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/* -*- C++ -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

/**

@defgroup datas Data Structures

This group describes the several data structures implemented in LEMON.

*/

/**

@defgroup graphs Graph Structures

@ingroup datas

\brief Graph structures implemented in LEMON.

The implementation of combinatorial algorithms heavily relies on 

efficient graph implementations. LEMON offers data structures which are 

planned to be easily used in an experimental phase of implementation studies, 

and thereafter the program code can be made efficient by small modifications. 

The most efficient implementation of diverse applications require the

usage of different physical graph implementations. These differences

appear in the size of graph we require to handle, memory or time usage

limitations or in the set of operations through which the graph can be

accessed.  LEMON provides several physical graph structures to meet

the diverging requirements of the possible users.  In order to save on

running time or on memory usage, some structures may fail to provide

some graph features like edge or node deletion.

Alteration of standard containers need a very limited number of 

operations, these together satisfy the everyday requirements. 

In the case of graph structures, different operations are needed which do 

not alter the physical graph, but gives another view. If some nodes or 

edges have to be hidden or the reverse oriented graph have to be used, then 

this is the case. It also may happen that in a flow implementation 

the residual graph can be accessed by another algorithm, or a node-set 

is to be shrunk for another algorithm. 

LEMON also provides a variety of graphs for these requirements called 

\ref graph_adaptors "graph adaptors". Adaptors cannot be used alone but only 

in conjunction with other graph representations. 

You are free to use the graph structure that fit your requirements

the best, most graph algorithms and auxiliary data structures can be used

with any graph structures. 

*/

/**

@defgroup semi_adaptors Semi-Adaptor Classes for Graphs

@ingroup graphs

\brief Graph types between real graphs and graph adaptors.

This group describes some graph types between real graphs and graph adaptors.

These classes wrap graphs to give new functionality as the adaptors do it. 

On the other hand they are not light-weight structures as the adaptors.

*/

/**

@defgroup maps Maps 

@ingroup datas

\brief Map structures implemented in LEMON.

This group describes the map structures implemented in LEMON.

LEMON provides several special purpose maps that e.g. combine

new maps from existing ones.

*/

/**

@defgroup graph_maps Graph Maps 

@ingroup maps

\brief Special Graph-Related Maps.

This group describes maps that are specifically designed to assign

values to the nodes and edges of graphs.

*/

/**

\defgroup map_adaptors Map Adaptors

\ingroup maps

\brief Tools to create new maps from existing ones

This group describes map adaptors that are used to create "implicit"

maps from other maps.

Most of them are \ref lemon::concepts::ReadMap "ReadMap"s. They can

make arithmetic operations between one or two maps (negation, scaling,

addition, multiplication etc.) or e.g. convert a map to another one

of different Value type.

The typical usage of this classes is passing implicit maps to

algorithms.  If a function type algorithm is called then the function

type map adaptors can be used comfortable. For example let's see the

usage of map adaptors with the \c graphToEps() function:

\code

  Color nodeColor(int deg) {

    if (deg >= 2) {

      return Color(0.5, 0.0, 0.5);

    } else if (deg == 1) {

      return Color(1.0, 0.5, 1.0);

    } else {

      return Color(0.0, 0.0, 0.0);

    }

  }

  Graph::NodeMap<int> degree_map(graph);

  graphToEps(graph, "graph.eps")

    .coords(coords).scaleToA4().undirected()

    .nodeColors(composeMap(functorMap(nodeColor), degree_map)) 

    .run();

\endcode 

The \c functorMap() function makes an \c int to \c Color map from the

\e nodeColor() function. The \c composeMap() compose the \e degree_map

and the previous created map. The composed map is proper function to

get color of each node.

The usage with class type algorithms is little bit harder. In this

case the function type map adaptors can not be used, because the

function map adaptors give back temporary objects.

\code

  Graph graph;

  typedef Graph::EdgeMap<double> DoubleEdgeMap;

  DoubleEdgeMap length(graph);

  DoubleEdgeMap speed(graph);

  typedef DivMap<DoubleEdgeMap, DoubleEdgeMap> TimeMap;

  TimeMap time(length, speed);

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

  dijkstra.run(source, target);

\endcode

We have a length map and a maximum speed map on a graph. The minimum

time to pass the edge 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 matrices Matrices 

@ingroup datas

\brief Two dimensional data storages implemented in LEMON.

This group describes two dimensional data storages implemented in LEMON.

*/

/**

@defgroup paths Path Structures

@ingroup datas

\brief Path structures implemented in LEMON.

This group describes 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 lemon::concepts::Path

*/

/**

@defgroup auxdat Auxiliary Data Structures

@ingroup datas

\brief Auxiliary data structures implemented in LEMON.

This group describes some data structures implemented in LEMON in

order to make it easier to implement combinatorial algorithms.

*/

/**

@defgroup algs Algorithms

\brief This group describes the several algorithms

implemented in LEMON.

This group describes the several algorithms

implemented in LEMON.

*/

/**

@defgroup search Graph Search

@ingroup algs

\brief Common graph search algorithms.

This group describes the common graph search algorithms like 

Breadth-first search (Bfs) and Depth-first search (Dfs).

*/

/**

@defgroup shortest_path Shortest Path algorithms

@ingroup algs

\brief Algorithms for finding shortest paths.

This group describes the algorithms for finding shortest paths in graphs.

*/

/** 

@defgroup max_flow Maximum Flow algorithms 

@ingroup algs 

\brief Algorithms for finding maximum flows.

This group describes the algorithms for finding maximum flows and

feasible circulations.

The maximum flow problem is to find a flow between a single source and

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

directed graph, an \f$c_a:A\rightarrow\mathbf{R}^+_0\f$ capacity

function and given \f$s, t \in V\f$ source and target node. The

maximum flow is the \f$f_a\f$ solution of the next optimization problem:

\f[ 0 \le f_a \le c_a \f]

\f[ \sum_{v\in\delta^{-}(u)}f_{vu}=\sum_{v\in\delta^{+}(u)}f_{uv} \qquad \forall u \in V \setminus \{s,t\}\f]

\f[ \max \sum_{v\in\delta^{+}(s)}f_{uv} - \sum_{v\in\delta^{-}(s)}f_{vu}\f]

LEMON contains several algorithms for solving maximum flow problems:

- \ref lemon::EdmondsKarp "Edmonds-Karp" 

- \ref lemon::Preflow "Goldberg's Preflow algorithm"

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

- \ref lemon::GoldbergTarjan "Preflow algorithm with dynamic trees"

In most cases the \ref lemon::Preflow "Preflow" algorithm provides the

fastest method to compute the maximum flow. All impelementations

provides functions to query the minimum cut, which is the dual linear

programming problem of the maximum flow.

*/

/**

@defgroup min_cost_flow Minimum Cost Flow algorithms

@ingroup algs

\brief Algorithms for finding minimum cost flows and circulations.

This group describes the algorithms for finding minimum cost flows and

circulations.  

*/

/**

@defgroup min_cut Minimum Cut algorithms 

@ingroup algs 

\brief Algorithms for finding minimum cut in graphs.

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

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

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

outgoing arcs. Formally, there is \f$G=(V,A)\f$ directed graph, an

\f$c_a:A\rightarrow\mathbf{R}^+_0\f$ 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}c_{uv}\f]

LEMON contains several algorithms related to minimum cut problems:

- \ref lemon::HaoOrlin "Hao-Orlin algorithm" to calculate minimum cut

  in directed graphs  

- \ref lemon::NagamochiIbaraki "Nagamochi-Ibaraki algorithm" to

  calculate minimum cut in undirected graphs

- \ref lemon::GomoryHuTree "Gomory-Hu tree computation" to calculate all

  pairs minimum cut in undirected graphs

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

please see the \ref max_flow "Maximum Flow page".

*/

/**

@defgroup graph_prop Connectivity and other graph properties

@ingroup algs

\brief Algorithms for discovering the graph properties

This group describes the algorithms for discovering the graph properties

like connectivity, bipartiteness, euler property, simplicity etc.

\image html edge_biconnected_components.png

\image latex edge_biconnected_components.eps "bi-edge-connected components" width=\textwidth

*/

/**

@defgroup planar Planarity embedding and drawing

@ingroup algs

\brief Algorithms for planarity checking, embedding and drawing

This group describes the algorithms for planarity checking, embedding and drawing.

\image html planar.png

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

*/

/**

@defgroup matching Matching algorithms 

@ingroup algs

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

This group contains algorithm objects and functions to calculate

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

finding a subset of the edges which does not shares common endpoints.

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 the finding maximum cardinality, maximum weight or minimum cost

matching. The search can be constrained to find perfect or

maximum cardinality matching.

Lemon contains the next algorithms:

- \ref lemon::MaxBipartiteMatching "MaxBipartiteMatching" Hopcroft-Karp 

  augmenting path algorithm for calculate maximum cardinality matching in 

  bipartite graphs

- \ref lemon::PrBipartiteMatching "PrBipartiteMatching" Push-Relabel 

  algorithm for calculate maximum cardinality matching in bipartite graphs 

- \ref lemon::MaxWeightedBipartiteMatching "MaxWeightedBipartiteMatching" 

  Successive shortest path algorithm for calculate maximum weighted matching 

  and maximum weighted bipartite matching in bipartite graph

- \ref lemon::MinCostMaxBipartiteMatching "MinCostMaxBipartiteMatching" 

  Successive shortest path algorithm for calculate minimum cost maximum 

  matching in bipartite graph

- \ref lemon::MaxMatching "MaxMatching" Edmond's blossom shrinking algorithm

  for calculate maximum cardinality matching in general graph

- \ref lemon::MaxWeightedMatching "MaxWeightedMatching" Edmond's blossom

  shrinking algorithm for calculate maximum weighted matching in general

  graph

- \ref lemon::MaxWeightedPerfectMatching "MaxWeightedPerfectMatching"

  Edmond's blossom shrinking algorithm for calculate maximum weighted

  perfect matching in general graph

\image html bipartite_matching.png

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

*/

/**

@defgroup spantree Minimum Spanning Tree algorithms

@ingroup algs

\brief Algorithms for finding a minimum cost spanning tree in a graph.

This group describes the algorithms for finding a minimum cost spanning

tree in a graph

*/

/**

@defgroup auxalg Auxiliary algorithms

@ingroup algs

\brief Auxiliary algorithms implemented in LEMON.

This group describes some algorithms implemented in LEMON

in order to make it easier to implement complex algorithms.

*/

/**

@defgroup approx Approximation algorithms

\brief Approximation algorithms.

This group describes the approximation and heuristic algorithms

implemented in LEMON.

*/

/**

@defgroup gen_opt_group General Optimization Tools

\brief This group describes some general optimization frameworks

implemented in LEMON.

This group describes 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 describes Lp and Mip solver interfaces for LEMON. The

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

interface.

*/

/** 

@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 describes 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 describes some simple basic graph utilities.

*/

/**

@defgroup misc Miscellaneous Tools

@ingroup utils

\brief Tools for development, debugging and testing.

This group describes 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 describes simple tools for measuring the performance

of algorithms.

*/

/**

@defgroup graphbits Tools for Graph Implementation

@ingroup utils

\brief Tools to make it easier to create graphs.

This group describes the tools that makes it easier to create graphs and

the maps that dynamically update with the graph changes.

*/

/**

@defgroup exceptions Exceptions

@ingroup utils

\brief Exceptions defined in LEMON.

This group describes the exceptions defined in LEMON.

*/

/**

@defgroup io_group Input-Output

\brief Graph Input-Output methods

This group describes the tools for importing and exporting graphs 

and graph related data. Now it supports the LEMON format, the

\c DIMACS format and the encapsulated postscript (EPS) format.

*/

/**

@defgroup lemon_io Lemon Input-Output

@ingroup io_group

\brief Reading and writing LEMON format

This group describes methods for reading and writing LEMON format. 

You can find more about this format on the \ref graph-io-page "Graph Input-Output"

tutorial pages.

*/

/**

@defgroup section_io Section readers and writers

@ingroup lemon_io

*/

/**

@ingroup lemon_io

*/

/**

@defgroup eps_io Postscript exporting

@ingroup io_group

\brief General \c EPS drawer and graph exporter

This group describes general \c EPS drawing methods and special

graph exporting tools. 

*/

/**

@defgroup concept Concepts

\brief Skeleton classes and concept checking classes

This group describes 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 describes the skeletons and concept checking classes of LEMON's

graph structures and helper classes used to implement these.

*/

/* --- Unused group

@defgroup experimental Experimental Structures and Algorithms

This group describes some Experimental structures and algorithms.

The stuff here is subject to change.

*/

/**

\anchor demoprograms

@defgroup demos Demo programs

Some demo programs are listed here. Their full source codes can be found in

the \c demo subdirectory of the source tree.

It order to compile them, use <tt>--enable-demo</tt> configure option when

build the library.

*/

/**

@defgroup tools Standalone utility applications

Some utility applications are listed here. 

The standard compilation procedure (<tt>./configure;make</tt>) will compile

them, as well. 

*/

/* -*- 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_TOLERANCE_H

#define LEMON_TOLERANCE_H

///\ingroup misc

///\file

///\brief A basic tool to handle the anomalies of calculation with

///floating point numbers.

///

///\todo It should be in a module like "Basic tools"

namespace lemon {

  /// \addtogroup misc

  /// @{

  ///\brief A class to provide a basic way to

  ///handle the comparison of numbers that are obtained

  ///as a result of a probably inexact computation.

  ///

  ///\ref Tolerance is a class to provide a basic way to

  ///handle the comparison of numbers that are obtained

  ///as a result of a probably inexact computation.

  ///

  ///

  ///\sa Tolerance<float>

  ///\sa Tolerance<double>

  ///\sa Tolerance<long double>

  ///\sa Tolerance<int>

  ///\sa Tolerance<long long int>

  ///\sa Tolerance<unsigned int>

  ///\sa Tolerance<unsigned long long int>

  template<class T>

  class Tolerance

  {

  public:

    typedef T Value;

    ///\name Comparisons

    ///The concept is that these bool functions return \c true only if

    ///the related comparisons hold even if some numerical error appeared

    ///during the computations.

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) {return false;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) {return false;}

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) {return false;}

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value a) {return false;}

    ///Returns \c true if \c a is \e surely non-zero

    static bool nonZero(Value a) {return false;}

    ///@}

    ///Returns the zero value.

    static Value zero() {return T();}

    //   static bool finite(Value a) {}

    //   static Value big() {}

    //   static Value negativeBig() {}

  };

  ///Float specialization of Tolerance.

  ///Float specialization of Tolerance.

  ///\sa Tolerance

  ///\relates Tolerance

  template<>

  class Tolerance<float>

  {

    static float def_epsilon;

    float _epsilon;

  public:

    ///\e

    typedef float Value;

    ///Constructor setting the epsilon tolerance to the default value.

    Tolerance() : _epsilon(def_epsilon) {}

    ///Constructor setting the epsilon tolerance to the given value.

    Tolerance(float e) : _epsilon(e) {}

    ///Returns the epsilon value.

    Value epsilon() const {return _epsilon;}

    ///Sets the epsilon value.

    void epsilon(Value e) {_epsilon=e;}

    ///Returns the default epsilon value.

    static Value defaultEpsilon() {return def_epsilon;}

    ///Sets the default epsilon value.

    static void defaultEpsilon(Value e) {def_epsilon=e;}

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    bool less(Value a,Value b) const {return a+_epsilon<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    bool different(Value a,Value b) const { return less(a,b)||less(b,a); }

    ///Returns \c true if \c a is \e surely positive

    bool positive(Value a) const { return _epsilon<a; }

    ///Returns \c true if \c a is \e surely negative

    bool negative(Value a) const { return -_epsilon>a; }

    ///Returns \c true if \c a is \e surely non-zero

    bool nonZero(Value a) const { return positive(a)||negative(a); }

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Double specialization of Tolerance.

  ///Double specialization of Tolerance.

  ///\sa Tolerance

  ///\relates Tolerance

  template<>

  class Tolerance<double>

  {

    static double def_epsilon;

    double _epsilon;

  public:

    ///\e

    typedef double Value;

    ///Constructor setting the epsilon tolerance to the default value.

    Tolerance() : _epsilon(def_epsilon) {}

    ///Constructor setting the epsilon tolerance to the given value.

    Tolerance(double e) : _epsilon(e) {}

    ///Returns the epsilon value.

    Value epsilon() const {return _epsilon;}

    ///Sets the epsilon value.

    void epsilon(Value e) {_epsilon=e;}

    ///Returns the default epsilon value.

    static Value defaultEpsilon() {return def_epsilon;}

    ///Sets the default epsilon value.

    static void defaultEpsilon(Value e) {def_epsilon=e;}

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    bool less(Value a,Value b) const {return a+_epsilon<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    bool different(Value a,Value b) const { return less(a,b)||less(b,a); }

    ///Returns \c true if \c a is \e surely positive

    bool positive(Value a) const { return _epsilon<a; }

    ///Returns \c true if \c a is \e surely negative

    bool negative(Value a) const { return -_epsilon>a; }

    ///Returns \c true if \c a is \e surely non-zero

    bool nonZero(Value a) const { return positive(a)||negative(a); }

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Long double specialization of Tolerance.

  ///Long double specialization of Tolerance.

  ///\sa Tolerance

  ///\relates Tolerance

  template<>

  class Tolerance<long double>

  {

    static long double def_epsilon;

    long double _epsilon;

  public:

    ///\e

    typedef long double Value;

    ///Constructor setting the epsilon tolerance to the default value.

    Tolerance() : _epsilon(def_epsilon) {}

    ///Constructor setting the epsilon tolerance to the given value.

    Tolerance(long double e) : _epsilon(e) {}

    ///Returns the epsilon value.

    Value epsilon() const {return _epsilon;}

    ///Sets the epsilon value.

    void epsilon(Value e) {_epsilon=e;}

    ///Returns the default epsilon value.

    static Value defaultEpsilon() {return def_epsilon;}

    ///Sets the default epsilon value.

    static void defaultEpsilon(Value e) {def_epsilon=e;}

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    bool less(Value a,Value b) const {return a+_epsilon<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    bool different(Value a,Value b) const { return less(a,b)||less(b,a); }

    ///Returns \c true if \c a is \e surely positive

    bool positive(Value a) const { return _epsilon<a; }

    ///Returns \c true if \c a is \e surely negative

    bool negative(Value a) const { return -_epsilon>a; }

    ///Returns \c true if \c a is \e surely non-zero

    bool nonZero(Value a) const { return positive(a)||negative(a); }

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Integer specialization of Tolerance.

  ///Integer specialization of Tolerance.

  ///\sa Tolerance

  template<>

  class Tolerance<int>

  {

  public:

    ///\e

    typedef int Value;

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value a) { return 0>a; }

    ///Returns \c true if \c a is \e surely non-zero

    static bool nonZero(Value a) { return a!=0; }

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Unsigned integer specialization of Tolerance.

  ///\sa Tolerance

  template<>

  class Tolerance<unsigned int>

  {

  public:

    ///\e

    typedef unsigned int Value;

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value) { return false; }

    ///Returns \c true if \c a is \e surely non-zero

    static bool nonZero(Value a) { return a!=0; }

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Long integer specialization of Tolerance.

  ///Long integer specialization of Tolerance.

  ///\sa Tolerance

  template<>

  class Tolerance<long int>

  {

  public:

    ///\e

    typedef long int Value;

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value a) { return 0>a; }

    ///Returns \c true if \c a is \e surely non-zero

    static bool nonZero(Value a) { return a!=0;}

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Unsigned long integer specialization of Tolerance.

  ///\sa Tolerance

  template<>

  class Tolerance<unsigned long int>

  {

  public:

    ///\e

    typedef unsigned long int Value;

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value) { return false; }

    ///Returns \c true if \c a is \e surely non-zero

    static bool nonZero(Value a) { return a!=0;}

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

#if defined __GNUC__ && !defined __STRICT_ANSI__

  ///Long long integer specialization of Tolerance.

  ///\warning This class (more exactly, type <tt>long long</tt>)

  ///is not ansi compatible.

  ///\sa Tolerance

  template<>

  class Tolerance<long long int>

  {

  public:

    ///\e

    typedef long long int Value;

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value a) { return 0>a; }

    ///Returns \c true if \c a is \e surely non-zero

    static bool nonZero(Value a) { return a!=0;}

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

  ///Unsigned long long integer specialization of Tolerance.

  ///\warning This class (more exactly, type <tt>unsigned long long</tt>)

  ///is not ansi compatible.

  ///\sa Tolerance

  template<>

  class Tolerance<unsigned long long int>

  {

  public:

    ///\e

    typedef unsigned long long int Value;

    ///\name Comparisons

    ///@{

    ///Returns \c true if \c a is \e surely strictly less than \c b

    static bool less(Value a,Value b) { return a<b;}

    ///Returns \c true if \c a is \e surely different from \c b

    static bool different(Value a,Value b) { return a!=b; }

    ///Returns \c true if \c a is \e surely positive

    static bool positive(Value a) { return 0<a; }

    ///Returns \c true if \c a is \e surely negative

    static bool negative(Value) { return false; }

    ///Returns \c true if \c a is \e surely non-zero

    static bool nonZero(Value a) { return a!=0;}

    ///@}

    ///Returns zero

    static Value zero() {return 0;}

  };

#endif

  /// @}

} //namespace lemon

#endif //LEMON_TOLERANCE_H