Testing local properties of arrays

TL;DR

This paper introduces a universal, efficient testing method for local properties of arrays in multiple dimensions, achieving optimal or near-optimal query complexities and covering many well-known properties.

Contribution

It presents a generic, property-independent testing approach with provably optimal query complexities for local array properties in any dimension.

Findings

Query complexity is $O(rac{1}{ ext{epsilon}}k ext{log}(rac{ ext{epsilon} n}{k}))$ for 1D arrays.

Query complexity for higher dimensions is $O(c_d rac{1}{ ext{epsilon}^{1/d}} k n^{d-1})$, matching lower bounds.

Provides the first sublinear upper bounds for some previously studied local properties.

Abstract

We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of $d$-dimensional arrays $f:[n]^d \to \Sigma$ is $k$-local if it can be defined by a family of $k \times \ldots \times k$ forbidden consecutive patterns. This definition captures numerous interesting properties. For example, monotonicity, Lipschitz continuity and submodularity are $2$-local; convexity is (usually) $3$-local; and many typical problems in computational biology and computer vision involve $o(n)$-local properties. In this work, we present a generic approach to test all local properties of arrays over any finite (and not necessarily bounded size) alphabet. We show that any $k$-local property of $d$-dimensional arrays is testable by a simple canonical one-sided error non-adaptive $\varepsilon$-test, whose query complexity is $O(\epsilon^{-1}k \log{\frac{\epsilon n}{k}})$ for $d = 1$ and $O(c_d \epsilon^{-1/d} k \cdot n^{d-1})$ for $d > 1$. The queries made by the canonical test constitute sphere-like structures of varying sizes, and are completely independent of the property and the alphabet $\Sigma$. The query complexity is optimal for a wide range of parameters: For $d=1$, this matches the query complexity of many previously investigated local properties, while for $d > 1$ we design and analyze new constructions of $k$-local properties whose one-sided non-adaptive query complexity matches our upper bounds. For some previously studied properties, our method provides the first known sublinear upper bound on the query complexity.