Constructing k-radius sequences

TL;DR

This paper investigates the minimal length of n-ary k-radius sequences, improving bounds using tilings and logarithms, with results applicable to many k values, especially when 2k+1 is prime.

Contribution

It provides new asymptotic estimates for the shortest length of k-radius sequences, leveraging tiling methods and logarithmic techniques, extending previous bounds.

Findings

f_k(n) ~ n^2/(2k) as n→∞ for certain k

Valid for infinitely many k, including all k<195

Sharper error terms when 2k+1 is prime

Abstract

An n-ary k-radius sequence is a finite sequence of elements taken from an alphabet of size n such that any two distinct elements of the alphabet occur within distance 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) be the shortest length of any k-radius sequence. We improve on earlier estimates for f_k(n) by using tilings and logarithms. The main result is that f_k(n) ~ n^2/(2k) as n tends to infinity whenever a certain tiling of Z^r exists. In particular this result holds for infinitely many k, including all k < 195 and all k such that k+1 or 2k+1 is prime. For certain k, in particular when 2k+1 is prime, we get a sharper error term using the theory of logarithms.