Streaming Algorithms for Extent Problems in High Dimensions

Pankaj K. Agarwal, R. Sharathkumar

Abstract: We present (single-pass) streaming algorithms for maintaining extent measures of a stream S of n points in $\mathbb{R} ^{d}$ . We focus on designing streaming algorithms whose working space is polynomial in d (poly(d)) and sub-linear in n. For the problems of computing diameter, width and minimum enclosing ball of S, we obtain lower bounds on the worst-case approximation ratio of any streaming algorithm that uses poly(d) space. On the positive side, we introduce the notion of blurred ball cover and use it for answering approximate farthest-point queries and maintaining approximate minimum enclosing ball and diameter of S. We describe a streaming algorithm for maintaining a blurred ball cover whose working space is linear in d and independent of n.