diff -r 70b199792735 -r ad40f7d32846 lemon/capacity_scaling.h
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/lemon/capacity_scaling.h	Sun Aug 11 15:28:12 2013 +0200
@@ -0,0 +1,990 @@
+/* -*- mode: C++; indent-tabs-mode: nil; -*-
+ *
+ * This file is a part of LEMON, a generic C++ optimization library.
+ *
+ * Copyright (C) 2003-2010
+ * 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 number type used for flow amounts, capacity bounds
+  /// and supply values. By default it is \c int.
+  /// \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" \ref amo93networkflows,
+  /// \ref edmondskarp72theoretical. 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 number type used for flow amounts, capacity bounds
+  /// and supply values in the algorithm. By default, it is \c int.
+  /// \tparam C The number type used for costs and potentials in the
+  /// algorithm. By default, it is the same as \c V.
+  /// \tparam TR The traits class that defines various types used by the
+  /// algorithm. By default, it is \ref CapacityScalingDefaultTraits
+  /// "CapacityScalingDefaultTraits<GR, V, C>".
+  /// In most cases, this parameter should not be set directly,
+  /// consider to use the named template parameters instead.
+  ///
+  /// \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
+      /// 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<Value> ValueVector;
+    typedef std::vector<Cost> CostVector;
+    typedef std::vector<char> BoolVector;
+    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
+
+  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;
+    };
+
+    /// @}
+
+  protected:
+
+    CapacityScaling() {}
+
+  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 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");
+
+      // Reset data structures
+      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).
+    ///
+    /// \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 given parameters
+    /// are kept for the next call, unless \ref resetParams() or \ref reset()
+    /// is used, thus only the modified parameters have to be set again.
+    /// If the underlying digraph was also modified after the construction
+    /// of the class (or the last \ref reset() call), then the \ref reset()
+    /// function must be called.
+    ///
+    /// \param 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
+    /// bounded over the feasible flows, but this algroithm cannot handle
+    /// these cases.
+    ///
+    /// \see ProblemType
+    /// \see resetParams(), reset()
+    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 \ref run() calls. Basically, all the given
+    /// parameters are kept for the next \ref run() call, unless
+    /// \ref resetParams() or \ref reset() is used.
+    /// If the underlying digraph was also modified after the construction
+    /// of the class or the last \ref reset() call, then the \ref reset()
+    /// function must be used, otherwise \ref resetParams() is sufficient.
+    ///
+    /// For example,
+    /// \code
+    ///   CapacityScaling<ListDigraph> cs(graph);
+    ///
+    ///   // First run
+    ///   cs.lowerMap(lower).upperMap(upper).costMap(cost)
+    ///     .supplyMap(sup).run();
+    ///
+    ///   // Run again with modified cost map (resetParams() is not called,
+    ///   // so only the cost map have to be set again)
+    ///   cost[e] += 100;
+    ///   cs.costMap(cost).run();
+    ///
+    ///   // Run again from scratch using resetParams()
+    ///   // (the lower bounds will be set to zero on all arcs)
+    ///   cs.resetParams();
+    ///   cs.upperMap(capacity).costMap(cost)
+    ///     .supplyMap(sup).run();
+    /// \endcode
+    ///
+    /// \return <tt>(*this)</tt>
+    ///
+    /// \see reset(), run()
+    CapacityScaling& resetParams() {
+      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;
+    }
+
+    /// \brief Reset the internal data structures and all the parameters
+    /// that have been given before.
+    ///
+    /// This function resets the internal data structures and all the
+    /// paramaters that have been given before using functions \ref lowerMap(),
+    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
+    ///
+    /// It is useful for multiple \ref run() calls. Basically, all the given
+    /// parameters are kept for the next \ref run() call, unless
+    /// \ref resetParams() or \ref reset() is used.
+    /// If the underlying digraph was also modified after the construction
+    /// of the class or the last \ref reset() call, then the \ref reset()
+    /// function must be used, otherwise \ref resetParams() is sufficient.
+    ///
+    /// See \ref resetParams() for examples.
+    ///
+    /// \return <tt>(*this)</tt>
+    ///
+    /// \see resetParams(), run()
+    CapacityScaling& reset() {
+      // 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
+      resetParams();
+      return *this;
+    }
+
+    /// @}
+
+    /// \name Query Functions
+    /// The results of the algorithm can be obtained using these
+    /// functions.\n
+    /// The \ref run() function must be called before using them.
+
+    /// @{
+
+    /// \brief Return the total cost of the found flow.
+    ///
+    /// This function returns the total cost of the found flow.
+    /// Its complexity is O(e).
+    ///
+    /// \note The return type of the function can be specified as a
+    /// template parameter. For example,
+    /// \code
+    ///   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 <= 1) 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, max_cap = 0;
+        for (int i = 0; i != _root; ++i) {
+          Value ex = _excess[i];
+          if ( ex > max_sup) max_sup =  ex;
+          if (-ex > max_dem) max_dem = -ex;
+          int last_out = _first_out[i+1] - 1;
+          for (int j = _first_out[i]; j != last_out; ++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