Streaming and dynamic algorithms for minimum enclosing balls in high dimensions

Abstract

At SODA'10, Agarwal and Sharathkumar presented a streaming algorithm for approximating the minimum enclosing ball of a set of points in d-dimensional Euclidean space. Their algorithm requires one pass, uses O(d) space, and was shown to have approximation factor at most (1 + √3)/2 + ε ≈ 1.3661. We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a (1 + √2)/2 ≈ 1.207 lower bound given by Agarwal and Sharathkumar. We also apply this technique to the dynamic version of the minimum enclosing ball problem (in the non-streaming setting). We give an O(dn)-space data structure that can maintain a 1.22-approximate minimum enclosing ball in O(d logn) expected amortized time per insertion/deletion. © 2011 Springer-Verlag.