Efficient Identification of k-Closed Strings

TLDR: This paper addresses the problem of identifying whether or not a given string of length n over an integer alphabet is k- closed and additionally specifying the border resulting in the string being k-closed, and presents an \(\mathcal {O}(kn)\)-time and \(\math CAL(n)-space algorithm to achieve this along with the pseudocode of an implementation.

Abstract: A closed string contains a proper factor occurring as both a prefix and a suffix but not elsewhere in the string. Closed strings were introduced by Fici (WORDS 2011) as objects of combinatorial interest. In this paper, we extend this definition to k-closed strings, for which a level of approximation is permitted up to a number of Hamming distance errors, set by the parameter k. We then address the problem of identifying whether or not a given string of length n over an integer alphabet is k-closed and additionally specifying the border resulting in the string being k-closed. Specifically, we present an \(\mathcal {O}(kn)\)-time and \(\mathcal {O}(n)\)-space algorithm to achieve this along with the pseudocode of an implementation.