Fast computation of abelian runs

Given a word w and a Parikh vector P , an abelian run of period P in w is a maximal occurrence of a substring of w having abelian period P . Our main result is an online algorithm that, given a word w of length n over an alphabet of cardinality σ and a Parikh vector P , returns all the abelian runs of period P in w in time O(n) and space O(σ + p), where p is the norm of P , i.e., the sum of its components. We also present an online algorithm that computes all the abelian runs with periods of norm p in w in time O(np), for any given norm p. Finally, we give an O(n)-time offline randomized algorithm for computing all the abelian runs of w. Its deterministic counterpart runs in O(n log σ) time.