Nearly Tight Bounds on Testing of Metric Properties

Given a non-negative $n\times n$ matrix viewed as a set of distances between $n$ points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree metrics and ultrametrics. For general metrics, our paper is the first to consider these questions. We prove an upper bound of $O(n^{2\mathrm{/}3}\mathrm{/}{\epsilon }^{4\mathrm{/}3})$ on the query complexity for this problem. Our algorithm is simple, but the analysis requires great care in bounding the variance on the number of violating triangles in a sample. When $\epsilon$ is a slowly decreasing function of $n$ (rather than a constant, as is standard), we prove a lower bound of matching dependence on $n$ of $\mathrm{\Omega }(n^{2\mathrm{/}3})$, ruling out any property testers with $o(n^{2\mathrm{/}3})$ query complexity unless their dependence on $1\mathrm{/}\epsilon$ is super-polynomial. Next, we turn to tree metrics and ultrametrics. While there were known upper and lower bounds, we considerably improve these bounds showing essentially tight bounds of $\tilde{O}(1\mathrm{/}\epsilon )$ on the sample complexity. We also show a lower bound of $\mathrm{\Omega }(1\mathrm{/}{\epsilon }^{4\mathrm{/}3})$ on the query complexity. Our upper bounds are derived by doing a more careful analysis of a natural, simple algorithm. For the lower bounds, we construct distributions on NO instances, where it is hard to find a witness showing that these are not ultrametrics.