_res_arc_num);
 		      _pi.resize(_node_num);
 		      _excess.resize(_node_num);
 		      _pred.resize(_node_num);
 		      // Copy the graph
 		      int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
 		      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
 		        _node_id[n] = i;
+		      }
 		      i = 0;
 		      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
 		        _first_out[i] = j;
 		        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
 		          _arc_idf[a] = j;
 		          _forward[j] = true;
 		          _source[j] = i;
 		          _target[j] = _node_id[_graph.runningNode(a)];
+		        }
 		        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
 		          _arc_idb[a] = j;
 		          _forward[j] = false;
 		          _source[j] = i;
 		          _target[j] = _node_id[_graph.runningNode(a)];
+		        }
 		        _forward[j] = false;
 		        _source[j] = i;
 		        _target[j] = _root;
 		        _reverse[j] = k;
 		        _forward[k] = true;
 		        _source[k] = _root;
 		        _target[k] = i;
 		        _reverse[k] = j;
 		        ++j; ++k;
+		      }
 		      _first_out[i] = j;
 		      _first_out[_node_num] = k;
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        int fi = _arc_idf[a];
 		        int bi = _arc_idb[a];
 		        _reverse[fi] = bi;
 		        _reverse[bi] = fi;
+		      }
 		      // Reset parameters
 		      reset();
+		    }
 		    /// \name Parameters
 		    /// The parameters of the algorithm can be specified using these
 		    /// functions.
 		    /// @{
 		    /// \brief Set the lower bounds on the arcs.
 		    ///
 		    /// This function sets the lower bounds on the arcs.
 		    /// If it is not used before calling \ref run(), the lower bounds
 		    /// will be set to zero on all arcs.
 		    ///
 		    /// \param map An arc map storing the lower bounds.
 		    /// Its \c Value type must be convertible to the \c Value type
 		    /// of the algorithm.
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    template <typename LowerMap>
 		    CapacityScaling& lowerMap(const LowerMap& map) {
 		      _have_lower = true;
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        _lower[_arc_idf[a]] = map[a];
 		        _lower[_arc_idb[a]] = map[a];
+		      }
 		      return *this;
+		    }
 		    /// \brief Set the upper bounds (capacities) on the arcs.
 		    ///
 		    /// This function sets the upper bounds (capacities) on the arcs.
 		    /// If it is not used before calling \ref run(), the upper bounds
 		    /// will be set to \ref INF on all arcs (i.e. the flow value will be
-		    /// unbounded from above on each arc).
 		    /// unbounded from above).
 		    ///
 		    /// \param map An arc map storing the upper bounds.
 		    /// Its \c Value type must be convertible to the \c Value type
 		    /// of the algorithm.
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    template<typename UpperMap>
 		    CapacityScaling& upperMap(const UpperMap& map) {
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        _upper[_arc_idf[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>
 		    CapacityScaling& costMap(const CostMap& map) {
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        _cost[_arc_idf[a]] =  map[a];
 		        _cost[_arc_idb[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>
 		    CapacityScaling& 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>
 		    CapacityScaling& 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;
+		    }
 		    /// @}
 		    /// \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().
 		    /// For example,
 		    /// \code
 		    ///   CapacityScaling<ListDigraph> cs(graph);
 		    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
 		    ///     .supplyMap(sup).run();
 		    /// \endcode
 		    ///
 		    /// This function can be called more than once. All the parameters
 		    /// that have been given are kept for the next call, unless
 		    /// \ref reset() is called, thus only the modified parameters
 		    /// have to be set again. See \ref reset() for examples.
 		    /// However the underlying digraph must not be modified after this
+		    /// However, the underlying digraph must not be modified after this
 		    /// class have been constructed, since it copies and extends the graph.
 		    ///
 		    /// \param factor The capacity scaling factor. It must be larger than
 		    /// one to use scaling. If it is less or equal to one, then scaling
 		    /// will be disabled.
 		    ///
 		    /// \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 digraph contains an arc of negative cost
 		    /// and infinite upper bound. It means that the objective function
 		    /// is unbounded on that arc, however note that it could actually be
+		    /// is unbounded on that arc, however, note that it could actually be
 		    /// bounded over the feasible flows, but this algroithm cannot handle
 		    /// these cases.
 		    ///
 		    /// \see ProblemType
 		    ProblemType run(int factor = 4) {
 		      _factor = factor;
 		      ProblemType pt = init();
 		      if (pt != OPTIMAL) return pt;
 		      return start();
+		    }
 		    /// \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().
 		    ///
 		    /// It is useful for multiple run() calls. If this function is not
 		    /// used, all the parameters given before are kept for the next
 		    /// \ref run() call.
 		    /// However, the underlying digraph must not be modified after this
 		    /// class have been constructed, since it copies and extends the graph.
 		    ///
 		    /// For example,
 		    /// \code
 		    ///   CapacityScaling<ListDigraph> cs(graph);
 		    ///
 		    ///   // First run
 		    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
 		    ///     .supplyMap(sup).run();
 		    ///
 		    ///   // Run again with modified cost map (reset() is not called,
 		    ///   // so only the cost map have to be set again)
 		    ///   cost[e] += 100;
 		    ///   cs.costMap(cost).run();
 		    ///
 		    ///   // Run again from scratch using reset()
 		    ///   // (the lower bounds will be set to zero on all arcs)
 		    ///   cs.reset();
 		    ///   cs.upperMap(capacity).costMap(cost)
 		    ///     .supplyMap(sup).run();
 		    /// \endcode
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    CapacityScaling& reset() {
 		      for (int i = 0; i != _node_num; ++i) {
 		        _supply[i] = 0;
+		      }
 		      for (int j = 0; j != _res_arc_num; ++j) {
 		        _lower[j] = 0;
 		        _upper[j] = INF;
 		        _cost[j] = _forward[j] ? 1 : -1;
+		      }
 		      _have_lower = false;
 		      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
 		    ///   cs.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_idb[a];
 		        c += static_cast<Number>(_res_cap[i]) *
 		             (-static_cast<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 _res_cap[_arc_idb[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, _res_cap[_arc_idb[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 the algorithm
 		    ProblemType init() {
 		      if (_node_num == 0) return INFEASIBLE;
 		      // Check the sum of supply values
 		      _sum_supply = 0;
 		      for (int i = 0; i != _root; ++i) {
 		        _sum_supply += _supply[i];
+		      }
 		      if (_sum_supply > 0) return INFEASIBLE;
 		      // Initialize vectors
 		      for (int i = 0; i != _root; ++i) {
 		        _pi[i] = 0;
 		        _excess[i] = _supply[i];
+		      }
 		      // Remove non-zero lower bounds
 		      const Value MAX = std::numeric_limits<Value>::max();
 		      int last_out;
 		      if (_have_lower) {
 		        for (int i = 0; i != _root; ++i) {
 		          last_out = _first_out[i+1];
 		          for (int j = _first_out[i]; j != last_out; ++j) {
 		            if (_forward[j]) {
 		              Value c = _lower[j];
 		              if (c >= 0) {
 		                _res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
 		              } else {
 		                _res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
+		              }
 		              _excess[i] -= c;
 		              _excess[_target[j]] += c;
 		            } else {
 		              _res_cap[j] = 0;
+		            }
+		          }
+		        }
 		      } else {
 		        for (int j = 0; j != _res_arc_num; ++j) {
 		          _res_cap[j] = _forward[j] ? _upper[j] : 0;
+		        }
+		      }
 		      // Handle negative costs
 		      for (int i = 0; i != _root; ++i) {
 		        last_out = _first_out[i+1] - 1;
 		        for (int j = _first_out[i]; j != last_out; ++j) {
 		          Value rc = _res_cap[j];
 		          if (_cost[j] < 0 && rc > 0) {
 		            if (rc >= MAX) return UNBOUNDED;
 		            _excess[i] -= rc;
 		            _excess[_target[j]] += rc;
 		            _res_cap[j] = 0;
 		            _res_cap[_reverse[j]] += rc;
+		          }
+		        }
+		      }
 		      // Handle GEQ supply type
 		      if (_sum_supply < 0) {
 		        _pi[_root] = 0;
 		        _excess[_root] = -_sum_supply;
 		        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
 		          int ra = _reverse[a];
 		          _res_cap[a] = -_sum_supply + 1;
 		          _res_cap[ra] = 0;
 		          _cost[a] = 0;
 		          _cost[ra] = 0;
+		        }
 		      } else {
 		        _pi[_root] = 0;
 		        _excess[_root] = 0;
 		        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
 		          int ra = _reverse[a];
 		          _res_cap[a] = 1;
 		          _res_cap[ra] = 0;
 		          _cost[a] = 0;
 		          _cost[ra] = 0;
+		        }
+		      }
 		      // Initialize delta value
 		      if (_factor > 1) {
 		        // With scaling
 		        Value max_sup = 0, max_dem = 0;
 		        for (int i = 0; i != _node_num; ++i) {
 		          Value ex = _excess[i];
 		          if ( ex > max_sup) max_sup =  ex;
 		          if (-ex > max_dem) max_dem = -ex;
+		        }
 		        Value max_cap = 0;
 		        for (int j = 0; j != _res_arc_num; ++j) {
 		          if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
+		        }
 		        max_sup = std::min(std::min(max_sup, max_dem), max_cap);
 		        for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
 		      } else {
 		        // Without scaling
 		        _delta = 1;
+		      }
 		      return OPTIMAL;
+		    }
 		    ProblemType start() {
 		      // Execute the algorithm
 		      ProblemType pt;
 		      if (_delta > 1)
 		        pt = startWithScaling();
 		      else
 		        pt = startWithoutScaling();
 		      // Handle non-zero lower bounds
 		      if (_have_lower) {
 		        int limit = _first_out[_root];
 		        for (int j = 0; j != limit; ++j) {
 		          if (!_forward[j]) _res_cap[j] += _lower[j];
+		        }
+		      }
 		      // Shift potentials if necessary
 		      Cost pr = _pi[_root];
 		      if (_sum_supply < 0 || pr > 0) {
 		        for (int i = 0; i != _node_num; ++i) {
 		          _pi[i] -= pr;
+		        }
+		      }
 		      return pt;
+		    }
 		    // Execute the capacity scaling algorithm
 		    ProblemType startWithScaling() {
 		      // Perform capacity scaling phases
 		      int s, t;
 		      ResidualDijkstra _dijkstra(*this);
 		      while (true) {
 		        // Saturate all arcs not satisfying the optimality condition
 		        int last_out;
 		        for (int u = 0; u != _node_num; ++u) {
 		          last_out = _sum_supply < 0 ?
 		            _first_out[u+1] : _first_out[u+1] - 1;
 		          for (int a = _first_out[u]; a != last_out; ++a) {
 		            int v = _target[a];
 		            Cost c = _cost[a] + _pi[u] - _pi[v];
 		            Value rc = _res_cap[a];
 		            if (c < 0 && rc >= _delta) {
 		              _excess[u] -= rc;
 		              _excess[v] += rc;
 		              _res_cap[a] = 0;
 		              _res_cap[_reverse[a]] += rc;
+		            }
+		          }
+		        }
 		        // Find excess nodes and deficit nodes
 		        _excess_nodes.clear();
 		        _deficit_nodes.clear();
 		        for (int u = 0; u != _node_num; ++u) {
 		          Value ex = _excess[u];
 		          if (ex >=  _delta) _excess_nodes.push_back(u);
 		          if (ex <= -_delta) _deficit_nodes.push_back(u);
+		        }
 		        int next_node = 0, next_def_node = 0;
 		        // Find augmenting shortest paths
 		        while (next_node < int(_excess_nodes.size())) {
 		          // Check deficit nodes
 		          if (_delta > 1) {
 		            bool delta_deficit = false;
 		            for ( ; next_def_node < int(_deficit_nodes.size());
 		                    ++next_def_node ) {
 		              if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
 		                delta_deficit = true;
 		                break;
+		              }
+		            }
 		            if (!delta_deficit) break;
+		          }
 		          // Run Dijkstra in the residual network
 		          s = _excess_nodes[next_node];
 		          if ((t = _dijkstra.run(s, _delta)) == -1) {
 		            if (_delta > 1) {
 		              ++next_node;
 		              continue;
+		            }
 		            return INFEASIBLE;
+		          }
 		          // Augment along a shortest path from s to t
 		          Value d = std::min(_excess[s], -_excess[t]);
 		          int u = t;
 		          int a;
 		          if (d > _delta) {
 		            while ((a = _pred[u]) != -1) {
 		              if (_res_cap[a] < d) d = _res_cap[a];
 		              u = _source[a];
+		            }
+		          }
 		          u = t;
 		          while ((a = _pred[u]) != -1) {
 		            _res_cap[a] -= d;
 		            _res_cap[_reverse[a]] += d;
 		            u = _source[a];
+		          }
 		          _excess[s] -= d;
 		          _excess[t] += d;
 		          if (_excess[s] < _delta) ++next_node;
+		        }
 		        if (_delta == 1) break;
 		        _delta = _delta <= _factor ? 1 : _delta / _factor;
+		      }
 		      return OPTIMAL;
+		    }
 		    // Execute the successive shortest path algorithm
 		    ProblemType startWithoutScaling() {
 		      // Find excess nodes
 		      _excess_nodes.clear();
 		      for (int i = 0; i != _node_num; ++i) {
 		        if (_excess[i] > 0) _excess_nodes.push_back(i);
+		      }
 		      if (_excess_nodes.size() == 0) return OPTIMAL;
 		      int next_node = 0;
 		      // Find shortest paths
 		      int s, t;
 		      ResidualDijkstra _dijkstra(*this);
 		      while ( _excess[_excess_nodes[next_node]] > 0 ||
 		              ++next_node < int(_excess_nodes.size()) )
+		      {
 		        // Run Dijkstra in the residual network
 		        s = _excess_nodes[next_node];
 		        if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
 		        // Augment along a shortest path from s to t
 		        Value d = std::min(_excess[s], -_excess[t]);
 		        int u = t;
 		        int a;
 		        if (d > 1) {
 		          while ((a = _pred[u]) != -1) {
 		            if (_res_cap[a] < d) d = _res_cap[a];
 		            u = _source[a];
+		          }
+		        }
 		        u = t;
 		        while ((a = _pred[u]) != -1) {
 		          _res_cap[a] -= d;
 		          _res_cap[_reverse[a]] += d;
 		          u = _source[a];
+		        }
 		        _excess[s] -= d;
 		        _excess[t] += d;
+		      }
 		      return OPTIMAL;
+		    }
 		  }; //class CapacityScaling
 		  ///@}
 		} //namespace lemon
 		#endif //LEMON_CAPACITY_SCALING_H

lemon/cost_scaling.h

Show inline comments

 		/* -*- 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_COST_SCALING_H
 		#define LEMON_COST_SCALING_H
 		/// \ingroup min_cost_flow_algs
 		/// \file
 		/// \brief Cost scaling algorithm for finding a minimum cost flow.
 		#include <vector>
 		#include <deque>
 		#include <limits>
 		#include <lemon/core.h>
 		#include <lemon/maps.h>
 		#include <lemon/math.h>
 		#include <lemon/static_graph.h>
 		#include <lemon/circulation.h>
 		#include <lemon/bellman_ford.h>
 		namespace lemon {
 		  /// \brief Default traits class of CostScaling algorithm.
 		  ///
 		  /// Default traits class of CostScaling algorithm.
 		  /// \tparam GR Digraph type.
-		  /// \tparam V The value type used for flow amounts, capacity bounds
+		  /// \tparam V The number type used for flow amounts, capacity bounds
 		  /// and supply values. By default it is \c int.
-		  /// \tparam C The value type used for costs and potentials.
+		  /// \tparam C The number type used for costs and potentials.
 		  /// By default it is the same as \c V.
 		#ifdef DOXYGEN
 		  template <typename GR, typename V = int, typename C = V>
 		#else
 		  template < typename GR, typename V = int, typename C = V,
 		             bool integer = std::numeric_limits<C>::is_integer >
 		#endif
 		  struct CostScalingDefaultTraits
+		  {
 		    /// The type of the digraph
 		    typedef GR Digraph;
 		    /// The type of the flow amounts, capacity bounds and supply values
 		    typedef V Value;
 		    /// The type of the arc costs
 		    typedef C Cost;
 		    /// \brief The large cost type used for internal computations
 		    ///
 		    /// The large cost type used for internal computations.
 		    /// It is \c long \c long if the \c Cost type is integer,
 		    /// otherwise it is \c double.
 		    /// \c Cost must be convertible to \c LargeCost.
 		    typedef double LargeCost;
 		  };
 		  // Default traits class for integer cost types
 		  template <typename GR, typename V, typename C>
 		  struct CostScalingDefaultTraits<GR, V, C, true>
+		  {
 		    typedef GR Digraph;
 		    typedef V Value;
 		    typedef C Cost;
 		#ifdef LEMON_HAVE_LONG_LONG
 		    typedef long long LargeCost;
 		#else
 		    typedef long LargeCost;
 		#endif
 		  };
 		  /// \addtogroup min_cost_flow_algs
 		  /// @{
 		  /// \brief Implementation of the Cost Scaling algorithm for
 		  /// finding a \ref min_cost_flow "minimum cost flow".
 		  ///
 		  /// \ref CostScaling implements a cost scaling algorithm that performs
 		  /// push/augment and relabel operations for finding a minimum cost
 		  /// flow. It is an efficient primal-dual solution method, which
 		  /// can be viewed as the generalization of the \ref Preflow
 		  /// "preflow push-relabel" algorithm for the maximum flow problem.
 		  ///
 		  /// Most of the parameters of the problem (except for the digraph)
 		  /// can be given using separate functions, and the algorithm can be
 		  /// executed using the \ref run() function. If some parameters are not
 		  /// specified, then default values will be used.
 		  ///
 		  /// \tparam GR The digraph type the algorithm runs on.
-		  /// \tparam V The value type used for flow amounts, capacity bounds
+		  /// \tparam V The number type used for flow amounts, capacity bounds
 		  /// and supply values in the algorithm. By default it is \c int.
-		  /// \tparam C The value type used for costs and potentials in the
+		  /// \tparam C The number type used for costs and potentials in the
 		  /// algorithm. By default it is the same as \c V.
 		  ///
-		  /// \warning Both value types must be signed and all input data must
+		  /// \warning Both number types must be signed and all input data must
 		  /// be integer.
 		  /// \warning This algorithm does not support negative costs for such
 		  /// arcs that have infinite upper bound.
 		  ///
 		  /// \note %CostScaling provides three different internal methods,
 		  /// from which the most efficient one is used by default.
 		  /// For more information, see \ref Method.
 		#ifdef DOXYGEN
 		  template <typename GR, typename V, typename C, typename TR>
 		#else
 		  template < typename GR, typename V = int, typename C = V,
 		             typename TR = CostScalingDefaultTraits<GR, V, C> >
 		#endif
 		  class CostScaling
+		  {
 		  public:
 		    /// The type of the digraph
 		    typedef typename TR::Digraph Digraph;
 		    /// The type of the flow amounts, capacity bounds and supply values
 		    typedef typename TR::Value Value;
 		    /// The type of the arc costs
 		    typedef typename TR::Cost Cost;
 		    /// \brief The large cost type
 		    ///
 		    /// The large cost type used for internal computations.
 		    /// Using the \ref CostScalingDefaultTraits "default traits class",
 		    /// it is \c long \c long if the \c Cost type is integer,
 		    /// otherwise it is \c double.
 		    typedef typename TR::LargeCost LargeCost;
 		    /// The \ref CostScalingDefaultTraits "traits class" of the algorithm
 		    typedef TR Traits;
 		  public:
 		    /// \brief Problem type constants for the \c run() function.
 		    ///
 		    /// Enum type containing the problem type constants that can be
 		    /// returned by the \ref run() function of the algorithm.
 		    enum ProblemType {
 		      /// The problem has no feasible solution (flow).
 		      INFEASIBLE,
 		      /// 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).
 		      OPTIMAL,
 		      /// The digraph contains an arc of negative cost and infinite
 		      /// upper bound. It means that the objective function is unbounded
 		      /// on that arc, however note that it could actually be bounded
+		      /// on that arc, however, note that it could actually be bounded
 		      /// over the feasible flows, but this algroithm cannot handle
 		      /// these cases.
 		      UNBOUNDED
 		    };
 		    /// \brief Constants for selecting the internal method.
 		    ///
 		    /// Enum type containing constants for selecting the internal method
 		    /// for the \ref run() function.
 		    ///
 		    /// \ref CostScaling provides three internal methods that differ mainly
 		    /// in their base operations, which are used in conjunction with the
 		    /// relabel operation.
 		    /// By default, the so called \ref PARTIAL_AUGMENT
 		    /// "Partial Augment-Relabel" method is used, which proved to be
 		    /// the most efficient and the most robust on various test inputs.
 		    /// However, the other methods can be selected using the \ref run()
 		    /// function with the proper parameter.
 		    enum Method {
 		      /// Local push operations are used, i.e. flow is moved only on one
 		      /// admissible arc at once.
 		      PUSH,
 		      /// Augment operations are used, i.e. flow is moved on admissible
 		      /// paths from a node with excess to a node with deficit.
 		      AUGMENT,
 		      /// Partial augment operations are used, i.e. flow is moved on
 		      /// admissible paths started from a node with excess, but the
 		      /// lengths of these paths are limited. This method can be viewed
 		      /// as a combined version of the previous two operations.
 		      PARTIAL_AUGMENT
 		    };
 		  private:
 		    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
 		    typedef std::vector<int> IntVector;
 		    typedef std::vector<char> BoolVector;
 		    typedef std::vector<Value> ValueVector;
 		    typedef std::vector<Cost> CostVector;
 		    typedef std::vector<LargeCost> LargeCostVector;
 		  private:
 		    template <typename KT, typename VT>
 		    class VectorMap {
 		    public:
 		      typedef KT Key;
 		      typedef VT Value;
 		      VectorMap(std::vector<Value>& v) : _v(v) {}
 		      const Value& operator[](const Key& key) const {
 		        return _v[StaticDigraph::id(key)];
+		      }
 		      Value& operator[](const Key& key) {
 		        return _v[StaticDigraph::id(key)];
+		      }
 		      void set(const Key& key, const Value& val) {
 		        _v[StaticDigraph::id(key)] = val;
+		      }
 		    private:
 		      std::vector<Value>& _v;
 		    };
 		    typedef VectorMap<StaticDigraph::Node, LargeCost> LargeCostNodeMap;
 		    typedef VectorMap<StaticDigraph::Arc, LargeCost> LargeCostArcMap;
 		  private:
 		    // Data related to the underlying digraph
 		    const GR &_graph;
 		    int _node_num;
 		    int _arc_num;
 		    int _res_node_num;
 		    int _res_arc_num;
 		    int _root;
 		    // Parameters of the problem
 		    bool _have_lower;
 		    Value _sum_supply;
 		    // Data structures for storing the digraph
 		    IntNodeMap _node_id;
 		    IntArcMap _arc_idf;
 		    IntArcMap _arc_idb;
 		    IntVector _first_out;
 		    BoolVector _forward;
 		    IntVector _source;
 		    IntVector _target;
 		    IntVector _reverse;
 		    // Node and arc data
 		    ValueVector _lower;
 		    ValueVector _upper;
 		    CostVector _scost;
 		    ValueVector _supply;
 		    ValueVector _res_cap;
 		    LargeCostVector _cost;
 		    LargeCostVector _pi;
 		    ValueVector _excess;
 		    IntVector _next_out;
 		    std::deque<int> _active_nodes;
 		    // Data for scaling
 		    LargeCost _epsilon;
 		    int _alpha;
 		    // Data for a StaticDigraph structure
 		    typedef std::pair<int, int> IntPair;
 		    StaticDigraph _sgr;
 		    std::vector<IntPair> _arc_vec;
 		    std::vector<LargeCost> _cost_vec;
 		    LargeCostArcMap _cost_map;
 		    LargeCostNodeMap _pi_map;
 		  public:
 		    /// \brief Constant for infinite upper bounds (capacities).
 		    ///
 		    /// Constant for infinite upper bounds (capacities).
 		    /// It is \c std::numeric_limits<Value>::infinity() if available,
 		    /// \c std::numeric_limits<Value>::max() otherwise.
 		    const Value INF;
 		  public:
 		    /// \name Named Template Parameters
 		    /// @{
 		    template <typename T>
 		    struct SetLargeCostTraits : public Traits {
 		      typedef T LargeCost;
 		    };
 		    /// \brief \ref named-templ-param "Named parameter" for setting
 		    /// \c LargeCost type.
 		    ///
 		    /// \ref named-templ-param "Named parameter" for setting \c LargeCost
 		    /// type, which is used for internal computations in the algorithm.
 		    /// \c Cost must be convertible to \c LargeCost.
 		    template <typename T>
 		    struct SetLargeCost
 		      : public CostScaling<GR, V, C, SetLargeCostTraits<T> > {
 		      typedef  CostScaling<GR, V, C, SetLargeCostTraits<T> > Create;
 		    };
 		    /// @}
 		  public:
 		    /// \brief Constructor.
 		    ///
 		    /// The constructor of the class.
 		    ///
 		    /// \param graph The digraph the algorithm runs on.
 		    CostScaling(const GR& graph) :
 		      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
 		      _cost_map(_cost_vec), _pi_map(_pi),
 		      INF(std::numeric_limits<Value>::has_infinity ?
 		          std::numeric_limits<Value>::infinity() :
 		          std::numeric_limits<Value>::max())
+		    {
-		      // Check the value types
+		      // Check the number types
 		      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
 		        "The flow type of CostScaling must be signed");
 		      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
 		        "The cost type of CostScaling must be signed");
 		      // Resize vectors
 		      _node_num = countNodes(_graph);
 		      _arc_num = countArcs(_graph);
 		      _res_node_num = _node_num + 1;
 		      _res_arc_num = 2 * (_arc_num + _node_num);
 		      _root = _node_num;
 		      _first_out.resize(_res_node_num + 1);
 		      _forward.resize(_res_arc_num);
 		      _source.resize(_res_arc_num);
 		      _target.resize(_res_arc_num);
 		      _reverse.resize(_res_arc_num);
 		      _lower.resize(_res_arc_num);
 		      _upper.resize(_res_arc_num);
 		      _scost.resize(_res_arc_num);
 		      _supply.resize(_res_node_num);
 		      _res_cap.resize(_res_arc_num);
 		      _cost.resize(_res_arc_num);
 		      _pi.resize(_res_node_num);
 		      _excess.resize(_res_node_num);
 		      _next_out.resize(_res_node_num);
 		      _arc_vec.reserve(_res_arc_num);
 		      _cost_vec.reserve(_res_arc_num);
 		      // Copy the graph
 		      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
 		      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
 		        _node_id[n] = i;
+		      }
 		      i = 0;
 		      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
 		        _first_out[i] = j;
 		        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
 		          _arc_idf[a] = j;
 		          _forward[j] = true;
 		          _source[j] = i;
 		          _target[j] = _node_id[_graph.runningNode(a)];
+		        }
 		        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
 		          _arc_idb[a] = j;
 		          _forward[j] = false;
 		          _source[j] = i;
 		          _target[j] = _node_id[_graph.runningNode(a)];
+		        }
 		        _forward[j] = false;
 		        _source[j] = i;
 		        _target[j] = _root;
 		        _reverse[j] = k;
 		        _forward[k] = true;
 		        _source[k] = _root;
 		        _target[k] = i;
 		        _reverse[k] = j;
 		        ++j; ++k;
+		      }
 		      _first_out[i] = j;
 		      _first_out[_res_node_num] = k;
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        int fi = _arc_idf[a];
 		        int bi = _arc_idb[a];
 		        _reverse[fi] = bi;
 		        _reverse[bi] = fi;
+		      }
 		      // Reset parameters
 		      reset();
+		    }
 		    /// \name Parameters
 		    /// The parameters of the algorithm can be specified using these
 		    /// functions.
 		    /// @{
 		    /// \brief Set the lower bounds on the arcs.
 		    ///
 		    /// This function sets the lower bounds on the arcs.
 		    /// If it is not used before calling \ref run(), the lower bounds
 		    /// will be set to zero on all arcs.
 		    ///
 		    /// \param map An arc map storing the lower bounds.
 		    /// Its \c Value type must be convertible to the \c Value type
 		    /// of the algorithm.
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    template <typename LowerMap>
 		    CostScaling& lowerMap(const LowerMap& map) {
 		      _have_lower = true;
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        _lower[_arc_idf[a]] = map[a];
 		        _lower[_arc_idb[a]] = map[a];
+		      }
 		      return *this;
+		    }
 		    /// \brief Set the upper bounds (capacities) on the arcs.
 		    ///
 		    /// This function sets the upper bounds (capacities) on the arcs.
 		    /// If it is not used before calling \ref run(), the upper bounds
 		    /// will be set to \ref INF on all arcs (i.e. the flow value will be
-		    /// unbounded from above on each arc).
 		    /// unbounded from above).
 		    ///
 		    /// \param map An arc map storing the upper bounds.
 		    /// Its \c Value type must be convertible to the \c Value type
 		    /// of the algorithm.
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    template<typename UpperMap>
 		    CostScaling& upperMap(const UpperMap& map) {
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        _upper[_arc_idf[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>
 		    CostScaling& costMap(const CostMap& map) {
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        _scost[_arc_idf[a]] =  map[a];
 		        _scost[_arc_idb[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>
 		    CostScaling& 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>
 		    CostScaling& stSupply(const Node& s, const Node& t, Value k) {
 		      for (int i = 0; i != _res_node_num; ++i) {
 		        _supply[i] = 0;
+		      }
 		      _supply[_node_id[s]] =  k;
 		      _supply[_node_id[t]] = -k;
 		      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().
 		    /// For example,
 		    /// \code
 		    ///   CostScaling<ListDigraph> cs(graph);
 		    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
 		    ///     .supplyMap(sup).run();
 		    /// \endcode
 		    ///
 		    /// This function can be called more than once. All the parameters
 		    /// that have been given are kept for the next call, unless
 		    /// \ref reset() is called, thus only the modified parameters
 		    /// have to be set again. See \ref reset() for examples.
 		    /// However, the underlying digraph must not be modified after this
 		    /// class have been constructed, since it copies and extends the graph.
 		    ///
 		    /// \param method The internal method that will be used in the
 		    /// algorithm. For more information, see \ref Method.
 		    /// \param factor The cost scaling factor. It must be larger than one.
 		    ///
 		    /// \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 digraph contains an arc of negative cost
 		    /// and infinite upper bound. It means that the objective function
 		    /// is unbounded on that arc, however note that it could actually be
+		    /// is unbounded on that arc, however, note that it could actually be
 		    /// bounded over the feasible flows, but this algroithm cannot handle
 		    /// these cases.
 		    ///
 		    /// \see ProblemType, Method
 		    ProblemType run(Method method = PARTIAL_AUGMENT, int factor = 8) {
 		      _alpha = factor;
 		      ProblemType pt = init();
 		      if (pt != OPTIMAL) return pt;
 		      start(method);
 		      return OPTIMAL;
+		    }
 		    /// \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().
 		    ///
 		    /// It is useful for multiple run() calls. If this function is not
 		    /// used, all the parameters given before are kept for the next
 		    /// \ref run() call.
 		    /// However the underlying digraph must not be modified after this
+		    /// However, the underlying digraph must not be modified after this
 		    /// class have been constructed, since it copies and extends the graph.
 		    ///
 		    /// For example,
 		    /// \code
 		    ///   CostScaling<ListDigraph> cs(graph);
 		    ///
 		    ///   // First run
 		    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
 		    ///     .supplyMap(sup).run();
 		    ///
 		    ///   // Run again with modified cost map (reset() is not called,
 		    ///   // so only the cost map have to be set again)
 		    ///   cost[e] += 100;
 		    ///   cs.costMap(cost).run();
 		    ///
 		    ///   // Run again from scratch using reset()
 		    ///   // (the lower bounds will be set to zero on all arcs)
 		    ///   cs.reset();
 		    ///   cs.upperMap(capacity).costMap(cost)
 		    ///     .supplyMap(sup).run();
 		    /// \endcode
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    CostScaling& reset() {
 		      for (int i = 0; i != _res_node_num; ++i) {
 		        _supply[i] = 0;
+		      }
 		      int limit = _first_out[_root];
 		      for (int j = 0; j != limit; ++j) {
 		        _lower[j] = 0;
 		        _upper[j] = INF;
 		        _scost[j] = _forward[j] ? 1 : -1;
+		      }
 		      for (int j = limit; j != _res_arc_num; ++j) {
 		        _lower[j] = 0;
 		        _upper[j] = INF;
 		        _scost[j] = 0;
 		        _scost[_reverse[j]] = 0;
+		      }
 		      _have_lower = false;
 		      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
 		    ///   cs.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_idb[a];
 		        c += static_cast<Number>(_res_cap[i]) *
 		             (-static_cast<Number>(_scost[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 _res_cap[_arc_idb[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, _res_cap[_arc_idb[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 static_cast<Cost>(_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, static_cast<Cost>(_pi[_node_id[n]]));
+		      }
+		    }
 		    /// @}
 		  private:
 		    // Initialize the algorithm
 		    ProblemType init() {
 		      if (_res_node_num == 0) return INFEASIBLE;
 		      // Check the sum of supply values
 		      _sum_supply = 0;
 		      for (int i = 0; i != _root; ++i) {
 		        _sum_supply += _supply[i];
+		      }
 		      if (_sum_supply > 0) return INFEASIBLE;
 		      // Initialize vectors
 		      for (int i = 0; i != _res_node_num; ++i) {
 		        _pi[i] = 0;
 		        _excess[i] = _supply[i];
+		      }
 		      // Remove infinite upper bounds and check negative arcs
 		      const Value MAX = std::numeric_limits<Value>::max();
 		      int last_out;
 		      if (_have_lower) {
 		        for (int i = 0; i != _root; ++i) {
 		          last_out = _first_out[i+1];
 		          for (int j = _first_out[i]; j != last_out; ++j) {
 		            if (_forward[j]) {
 		              Value c = _scost[j] < 0 ? _upper[j] : _lower[j];
 		              if (c >= MAX) return UNBOUNDED;
 		              _excess[i] -= c;
 		              _excess[_target[j]] += c;
+		            }
+		          }
+		        }
 		      } else {
 		        for (int i = 0; i != _root; ++i) {
 		          last_out = _first_out[i+1];
 		          for (int j = _first_out[i]; j != last_out; ++j) {
 		            if (_forward[j] && _scost[j] < 0) {
 		              Value c = _upper[j];
 		              if (c >= MAX) return UNBOUNDED;
 		              _excess[i] -= c;
 		              _excess[_target[j]] += c;
+		            }
+		          }
+		        }
+		      }
 		      Value ex, max_cap = 0;
 		      for (int i = 0; i != _res_node_num; ++i) {
 		        ex = _excess[i];
 		        _excess[i] = 0;
 		        if (ex < 0) max_cap -= ex;
+		      }
 		      for (int j = 0; j != _res_arc_num; ++j) {
 		        if (_upper[j] >= MAX) _upper[j] = max_cap;
+		      }
 		      // Initialize the large cost vector and the epsilon parameter
 		      _epsilon = 0;
 		      LargeCost lc;
 		      for (int i = 0; i != _root; ++i) {
 		        last_out = _first_out[i+1];
 		        for (int j = _first_out[i]; j != last_out; ++j) {
 		          lc = static_cast<LargeCost>(_scost[j]) * _res_node_num * _alpha;
 		          _cost[j] = lc;
 		          if (lc > _epsilon) _epsilon = lc;
+		        }
+		      }
 		      _epsilon /= _alpha;
 		      // Initialize maps for Circulation and remove non-zero lower bounds
 		      ConstMap<Arc, Value> low(0);
 		      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
 		      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
 		      ValueArcMap cap(_graph), flow(_graph);
 		      ValueNodeMap sup(_graph);
 		      for (NodeIt n(_graph); n != INVALID; ++n) {
 		        sup[n] = _supply[_node_id[n]];
+		      }
 		      if (_have_lower) {
 		        for (ArcIt a(_graph); a != INVALID; ++a) {
 		          int j = _arc_idf[a];
 		          Value c = _lower[j];
 		          cap[a] = _upper[j] - c;
 		          sup[_graph.source(a)] -= c;
 		          sup[_graph.target(a)] += c;
+		        }
 		      } else {
 		        for (ArcIt a(_graph); a != INVALID; ++a) {
 		          cap[a] = _upper[_arc_idf[a]];
+		        }
+		      }
 		      // Find a feasible flow using Circulation
 		      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
 		        circ(_graph, low, cap, sup);
 		      if (!circ.flowMap(flow).run()) return INFEASIBLE;
 		      // Set residual capacities and handle GEQ supply type
 		      if (_sum_supply < 0) {
 		        for (ArcIt a(_graph); a != INVALID; ++a) {
 		          Value fa = flow[a];
 		          _res_cap[_arc_idf[a]] = cap[a] - fa;
 		          _res_cap[_arc_idb[a]] = fa;
 		          sup[_graph.source(a)] -= fa;
 		          sup[_graph.target(a)] += fa;
+		        }
 		        for (NodeIt n(_graph); n != INVALID; ++n) {
 		          _excess[_node_id[n]] = sup[n];
+		        }
 		        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
 		          int u = _target[a];
 		          int ra = _reverse[a];
 		          _res_cap[a] = -_sum_supply + 1;
 		          _res_cap[ra] = -_excess[u];
 		          _cost[a] = 0;
 		          _cost[ra] = 0;
 		          _excess[u] = 0;
+		        }
 		      } else {
 		        for (ArcIt a(_graph); a != INVALID; ++a) {
 		          Value fa = flow[a];
 		          _res_cap[_arc_idf[a]] = cap[a] - fa;
 		          _res_cap[_arc_idb[a]] = fa;
+		        }
 		        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
 		          int ra = _reverse[a];
 		          _res_cap[a] = 1;
 		          _res_cap[ra] = 0;
 		          _cost[a] = 0;
 		          _cost[ra] = 0;
+		        }
+		      }
 		      return OPTIMAL;
+		    }
 		    // Execute the algorithm and transform the results
 		    void start(Method method) {
 		      // Maximum path length for partial augment
 		      const int MAX_PATH_LENGTH = 4;
 		      // Execute the algorithm
 		      switch (method) {
 		        case PUSH:
 		          startPush();
 		          break;
 		        case AUGMENT:
 		          startAugment();
 		          break;
 		        case PARTIAL_AUGMENT:
 		          startAugment(MAX_PATH_LENGTH);
 		          break;
+		      }
 		      // Compute node potentials for the original costs
 		      _arc_vec.clear();
 		      _cost_vec.clear();
 		      for (int j = 0; j != _res_arc_num; ++j) {
 		        if (_res_cap[j] > 0) {
 		          _arc_vec.push_back(IntPair(_source[j], _target[j]));
 		          _cost_vec.push_back(_scost[j]);
+		        }
+		      }
 		      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
 		      typename BellmanFord<StaticDigraph, LargeCostArcMap>
 		        ::template SetDistMap<LargeCostNodeMap>::Create bf(_sgr, _cost_map);
 		      bf.distMap(_pi_map);
 		      bf.init(0);
 		      bf.start();
 		      // Handle non-zero lower bounds
 		      if (_have_lower) {
 		        int limit = _first_out[_root];
 		        for (int j = 0; j != limit; ++j) {
 		          if (!_forward[j]) _res_cap[j] += _lower[j];
+		        }
+		      }
+		    }
 		    /// Execute the algorithm performing augment and relabel operations
 		    void startAugment(int max_length = std::numeric_limits<int>::max()) {
 		      // Paramters for heuristics
 		      const int BF_HEURISTIC_EPSILON_BOUND = 1000;
 		      const int BF_HEURISTIC_BOUND_FACTOR  = 3;
 		      // Perform cost scaling phases
 		      IntVector pred_arc(_res_node_num);
 		      std::vector<int> path_nodes;
 		      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
 : _epsilon / _alpha )
+		      {
 		        // "Early Termination" heuristic: use Bellman-Ford algorithm
 		        // to check if the current flow is optimal
 		        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
 		          _arc_vec.clear();
 		          _cost_vec.clear();
 		          for (int j = 0; j != _res_arc_num; ++j) {
 		            if (_res_cap[j] > 0) {
 		              _arc_vec.push_back(IntPair(_source[j], _target[j]));
 		              _cost_vec.push_back(_cost[j] + 1);
+		            }
+		          }
 		          _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
 		          BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
 		          bf.init(0);
 		          bool done = false;
 		          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
 		          for (int i = 0; i < K && !done; ++i)
 		            done = bf.processNextWeakRound();
 		          if (done) break;
+		        }
 		        // Saturate arcs not satisfying the optimality condition
 		        for (int a = 0; a != _res_arc_num; ++a) {
 		          if (_res_cap[a] > 0 &&
 		              _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
 		            Value delta = _res_cap[a];
 		            _excess[_source[a]] -= delta;
 		            _excess[_target[a]] += delta;
 		            _res_cap[a] = 0;
 		            _res_cap[_reverse[a]] += delta;
+		          }
+		        }
 		        // Find active nodes (i.e. nodes with positive excess)
 		        for (int u = 0; u != _res_node_num; ++u) {
 		          if (_excess[u] > 0) _active_nodes.push_back(u);
+		        }
 		        // Initialize the next arcs
 		        for (int u = 0; u != _res_node_num; ++u) {
 		          _next_out[u] = _first_out[u];
+		        }
 		        // Perform partial augment and relabel operations
 		        while (true) {
 		          // Select an active node (FIFO selection)
 		          while (_active_nodes.size() > 0 &&
 		                 _excess[_active_nodes.front()] <= 0) {
 		            _active_nodes.pop_front();
+		          }
 		          if (_active_nodes.size() == 0) break;
 		          int start = _active_nodes.front();
 		          path_nodes.clear();
 		          path_nodes.push_back(start);
 		          // Find an augmenting path from the start node
 		          int tip = start;
 		          while (_excess[tip] >= 0 &&
 		                 int(path_nodes.size()) <= max_length) {
 		            int u;
 		            LargeCost min_red_cost, rc;
 		            int last_out = _sum_supply < 0 ?
 		              _first_out[tip+1] : _first_out[tip+1] - 1;
 		            for (int a = _next_out[tip]; a != last_out; ++a) {
 		              if (_res_cap[a] > 0 &&
 		                  _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
 		                u = _target[a];
 		                pred_arc[u] = a;
 		                _next_out[tip] = a;
 		                tip = u;
 		                path_nodes.push_back(tip);
 		                goto next_step;
+		              }
+		            }
 		            // Relabel tip node
 		            min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
 		            for (int a = _first_out[tip]; a != last_out; ++a) {
 		              rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
 		              if (_res_cap[a] > 0 && rc < min_red_cost) {
 		                min_red_cost = rc;
+		              }
+		            }
 		            _pi[tip] -= min_red_cost + _epsilon;
 		            // Reset the next arc of tip
 		            _next_out[tip] = _first_out[tip];
 		            // Step back
 		            if (tip != start) {
 		              path_nodes.pop_back();
 		              tip = path_nodes.back();
+		            }
 		          next_step: ;
+		          }
 		          // Augment along the found path (as much flow as possible)
 		          Value delta;
 		          int u, v = path_nodes.front(), pa;
 		          for (int i = 1; i < int(path_nodes.size()); ++i) {
 		            u = v;
 		            v = path_nodes[i];
 		            pa = pred_arc[v];
 		            delta = std::min(_res_cap[pa], _excess[u]);
 		            _res_cap[pa] -= delta;
 		            _res_cap[_reverse[pa]] += delta;
 		            _excess[u] -= delta;
 		            _excess[v] += delta;
 		            if (_excess[v] > 0 && _excess[v] <= delta)
 		              _active_nodes.push_back(v);
+		          }
+		        }
+		      }
+		    }
 		    /// Execute the algorithm performing push and relabel operations
 		    void startPush() {
 		      // Paramters for heuristics
 		      const int BF_HEURISTIC_EPSILON_BOUND = 1000;
 		      const int BF_HEURISTIC_BOUND_FACTOR  = 3;
 		      // Perform cost scaling phases
 		      BoolVector hyper(_res_node_num, false);
 		      for ( ; _epsilon >= 1; _epsilon = _epsilon < _alpha && _epsilon > 1 ?
 : _epsilon / _alpha )
+		      {
 		        // "Early Termination" heuristic: use Bellman-Ford algorithm
 		        // to check if the current flow is optimal
 		        if (_epsilon <= BF_HEURISTIC_EPSILON_BOUND) {
 		          _arc_vec.clear();
 		          _cost_vec.clear();
 		          for (int j = 0; j != _res_arc_num; ++j) {
 		            if (_res_cap[j] > 0) {
 		              _arc_vec.push_back(IntPair(_source[j], _target[j]));
 		              _cost_vec.push_back(_cost[j] + 1);
+		            }
+		          }
 		          _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
 		          BellmanFord<StaticDigraph, LargeCostArcMap> bf(_sgr, _cost_map);
 		          bf.init(0);
 		          bool done = false;
 		          int K = int(BF_HEURISTIC_BOUND_FACTOR * sqrt(_res_node_num));
 		          for (int i = 0; i < K && !done; ++i)
 		            done = bf.processNextWeakRound();
 		          if (done) break;
+		        }
 		        // Saturate arcs not satisfying the optimality condition
 		        for (int a = 0; a != _res_arc_num; ++a) {
 		          if (_res_cap[a] > 0 &&
 		              _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
 		            Value delta = _res_cap[a];
 		            _excess[_source[a]] -= delta;
 		            _excess[_target[a]] += delta;
 		            _res_cap[a] = 0;
 		            _res_cap[_reverse[a]] += delta;
+		          }
+		        }
 		        // Find active nodes (i.e. nodes with positive excess)
 		        for (int u = 0; u != _res_node_num; ++u) {
 		          if (_excess[u] > 0) _active_nodes.push_back(u);
+		        }
 		        // Initialize the next arcs
 		        for (int u = 0; u != _res_node_num; ++u) {
 		          _next_out[u] = _first_out[u];
+		        }
 		        // Perform push and relabel operations
 		        while (_active_nodes.size() > 0) {
 		          LargeCost min_red_cost, rc;
 		          Value delta;
 		          int n, t, a, last_out = _res_arc_num;
 		          // Select an active node (FIFO selection)
 		        next_node:
 		          n = _active_nodes.front();
 		          last_out = _sum_supply < 0 ?
 		            _first_out[n+1] : _first_out[n+1] - 1;
 		          // Perform push operations if there are admissible arcs
 		          if (_excess[n] > 0) {
 		            for (a = _next_out[n]; a != last_out; ++a) {
 		              if (_res_cap[a] > 0 &&
 		                  _cost[a] + _pi[_source[a]] - _pi[_target[a]] < 0) {
 		                delta = std::min(_res_cap[a], _excess[n]);
 		                t = _target[a];
 		                // Push-look-ahead heuristic
 		                Value ahead = -_excess[t];
 		                int last_out_t = _sum_supply < 0 ?
 		                  _first_out[t+1] : _first_out[t+1] - 1;
 		                for (int ta = _next_out[t]; ta != last_out_t; ++ta) {
 		                  if (_res_cap[ta] > 0 &&
 		                      _cost[ta] + _pi[_source[ta]] - _pi[_target[ta]] < 0)
 		                    ahead += _res_cap[ta];
 		                  if (ahead >= delta) break;
+		                }
 		                if (ahead < 0) ahead = 0;
 		                // Push flow along the arc
 		                if (ahead < delta) {
 		                  _res_cap[a] -= ahead;
 		                  _res_cap[_reverse[a]] += ahead;
 		                  _excess[n] -= ahead;
 		                  _excess[t] += ahead;
 		                  _active_nodes.push_front(t);
 		                  hyper[t] = true;
 		                  _next_out[n] = a;
 		                  goto next_node;
 		                } else {
 		                  _res_cap[a] -= delta;
 		                  _res_cap[_reverse[a]] += delta;
 		                  _excess[n] -= delta;
 		                  _excess[t] += delta;
 		                  if (_excess[t] > 0 && _excess[t] <= delta)
 		                    _active_nodes.push_back(t);
+		                }
 		                if (_excess[n] == 0) {
 		                  _next_out[n] = a;
 		                  goto remove_nodes;
+		                }
+		              }
+		            }
 		            _next_out[n] = a;
+		          }
 		          // Relabel the node if it is still active (or hyper)
 		          if (_excess[n] > 0 || hyper[n]) {
 		            min_red_cost = std::numeric_limits<LargeCost>::max() / 2;
 		            for (int a = _first_out[n]; a != last_out; ++a) {
 		              rc = _cost[a] + _pi[_source[a]] - _pi[_target[a]];
 		              if (_res_cap[a] > 0 && rc < min_red_cost) {
 		                min_red_cost = rc;
+		              }
+		            }
 		            _pi[n] -= min_red_cost + _epsilon;
 		            hyper[n] = false;
 		            // Reset the next arc
 		            _next_out[n] = _first_out[n];
+		          }
 		          // Remove nodes that are not active nor hyper
 		        remove_nodes:
 		          while ( _active_nodes.size() > 0 &&
 		                  _excess[_active_nodes.front()] <= 0 &&
 		                  !hyper[_active_nodes.front()] ) {
 		            _active_nodes.pop_front();
+		          }
+		        }
+		      }
+		    }
 		  }; //class CostScaling
 		  ///@}
 		} //namespace lemon
 		#endif //LEMON_COST_SCALING_H

lemon/network_simplex.h

Show inline comments

 		/* -*- mode: C++; indent-tabs-mode: nil; -*-
+		 *
 		 * This file is a part of LEMON, a generic C++ optimization library.
+		 *
 		 * Copyright (C) 2003-2009
 		 * 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_algs
 		///
 		/// \file
 		/// \brief Network Simplex algorithm for finding a minimum cost flow.
 		#include <vector>
 		#include <limits>
 		#include <algorithm>
 		#include <lemon/core.h>
 		#include <lemon/math.h>
 		namespace lemon {
 		  /// \addtogroup min_cost_flow_algs
 		  /// @{
 		  /// \brief Implementation of the primal Network Simplex algorithm
 		  /// for finding a \ref min_cost_flow "minimum cost flow".
 		  ///
 		  /// \ref NetworkSimplex implements the primal Network Simplex algorithm
 		  /// for finding a \ref min_cost_flow "minimum cost flow"
 		  /// \ref amo93networkflows, \ref dantzig63linearprog,
 		  /// \ref kellyoneill91netsimplex.
 		  /// This algorithm is a specialized version of the linear programming
 		  /// simplex method directly for the minimum cost flow problem.
-		  /// It is one of the most efficient solution methods.
+		  /// This algorithm is a highly efficient specialized version of the
 		  /// linear programming simplex method directly for the minimum cost
 		  /// flow problem.
 		  ///
 		  /// In general this class is the fastest implementation available
 		  /// in LEMON for the minimum cost flow problem.
-		  /// Moreover it supports both directions of the supply/demand inequality
+		  /// In general, %NetworkSimplex is the fastest implementation available
 		  /// in LEMON for this problem.
 		  /// Moreover, it supports both directions of the supply/demand inequality
 		  /// constraints. For more information, see \ref SupplyType.
 		  ///
 		  /// Most of the parameters of the problem (except for the digraph)
 		  /// can be given using separate functions, and the algorithm can be
 		  /// executed using the \ref run() function. If some parameters are not
 		  /// specified, then default values will be used.
 		  ///
 		  /// \tparam GR The digraph type the algorithm runs on.
-		  /// \tparam V The value type used for flow amounts, capacity bounds
+		  /// \tparam V The number type used for flow amounts, capacity bounds
 		  /// and supply values in the algorithm. By default, it is \c int.
-		  /// \tparam C The value type used for costs and potentials in the
+		  /// \tparam C The number type used for costs and potentials in the
 		  /// algorithm. By default, it is the same as \c V.
 		  ///
-		  /// \warning Both value types must be signed and all input data must
+		  /// \warning Both number types must be signed and all input data must
 		  /// be integer.
 		  ///
 		  /// \note %NetworkSimplex provides five different pivot rule
 		  /// implementations, from which the most efficient one is used
 		  /// by default. For more information, see \ref PivotRule.
 		  template <typename GR, typename V = int, typename C = V>
 		  class NetworkSimplex
+		  {
 		  public:
 		    /// The type of the flow amounts, capacity bounds and supply values
 		    typedef V Value;
 		    /// The type of the arc costs
 		    typedef C Cost;
 		  public:
 		    /// \brief Problem type constants for the \c run() function.
 		    ///
 		    /// Enum type containing the problem type constants that can be
 		    /// returned by the \ref run() function of the algorithm.
 		    enum ProblemType {
 		      /// The problem has no feasible solution (flow).
 		      INFEASIBLE,
 		      /// 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).
 		      OPTIMAL,
 		      /// The objective function of the problem is unbounded, i.e.
 		      /// there is a directed cycle having negative total cost and
 		      /// infinite upper bound.
 		      UNBOUNDED
 		    };
 		    /// \brief Constants for selecting the type of the supply constraints.
 		    ///
 		    /// Enum type containing constants for selecting the supply type,
 		    /// i.e. the direction of the inequalities in the supply/demand
 		    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
 		    ///
 		    /// The default supply type is \c GEQ, the \c LEQ type can be
 		    /// selected using \ref supplyType().
 		    /// The equality form is a special case of both supply types.
 		    enum SupplyType {
 		      /// This option means that there are <em>"greater or equal"</em>
 		      /// supply/demand constraints in the definition of the problem.
 		      GEQ,
 		      /// This option means that there are <em>"less or equal"</em>
 		      /// supply/demand constraints in the definition of the problem.
 		      LEQ
 		    };
 		    /// \brief Constants for selecting the pivot rule.
 		    ///
 		    /// Enum type containing constants for selecting the pivot rule for
 		    /// the \ref run() function.
 		    ///
 		    /// \ref NetworkSimplex provides five different pivot rule
 		    /// implementations that significantly affect the running time
 		    /// of the algorithm.
 		    /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
 		    /// proved to be the most efficient and the most robust on various
-		    /// test inputs according to our benchmark tests.
 		    /// test inputs.
 		    /// However, another pivot rule can be selected using the \ref run()
 		    /// function with the proper parameter.
 		    enum PivotRule {
 		      /// The \e First \e Eligible pivot rule.
 		      /// The next eligible arc is selected in a wraparound fashion
 		      /// in every iteration.
 		      FIRST_ELIGIBLE,
 		      /// The \e Best \e Eligible pivot rule.
 		      /// The best eligible arc is selected in every iteration.
 		      BEST_ELIGIBLE,
 		      /// The \e Block \e Search pivot rule.
 		      /// A specified number of arcs are examined in every iteration
 		      /// in a wraparound fashion and the best eligible arc is selected
 		      /// from this block.
 		      BLOCK_SEARCH,
 		      /// The \e Candidate \e List pivot rule.
 		      /// In a major iteration a candidate list is built from eligible arcs
 		      /// in a wraparound fashion and in the following minor iterations
 		      /// the best eligible arc is selected from this list.
 		      CANDIDATE_LIST,
 		      /// The \e Altering \e Candidate \e List pivot rule.
 		      /// It is a modified version of the Candidate List method.
 		      /// It keeps only the several best eligible arcs from the former
 		      /// candidate list and extends this list in every iteration.
 		      ALTERING_LIST
 		    };
 		  private:
 		    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
 		    typedef std::vector<int> IntVector;
 		    typedef std::vector<char> CharVector;
 		    typedef std::vector<Value> ValueVector;
 		    typedef std::vector<Cost> CostVector;
 		    // State constants for arcs
 		    enum ArcStateEnum {
 		      STATE_UPPER = -1,
 		      STATE_TREE  =  0,
 		      STATE_LOWER =  1
 		    };
 		  private:
 		    // Data related to the underlying digraph
 		    const GR &_graph;
 		    int _node_num;
 		    int _arc_num;
 		    int _all_arc_num;
 		    int _search_arc_num;
 		    // Parameters of the problem
 		    bool _have_lower;
 		    SupplyType _stype;
 		    Value _sum_supply;
 		    // Data structures for storing the digraph
 		    IntNodeMap _node_id;
 		    IntArcMap _arc_id;
 		    IntVector _source;
 		    IntVector _target;
 		    // Node and arc data
 		    ValueVector _lower;
 		    ValueVector _upper;
 		    ValueVector _cap;
 		    CostVector _cost;
 		    ValueVector _supply;
 		    ValueVector _flow;
 		    CostVector _pi;
 		    // Data for storing the spanning tree structure
 		    IntVector _parent;
 		    IntVector _pred;
 		    IntVector _thread;
 		    IntVector _rev_thread;
 		    IntVector _succ_num;
 		    IntVector _last_succ;
 		    IntVector _dirty_revs;
 		    CharVector _forward;
 		    CharVector _state;
 		    int _root;
 		    // Temporary data used in the current pivot iteration
 		    int in_arc, join, u_in, v_in, u_out, v_out;
 		    int first, second, right, last;
 		    int stem, par_stem, new_stem;
 		    Value delta;
 		    const Value MAX;
 		  public:
 		    /// \brief Constant for infinite upper bounds (capacities).
 		    ///
 		    /// Constant for infinite upper bounds (capacities).
 		    /// It is \c std::numeric_limits<Value>::infinity() if available,
 		    /// \c std::numeric_limits<Value>::max() otherwise.
 		    const Value INF;
 		  private:
 		    // Implementation of the First Eligible pivot rule
 		    class FirstEligiblePivotRule
+		    {
 		    private:
 		      // References to the NetworkSimplex class
 		      const IntVector  &_source;
 		      const IntVector  &_target;
 		      const CostVector &_cost;
 		      const CharVector &_state;
 		      const CostVector &_pi;
 		      int &_in_arc;
 		      int _search_arc_num;
 		      // Pivot rule data
 		      int _next_arc;
 		    public:
 		      // Constructor
 		      FirstEligiblePivotRule(NetworkSimplex &ns) :
 		        _source(ns._source), _target(ns._target),
 		        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
 		        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
 		        _next_arc(0)
 		      {}
 		      // Find next entering arc
 		      bool findEnteringArc() {
 		        Cost c;
 		        for (int e = _next_arc; e < _search_arc_num; ++e) {
 		          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (c < 0) {
 		            _in_arc = e;
 		            _next_arc = e + 1;
 		            return true;
+		          }
+		        }
 		        for (int e = 0; e < _next_arc; ++e) {
 		          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (c < 0) {
 		            _in_arc = e;
 		            _next_arc = e + 1;
 		            return true;
+		          }
+		        }
 		        return false;
+		      }
 		    }; //class FirstEligiblePivotRule
 		    // Implementation of the Best Eligible pivot rule
 		    class BestEligiblePivotRule
+		    {
 		    private:
 		      // References to the NetworkSimplex class
 		      const IntVector  &_source;
 		      const IntVector  &_target;
 		      const CostVector &_cost;
 		      const CharVector &_state;
 		      const CostVector &_pi;
 		      int &_in_arc;
 		      int _search_arc_num;
 		    public:
 		      // Constructor
 		      BestEligiblePivotRule(NetworkSimplex &ns) :
 		        _source(ns._source), _target(ns._target),
 		        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
 		        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
 		      {}
 		      // Find next entering arc
 		      bool findEnteringArc() {
 		        Cost c, min = 0;
 		        for (int e = 0; e < _search_arc_num; ++e) {
 		          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (c < min) {
 		            min = c;
 		            _in_arc = e;
+		          }
+		        }
 		        return min < 0;
+		      }
 		    }; //class BestEligiblePivotRule
 		    // Implementation of the Block Search pivot rule
 		    class BlockSearchPivotRule
+		    {
 		    private:
 		      // References to the NetworkSimplex class
 		      const IntVector  &_source;
 		      const IntVector  &_target;
 		      const CostVector &_cost;
 		      const CharVector &_state;
 		      const CostVector &_pi;
 		      int &_in_arc;
 		      int _search_arc_num;
 		      // Pivot rule data
 		      int _block_size;
 		      int _next_arc;
 		    public:
 		      // Constructor
 		      BlockSearchPivotRule(NetworkSimplex &ns) :
 		        _source(ns._source), _target(ns._target),
 		        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
 		        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
 		        _next_arc(0)
+		      {
 		        // The main parameters of the pivot rule
 		        const double BLOCK_SIZE_FACTOR = 0.5;
 		        const int MIN_BLOCK_SIZE = 10;
 		        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
 		                                    std::sqrt(double(_search_arc_num))),
 		                                MIN_BLOCK_SIZE );
+		      }
 		      // Find next entering arc
 		      bool findEnteringArc() {
 		        Cost c, min = 0;
 		        int cnt = _block_size;
 		        int e;
 		        for (e = _next_arc; e < _search_arc_num; ++e) {
 		          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (c < min) {
 		            min = c;
 		            _in_arc = e;
+		          }
 		          if (--cnt == 0) {
 		            if (min < 0) goto search_end;
 		            cnt = _block_size;
+		          }
+		        }
 		        for (e = 0; e < _next_arc; ++e) {
 		          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (c < min) {
 		            min = c;
 		            _in_arc = e;
+		          }
 		          if (--cnt == 0) {
 		            if (min < 0) goto search_end;
 		            cnt = _block_size;
+		          }
+		        }
 		        if (min >= 0) return false;
 		      search_end:
 		        _next_arc = e;
 		        return true;
+		      }
 		    }; //class BlockSearchPivotRule
 		    // Implementation of the Candidate List pivot rule
 		    class CandidateListPivotRule
+		    {
 		    private:
 		      // References to the NetworkSimplex class
 		      const IntVector  &_source;
 		      const IntVector  &_target;
 		      const CostVector &_cost;
 		      const CharVector &_state;
 		      const CostVector &_pi;
 		      int &_in_arc;
 		      int _search_arc_num;
 		      // Pivot rule data
 		      IntVector _candidates;
 		      int _list_length, _minor_limit;
 		      int _curr_length, _minor_count;
 		      int _next_arc;
 		    public:
 		      /// Constructor
 		      CandidateListPivotRule(NetworkSimplex &ns) :
 		        _source(ns._source), _target(ns._target),
 		        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
 		        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
 		        _next_arc(0)
+		      {
 		        // The main parameters of the pivot rule
 		        const double LIST_LENGTH_FACTOR = 0.25;
 		        const int MIN_LIST_LENGTH = 10;
 		        const double MINOR_LIMIT_FACTOR = 0.1;
 		        const int MIN_MINOR_LIMIT = 3;
 		        _list_length = std::max( int(LIST_LENGTH_FACTOR *
 		                                     std::sqrt(double(_search_arc_num))),
 		                                 MIN_LIST_LENGTH );
 		        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
 		                                 MIN_MINOR_LIMIT );
 		        _curr_length = _minor_count = 0;
 		        _candidates.resize(_list_length);
+		      }
 		      /// Find next entering arc
 		      bool findEnteringArc() {
 		        Cost min, c;
 		        int e;
 		        if (_curr_length > 0 && _minor_count < _minor_limit) {
 		          // Minor iteration: select the best eligible arc from the
 		          // current candidate list
 		          ++_minor_count;
 		          min = 0;
 		          for (int i = 0; i < _curr_length; ++i) {
 		            e = _candidates[i];
 		            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		            if (c < min) {
 		              min = c;
 		              _in_arc = e;
+		            }
 		            else if (c >= 0) {
 		              _candidates[i--] = _candidates[--_curr_length];
+		            }
+		          }
 		          if (min < 0) return true;
+		        }
 		        // Major iteration: build a new candidate list
 		        min = 0;
 		        _curr_length = 0;
 		        for (e = _next_arc; e < _search_arc_num; ++e) {
 		          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (c < 0) {
 		            _candidates[_curr_length++] = e;
 		            if (c < min) {
 		              min = c;
 		              _in_arc = e;
+		            }
 		            if (_curr_length == _list_length) goto search_end;
+		          }
+		        }
 		        for (e = 0; e < _next_arc; ++e) {
 		          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (c < 0) {
 		            _candidates[_curr_length++] = e;
 		            if (c < min) {
 		              min = c;
 		              _in_arc = e;
+		            }
 		            if (_curr_length == _list_length) goto search_end;
+		          }
+		        }
 		        if (_curr_length == 0) return false;
 		      search_end:
 		        _minor_count = 1;
 		        _next_arc = e;
 		        return true;
+		      }
 		    }; //class CandidateListPivotRule
 		    // Implementation of the Altering Candidate List pivot rule
 		    class AlteringListPivotRule
+		    {
 		    private:
 		      // References to the NetworkSimplex class
 		      const IntVector  &_source;
 		      const IntVector  &_target;
 		      const CostVector &_cost;
 		      const CharVector &_state;
 		      const CostVector &_pi;
 		      int &_in_arc;
 		      int _search_arc_num;
 		      // Pivot rule data
 		      int _block_size, _head_length, _curr_length;
 		      int _next_arc;
 		      IntVector _candidates;
 		      CostVector _cand_cost;
 		      // Functor class to compare arcs during sort of the candidate list
 		      class SortFunc
+		      {
 		      private:
 		        const CostVector &_map;
 		      public:
 		        SortFunc(const CostVector &map) : _map(map) {}
 		        bool operator()(int left, int right) {
 		          return _map[left] > _map[right];
+		        }
 		      };
 		      SortFunc _sort_func;
 		    public:
 		      // Constructor
 		      AlteringListPivotRule(NetworkSimplex &ns) :
 		        _source(ns._source), _target(ns._target),
 		        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
 		        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
 		        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
+		      {
 		        // The main parameters of the pivot rule
 		        const double BLOCK_SIZE_FACTOR = 1.0;
 		        const int MIN_BLOCK_SIZE = 10;
 		        const double HEAD_LENGTH_FACTOR = 0.1;
 		        const int MIN_HEAD_LENGTH = 3;
 		        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
 		                                    std::sqrt(double(_search_arc_num))),
 		                                MIN_BLOCK_SIZE );
 		        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
 		                                 MIN_HEAD_LENGTH );
 		        _candidates.resize(_head_length + _block_size);
 		        _curr_length = 0;
+		      }
 		      // Find next entering arc
 		      bool findEnteringArc() {
 		        // Check the current candidate list
 		        int e;
 		        for (int i = 0; i < _curr_length; ++i) {
 		          e = _candidates[i];
 		          _cand_cost[e] = _state[e] *
 		            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (_cand_cost[e] >= 0) {
 		            _candidates[i--] = _candidates[--_curr_length];
+		          }
+		        }
 		        // Extend the list
 		        int cnt = _block_size;
 		        int limit = _head_length;
 		        for (e = _next_arc; e < _search_arc_num; ++e) {
 		          _cand_cost[e] = _state[e] *
 		            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (_cand_cost[e] < 0) {
 		            _candidates[_curr_length++] = e;
+		          }
 		          if (--cnt == 0) {
 		            if (_curr_length > limit) goto search_end;
 		            limit = 0;
 		            cnt = _block_size;
+		          }
+		        }
 		        for (e = 0; e < _next_arc; ++e) {
 		          _cand_cost[e] = _state[e] *
 		            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
 		          if (_cand_cost[e] < 0) {
 		            _candidates[_curr_length++] = e;
+		          }
 		          if (--cnt == 0) {
 		            if (_curr_length > limit) goto search_end;
 		            limit = 0;
 		            cnt = _block_size;
+		          }
+		        }
 		        if (_curr_length == 0) return false;
 		      search_end:
 		        // Make heap of the candidate list (approximating a partial sort)
 		        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
 		                   _sort_func );
 		        // Pop the first element of the heap
 		        _in_arc = _candidates[0];
 		        _next_arc = e;
 		        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
 		                  _sort_func );
 		        _curr_length = std::min(_head_length, _curr_length - 1);
 		        return true;
+		      }
 		    }; //class AlteringListPivotRule
 		  public:
 		    /// \brief Constructor.
 		    ///
 		    /// The constructor of the class.
 		    ///
 		    /// \param graph The digraph the algorithm runs on.
 		    /// \param arc_mixing Indicate if the arcs have to be stored in a
 		    /// mixed order in the internal data structure.
 		    /// In special cases, it could lead to better overall performance,
 		    /// but it is usually slower. Therefore it is disabled by default.
 		    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
 		      _graph(graph), _node_id(graph), _arc_id(graph),
 		      MAX(std::numeric_limits<Value>::max()),
 		      INF(std::numeric_limits<Value>::has_infinity ?
 		          std::numeric_limits<Value>::infinity() : MAX)
+		    {
-		      // Check the value types
+		      // Check the number types
 		      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
 		        "The flow type of NetworkSimplex must be signed");
 		      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
 		        "The cost type of NetworkSimplex must be signed");
 		      // 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
 		      reset();
+		    }
 		    /// \name Parameters
 		    /// The parameters of the algorithm can be specified using these
 		    /// functions.
 		    /// @{
 		    /// \brief Set the lower bounds on the arcs.
 		    ///
 		    /// This function sets the lower bounds on the arcs.
 		    /// If it is not used before calling \ref run(), the lower bounds
 		    /// will be set to zero on all arcs.
 		    ///
 		    /// \param map An arc map storing the lower bounds.
 		    /// Its \c Value type must be convertible to the \c Value type
 		    /// of the algorithm.
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    template <typename LowerMap>
 		    NetworkSimplex& lowerMap(const LowerMap& map) {
 		      _have_lower = true;
 		      for (ArcIt a(_graph); a != INVALID; ++a) {
 		        _lower[_arc_id[a]] = map[a];
+		      }
 		      return *this;
+		    }
 		    /// \brief Set the upper bounds (capacities) on the arcs.
 		    ///
 		    /// This function sets the upper bounds (capacities) on the arcs.
 		    /// If it is not used before calling \ref run(), the upper bounds
 		    /// will be set to \ref INF on all arcs (i.e. the flow value will be
-		    /// unbounded from above on each arc).
 		    /// unbounded from above).
 		    ///
 		    /// \param map An arc map storing the upper bounds.
 		    /// Its \c Value type must be convertible to the \c Value type
 		    /// of the algorithm.
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    template<typename UpperMap>
 		    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 parameters
 		    /// that have been given are kept for the next call, unless
 		    /// \ref reset() is called, thus only the modified parameters
 		    /// have to be set again. See \ref reset() for examples.
 		    /// However, the underlying digraph must not be modified after this
 		    /// class have been constructed, since it copies and extends the graph.
 		    ///
 		    /// \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
 		    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 run() calls. If this function is not
 		    /// used, all the parameters given before are kept for the next
 		    /// \ref run() call.
 		    /// However, the underlying digraph must not be modified after this
 		    /// class have been constructed, since it copies and extends the graph.
 		    ///
 		    /// 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 (reset() 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 reset()
 		    ///   // (the lower bounds will be set to zero on all arcs)
 		    ///   ns.reset();
 		    ///   ns.upperMap(capacity).costMap(cost)
 		    ///     .supplyMap(sup).run();
 		    /// \endcode
 		    ///
 		    /// \return <tt>(*this)</tt>
 		    NetworkSimplex& reset() {
 		      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;
+		    }
 		    /// @}
 		    /// \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 {
 		        ART_COST = std::numeric_limits<Cost>::min();
 		        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;
+		      }
+		    }
 		    // 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);
 		      // 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) {
 		          Cost max_pot = std::numeric_limits<Cost>::min();
 		          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

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