Constructing 𝑘-radius sequences

Abstract

An n n -ary k k -radius sequence is a finite sequence of elements taken from an alphabet of size n n such that any two distinct elements of the alphabet occur within distance k k of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc to model a caching strategy for computing certain functions on large data sets such as medical images. Let f k ( n ) f_k(n) be the shortest length of any k k -radius sequence. We improve on earlier estimates for f k ( n ) f_k(n) by using tilings and logarithms. The main result is that f k ( n ) ∼ 1 k ( n 2 ) f_k(n)\sim \frac {1}{k}\binom {n}{2} as n → ∞ n\rightarrow \infty whenever there exists a tiling of Z π ( k ) \mathbb {Z}^{\pi (k)} by a certain cluster of k k hypercubes. In particular this result holds for infinitely many k k , including all k ≤ 194 k\le 194 and all k k such that k + 1 k+1 or 2 k + 1 2k+1 is prime. For certain k k , in particular when 2 k + 1 2k+1 is prime, we get a sharper error term using the theory of logarithms.