Shortest cover after edit

This paper investigates the (quasi-)periodicity of a string when the string is edited. A string C is called a cover (as known as a quasi-period) of a string T if each character of T lies within some occurrence of C. By definition, a cover of T must be a border of T; that is, it occurs both as a prefix and as a suffix of T. In this paper, we focus on the changes in the longest border and the shortest cover of a string when the string is edited only once. We propose a data structure of size O(n) that computes the longest border and the shortest cover of the string in O(ℓ + log n) time after an edit operation (either insertion, deletion, or substitution of some string) is applied to the input string T of length n, where ℓ is the length of the string being inserted or substituted. The data structure can be constructed in O(n) time given string T.