Computing the Antiperiod(s) of a String

A string S[1, n] is a power (or repetition or tandem repeat) of order k and period n/k, if it can be decomposed into k consecutive identical blocks each n/k letters in length. Powers and periods are fundamental structures in the study of strings and algorithms to compute them efficiently have been widely studied. Recently, Fici et al. (Proc. ICALP 2016) introduced an antipower of order k to be a string composed of k distinct blocks of the same length, n/k, called the antiperiod. Antipowers are a natural converse to powers, and are objects of combinatorial interest in their own right. In this paper, we describe efficient algorithm for computing the smallest antiperiod t of a string S of length n in O(n log∗ t) time. We also describe an algorithm to compute all the antiperiods of S that runs in O(n log n) time.