Computing Order Statistics in the Farey Sequence

We study the problem of computing the k-th term of the Farey sequence of order n, for given n and k. Several methods for generating the entire Farey sequence are known. However, these algorithms require at least quadratic time, since the Farey sequence has Θ(n2 ) elements. For the problem of finding the k-th element, we obtain an algorithm that runs in time O(n lg n) and uses space O( √ n). The same bounds hold for the problem of determining the rank in the Farey sequence of a given fraction. A more complicated solution can reduce the space to O(n1/3 (lg lg n)2/3 ), and, for the problem of determining the rank of a fraction, reduce the time to O(n). We also argue that an algorithm with running time O(poly(lg n)) is unlikely to exist, since that would give a polynomial-time algorithm for integer factorization.

• Type: Conference
• Year: 2004
• Where: ANTS
• Authors: Corina E. Patrascu, Mihai Patrascu
• Keywords: Algorithm, Algorithms, ANTS 2004, Entire Farey Sequence, Farey Sequence