Testing properties of sets of points in metric spaces

Abstract

Given query access to a set of points in a metric space, we wish to quickly check if it has a specific property. More precisely, we wish to distinguish sets of points that have the property from those that need to have at least an ε fraction of points modified to achieve it. We show one-sided error testers that immediately follow from known characterizations of metric spaces. Among other things, we give testers for tree metrics and ultrametrics which are optimal among one-sided error testers. Our tester for embeddability into the line is optimal even among two-sided error testers, and runs in sublinear time. We complement our algorithms with several lower bounds. For instance, we present lower bounds for testing dimensionality reduction in the ℓ1 and ℓ∞ metrics, which improve upon lower bounds given by Krauthgamer and Sasson (SODA 2003). All our lower bounds are constructed by using a generic approach. We also look at the problem from a streaming perspective, and give a method for converting each of our property testers into a streaming tester. © 2008 Springer-Verlag.