Efficient Representation and Counting of Antipower Factors in Words

A k-antipower (for \(k \ge 2\)) 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 \(\mathcal {O}(nk \log k)\) time and \(\mathcal {O}(nk \log k\,+\,C)\) time, respectively, where C is the number of reported factors. For \(k=o(\sqrt{n/\log n})\), this improves the time complexity of \(\mathcal {O}(n^2/k)\) 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.