Abstract:
An $\epsilon$-test for a property $P$ of functions from ${\cal D}=\{1,\ldots,d\}$ to the positive integers is a randomized algorithm, which makes queries on the value of an input function at specified locations, and distinguishes with high probability between the case of the function satisfying $P$, and the case that it has to be modified in more than $\epsilon d$ places to make it satisfy $P$. We prove that an $\epsilon$-test for any property defined in terms of the order relation, such as the property of being a monotone nondecreasing sequence, cannot perform less queries (in the worst case) than the best $\epsilon$-test which uses only comparisons between the queried values. In particular, we show that an adaptive algorithm for testing that a sequence is monotone nondecreasing performs no better than the best non-adaptive one, with respect to query complexity. From this follows a tight lower bound on tests for this property.