Efficient Representation and Counting of Antipower Factors in Words

Abstract

A k-antipower (for (Formula Presented) ) is a concatenation of k pairwise distinct words of the same length. The study of antipower factors of a word was initiated by Fici et al. (ICALP 2016) and first algorithms for computing antipower factors were presented by Badkobeh et al. (Inf. Process. Lett., 2018). We address two open problems posed by Badkobeh et al. Our main results are algorithms for counting and reporting factors of a word which are k-antipowers. They work in (Formula Presented) time and (Formula Presented) time, respectively, where C is the number of reported factors. For (Formula Presented)(Formula Presented)(Formula Presented)(Formula Presented), this improves the time complexity of (Formula Presented) of the solution by Badkobeh et al. Our main algorithmic tools are runs and gapped repeats. We also present an improved data structure that checks, for a given factor of a word and an integer k, if the factor is a k-antipower.