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 \((1+\sqrt{3})/2 +\varepsilon \approx 1.3661\). We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a \((1+\sqrt{2})/2 \approx 1.207\) 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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. Journal of the ACM 51, 606–635 (2004)
Agarwal, P.K., Sharathkumar, R.: Streaming algorithms for extent problems in high dimensions. In: Proc. 21st ACM–SIAM Sympos. Discrete Algorithms, pp. 1481–1489 (2010)
Agarwal, P.K., Yu, H.: A space-optimal data-stream algorithm for coresets in the plane. In: Proc. 23rd Sympos. Comput. Geom., pp. 1–10 (2007)
Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. J. Algorithms 38, 91–109 (2001)
Bentley, J.L., Saxe, J.B.: Decomposable searching problems I: Static-to-dynamic transformations. J. Algorithms 1, 301–358 (1980)
Bădoiu, M., Clarkson, K.L.: Smaller core-sets for balls. In: Proc. 14th ACM-SIAM Sympos. Discrete Algorithms, pp. 801–802 (2003)
Bădoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proc. 34th ACM Sympos. Theory Comput., pp. 250–257 (2002)
Chan, T.M.: Faster core-set constructions and data stream algorithms in fixed dimensions. Comput. Geom. Theory Appl. 35, 20–35 (2006)
Chan, T.M.: Dynamic coresets. Discrete Comput. Geom. 42, 469–488 (2009)
Kumar, P., Mitchell, J.S.B., Yildirim, E.A.: Approximating minimum enclosing balls in high dimensions using core-sets. ACM J. Experimental Algorithmics 8, 1.1 (2003)
Zarrabi-Zadeh, H.: An almost space-optimal streaming algorithm for coresets in fixed dimensions. In: Halperin, D., Mehlhorn, K. (eds.) Esa 2008. LNCS, vol. 5193, pp. 817–829. Springer, Heidelberg (2008)
Zarrabi-Zadeh, H., Chan, T.M.: A simple streaming algorithm for minimum enclosing balls. In: Proc. 18th Canad. Conf. Comput. Geom., pp. 139–142 (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chan, T.M., Pathak, V. (2011). Streaming and Dynamic Algorithms for Minimum Enclosing Balls in High Dimensions. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-22300-6_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22299-3
Online ISBN: 978-3-642-22300-6
eBook Packages: Computer ScienceComputer Science (R0)