Helly-Type Theorems in Property Testing [article]

Sourav Chakraborty, Rameshwar Pratap, Sasanka Roy, Shubhangi Saraf
2013 arXiv   pre-print
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 R^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
more » ... ish between two cases: when S is (k,G)-clusterable, and when it is ϵ-far from being (k,G)-clusterable. A set S is ϵ-far (0<ϵ≤1) from being (k,G)-clusterable if at least ϵ n points need to be removed from S to make it (k,G)-clusterable. We solve this problem for k=1 and when 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 clustering with outliers, we show that one can find the approximate clusters by querying a constant size sample, with high probability.
arXiv:1307.8268v2 fatcat:tgjkwnrz45eppo4ebk73xuqd34