An optimal algorithm for online square detection

Abstract

A square is the concatenation of two identical non-empty strings. Let S be the input string which is given character by character. Let m be the (unknown) smallest integer such that the m-th prefix of S contains a square. The online square detection problem is to determine m as soon as the m-th character of S is read. The best previously known algorithm of the online square detection problem, due to Leung, Peng, and Ting, runs in O(m log2 m) time. We improve the time complexity to O(m log β), where β is the number of distinct characters in the m-th prefix of the input string. It is not difficult to implement our algorithm to run in expected O(m) time. © Springer-Verlag Berlin Heidelberg 2005.