Classes which inherit from this base class are used in implementing bounds on the length of the shortest nonzero vector in a lattice. More...

#include <latticetester/Normalizer.h>

Inherited by LatticeTester::NormaBestBound< RedDbl >, LatticeTester::NormaBestLat< RedDbl >, LatticeTester::NormaLaminated< RedDbl >, LatticeTester::NormaMinkL1< RedDbl >, LatticeTester::NormaMinkowski< RedDbl >, LatticeTester::NormaPalpha< Int, RedDbl >, and LatticeTester::NormaRogers< RedDbl >.

Public Member Functions
	Normalizer (RedDbl &logDensity, int t, std::string Name, NormType norm=L2NORM, double beta=1)
	Complete constructor for a `Normalizer`. More...

	Normalizer (int t, std::string Name, NormType norm=L2NORM, double beta=1)
	Constructor that does not take the density as an argument. More...

virtual	~Normalizer ()
	Destructor. More...

virtual void	init (RedDbl &logDensity, double beta)
	This is a method that will initialize the bounds this normalizer can return. More...

std::string	ToString () const
	Returns a string that describes this object. More...

NormType	getNorm () const
	Returns the norm associated with this object. More...

void	setLogDensity (RedDbl logDensity)
	Sets the log-density associated with this object to `logDensity`. More...

RedDbl	getLogDensity () const
	Returns the `logDensity` associated with this object. More...

void	setNorm (NormType norm)
	Sets the norm associated with this object to `norm`. More...

int	getDim () const
	Returns the maximal dimension for this object. More...

double	getPreComputedBound (int j) const
	Returns the bound for dimension `j` as computed in Normalizer::init(). More...

virtual RedDbl	getBound (int j) const
	Calculates and returns the bound on the length of the shortest nonzero vector in dimension `j`. More...

virtual double	getGamma (int j) const
	Returns the value of a lattice constant \(\gamma\) in dimension \(j\). More...

Static Public Attributes
static const int	MAX_DIM = 48
	The maximum dimension of the lattices for which this class can give an upper bound. More...

Protected Attributes
std::string	m_name
	Name of the normalizer. More...

NormType	m_norm
	Norm associated with this object. More...

RedDbl	m_logDensity
	log of the density, ie log of the number of points of the lattice per unit of volume. More...

int	m_maxDim
	Only elements 1 to `m_maxDim` (inclusive) of m_bounds bellow will be pre-computed. More...

double	m_beta
	Beta factor used to give more or less importance to some of the dimensions. More...

double *	m_bounds
	Contains the bounds on the length of the shortest nonzero vector in the lattice in each dimension. More...

Detailed Description

template<typename RedDbl>
class LatticeTester::Normalizer< RedDbl >

Classes which inherit from this base class are used in implementing bounds on the length of the shortest nonzero vector in a lattice.

Given a lattice in dimension \(t\), it is possible to center non-interlapping spheres of radius \( d_t \) where \( d_t \) is the lenght of the shortest vector in the lattice. It turns out that the proportion of the space covered by these spheres can be used as a figure of merit for both the dual and the primal lattice. Suppose \( V \) contains the basis of the lattice in its lines. Then the density of the lattice is defined by \( \frac{1}{|\text{det}(V)|} \). It is the number of points that the (not rescalled) lattice will have in \([0,1)^t\). It is easy to see that the amount of space that will be covered by the spheres is

\[ \text{Volume of an }n\text{-sphere of radius }d_n\times\text{Density of the lattice}}. \]

There are known upper bounds on the greatest proportion of the space that can be covered in that way [4]. If we divide the proportion of the space covered in the current lattice by such an upper bound, the number obtained is what we call normalized. This means that the number we will get will be somewhere between 0 and 1 and that it can be used to compare the distribution of the points of the lattice with the points of other lattices. Having a value closer to 1 means a more evenly distributed lattice in some way (see the section on the figures of merit to see which tests use this in the intro). This class is intended as an interface to implement such bounds. It is possible that subclasses of this one implement bounds for a different interpretation of this problem, please look at their documentation before using them. The bounds also sometimes use different norms.

This base classe initializes the bounds at 1 and can be used if no normalization can be done, or has to be done. This can be usefull in a few implementations if there is a switch at runtime to instanciate a Normalizer subclass because there will be no need to duplicate the code for the case with no normalization.

To create a subclass of this one, it is important to implement the getGamma(int) const, getBound(int) const and init() methods and to call the init() method in the constructor to pre-compute the bounds this class will use for normalization.

To instanciate this class or its subclasses, the usage of the copy constructor or of the assignment = is prohibited by the fact that these methods are private. The prefered usage for this class is to declare a pointer to a Normalizer and to instanciate subclasses with dynamically allocated memory:

Normalizer<RScal>* norma;
norma = new NormaBestLat<RScal>(logDensity, t);
delete norma;

Constructor & Destructor Documentation

◆ Normalizer() [1/2]

template<typename RedDbl >

LatticeTester::Normalizer< RedDbl >::Normalizer	(	RedDbl &	logDensity,
		int	t,
		std::string	Name,
		NormType	norm = `L2NORM`,
		double	beta = `1`
	)

Complete constructor for a Normalizer.

This will create a normalizer for lattices with a density of \(\exp(\text{logDensity})\) in all dimensions \(\leq t\). Ususally, such a normalizer will return bounds that take into a account the density. Name allows the user to give a name to a normalizer object. This name will be printed by the ToString() method. It serves no purpose implementation-wise, but this can be usefull while debugging code. norm is the NormType that will be used by this object. It cannot be changed. The usage of beta is deprecated. This is a bias factor that can be usefull in the case where a figure of merit with numerous projections is computed. It can be used to give more weight to the first dimensions by taking \(\beta< 1\). It inflates the figures of merit by \((1/\beta)^t\), thus weakening the requirements for good results in large dimensions in a worst-case figure of merit. One normally uses \(\beta= 1\).

Note that the log value of the density is stored (instead of the density itself) so it is easier to manipulate really large values of density.

Remarks: Richard: Je crois que ce facteur beta devrait disparaître car des poids beaucoup plus généraux sont maintenant implantés dans les classes *Weights. Marc-Antoine: Je crois que le design de cette classe est à repenser. Il est probablement intéressant de considérer la logDensity en tableaux parce que dans certaines applications la densité change en fonction de la dimension.

◆ Normalizer() [2/2]

template<typename RedDbl >

LatticeTester::Normalizer< RedDbl >::Normalizer	(	int	t,
		std::string	Name,
		NormType	norm = `L2NORM`,
		double	beta = `1`
	)

Constructor that does not take the density as an argument.

The fields are essentially the same as for Normalizer(RedDbl, int, std::string, NormType, double)

This is only used in the case of rank 1 lattices in the NormaPalpha class with a prime density.

◆ ~Normalizer()

template<typename RedDbl >

virtual LatticeTester::Normalizer< RedDbl >::~Normalizer ( )

inlinevirtual

Destructor.

Member Function Documentation

◆ getBound()

template<typename RedDbl >

RedDbl LatticeTester::Normalizer< RedDbl >::getBound ( int j ) const

virtual

Calculates and returns the bound on the length of the shortest nonzero vector in dimension j.

◆ getDim()

template<typename RedDbl >

int LatticeTester::Normalizer< RedDbl >::getDim ( ) const

inline

Returns the maximal dimension for this object.

This is the t parameter of the constructors.

◆ getGamma()

template<typename RedDbl >

double LatticeTester::Normalizer< RedDbl >::getGamma ( int j ) const

virtual

Returns the value of a lattice constant \(\gamma\) in dimension \(j\).

These constants can be used by subclasses to implement the init() and the getBound(int) const methods. For this base class, always returns 1.

Reimplemented in LatticeTester::NormaRogers< RedDbl >, LatticeTester::NormaLaminated< RedDbl >, LatticeTester::NormaBestLat< RedDbl >, LatticeTester::NormaBestBound< RedDbl >, LatticeTester::NormaMinkL1< RedDbl >, and LatticeTester::NormaMinkowski< RedDbl >.

◆ getLogDensity()

template<typename RedDbl >

RedDbl LatticeTester::Normalizer< RedDbl >::getLogDensity ( ) const

inline

Returns the logDensity associated with this object.

◆ getNorm()

template<typename RedDbl >

NormType LatticeTester::Normalizer< RedDbl >::getNorm ( ) const

inline

Returns the norm associated with this object.

◆ getPreComputedBound()

template<typename RedDbl >

double LatticeTester::Normalizer< RedDbl >::getPreComputedBound ( int j ) const

Returns the bound for dimension j as computed in Normalizer::init().

◆ init()

template<typename RedDbl >

void LatticeTester::Normalizer< RedDbl >::init	(	RedDbl &	logDensity,
		double	beta
	)

virtual

This is a method that will initialize the bounds this normalizer can return.

This will change the logDensity and the beta variables that are stored in this object. This will compute bounds for all dimensions smaller than t (the parameter passed to the constructors) that can be retrived with the getPreComputedBounds(int) method.

◆ setLogDensity()

template<typename RedDbl >

void LatticeTester::Normalizer< RedDbl >::setLogDensity ( RedDbl logDensity )

inline

Sets the log-density associated with this object to logDensity.

◆ setNorm()

template<typename RedDbl >

void LatticeTester::Normalizer< RedDbl >::setNorm ( NormType norm )

inline

Sets the norm associated with this object to norm.

◆ ToString()

template<typename RedDbl >

std::string LatticeTester::Normalizer< RedDbl >::ToString ( ) const

Returns a string that describes this object.

Member Data Documentation

◆ m_beta

template<typename RedDbl >

double LatticeTester::Normalizer< RedDbl >::m_beta

protected

Beta factor used to give more or less importance to some of the dimensions.

◆ m_bounds

template<typename RedDbl >

double* LatticeTester::Normalizer< RedDbl >::m_bounds

protected

Contains the bounds on the length of the shortest nonzero vector in the lattice in each dimension.

This array is initialized by the init() method, and it's values are returned with getPreComputedBound(int).

◆ m_logDensity

template<typename RedDbl >

RedDbl LatticeTester::Normalizer< RedDbl >::m_logDensity

protected

log of the density, ie log of the number of points of the lattice per unit of volume.

◆ m_maxDim

template<typename RedDbl >

int LatticeTester::Normalizer< RedDbl >::m_maxDim

protected

Only elements 1 to m_maxDim (inclusive) of m_bounds bellow will be pre-computed.

This stores the t parameter of the constructors.

◆ m_name

template<typename RedDbl >

std::string LatticeTester::Normalizer< RedDbl >::m_name

protected

Name of the normalizer.

◆ m_norm

template<typename RedDbl >

NormType LatticeTester::Normalizer< RedDbl >::m_norm

protected

Norm associated with this object.

◆ MAX_DIM

template<typename RedDbl >

const int LatticeTester::Normalizer< RedDbl >::MAX_DIM = 48

static

The maximum dimension of the lattices for which this class can give an upper bound.

The documentation for this class was generated from the following file:

include/latticetester/Normalizer.h

Public Member Functions

Static Public Attributes

Protected Attributes

Detailed Description

template<typename RedDbl> class LatticeTester::Normalizer< RedDbl >

Constructor & Destructor Documentation

◆ Normalizer() [1/2]

◆ Normalizer() [2/2]

◆ ~Normalizer()

Member Function Documentation

◆ getBound()

◆ getDim()

◆ getGamma()

◆ getLogDensity()

◆ getNorm()

◆ getPreComputedBound()

◆ init()

◆ setLogDensity()

◆ setNorm()

◆ ToString()

Member Data Documentation

◆ m_beta

◆ m_bounds

◆ m_logDensity

◆ m_maxDim

◆ m_name

◆ m_norm

◆ MAX_DIM

template<typename RedDbl>
class LatticeTester::Normalizer< RedDbl >