@article{Rajagopalan2000VerificationOI,
title={Verification of Identities},
author={Sridhar Rajagopalan and Leonard J. Schulman},
journal={SIAM J. Comput.},
year={2000},
volume={29},
pages={1155-1163}
}

We provide an O(n2 log 1 δ ) time randomized algorithm to check whether a given operation ◦ : S × S → S is associative (where n = |S| and δ > 0 is the error probability required of the algorithm). We prove that (for any constant δ) this performance is optimal up to a constant factor, even if the operation is “cancellative.” No sub-n3 time algorithm was previously known for this task. More generally we give an O(nc) time randomized algorithm to check whether a collection of c-ary operations… CONTINUE READING