Testing local properties of arrays

We study testing of local properties in one-dimensional and multi-dimensional arrays. A property of d-dimensional arrays f : [n]^d → Σ is k-local if it can be defined by a family of k x . . . x 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 ddimensional arrays is testable by a simple canonical one-sided error non-adaptive ϵ-test, whose query complexity is O(ϵ⁻¹k log ϵn/k ) for d = 1 and O(c_dϵ^−1/dk · 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 Σ. 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.