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An optimal $O(\log\log n)$-time concurrent-read concurrent-write parallel algorithm for detecting all squares in a string is presented. A tight lower bound shows that over general alphabets, this is the fastest possible optimal algorithm. When $p$ processors are available, the bounds become $\Theta(\lceil{({n\log n})/ p}\rceil + \log\log_{\lceil 1+p/n \rceil} 2p)$. The algorithm uses an optimal parallel string-matching algorithm together with periodicity properties to locate the squares within the input string.

An Optimal $O(\log \log N)$-Time Parallel Algorithm for Detecting all Squares in a String