Testing forbidden order-pattern properties on hypergrids

TL;DR

This work initiates a systematic study of order-pattern freeness for real-valued functions on hypergrids, focusing on patterns of length three. It develops sublinear, sometimes adaptive testers for π-freeness on [n]^2, including an adaptive one-sided tester for all π ∈ S_3 with complexity O(n^{4/5+o(1)}) and nonadaptive polylog testers for monotone 3-patterns, while establishing near-tight lower bounds that separate adaptive and nonadaptive regimes. A key technical contribution is erasure-resilient monotonicity testing in high dimensions, which underpins the π-freeness testers via reductions to partial and erasure-prone problems, plus a 3D gridding framework that generalizes layerings and box decompositions. The paper also proves that Hamming and deletion distances coincide for patterns of length at most 3 on hypergrids, but can diverge for certain 4-patterns, and it provides lower bounds for the (1,3,2)-freeness problem through a reduction from an intersection-search problem, demonstrating fundamental limits of sublinear testing in higher dimensions. Overall, the results reveal new phenomena and complexity separations in higher-dimensional pattern freeness testing and map out open directions for larger patterns and higher dimensions.

Abstract

We study testing $π$-freeness of functions $f:[n]^d\to\mathbb{R}$, where $f$ is $π$-free if there there are no $k$ indices $x_1\prec\cdots\prec x_k\in [n]^d$ such that $f(x_i)<f(x_j)$ and $π(i) < π(j)$ for all $i,j \in [k]$, where $\prec$ is the natural partial order over $[n]^d$. Given $ε\in(0,1)$, $ε$-testing $π$-freeness asks to distinguish $π$-free functions from those which are $ε$-far -- meaning at least $εn^d$ function values must be modified to make it $π$-free. While $k=2$ coincides with monotonicity testing, far less is known for $k>2$. We initiate a systematic study of pattern freeness on higher-dimensional grids. For $d=2$ and all permutations of size $k=3$, we design an adaptive one-sided tester with query complexity $O(n^{4/5+o(1)})$. We also prove general lower bounds for $k=3$: every nonadaptive tester requires $Ω(n)$ queries, and every adaptive tester requires $Ω(\sqrt{n})$ queries, yielding the first super-logarithmic lower bounds for $π$-freeness. For the monotone patterns $π=(1,2,3)$ and $(3,2,1)$, we present a nonadaptive tester with polylogarithmic query complexity, giving an exponential separation between monotone and nonmonotone patterns (unlike the one-dimensional case). A key ingredient in our $π$-freeness testers is new erasure-resilient ($δ$-ER) $ε$-testers for monotonicity over $[n]^d$ with query complexity $O(\log^{O(d)}n/(ε(1-δ)))$, where $0<δ<1$ is an upper bound on the fraction of erasures. Prior ER testers worked only for $δ=O(ε/d)$. Our nonadaptive monotonicity tester is nearly optimal via a matching lower bound due to Pallavoor, Raskhodnikova, and Waingarten (Random Struct. Algorithms, 2022). Finally, we show that current techniques cannot yield sublinear-query testers for patterns of length $4$ even on two-dimensional hypergrids.

Figures (4)

• Figure 1: (a) $M_{11}$: $\Delta_y(2\text{-leg},3\text{-leg}) \geq \Delta_y(1\text{-leg},2\text{-leg})$ (b) $M_{12}$: $\Delta_y(2\text{-leg},3\text{-leg}) < \Delta_y(1\text{-leg},2\text{-leg})$
• Figure 2: $M_3$: (a) $(1,2,3)$-appearance in a $\Gamma$-shape (b) $(1,2,3)$-appearance forming an inverted $\Gamma$-shape
• Figure 3: (a), (b) The harder cases -- Boxes in $G_m^{(2)}$ containing the legs of a $(1,3,2)$-appearance forming a (a) $\Gamma$-shape, and (b) an inverted $\Gamma$-shape. (c), (d) The easier cases -- Box in $G_m^{(2)}$ containing the $2$-leg is in a row and column different from that of the box containing the $3$-leg.
• Figure 7: (a), (b) $M_{22}$: Boxes containing legs of $(1,2,3)$-appearances spanning $2$ rows of $G_m^{(2)}$ (c) $M_{23}$: Boxes containing legs of $(1,2,3)$-appearances spanning $3$ rows and $3$ columns of $G_m^{(2)}$

Theorems & Definitions (64)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1: Deletion and Hamming distances
• Claim 2.2
• Corollary 2.3
• proof
• Claim 2.4