#### 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 ≤ 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.