Improved Bounds for Testing Forbidden Order Patterns

A sequence $f\colon{1,\dots,n}\to\mathbb{R}$ contains a permutation $\pi$ of length $k$ if there exist $i1<\dots<ik$ such that, for all $x,y$, $f(ix)<f(iy)$ if and only if $\pi(x)<\pi(y)$; otherwise, $f$ is said to be $\pi$-free. In this work, we consider the problem of testing for $\pi$-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 $\varepsilon$-test for $\pi$-freeness making $\Theta(\varepsilon^{-1/k} n^{1-1/k})$ queries, our lower bounds imply that this is almost optimal for most permutations! Specifically, for most permutations $\pi$ of length $k$, any non-adaptive one-sided $\varepsilon$-test requires $\varepsilon^{{-1/(k-\Theta(1))}n{1-1/(k-\Theta(1))}$} queries; furthermore, the permutations that are hardest to test require $\Theta(\varepsilon^{{-1/(k-1)}n{1-1/(k-1)})$} queries, which is tight in $n$ and $\varepsilon$. Additionally, we show two hierarchical behaviors here. First, for any $k$ and $l\leq k-1$, there exists some $\pi$ of length $k$ that requires $\tilde{\Theta}_{\varepsilon}(n^{1-1/l})$ non-adaptive queries. Second, we show an adaptivity hierarchy for $\pi=(1,3,2)$ by proving upper and lower bounds for (one- and two-sided) testing of $\pi$-freeness with $r$ rounds of adaptivity. The results answer open questions of Newman et al. and [Canonne and Gur, CCC'17].