Improved Bounds for Testing Forbidden Order Patterns

A sequence f : {1, . . . , n} → R contains a permutation π of length k if there exist i1 < . . . < ik such that f(ix) < f(iy) iff π(x) < π(y); otherwise, f is said to be π-free. As a simple special case, f is weakly monotone increasing iff it is (2, 1)-free. The problem of (one-sided) testing π-freeness in a sequence (or equivalently, efficiently finding a π-copy in a sequence far from π-freeness), was proposed by Newman, Rabinovich, Rajendraprasad, and Sohler [SODA’17], who mainly focused on the study of several special cases. However, the general problem of understanding the behavior of optimal non-adaptive and adaptive one-sided tests for π-freeness, for any given permutation π, has remained wide open. In this work, we improve the understanding of testing π-freeness, demonstrating several fascinating aspects of this problem in the non-adaptive and the partially-adaptive settings. Some of our results are given below. • For any permutation π of length k ≥ 3, π-freeness has a one-sided non-adaptive ε-test making O(ε− 1 k−1n1− 1 k−1 ) queries. This improves upon the previously known upper bounds for all non-monotone permutations. • The upper bound is tight in both n and ε: Any one-sided non-adaptive test for any permutation π of length k ≥ 3 in which |π−1(1)− π−1(k)| = 1 must make Ω(ε− 1 k−1n1− 1 k−1 ) queries. • This lower bound can be extended to obtain a permutation-dependent lower bound, implying that for most permutations π of length k, an optimal one-sided non-adaptive test requires ε 1 k−Θ(1)n 1 k−Θ(1) queries. Thus, for most permutations, the trivial non-adaptive sample-based one-sided test (making Θ(ε−1/kn1−1/k) queries) is almost optimal! • For any k and 1 ≤ ` ≤ k − 1, there is a permutation of length k requiring Θ̃ε(n) queries. This resolves an open question of Newman et al., who asked whether there exist infinitely many non-monotone permutations that are testable with O(n0.99) queries. • Let π = (1, 3, 2). We obtain lower and upper bounds for (one-sided and two-sided) testing of π-freeness using r rounds of adaptivity, for any constant r, providing the first known example of a natural property that has an adaptivity-hierarchical behavior. This settles an open question of Canonne and Gur [CCC’17]. ∗Blavatnik School of Computer Science, Tel-Aviv University. Email: [email protected]. †Columbia University. Email: [email protected]. Research supported by NSF grants CCF-1115703 and NSF CCF-1319788.