Inferring Strings from Full Abelian Periods

Strings u, v are said to be Abelian equivalent if u is a permutation of the characters appearing in v. A string w is said to have a full Abelian periodp if \(w = w_1 \cdots w_k\), where all \(w_i\)’s are of length p each and are all Abelian equivalent. This paper studies reverse-engineering problems on full Abelian periods. Given a positive integer n and a set D of divisors of n, we show how to compute in O(n) time the lexicographically smallest string of length n which has all elements of D as its full Abelian periods and has the minimum number of full Abelian periods not in D. Moreover, we give an algorithm to enumerate all such strings in amortized constant time per output after O(n)-time preprocessing. Also, we show how to enumerate the strings which have all elements of D as its full Abelian periods in amortized constant time per output after O(n)-time preprocessing.