Improved Bounds for Testing Forbidden Order Patterns

Omri Ben-Eliezer, Clément L. Canonne

Abstract: A sequence f: {1, …, n} → R contains a permutation π of length k if there exist i1 < · · · < ik such that, for all x, y, f (ix) < f (iy) 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/kn1–1/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))n1–1/(k–Θ(1)) queries; furthermore, the permutations that are hardest to test require Θ(ε–1/(k–1)n1–1/(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 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].