On Periodicity Lemma for Partial Words

TL;DR

This paper studies the threshold function related to periodicity in partial words with holes, providing efficient evaluation algorithms and a unified structural understanding of the formulae for various hole counts.

Contribution

It introduces an efficient $O(\log p + \log q)$ algorithm for evaluating the threshold function and reveals its piecewise-linear structure for any number of holes.

Findings

Efficient evaluation algorithm for the threshold function.

Structural characterization of the threshold function as piecewise-linear.

Generalization of formulae for any number of holes.

Abstract

We investigate the function $L(h,p,q)$, called here the threshold function, related to periodicity of partial words (words with holes). The value $L(h,p,q)$ is defined as the minimum length threshold which guarantees that a natural extension of the periodicity lemma is valid for partial words with $h$ holes and (strong) periods $p,q$. We show how to evaluate the threshold function in $O(\log p + \log q)$ time, which is an improvement upon the best previously known $O(p+q)$-time algorithm. In a series of papers, the formulae for the threshold function, in terms of $p$ and $q$, were provided for each fixed $h \le 7$. We demystify the generic structure of such formulae, and for each value $h$ we express the threshold function in terms of a piecewise-linear function with $O(h)$ pieces.