RhodeCode

  Dijkstra<Digraph, TimeMap> dijkstra(graph, time);

  dijkstra.run(source, target);

\endcode

We have a length map and a maximum speed map on the arcs of a digraph.

The minimum time to pass the arc can be calculated as the division of

the two maps which can be done implicitly with the \c DivMap template

class. We use the implicit minimum time map as the length map of the

\c Dijkstra algorithm.

*/

/**

@defgroup paths Path Structures

@ingroup datas

\brief %Path structures implemented in LEMON.

This group contains the path structures implemented in LEMON.

LEMON provides flexible data structures to work with paths.

All of them have similar interfaces and they can be copied easily with

assignment operators and copy constructors. This makes it easy and

efficient to have e.g. the Dijkstra algorithm to store its result in

any kind of path structure.

\sa \ref concepts::Path "Path concept"

*/

/**

@defgroup heaps Heap Structures

@ingroup datas

\brief %Heap structures implemented in LEMON.

This group contains the heap structures implemented in LEMON.

LEMON provides several heap classes. They are efficient implementations

of the abstract data type \e priority \e queue. They store items with

specified values called \e priorities in such a way that finding and

removing the item with minimum priority are efficient.

The basic operations are adding and erasing items, changing the priority

of an item, etc.

Heaps are crucial in several algorithms, such as Dijkstra and Prim.

The heap implementations have the same interface, thus any of them can be

used easily in such algorithms.

\sa \ref concepts::Heap "Heap concept"

*/

/**

@defgroup matrices Matrices

@ingroup datas

\brief Two dimensional data storages implemented in LEMON.

This group contains two dimensional data storages implemented in LEMON.

*/

/**

@defgroup auxdat Auxiliary Data Structures

@ingroup datas

\brief Auxiliary data structures implemented in LEMON.

This group contains some data structures implemented in LEMON in

order to make it easier to implement combinatorial algorithms.

*/

/**

@defgroup geomdat Geometric Data Structures

@ingroup auxdat

\brief Geometric data structures implemented in LEMON.

This group contains geometric data structures implemented in LEMON.

 - \ref lemon::dim2::Point "dim2::Point" implements a two dimensional

   vector with the usual operations.

 - \ref lemon::dim2::Box "dim2::Box" can be used to determine the

   rectangular bounding box of a set of \ref lemon::dim2::Point

   "dim2::Point"'s.

*/

/**

@defgroup matrices Matrices

@ingroup auxdat

\brief Two dimensional data storages implemented in LEMON.

This group contains two dimensional data storages implemented in LEMON.

*/

/**

@defgroup algs Algorithms

\brief This group contains the several algorithms

implemented in LEMON.

This group contains the several algorithms

implemented in LEMON.

*/

/**

@defgroup search Graph Search

@ingroup algs

\brief Common graph search algorithms.

This group contains the common graph search algorithms, namely

\e breadth-first \e search (BFS) and \e depth-first \e search (DFS)

\ref clrs01algorithms.

*/

/**

@defgroup shortest_path Shortest Path Algorithms

@ingroup algs

\brief Algorithms for finding shortest paths.

This group contains the algorithms for finding shortest paths in digraphs

\ref clrs01algorithms.

 - \ref Dijkstra algorithm for finding shortest paths from a source node

   when all arc lengths are non-negative.

 - \ref BellmanFord "Bellman-Ford" algorithm for finding shortest paths

   from a source node when arc lenghts can be either positive or negative,

   but the digraph should not contain directed cycles with negative total

   length.

 - \ref FloydWarshall "Floyd-Warshall" and \ref Johnson "Johnson" algorithms

   for solving the \e all-pairs \e shortest \e paths \e problem when arc

   lenghts can be either positive or negative, but the digraph should

   not contain directed cycles with negative total length.

 - \ref Suurballe A successive shortest path algorithm for finding

   arc-disjoint paths between two nodes having minimum total length.

*/

/**

@defgroup spantree Minimum Spanning Tree Algorithms

@ingroup algs

\brief Algorithms for finding minimum cost spanning trees and arborescences.

This group contains the algorithms for finding minimum cost spanning

trees and arborescences \ref clrs01algorithms.

*/

/**

@defgroup max_flow Maximum Flow Algorithms

@ingroup algs

\brief Algorithms for finding maximum flows.

This group contains the algorithms for finding maximum flows and

feasible circulations \ref clrs01algorithms, \ref amo93networkflows.

The \e maximum \e flow \e problem is to find a flow of maximum value between

a single source and a single target. Formally, there is a \f$G=(V,A)\f$

digraph, a \f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function and

\f$s, t \in V\f$ source and target nodes.

A maximum flow is an \f$f: A\rightarrow\mathbf{R}^+_0\f$ solution of the

following optimization problem.

\f[ \max\sum_{sv\in A} f(sv) - \sum_{vs\in A} f(vs) \f]

\f[ \sum_{uv\in A} f(uv) = \sum_{vu\in A} f(vu)

    \quad \forall u\in V\setminus\{s,t\} \f]

\f[ 0 \leq f(uv) \leq cap(uv) \quad \forall uv\in A \f]

LEMON contains several algorithms for solving maximum flow problems:

- \ref EdmondsKarp Edmonds-Karp algorithm

  \ref edmondskarp72theoretical.

- \ref Preflow Goldberg-Tarjan's preflow push-relabel algorithm

  \ref goldberg88newapproach.

- \ref DinitzSleatorTarjan Dinitz's blocking flow algorithm with dynamic trees

  \ref dinic70algorithm, \ref sleator83dynamic.

- \ref GoldbergTarjan !Preflow push-relabel algorithm with dynamic trees

  \ref goldberg88newapproach, \ref sleator83dynamic.

In most cases the \ref Preflow algorithm provides the

fastest method for computing a maximum flow. All implementations

also provide functions to query the minimum cut, which is the dual

problem of maximum flow.

\ref Circulation is a preflow push-relabel algorithm implemented directly 

for finding feasible circulations, which is a somewhat different problem,

but it is strongly related to maximum flow.

For more information, see \ref Circulation.

*/

/**

@defgroup min_cost_flow_algs Minimum Cost Flow Algorithms

@ingroup algs

\brief Algorithms for finding minimum cost flows and circulations.

This group contains the algorithms for finding minimum cost flows and

circulations \ref amo93networkflows. For more information about this

problem and its dual solution, see \ref min_cost_flow

"Minimum Cost Flow Problem".

LEMON contains several algorithms for this problem.

 - \ref NetworkSimplex Primal Network Simplex algorithm with various

   pivot strategies \ref dantzig63linearprog, \ref kellyoneill91netsimplex.

   \ref bunnagel98efficient.

In general NetworkSimplex is the most efficient implementation,

but in special cases other algorithms could be faster.

For example, if the total supply and/or capacities are rather small,

CapacityScaling is usually the fastest algorithm (without effective scaling).

*/

/**

@defgroup min_cut Minimum Cut Algorithms

@ingroup algs

\brief Algorithms for finding minimum cut in graphs.

This group contains the algorithms for finding minimum cut in graphs.

The \e minimum \e cut \e problem is to find a non-empty and non-complete

\f$X\f$ subset of the nodes with minimum overall capacity on

outgoing arcs. Formally, there is a \f$G=(V,A)\f$ digraph, a

\f$cap: A\rightarrow\mathbf{R}^+_0\f$ capacity function. The minimum

cut is the \f$X\f$ solution of the next optimization problem:

\f[ \min_{X \subset V, X\not\in \{\emptyset, V\}}

    \sum_{uv\in A: u\in X, v\not\in X}cap(uv) \f]

LEMON contains several algorithms related to minimum cut problems:

- \ref HaoOrlin "Hao-Orlin algorithm" for calculating minimum cut

  in directed graphs.

- \ref NagamochiIbaraki "Nagamochi-Ibaraki algorithm" for

  calculating minimum cut in undirected graphs.

- \ref GomoryHu "Gomory-Hu tree computation" for calculating

  all-pairs minimum cut in undirected graphs.

If you want to find minimum cut just between two distinict nodes,

see the \ref max_flow "maximum flow problem".

*/

/**

@defgroup min_mean_cycle Minimum Mean Cycle Algorithms

@ingroup algs

\brief Algorithms for finding minimum mean cycles.

This group contains the algorithms for finding minimum mean cycles

\ref clrs01algorithms, \ref amo93networkflows.

The \e minimum \e mean \e cycle \e problem is to find a directed cycle

of minimum mean length (cost) in a digraph.

The mean length of a cycle is the average length of its arcs, i.e. the

ratio between the total length of the cycle and the number of arcs on it.

This problem has an important connection to \e conservative \e length

\e functions, too. A length function on the arcs of a digraph is called

conservative if and only if there is no directed cycle of negative total

length. For an arbitrary length function, the negative of the minimum

cycle mean is the smallest \f$\epsilon\f$ value so that increasing the

arc lengths uniformly by \f$\epsilon\f$ results in a conservative length

function.

LEMON contains three algorithms for solving the minimum mean cycle problem:

- \ref Karp "Karp"'s original algorithm \ref amo93networkflows,

  \ref dasdan98minmeancycle.

- \ref HartmannOrlin "Hartmann-Orlin"'s algorithm, which is an improved

  version of Karp's algorithm \ref dasdan98minmeancycle.

- \ref Howard "Howard"'s policy iteration algorithm

  \ref dasdan98minmeancycle.

In practice, the Howard algorithm proved to be by far the most efficient

one, though the best known theoretical bound on its running time is

exponential.

Both Karp and HartmannOrlin algorithms run in time O(ne) and use space

O(n<sup>2</sup>+e), but the latter one is typically faster due to the

applied early termination scheme.

*/

/**

@defgroup matching Matching Algorithms

@ingroup algs

\brief Algorithms for finding matchings in graphs and bipartite graphs.

This group contains the algorithms for calculating

matchings in graphs and bipartite graphs. The general matching problem is

finding a subset of the edges for which each node has at most one incident

edge.

There are several different algorithms for calculate matchings in

graphs.  The matching problems in bipartite graphs are generally

easier than in general graphs. The goal of the matching optimization

can be finding maximum cardinality, maximum weight or minimum cost

matching. The search can be constrained to find perfect or

maximum cardinality matching.

The matching algorithms implemented in LEMON:

- \ref MaxBipartiteMatching Hopcroft-Karp augmenting path algorithm

  for calculating maximum cardinality matching in bipartite graphs.

- \ref PrBipartiteMatching Push-relabel algorithm

  for calculating maximum cardinality matching in bipartite graphs.

- \ref MaxWeightedBipartiteMatching

  Successive shortest path algorithm for calculating maximum weighted

  matching and maximum weighted bipartite matching in bipartite graphs.

- \ref MinCostMaxBipartiteMatching

  Successive shortest path algorithm for calculating minimum cost maximum

  matching in bipartite graphs.

- \ref MaxMatching Edmond's blossom shrinking algorithm for calculating

  maximum cardinality matching in general graphs.

- \ref MaxWeightedMatching Edmond's blossom shrinking algorithm for calculating

  maximum weighted matching in general graphs.

- \ref MaxWeightedPerfectMatching

  Edmond's blossom shrinking algorithm for calculating maximum weighted

  perfect matching in general graphs.

\image html bipartite_matching.png

\image latex bipartite_matching.eps "Bipartite Matching" width=\textwidth

*/

/**

@defgroup graph_properties Connectivity and Other Graph Properties

@ingroup algs

\brief Algorithms for discovering the graph properties

This group contains the algorithms for discovering the graph properties

like connectivity, bipartiteness, euler property, simplicity etc.

\image html connected_components.png

\image latex connected_components.eps "Connected components" width=\textwidth

*/

/**

@defgroup planar Planarity Embedding and Drawing

@ingroup algs

\brief Algorithms for planarity checking, embedding and drawing

This group contains the algorithms for planarity checking,

embedding and drawing.

\image html planar.png

\image latex planar.eps "Plane graph" width=\textwidth

*/

/**

@defgroup approx Approximation Algorithms

@ingroup algs

\brief Approximation algorithms.

This group contains the approximation and heuristic algorithms

implemented in LEMON.

*/

/**

@defgroup auxalg Auxiliary Algorithms

@ingroup algs

\brief Auxiliary algorithms implemented in LEMON.

This group contains some algorithms implemented in LEMON

in order to make it easier to implement complex algorithms.

*/

/**

@defgroup gen_opt_group General Optimization Tools

\brief This group contains some general optimization frameworks

implemented in LEMON.

This group contains some general optimization frameworks

implemented in LEMON.

*/

/**

@defgroup lp_group LP and MIP Solvers

@ingroup gen_opt_group

\brief LP and MIP solver interfaces for LEMON.

This group contains LP and MIP solver interfaces for LEMON.

Various LP solvers could be used in the same manner with this

high-level interface.

The currently supported solvers are \ref glpk, \ref clp, \ref cbc,

\ref cplex, \ref soplex.

*/

/**

@defgroup lp_utils Tools for Lp and Mip Solvers

@ingroup lp_group

\brief Helper tools to the Lp and Mip solvers.

This group adds some helper tools to general optimization framework

implemented in LEMON.

*/

/**

@defgroup metah Metaheuristics

@ingroup gen_opt_group

\brief Metaheuristics for LEMON library.

/* -*- 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 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

  /// 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.

  ///

  /// \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<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();

/* -*- 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 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.

#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

  /// 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 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.

  ///

  /// \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

      /// 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,