Helly-Type Theorems in Property Testing
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 distinguish 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 εn 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.
Unable to display preview. Download preview PDF.
- 2.Anderberg, M.R.: Cluster Analysis for Applications. Academic Press (1973)Google Scholar
- 3.Chakraborty, S., Pratap, R., Roy, S., Saraf, S.: Helly-type theorems in property testing. CoRR, abs/1307.8268 (2013)Google Scholar
- 4.Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers, pp. 642–651 (2001)Google Scholar
- 9.Goldreich, O.: Combinatorial property testing (a survey). Electronic Colloquium on Computational Complexity (ECCC) 4(56) (1997)Google Scholar
- 11.Helly, E.: Über Mengen konvexer Köper mit gemeinschaftlichen Punkten (germen). Jahresber. Deutsch.Math. Verein (32), 175–176 (1923)Google Scholar
- 12.Jain, A.K., Dubes, R.C.: Algorithms for Clustering. Prentice-Hall (1988)Google Scholar
- 13.Kalai, G.: Intersection patterns of convex sets. Israel J. Math. (48), 161–174 (1984)Google Scholar
- 14.Katchalski, M., Nashtir, D.: On a conjecture of danzer and grunbaum. Proc. A.M.S (124), 3213–3218 (1996)Google Scholar
- 15.Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data: An Introduction to Cluster Analysis. John Wiley (1990)Google Scholar
- 16.Katchalski, M., Liu, A.: A problem of geometry in ℝn. Proc. A.M.S (75), 284–288 (1979)Google Scholar