Fast algorithms for Abelian periods in words and greatest common divisor queries

We present efficient algorithms computing all Abelian periods of two types in a word. Regular Abelian periods are computed in O(nlog⁡log⁡n) randomized time which improves over the best previously known algorithm by almost a factor of n. The other algorithm, for full Abelian periods, works in O(n) time. As a tool we develop an O(n)-time construction of a data structure that allows O(1)-time gcd⁡(i,j) queries for all 1≤i,j≤n. This is a result of independent interest.