Inferring Strings from Lyndon Factorization

The Lyndon factorization of a string w is a unique factorization \(\ell_1^{p_1}, \ldots, \ell_m^{p_m}\) of w s.t. l1, …, l m is a sequence of Lyndon words that is monotonically decreasing in lexicographic order. In this paper, we consider the reverse-engineering problem on Lyndon factorization: Given a sequence S = ((s 1, p 1), …, (s m , p m )) of ordered pairs of positive integers, find a string w whose Lyndon factorization corresponds to the input sequence S, i.e., the Lyndon factorization of w is in a form of \(\ell_1^{p_1}, \ldots, \ell_m^{p_m}\) with |l i | = s i for all 1 ≤ i ≤ m. Firstly, we show that there exists a simple O(n)-time algorithm if the size of the alphabet is unbounded, where n is the length of the output string. Secondly, we present an O(n)-time algorithm to compute a string over an alphabet of the smallest size. Thirdly, we show how to compute only the size of the smallest alphabet in O(m) time. Fourthly, we give an O(m)-time algorithm to compute an O(m)-size representation of a string over an alphabet of the smallest size. Finally, we propose an efficient algorithm to enumerate all strings whose Lyndon factorizations correspond to S.