Let n and k be positive integers, and let F be an alphabet of size n.
A sequence over F of length m is a \emph{k-radius sequence} if any two
distinct elements of F occur within distance k of each other somewhere in
the sequence. These sequences were introduced by Jaromczyk and Lonc in 2004, in
order to produce an efficient caching strategy when computing certain functions
on large data sets such as medical images.
Let fkβ(n) be the length of the shortest n-ary k-radius sequence. The
paper shows, using a probabilistic argument, that whenever k is fixed and
nββfkβ(n)βΌk1β(2nβ).
The paper observes that the same argument generalises to the situation when
we require the following stronger property for some integer t such that
2β€tβ€k+1: any t distinct elements of F must simultaneously occur
within a distance k of each other somewhere in the sequence.Comment: 8 pages. More papers cited, and a minor reorganisation of the last
section, since last version. Typo corrected in the statement of Theorem