Order Statistics in the Farey Sequences in Sublinear Time and Counting Primitive Lattice Points in Polygons

Abstract

We present the rst sublinear-time algorithms for computing order statistics in the Farey sequence and for the related problem of ranking. Our algorithms achieve a running times of nearly O(n 2/3 ), which is a signicant improvement over the previous algorithms taking time O(n).We also initiate the study of a more general problem: counting primitive lattice points inside planar shapes. For rational polygons containing the origin, we obtain a running time proportional to D 6/7 , where D is the diameter of the polygon.

On Farey table and its compression for space optimization with guaranteed error bounds

“…[22,23]. It is worth mentioning here that runtime computation of the fraction ranks in a real-time application cannot serve the desired purpose, since even the best-known algorithm can compute the rank of a fraction in a time longer than O(n 2/3 log 1/3 n) [21]. In a Farey table designed by us, the row indices correspond to the fraction numerators, and the column indices to their denominators.…”

“…Given the order n, the rank problem is to find the rank of a given fraction in F n , whereas the order statistics problem deals with finding the fraction in F n with some given rank. Efficient solutions of these two problems may be seen in [20,21].…”

On the Farey sequence and its augmentation for applications to image analysis

“…For example, the first five sequences are The Farey sequences are named after John Farey, who first conjectured in 1816 that F n can be obtained from F n−1 ; a brief history following that can be found in the work of Hardy and Wright (1968). There are several works related with the Farey sequence, which mostly concern the theory of fractions (Hardy and Wright, 1968;Graham et al, 1994;Neville, 1950;Pȃtraşcu and Pȃtraşcu, 2004;Pawlewicz and Pȃtraşcu, 2009;Schroeder, 2006). However, from the viewpoint of algorithms or computation, limited work has been done so far.…”

“…However, from the viewpoint of algorithms or computation, limited work has been done so far. A computationally interesting problem addressed in recent time is the rank problem and its associated order statistic problem (Pawlewicz and Pȃtraşcu, 2009). The rank r n (x) of a fraction x in a Farey sequence F n is the number of fractions less than or equal to x in F n , i.e., r n (x) = |{y : (y ≤ x) ∧ (y ∈ F n )}|.…”

“…It becomes a natural choice in our applications, since the best-known algorithm can compute the rank of a fraction in no sooner than O(n 2/3 log 1/3 n) of time (Pawlewicz and Pȃtraşcu, 2009). Using the AFT, the fraction ranks are fetched almost instantaneously as they are queried very frequently in different procedural steps, such as comparing the slopes of two line segments with integer endpoints, checking the collinearity of three or more integer points, determining the type of turn (left or right) for three non-collinear integer points, etc.…”