Constructing k-radius sequences
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.
Cite
@article{arxiv.1006.5812,
title = {Constructing k-radius sequences},
author = {Simon R. Blackburn and James F. McKee},
journal= {arXiv preprint arXiv:1006.5812},
year = {2010}
}
Comments
21 pages, 1 figure. Revision after reviewer comments. Minor changes of phrasing and a few lines of extra explanation added at some points; update of status of an open problem