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Online Square Detection

The online square detection problem is to detect the first occurrence of a square in a string whose characters are provided as input one at a time. Recall that a square is a string that is a concatenation of two identical strings. In this paper we present an algorithm solving this problem in $O(n\logσ)$ time and linear space on ordered alphabet, where $σ$ is the number of different letters in the input string. Our solution is relatively simple and does not require much memory unlike the previously known online algorithm with the same working time. Also we present an algorithm working in $O(n\log n)$ time and linear space on unordered alphabet, though this solution does not outperform the previously known result with the same time bound.