The existence of k-radius sequences

TL;DR

This paper investigates the minimal length of sequences over an alphabet where every pair or t-tuple of elements appears within a certain distance, providing asymptotic results for large alphabets using probabilistic methods.

Contribution

It establishes the asymptotic length of shortest k-radius sequences for large n and extends the analysis to sequences where t-element subsets also appear within the distance k.

Findings

Asymptotic length of n-ary k-radius sequences is (1/k) times the binomial coefficient of n choose 2.

Probabilistic argument used to derive the asymptotic result.

Generalization to sequences where t-element subsets occur within distance k.

Abstract

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 $f_k(n)$ be the length of the shortest $n$-ary $k$-radius sequence. The paper shows, using a probabilistic argument, that whenever $k$ is fixed and $n\rightarrow\infty$ \[ f_k(n)\sim \frac{1}{k}\binom{n}{2}. \] The paper observes that the same argument generalises to the situation when we require the following stronger property for some integer $t$ such that $2\leq t\leq k+1$: any $t$ distinct elements of $F$ must simultaneously occur within a distance $k$ of each other somewhere in the sequence.