The Power and Limitations of Uniform Samples in Testing Properties of Figures

We investigate testing of properties of 2-dimensional figures that consist of a black object on a white background. Given a parameter $${\epsilon }\in (0,1/2)$$ϵ∈(0,1/2), a tester for a specified property has to accept with probability at least 2/3 if the input figure satisfies the property and reject with probability at least 2/3 if it is $${\epsilon }$$ϵ-far from satisfying the property. In general, property testers can query the color of any point in the input figure. We study the power of testers that get access only to uniform samples from the input figure. We show that for the property of being a half-plane, the uniform testers are as powerful as general testers: they require only $$O({\epsilon }^{-1})$$O(ϵ-1) samples. In contrast, we prove that convexity can be tested with $$O({\epsilon }^{-1})$$O(ϵ-1) queries by testers that can make queries of their choice while uniform testers for this property require $$\varOmega ({\epsilon }^{-5/4})$$Ω(ϵ-5/4) samples. Previously, the fastest known tester for convexity needed $$\varTheta ({\epsilon }^{-4/3})$$Θ(ϵ-4/3) queries.