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### Paper:

TR01-008 | 6th November 2000 00:00

#### On the strength of comparisons in property testing

TR01-008
Authors: Eldar Fischer
Publication: 10th January 2001 16:00
Keywords:

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.

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