Improved Bounds for Testing Forbidden Order Patterns
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A sequence f{1,...,n}→R contains a permutation π of length k if there exist i_1<...<i_k such that, for all x,y, f(i_x)<f(i_y) if and only if π(x)<π(y); otherwise, f is said to be π-free. In this work, we consider the problem of testing for π-freeness with one-sided error, continuing the investigation of [Newman et al., SODA'17]. We demonstrate a surprising behavior for non-adaptive tests with one-sided error: While a trivial sampling-based approach yields an ε-test for π-freeness making Θ(ε^-1/k n^1-1/k) queries, our lower bounds imply that this is almost optimal for most permutations! Specifically, for most permutations π of length k, any non-adaptive one-sided ε-test requires ε^-1/(k-Θ(1))n^1-1/(k-Θ(1)) queries; furthermore, the permutations that are hardest to test require Θ(ε^-1/(k-1)n^1-1/(k-1)) queries, which is tight in n and ε. Additionally, we show two hierarchical behaviors here. First, for any k and l≤ k-1, there exists some π of length k that requires Θ̃_ε(n^1-1/l) non-adaptive queries. Second, we show an adaptivity hierarchy for π=(1,3,2) by proving upper and lower bounds for (one- and two-sided) testing of π-freeness with r rounds of adaptivity. The results answer open questions of Newman et al. and [Canonne and Gur, CCC'17].
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