DGtal
0.9.3

Part of the Arithmetic package.
This part of the manual describes how to create and use irreducible fractions. More precisely, several representations of irreducible fractions are provided. They are based on the SternBrocot tree structure. With these fractions, amortized constant time operations are provided for computing reduced fractions.
Part of the code is a backport from ImaGene. [39]
A positive irreducible fraction is a fraction p / q, such that p and q are positive integers and gcd(p, q)=1. The fractions \(\frac{0}{1}\) (zero) and \(\frac{1}{0}\) (infinite) are added for convenience. Standard operations like '+', '', '*', or '/' can be defined over these fractions: it is a matter of playing with numerators and denominators.
However, these fractions have other properties that are useful in the context of digital geometry, especially in digital straightness. These other properties are related to Euclid algorithm, the SternBrocot tree, and simple continued fractions.
The wellknown Euclid algorithm is useful to compute the greatest common divisor of two integers p and q. However, its intermediate computations are also useful. Let us play with 8/3 (see IntegerComputer::gcd).
p  =  u  *  q  +  r 

8  =  2  *  3  +  2 
3  =  1  *  2  +  1 
2  =  2  *  1  +  0 
1  *  0 
At each iteration, p takes the former value of q, q takes the former value of r. The algorithm stops when q is null. Then p is the gcd (1 here).
The values of u are called the quotients of \(\frac{p}{q}\), and are numbered from 0. A unique way to describe \(\frac{p}{q}\) is the sequence \(u_0,u_1,\ldots,u_k\), which is in fact denoted as \([u_0;u_1,\ldots,u_k]\). This sequence characterizes the fraction.
In fact, one can prove that \(\frac{p}{q}=u_0+\frac{1}{u_1+\frac{1}{\ldots+\frac{1}{u_k}}}\), which is the simple continued fraction of p / q.
The successive fractions \(\frac{p_i}{q_i}=[u_0;u_1,\ldots,u_i]\), for \(k < i\) are called the principal convergents of \(\frac{p}{q}\), and are the best approximations of this fraction for the size of their denominator.
A ireduced of a continued fraction is its k  ith convergent. The depth of the fraction is k.
A nice property of convergents with respect to their fraction is that even convergents are smaller than the fraction while odd convergents are greater.
Another (but onetoone) way of seeing irreducible fractions is the SternBrocot tree. Place the fractions 0/1 and 1/0 at the top, then for each pair of consecutive fractions \(\frac{p}{q}\) and \(\frac{p'}{q'}\) compute its mediant \(\frac{p+p'}{q+q'}\). The fraction 1/1 appears first, then 1/2 and 2/1, then 1/3, 2/3, 3/2, 3/1, etc.
Now the sequence of quotients \(u_0, u_1,\ldots, u_k \) is exactly the sequence of rightthenleft moves in the SternBrocot tree from node 1/1 when \( u_0 \ge 1 \), except for the last \( u_k \) where there is one less movement. Otherwise, when \( u_0 = 0 \), the sequence \( u_1,\ldots, u_k \) is exactly the sequence of leftthenright moves in the SternBrocot tree from node 1/1, except for the last \( u_k \) where there is one less movement.
Looking back at \( 8/3=[2;1,2] \) gives 2 right moves, 1 left move, then 21 right move.
When playing with irreducible fractions, computing reduced fractions or partial quotients is one of the main task. For instance, splitting formula and Berstel splitting formula is intensively used by algorithms related to digital straightness. Therefore, we must have representations of irreducible fractions which are efficient for that kind of requests.
Several representations have been implemented in the Arithmetic package. Their interface are common (see Irreducible fraction concept) so that a user can choose which representation suits him best.
Irreducible fractions are first described abstractly with the concept CPositiveIrreducibleFraction. Note that the following operations are defined for each irreducible fraction.
Name  Expression  Type requirements  Return type  Precondition  Semantics  Post condition  Complexity 

Constructor  Fraction ( p, q )  X  creates the fraction p'/q', where p'=p/g, q'=q/g, g=gcd(p,q)  o(p+q)  
numerator  x.p()  Integer  ! x.null()  returns the numerator  O(1)  
denominator  x.q()  Integer  ! x.null()  returns the denominator  O(1)  
quotient  x.u()  Size  ! x.null()  returns the quotient \(u_k\)  O(1)  
depth  x.k()  Size  ! x.null()  returns the depth k  O(1)  
null test  x.null()  bool  returns 'true' if the fraction is null 0/0 (default fraction)  O(1)  
even parity  x.even()  bool  ! x.null()  returns 'true' iff the fraction is even, i.e. k is even  O(1)  
odd parity  x.odd()  bool  ! x.null()  returns 'true' iff the fraction is odd, i.e. k is odd  O(1)  
left descendant  x.left()  X  ! x.null()  returns the left descendant of p/q in the SternBrocot tree  O(1)  
right descendant  x.right()  X  ! x.null()  returns the right descendant of p/q in the SternBrocot tree  O(1)  
father  x.father()  X  ! x.null()  returns the father of this fraction, ie \([u_0,...,u_k  1]\)  O(1)  
mfather  x.father (m)  X  ! x.null() , m>=0  returns the mfather of this fraction, ie \([u_0,...,u_{k1}, m]\)  O( m)  
previousPartial  x.previousPartial()  X  ! x.null()  returns the previous partial of this fraction, ie \([u_0,...,u_{k1}]\)  O(1)  
inverse  x.inverse()  X  ! x.null()  returns the inverse of this fraction, ie \([0,u_0,...,u_k]\) if \(u_0 \neq 0 \) or \([u_1,...,u_k]\) otherwise  O(1)  
mth partial  x.partial(m)  X  ! x.null()  returns the mth partial of this fraction, ie \([u_0,...,u_m]\)  O(1)  
mth reduced  x.reduced(m)  X  ! x.null()  returns the mth reduced of this fraction, equivalently the \(km\) partial, ie \([u_0,...,u_{km}]\)  O(1)  
splitting formula  x.getSplit (x1, x2)  void  ! x.null()  modifies fractions x1 and x2 such that \( x1 \oplus x2 = x \)  O(1)  
Berstel splitting formula  x.getSplitBerstel (x1, n1, x2, n2)  void  ! x.null()  modifies fractions x1 and x2 and integers n1 and n2 such that \( (x1)^{n1} \oplus (x2)^{n2} = x \)  O(1)  
Continued fraction coefficients  x.getCFrac (quots)  void  modifies the vector quots such that it contains the quotients \(u_0,u_1,...,u_k \)  O(k)  
equality  x.equals (p, q)  bool  returns 'true' iff the fraction is equal to \( p / q \).  O(1)  
less than  x.lessThan (p, q)  bool  returns 'true' iff the fraction is inferior to \( p / q \).  O(1)  
more than  x.moreThan (p, q)  bool  returns 'true' iff the fraction is superior to \( p / q \).  O(1)  
equality ==  x == y  bool  returns 'true' iff the fraction is equal to y.  O(1)  
inequality !=  x != y  bool  returns 'true' iff the fraction is different from y.  O(1)  
less than <  x < y  bool  returns 'true' iff the fraction is inferior to y.  O(1)  
more than >  x > y  bool  returns 'true' iff the fraction is superior to y.  O(1)  
Next continued fraction  x.pushBack( pair )  transforms this fraction \([0,u_0,...,u_k]\) into \([0,u_0,...,u_k,m]\), where pair is \((m,k+1)\)  O(m)  
Next continued fraction  x.push_back( pair )  transforms this fraction \([0,u_0,...,u_k]\) into \([0,u_0,...,u_k,m]\), where pair is \((m,k+1)\)  O(m)  
Begin visiting quotients  x.begin()  ConstIterator  returns a forward iterator on the beginning of the sequence of quotients \([u_0,...,u_k]\)  
End visiting quotients  x.end()  ConstIterator  returns a forward iterator after the end of the sequence of quotients \([u_0,...,u_k]\) 
Inner types are:
std::pair<Size,Size>
, useful to create back insertion sequence.Notations are:
std::vector<Size>
std::pair<Size,Size>
, here (m,k+1)A naive approach is to represent an irreducible fraction with three arrays of integers: the numerators (p_i), the denominators (q_i), the quotients (u_i).
However, each time we duplicate the fraction, or each time we compute a reduced fraction, the complexity of the operation is proportional to the depth of the fraction, i.e. in O(k).
We wish to have amortized O(1) complexity for these operations. Therefore, we take another path.
This approach is implemented in the class SternBrocot::Fraction (see also class SternBrocot). We construct on demand parts of the SternBrocot tree structure. For instance, if one wish to manipule fraction 8/3, the following nodes of the tree are computed once and for all: 0/1, 1/0, 1/1, 2/1, 1/2, 3/1, 1/3, 5/2, 2/5, 8/3, 3/8.
Each node in the SternBrocot is created once. The node stores information on the irreducible fraction itself (p/q, the partial quotient u, the depth k), but also pointers to ascendants, descendants and inverse in the SternBrocot tree. Nodes are constructed on demand, when the user ask for descendant or for a specific fraction.
A fraction SternBrocot::Fraction is just a pointer to the corresponding node in the SternBrocot tree. Looking for a reduced partial is just moving from node to node.
You may have a look at testSternBrocot.cpp to see how to use these fractions.
Also inverses and reduced fractions are quickly computed, this representation has two disadvantages:
This approach is implemented in the class LighterSternBrocot::Fraction (see also class LighterSternBrocot).
There are two main differences with the class SternBrocot. The first one is that inverses are not stored. With this optimization, there are twice less nodes and each node is lighter. The second one lies in the access to the children of a node. Here, a map type M is provided so that a node \( [u_0; u_1, ..., u_n] \) can access its child node \( [u_0; u_1, ..., u_n, k] \) in the time of the operation M::operator[]. This representation is also different from LightSternBrocot in the sense that nodes have only one set of child nodes and that only fractions greater than 1/1 are stored.
In this representation, the fraction 1/1 has depth 0, like 2/1, 3/1, etc. Furthermore, each fraction \( [u_0,...,u_n] \) has an origin which is the fraction \( [u_0,...,u_{n1},1] \). It is the top extremity of this branch. The origin has depth n1 since \( [u_0,...,u_{n1},1] \) = \( [u_0,...,u_{n1}+1] \). Inversely a kchild of \( [u_0,...,u_n] \), for k >= 2, is the fraction \( [u_0,...,u_n  1, k] \). A 1child of a fraction f is itself, except for the fraction 1/0 where its 1child is 1/1 by convention.
In practice, also this class has supposedly a better complexity than SternBrocot, it is 1% slower for integers smaller than 10^9 and 5% slower for integers smaller than 10^4. Note however that it takes like 7 times less memory (and asymptotically less when the number of computations tends toward infinity).
This class is not to be instantiated, since it is useless to duplicate it. Use static method LighterSternBrocot::fraction or constructor of LighterSternBrocot::Fraction to obtain your fractions.
You may have a look at testLighterSternBrocot.cpp to see how to use these fractions.
Available fractions allows you to choose the ultimate precision you wish to have for your fractions.
You have two different integral types for irreducible fractions: TInteger and TSize. The type TInteger defines the integral type for the numerators and the denominators. The type TSize defines the integral type for the quotients and for the depth of the fraction.
You may for instance choose your fractions like:
should be valid.
You may instantiate a fraction directly by giving the numerator p and denominator q. Note that if \( g = gcd(p,q) \neq 1 \), the instantiated fraction is \( \frac{p'}{q'} \) where \( p'=p/g, q'=q/g \). We follow the example convergents.cpp.
First we give the correct includes:
Then the useful types:
The program is given p and q in arguments argv. The fraction is instantiated as follows:
You may visit the quotients of a fraction by using the singlepass iterators provided by the class. A fraction is indeed a model of CSinglePassConstRange. The quotients are computed and stored with the call to begin()
. We therefore advise you to store the begin() iterator in a variable. Then, all other operations with iterators (copy, displacement, comparison, dereferenciation) are O(1). We display the quotients of the fraction f as follows.
std::pair<Size,Size>
. This is done for consistency with push_back
method.One way of obtaining all the convergents of a fraction is simply to create the fraction progressively by adding quotients one at a time. You may use the methods pushBack
or push_back
to do so. This can be done as follows:
You may now execute this program to get the quotients and the convergents of a given fraction. We give below several examples:
# A Fibonacci fraction $ ./examples/arithmetic/convergents 21 34 z = [0,1,1,1,1,1,1,2] z_0 = 0 / 1 z_1 = 1 / 1 z_2 = 1 / 2 z_3 = 2 / 3 z_4 = 3 / 5 z_5 = 5 / 8 z_6 = 8 / 13 z_7 = 21 / 34 # Approximations of pi $ ./examples/arithmetic/convergents 103993 33102 z = [3,7,15,1,292] z_0 = 3 / 1 z_1 = 22 / 7 z_2 = 333 / 106 z_3 = 355 / 113 z_4 = 103993 / 33102 # A huge fraction $ ./examples/arithmetic/convergentsbiginteger 243224233245235253407096734543059 4324213412343432913758138673203834 z = [0,17,1,3,1,1,12,1,2,33,2,1,1,1,1,49,1,1,1,1,17,34,1,1,304,1,2,1,1,1,2,1,48,1,20,2,3,5,1,1,16,9,1,1,5,1,2,2,7,4,3,1,7,1,1,17,1,1,29,1,12,2,5] z_0 = 0 / 1 z_1 = 1 / 17 z_2 = 1 / 18 z_3 = 4 / 71 z_4 = 5 / 89 ... z_18 = 23610961 / 419772458 ... z_40 = 832739221613445323225 / 14805030169237188131024 ... z_62 = 243224233245235253407096734543059 / 4324213412343432913758138673203834
Since fractions have a push_back
method, you can use std::back_inserter
to create a std::back_insert_iterator
on your fraction. The code for computing a fraction from its quotients looks like:
which gives
# More precise approximation of pi $ ./examples/arithmetic/fraction 3 7 15 1 292 1 1 1 2 1 3 1 14 z = 80143857 / 25510582
If \( z_k \) is a fraction, its 1reduced is \( z_{k1} \), its 2reduced is \( z_{k2} \) and so on. Reduced are in particular used for computing Bézout vectors, or in splitting formulas. You may use the methods reduced
(SternBrocot::Fraction::reduced, LighterSternBrocot::Fraction::reduced) or partial
(SternBrocot::Fraction::partial, LighterSternBrocot::Fraction::partial) to obtain them.
Assume f is the fraction \( 103993 / 33102 = [3;7,15,1,292] \). Then
You dispose also of the methods getSplit
and getSplitBerstel
to obtain directly the decomposition of a fraction into two fractions. Their "mediant sum" gives the fraction itself. Fractions are seen as vectors in this case.
Convergents or partials, and reduced fractions are ancestors of a given fractions. There exist inbetween ancestors which are not given by these methods. We list them below with their exact definition, assuming f is \( [u_0,...,u_{k1},u_k] \):
partial
, SternBrocot::Fraction::partial, LighterSternBrocot::Fraction::partial.reduced
, SternBrocot::Fraction::reduced, LighterSternBrocot::Fraction::reduced.father
, SternBrocot::Fraction::father, LighterSternBrocot::Fraction::father.father(Size m)
, SternBrocot::Fraction::father(Size m) const, LighterSternBrocot::Fraction::father(Size m) const.origin
, SternBrocot::Fraction::origin, LighterSternBrocot::Fraction::origin.You can get the inverse of a fraction in O(1) with the method inverse
(SternBrocot::Fraction::inverse, LighterSternBrocot::Fraction::inverse).
A simple way to get rational approximation of a "real" number is to use the mapping: \( G(x)=\frac{1}{x \lfloor x \rfloor} \). The successive \( \lfloor x \rfloor \) gives the successive quotients of the rational approximation. You may have a look at example approximation.cpp.
There are not implemented yet and continued fractions are not the best representation for doing them. Here, you must use numerators and denominators to perform computations with these fractions.