Helly-Type Theorems in Property Testing
International Journal of Computational Geometry and Applications
Helly's theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If S is a set of n points in d, we say that S is (k,G)-clusterable if it can be partitioned into k clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object G. In this paper, as an application of Helly's theorem, by taking a constant size sample from S, we present a testing algorithm for (k,G)-clustering, i.e., to distinguish between the following two cases: when S is (k,G)-clusterable, and when it is -far from being (k,G)-clusterable. A set S is -far (0 < ≤ 1) from being (k,G)-clusterable if at least n points need to be removed from S in order to make it (k,G)-clusterable. We solve this problem when k = 1, and G is a symmetric convex object. For k > 1, we solve a weaker version of this problem. Finally, as an application of our testing result, in the case of clustering with outliers, we show that with high probability one can find the approximate clusters by querying only a constant size sample.
Chakraborty, Sourav; Pratap, Rameshwar; Roy, Sasanka; and Saraf, Shubhangi, "Helly-Type Theorems in Property Testing" (2018). Journal Articles. 1111.