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Streaming Algorithms for Extent Problems in High Dimensions

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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.

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Notes

  1. We assume d>8.

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Correspondence to R. Sharathkumar.

Additional information

This work is supported by NSF under grants CCF-09-40671, CCF-10-12254, and CCF-11-61359, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, and by an ERDC contract W9132V-11-C-0003.

A preliminary version of this paper appeared in Proc. 44th Annual ACM Sympos. on Discret. Algorithms., pp. 1481–1489 (2010).

This work was done while R. Sharathkumar was a Ph.D. student at Duke University.

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Agarwal, P.K., Sharathkumar, R. Streaming Algorithms for Extent Problems in High Dimensions. Algorithmica 72, 83–98 (2015). https://doi.org/10.1007/s00453-013-9846-4

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