Locating maximal approximate runs in a string

An exact run in a string T is a non-empty substring of T that is a repetition of a smaller substring possibly followed by a prefix of it. Finding maximal exact runs in strings is an important problem and therefore a well-studied one in the area of stringology. For a given string T of length n, finding all maximal exact runs in the string can be done in O(nlog⁡n) time on general ordered alphabets or O(n) time on integer alphabets. In this paper, we investigate the maximal approximate runs problem: for a given string T and a number k, find non-empty substrings T^′ of T such that changing at most k letters in T^′ transforms them into a maximal exact run. We present an O(nk²log²⁡k+occ) algorithm to solve this problem, where occ is the number of substrings found.