Helly-Type Theorems in Property Testing [chapter]

Sourav Chakraborty, Rameshwar Pratap, Sasanka Roy, Shubhangi Saraf
2014 Lecture Notes in Computer Science  
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 » ... guish between 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 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.
doi:10.1007/978-3-642-54423-1_27 fatcat:zjpkcacguveh7hvsa55jlicbcy