This fraction is a model of CPositiveIrreducibleFraction. More...

#include <DGtal/arithmetic/SternBrocot.h>

Public Types
typedef TInteger	Integer

typedef TQuotient	Quotient

typedef SternBrocot< TInteger, TQuotient >	SternBrocotTree

typedef SternBrocotTree::Fraction	Self

typedef NumberTraits< Integer >::UnsignedVersion	UnsignedInteger

typedef std::pair< Quotient, Quotient >	Value

typedef std::vector< Quotient >	CFracSequence

typedef InputIteratorWithRankOnSequence< CFracSequence, Quotient >	ConstIterator

typedef Value	value_type

typedef ConstIterator	const_iterator

typedef const value_type &	const_reference

Public Member Functions
	Fraction (Integer aP, Integer aQ, Fraction ancestor=SternBrocotTree::zeroOverOne())

	Fraction (Node *sb_node=0)

	Fraction (const Self &other)

Self &	operator= (const Self &other)

bool	null () const

Integer	p () const

Integer	q () const

Quotient	u () const

Quotient	k () const

Fraction	left () const

Fraction	right () const

bool	even () const

bool	odd () const

Fraction	father () const

Fraction	father (Quotient m) const

Fraction	previousPartial () const

Fraction	inverse () const

Fraction	partial (Quotient kp) const

Fraction	reduced (Quotient i) const

void	push_back (const std::pair< Quotient, Quotient > &quotient)

void	pushBack (const std::pair< Quotient, Quotient > &quotient)

void	getSplit (Fraction &f1, Fraction &f2) const

void	getSplitBerstel (Fraction &f1, Quotient &nb1, Fraction &f2, Quotient &nb2) const

void	getCFrac (std::vector< Quotient > &quotients) const

bool	equals (Integer p1, Integer q1) const

bool	lessThan (Integer p1, Integer q1) const

bool	moreThan (Integer p1, Integer q1) const

bool	operator== (const Fraction &other) const

bool	operator!= (const Fraction &other) const

bool	operator< (const Fraction &other) const

bool	operator> (const Fraction &other) const

void	selfDisplay (std::ostream &out) const

ConstIterator	begin () const

ConstIterator	end () const

Fraction	median (const Fraction &g) const

Fraction	simplestFractionInBetween (const Fraction &other) const

Private Attributes
Node *	myNode

Detailed Description

template<typename TInteger, typename TQuotient = int32_t>
class DGtal::SternBrocot< TInteger, TQuotient >::Fraction

This fraction is a model of CPositiveIrreducibleFraction.

It represents a positive irreducible fraction, i.e. some p/q qith gcd(p,q)=1. It is an inner class of SternBrocot. This representation of a fraction is simply a pointer to the corresponding node in this tree.

Definition at line 148 of file SternBrocot.h.

Member Typedef Documentation

◆ CFracSequence

template<typename TInteger , typename TQuotient = int32_t>

typedef std::vector<Quotient> DGtal::SternBrocot< TInteger, TQuotient >::Fraction::CFracSequence

Definition at line 156 of file SternBrocot.h.

◆ const_iterator

template<typename TInteger , typename TQuotient = int32_t>

typedef ConstIterator DGtal::SternBrocot< TInteger, TQuotient >::Fraction::const_iterator

Definition at line 161 of file SternBrocot.h.

◆ const_reference

template<typename TInteger , typename TQuotient = int32_t>

typedef const value_type& DGtal::SternBrocot< TInteger, TQuotient >::Fraction::const_reference

Definition at line 162 of file SternBrocot.h.

◆ ConstIterator

template<typename TInteger , typename TQuotient = int32_t>

typedef InputIteratorWithRankOnSequence<CFracSequence,Quotient> DGtal::SternBrocot< TInteger, TQuotient >::Fraction::ConstIterator

Definition at line 157 of file SternBrocot.h.

◆ Integer

template<typename TInteger , typename TQuotient = int32_t>

typedef TInteger DGtal::SternBrocot< TInteger, TQuotient >::Fraction::Integer

Definition at line 150 of file SternBrocot.h.

◆ Quotient

template<typename TInteger , typename TQuotient = int32_t>

typedef TQuotient DGtal::SternBrocot< TInteger, TQuotient >::Fraction::Quotient

Definition at line 151 of file SternBrocot.h.

◆ Self

template<typename TInteger , typename TQuotient = int32_t>

typedef SternBrocotTree::Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::Self

Definition at line 153 of file SternBrocot.h.

◆ SternBrocotTree

template<typename TInteger , typename TQuotient = int32_t>

typedef SternBrocot<TInteger,TQuotient> DGtal::SternBrocot< TInteger, TQuotient >::Fraction::SternBrocotTree

Definition at line 152 of file SternBrocot.h.

◆ UnsignedInteger

template<typename TInteger , typename TQuotient = int32_t>

typedef NumberTraits<Integer>::UnsignedVersion DGtal::SternBrocot< TInteger, TQuotient >::Fraction::UnsignedInteger

Definition at line 154 of file SternBrocot.h.

◆ Value

template<typename TInteger , typename TQuotient = int32_t>

typedef std::pair<Quotient, Quotient> DGtal::SternBrocot< TInteger, TQuotient >::Fraction::Value

Definition at line 155 of file SternBrocot.h.

◆ value_type

template<typename TInteger , typename TQuotient = int32_t>

typedef Value DGtal::SternBrocot< TInteger, TQuotient >::Fraction::value_type

Definition at line 160 of file SternBrocot.h.

Constructor & Destructor Documentation

◆ Fraction() [1/3]

template<typename TInteger , typename TQuotient = int32_t>

DGtal::SternBrocot< TInteger, TQuotient >::Fraction::Fraction	(	Integer	aP,
		Integer	aQ,
		Fraction	ancestor = SternBrocotTree::zeroOverOne() )

Any fraction p/q. Complexity is in \( \sum_i u_i \), where u_i are the partial quotients of p/q.

Parameters

aP	the numerator (>=0)
aQ	the denominator (>=0)
ancestor	(optional) any ancestor of aP/aQ in the tree (for speed-up).

Construct the corresponding fraction in the Stern-Brocot tree.

NB: Complexity is bounded by \( 2 \sum_i u_i \), where u_i are the partial quotients of aP/aQ.

◆ Fraction() [2/3]

template<typename TInteger , typename TQuotient = int32_t>

DGtal::SternBrocot< TInteger, TQuotient >::Fraction::Fraction ( Node * sb_node = 0 )

Default constructor.

Parameters

sb_node the associated node (or 0 for null fraction).

◆ Fraction() [3/3]

template<typename TInteger , typename TQuotient = int32_t>

DGtal::SternBrocot< TInteger, TQuotient >::Fraction::Fraction ( const Self & other )

Copy constructor.

Parameters

other the object to clone.

Member Function Documentation

◆ begin()

template<typename TInteger , typename TQuotient = int32_t>

ConstIterator DGtal::SternBrocot< TInteger, TQuotient >::Fraction::begin ( ) const

Returns: a const iterator pointing on the beginning of the sequence of quotients of this fraction. NB: \( O(\sum_i u_i) \) operation.

◆ end()

template<typename TInteger , typename TQuotient = int32_t>

ConstIterator DGtal::SternBrocot< TInteger, TQuotient >::Fraction::end ( ) const

Returns: a const iterator pointing after the end of the sequence of quotients of this fraction. NB: O(1) operation.

◆ equals()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::equals	(	Integer	p1,
		Integer	q1 ) const

Parameters

p1	a numerator.
q1	a denominator.

Returns: 'true' if this is the fraction p1/q1.

◆ even()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::even ( ) const

Returns: 'true' if it is an even fraction, i.e. its depth k() is even.

◆ father() [1/2]

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::father ( ) const

Returns: the father of this fraction in O(1), ie [u0,...,uk] => [u0,...,uk - 1]

◆ father() [2/2]

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::father ( Quotient m ) const

Parameters

m	a quotient between 1 and uk-1.

Returns: a given father of this fraction in O(uk - m), ie [u0,...,uk] => [u0,...,m]

Todo: Do it in O(1)... but require to change the data structure.

◆ getCFrac()

template<typename TInteger , typename TQuotient = int32_t>

void DGtal::SternBrocot< TInteger, TQuotient >::Fraction::getCFrac ( std::vector< Quotient > & quotients ) const

Parameters

quotients (returns) the coefficients of the continued fraction of 'this'.

◆ getSplit()

template<typename TInteger , typename TQuotient = int32_t>

void DGtal::SternBrocot< TInteger, TQuotient >::Fraction::getSplit	(	Fraction &	f1,
		Fraction &	f2 ) const

Splitting formula, O(1) time complexity. This fraction should not be 0/1 or 1/0. NB: 'this' = [f1] \(\oplus\) [f2].

Parameters

f1	(returns) the left part of the split.
f2	(returns) the right part of the split.

◆ getSplitBerstel()

template<typename TInteger , typename TQuotient = int32_t>

void DGtal::SternBrocot< TInteger, TQuotient >::Fraction::getSplitBerstel	(	Fraction &	f1,
		Quotient &	nb1,
		Fraction &	f2,
		Quotient &	nb2 ) const

Berstel splitting formula, O(1) time complexity. This fraction should not be 0/1 or 1/0. NB: 'this' = nb1*[f1] \(\oplus\) nb2*[f2]. Also, if 'this->k' is even then nb1=1, otherwise nb2=1.

Parameters

f1	(returns) the left part of the split (left pattern).
nb1	(returns) the number of repetition of the left pattern
f2	(returns) the right part of the split (right pattern).
nb2	(returns) the number of repetition of the right pattern

◆ inverse()

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::inverse ( ) const

Returns: the inverse of this fraction in O(1), ie [u0,...,uk] => [0,u0,...,uk] or [0,u0,...,uk] => [u0,...,uk].

◆ k()

template<typename TInteger , typename TQuotient = int32_t>

Quotient DGtal::SternBrocot< TInteger, TQuotient >::Fraction::k ( ) const

Returns: its depth (1+number of coefficients of its continued fraction).

◆ left()

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::left ( ) const

Returns: its left descendant (construct it if it does not exist yet).

◆ lessThan()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::lessThan	(	Integer	p1,
		Integer	q1 ) const

Parameters

p1	a numerator.
q1	a denominator.

Returns: 'true' if this is < to the fraction p/q.

◆ median()

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::median ( const Fraction & g ) const

Parameters

g	any fraction

Returns: the median of 'this' and g

◆ moreThan()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::moreThan	(	Integer	p1,
		Integer	q1 ) const

Parameters

p1	a numerator.
q1	a denominator.

Returns: 'true' if this is > to the fraction p1/q1.

◆ null()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::null ( ) const

Returns: 'true' iff it is the null fraction 0/0.

◆ odd()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::odd ( ) const

Returns: 'true' if it is an odd fraction, i.e. its depth k() is odd.

◆ operator!=()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::operator!= ( const Fraction & other ) const

Parameters

other any fraction.

Returns: 'true' iff this is different from other.

◆ operator<()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::operator< ( const Fraction & other ) const

Parameters

other any fraction.

Returns: 'true' iff this is < to other.

◆ operator=()

template<typename TInteger , typename TQuotient = int32_t>

Self & DGtal::SternBrocot< TInteger, TQuotient >::Fraction::operator= ( const Self & other )

Assignment

Parameters

other the object to clone.

Returns: a reference to 'this'.

◆ operator==()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::operator== ( const Fraction & other ) const

Parameters

other any fraction.

Returns: 'true' iff this is equal to other.

◆ operator>()

template<typename TInteger , typename TQuotient = int32_t>

bool DGtal::SternBrocot< TInteger, TQuotient >::Fraction::operator> ( const Fraction & other ) const

Parameters

other any fraction.

Returns: 'true' iff this is > to other.

◆ p()

template<typename TInteger , typename TQuotient = int32_t>

Integer DGtal::SternBrocot< TInteger, TQuotient >::Fraction::p ( ) const

Returns: its numerator;

◆ partial()

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::partial ( Quotient kp ) const

Parameters

kp	the chosen depth of the partial fraction (kp <= k()).

Returns: the partial fraction of depth kp, ie. [u0,...,uk] => [u0,...,ukp]

◆ previousPartial()

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::previousPartial ( ) const

Returns: the previous partial of this fraction in O(1), ie [u0,...,u{k-1},uk] => [u0,...,u{k-1}]. Otherwise said, it is its ascendant with a smaller depth.

◆ push_back()

template<typename TInteger , typename TQuotient = int32_t>

void DGtal::SternBrocot< TInteger, TQuotient >::Fraction::push_back ( const std::pair< Quotient, Quotient > & quotient )

Modifies this fraction \([u_0,...,u_k]\) to obtain the fraction \([u_0,...,u_k,m]\). The depth of the quotient must be given, since continued fractions have two writings \([u_0,...,u_k]\) and \([u_0,...,u_k - 1, 1]\).

Useful to create output iterators, for instance with

typedef ... Fraction; 
Fraction f;
std::back_insert_iterator<Fraction> itout = std::back_inserter( f );

Parameters

quotient the pair \((m,k+1)\).

◆ pushBack()

template<typename TInteger , typename TQuotient = int32_t>

void DGtal::SternBrocot< TInteger, TQuotient >::Fraction::pushBack ( const std::pair< Quotient, Quotient > & quotient )

Modifies this fraction \([u_0,...,u_k]\) to obtain the fraction \([u_0,...,u_k,m]\). The depth of the quotient must be given, since continued fractions have two writings \([u_0,...,u_k]\) and \([u_0,...,u_k - 1, 1]\).

See push_back for creating output iterators.

Parameters

quotient the pair \((m,k+1)\).

◆ q()

template<typename TInteger , typename TQuotient = int32_t>

Integer DGtal::SternBrocot< TInteger, TQuotient >::Fraction::q ( ) const

Returns: its denominator;

◆ reduced()

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::reduced ( Quotient i ) const

Parameters

i	a positive integer smaller or equal to k()+2.

Returns: the partial fraction of depth k()-i, ie. [u0,...,uk] => [u0,...,u{k-i}]

◆ right()

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::right ( ) const

Returns: its right descendant (construct it if it does not exist yet).

◆ selfDisplay()

template<typename TInteger , typename TQuotient = int32_t>

void DGtal::SternBrocot< TInteger, TQuotient >::Fraction::selfDisplay ( std::ostream & out ) const

Writes/Displays the fraction on an output stream.

Parameters

out	the output stream where the object is written.

◆ simplestFractionInBetween()

template<typename TInteger , typename TQuotient = int32_t>

Fraction DGtal::SternBrocot< TInteger, TQuotient >::Fraction::simplestFractionInBetween ( const Fraction & other ) const

Compute the fraction of smallest denominator strictly between this fraction and other fraction. Assumes that "this" fraction is smaller than "other" fraction.

Parameters

other any fraction

Returns: a fraction NB: \( O(k) \) where \( k \) is the depth of the output fraction.

◆ u()

template<typename TInteger , typename TQuotient = int32_t>

Quotient DGtal::SternBrocot< TInteger, TQuotient >::Fraction::u ( ) const

Returns: its quotient (last coefficient of its continued fraction).

Field Documentation

◆ myNode

template<typename TInteger , typename TQuotient = int32_t>

Node* DGtal::SternBrocot< TInteger, TQuotient >::Fraction::myNode

private

Definition at line 165 of file SternBrocot.h.

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

SternBrocot.h

Public Types

Public Member Functions

Private Attributes

Detailed Description

Member Typedef Documentation

◆ CFracSequence

◆ const_iterator

◆ const_reference

◆ ConstIterator

◆ Integer

◆ Quotient

◆ Self

◆ SternBrocotTree

◆ UnsignedInteger

◆ Value

◆ value_type

Constructor & Destructor Documentation

◆ Fraction() [1/3]

◆ Fraction() [2/3]

◆ Fraction() [3/3]

Member Function Documentation

◆ begin()

◆ end()

◆ equals()

◆ even()

◆ father() [1/2]

◆ father() [2/2]

◆ getCFrac()

◆ getSplit()

◆ getSplitBerstel()

◆ inverse()

◆ k()

◆ left()

◆ lessThan()

◆ median()

◆ moreThan()

◆ null()

◆ odd()

◆ operator!=()

◆ operator<()

◆ operator=()

◆ operator==()

◆ operator>()

◆ p()

◆ partial()

◆ previousPartial()

◆ push_back()

◆ pushBack()

◆ q()

◆ reduced()

◆ right()

◆ selfDisplay()

◆ simplestFractionInBetween()

◆ u()

Field Documentation

◆ myNode