A periodicity lemma for partial words

Abstract We investigate the function L ( h , p , q ) , called here the length function, such that L ( h , p , q ) is the minimum length which guarantees that a natural extension of the periodicity lemma is valid for partial words with h holes and (so-called strong) periods p , q . In a series of papers, the formulae for the length function, in terms of p and q, were provided for each fixed h ≤ 7 . We demystify the generic structure of such formulae and give a complete characterization of the length function for any parameter h expressed in terms of a piecewise-linear function with O ( h ) pieces. We also show how to evaluate the length function in O ( log ⁡ p + log ⁡ q ) time, which is an improvement upon the best previously known O ( p + q ) -time algorithm.