Efficient on-line repetition detection

A repetition is a nonempty string of the form X-q, where q >= 2. Given a string S character by character and the value of q, the on-line repetition detection problem is to detect and report the first repetition in S, if it exists, in an on-line manner. Leung, Peng and Ting first studied the problem for q = 2 and gave an O(m log(2) m) time algorithm (refer to [H.-F. Leung, Z. Peng, H.-F. Ting, An efficient algorithm for online square detection, Theoretical Computer Science 363 (1) (2006) 69-75]), where in is the ending position of the first repetition in S. In this paper, we improve the above cited work by reducing the time complexity to O(m log beta), where beta is the number of distinct characters in the first in characters of S. Moreover, we also solve the problem for q >= 3 with the same time complexity. (C) 2008 Elsevier B.V. All rights reserved.