RhodeCode

/* -*- 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.

 *

 */

/*!

\page coding_style LEMON Coding Style

\section naming_conv Naming Conventions

In order to make development easier we have made some conventions

according to coding style. These include names of types, classes,

functions, variables, constants and exceptions. If these conventions

are met in one's code then it is easier to read and maintain

it. Please comply with these conventions if you want to contribute

developing LEMON library.

\note When the coding style requires the capitalization of an abbreviation,

only the first letter should be upper case.

\code

XmlReader

\endcode

\warning In some cases we diverge from these rules.

This is primary done because STL uses different naming convention and

in certain cases

it is beneficial to provide STL compatible interface.

\subsection cs-files File Names

The header file names should look like the following.

\code

header_file.h

\endcode

Note that all standard LEMON headers are located in the \c lemon subdirectory,

so you should include them from C++ source like this:

\code

#include <lemon/header_file.h>

\endcode

The source code files use the same style and they have '.cc' extension.

\code

source_code.cc

\endcode

\subsection cs-class Classes and other types

The name of a class or any type should look like the following.

\code

AllWordsCapitalizedWithoutUnderscores

\endcode

\subsection cs-func Methods and other functions

The name of a function should look like the following.

\code

firstWordLowerCaseRestCapitalizedWithoutUnderscores

\endcode

\subsection cs-funcs Constants, Macros

The names of constants and macros should look like the following.

\code

ALL_UPPER_CASE_WITH_UNDERSCORES

\endcode

\subsection cs-loc-var Class and instance member variables, auto variables

The names of class and instance member variables and auto variables

(=variables used locally in methods) should look like the following.

\code

all_lower_case_with_underscores

\endcode

\subsection pri-loc-var Private member variables

\code

\endcode

\subsection cs-excep Exceptions

When writing exceptions please comply the following naming conventions.

\code

ClassNameEndsWithException

\endcode

or

\code

ClassNameEndsWithError

\endcode

\section header-template Template Header File

Each LEMON header file should look like this:

\include template.h

*/

  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 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 CostScaling Cost Scaling algorithm based on push/augment and

   relabel operations \ref goldberg90approximation, \ref goldberg97efficient,

   \ref bunnagel98efficient.

 - \ref CapacityScaling Capacity Scaling algorithm based on the successive

   shortest path method \ref edmondskarp72theoretical.

 - \ref CycleCanceling Cycle-Canceling algorithms, two of which are

   strongly polynomial \ref klein67primal, \ref goldberg89cyclecanceling.

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

*/

/**

@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 KarpMmc Karp's original algorithm \ref amo93networkflows,

  \ref dasdan98minmeancycle.

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

  version of Karp's algorithm \ref dasdan98minmeancycle.

- \ref HowardMmc Howard's policy iteration algorithm

  \ref dasdan98minmeancycle.

most efficient one, though the best known theoretical bound on its running

time is exponential.

Both \ref KarpMmc "Karp" and \ref HartmannOrlinMmc "Hartmann-Orlin" 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.

- \ref MaxFractionalMatching Push-relabel algorithm for calculating

  maximum cardinality fractional matching in general graphs.

- \ref MaxWeightedFractionalMatching Augmenting path algorithm for calculating

  maximum weighted fractional matching in general graphs.

- \ref MaxWeightedPerfectFractionalMatching

  Augmenting path algorithm for calculating maximum weighted

  perfect fractional matching in general graphs.

\image html matching.png

\image latex matching.eps "Min Cost Perfect 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

*/

/**

@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_algs Approximation Algorithms

@ingroup algs

\brief Approximation algorithms.

This group contains the approximation and heuristic algorithms

implemented in LEMON.

<b>Maximum Clique Problem</b>

  - \ref GrossoLocatelliPullanMc An efficient heuristic algorithm of

    Grosso, Locatelli, and Pullan.

*/

/**

@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.

This group contains some metaheuristic optimization tools.

*/

/**

@defgroup utils Tools and Utilities

\brief Tools and utilities for programming in LEMON

Tools and utilities for programming in LEMON.

*/

/**

@defgroup gutils Basic Graph Utilities

@ingroup utils

\brief Simple basic graph utilities.

This group contains some simple basic graph utilities.

*/

/**

@defgroup misc Miscellaneous Tools

@ingroup utils

\brief Tools for development, debugging and testing.

This group contains several useful tools for development,

debugging and testing.

*/

/**

@defgroup timecount Time Measuring and Counting

@ingroup misc

\brief Simple tools for measuring the performance of algorithms.

This group contains simple tools for measuring the performance

of algorithms.

*/

/**

@defgroup exceptions Exceptions

@ingroup utils

\brief Exceptions defined in LEMON.

This group contains the exceptions defined in LEMON.

*/

/**

@defgroup io_group Input-Output

\brief Graph Input-Output methods

This group contains the tools for importing and exporting graphs

and graph related data. Now it supports the \ref lgf-format

"LEMON Graph Format", the \c DIMACS format and the encapsulated

postscript (EPS) format.

*/

/**

@defgroup lemon_io LEMON Graph Format

@ingroup io_group

\brief Reading and writing LEMON Graph Format.

This group contains methods for reading and writing

\ref lgf-format "LEMON Graph Format".

*/

/**

@defgroup eps_io Postscript Exporting

@ingroup io_group

\brief General \c EPS drawer and graph exporter

This group contains general \c EPS drawing methods and special

graph exporting tools.

*/

/**

@defgroup dimacs_group DIMACS Format

@ingroup io_group

\brief Read and write files in DIMACS format

Tools to read a digraph from or write it to a file in DIMACS format data.

*/

/**

@defgroup nauty_group NAUTY Format

@ingroup io_group

\brief Read \e Nauty format

Tool to read graphs from \e Nauty format data.

*/

/**

@defgroup concept Concepts

\brief Skeleton classes and concept checking classes

This group contains the data/algorithm skeletons and concept checking

classes implemented in LEMON.

The purpose of the classes in this group is fourfold.

- These classes contain the documentations of the %concepts. In order

  to avoid document multiplications, an implementation of a concept

  simply refers to the corresponding concept class.

- These classes declare every functions, <tt>typedef</tt>s etc. an

  implementation of the %concepts should provide, however completely

  without implementations and real data structures behind the

  interface. On the other hand they should provide nothing else. All

  the algorithms working on a data structure meeting a certain concept

  should compile with these classes. (Though it will not run properly,

  of course.) In this way it is easily to check if an algorithm

  doesn't use any extra feature of a certain implementation.

- The concept descriptor classes also provide a <em>checker class</em>

  that makes it possible to check whether a certain implementation of a

  concept indeed provides all the required features.

- Finally, They can serve as a skeleton of a new implementation of a concept.

*/

/**

@defgroup graph_concepts Graph Structure Concepts

@ingroup concept

\brief Skeleton and concept checking classes for graph structures

This group contains the skeletons and concept checking classes of

graph structures.

/* -*- 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 \c V and \c C must be signed number types.

  /// \warning All input data (capacities, supply values, and costs) must

  /// be integer.

#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()

    /// 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;

    }

    /// @}