Abstract
A ball spans a set of n points when none of the points lie outside it. In Zarrabi-Zadeh and Chan (Proceedings of the 18th Canadian conference on computational geometry (CCCG’06), pp 139–142, 2006) proposed an algorithm to compute an approximate spanning ball in the streaming model of computation, and showed that the radius of the approximate ball is within 3/2 of the minimum. Spurred by this, in this paper we consider the 2-dimensional extension of this result: computation of spanning ellipses. The ball algorithm is simple to the point of being trivial, but the extension of the algorithm to ellipses is non-trivial. Surprisingly, the area of the approximate ellipse computed by this approach is not within a constant factor of the minimum and we provide an elegant proof of this. We have implemented this algorithm, and experiments with a variety of inputs, except for a very pathological one, show that it can nevertheless serve as a good heuristic for computing an approximate ellipse.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bagchi, A., Chaudhary, A., Eppstein, D., Goodrich, M.T.: Deterministic sampling and range counting in geometric data streams. In: ACM Transactions on Algorithms (TALG), vol. 3 (2007)
Ben-Tal A., Nemirovski A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, volume 2 of MPS/SIAM Series on Optimization. SIAM, Philadelphia (2001)
Dyer, M.E.: A class of convex programs with applications to computational geometry. In: Symposium on Computational Geometry, pp. 9–15 (1992)
Feigenbaum J., Kannan S., Zhang J.: Computing diameter in the streaming and sliding-window models. Algorithmica 41(1), 25–41 (2004)
Henzinger, M., Raghavan, P., Rajagopalan S.: Computing on Data Streams. Technical Report SRC-TN-1998-011, Hewlett Packard Laboratories (1998)
Hershberger, J., Suri S.: Adaptive sampling for geometric problems over data streams. In: PODS: 23th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems (2004)
Isenburg, L., Shewchuk, J., Snoeyink, J.: Illustrating the streaming construction of 2D delaunay triangulations (short). In: COMPGEOM: Annual ACM Symposium on Computational Geometry (2006)
Muthukrishnan S.: Data streams: algorithms and applications. Found. Trends Theor. Comput. Sci. 1(2), 117–236 (2005)
Post, M.J.: A minimum spanning ellipse algorithm. In: FOCS, pp. 115–122, IEEE (1981)
Silverman B.W., Titterington D.M.: Minimum covering ellipses. SIAM J. Sci. Stat. Comput. 1(4), 401–409 (1980)
Weisstein, E.W.: Ellipse. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Ellipse.html.
Welzl, E.: Smallest enclosing disks (balls and ellipsoids). Lecture Notes in Computer Science, vol. 555, pp. 359–370 (1991)
Zarrabi-Zadeh H., Chan T.: A simple streaming algorithm for minimum enclosing balls. In: Proceedings of the 18th Canadian Conference on Computational Geometry (CCCG’06), pp. 139–142 (2006)
Zarrabi-Zadeh, H., Mukhopadhyay, A.: Streaming 1-center with outliers in high dimensions. In: Proceedings of the 21st Canadian Conference on Computational Geometry (CCCG’09), pp. 83–86 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mukhopadhyay, A., Greene, E., Sarker, A. et al. From approximate balls to approximate ellipses. J Glob Optim 56, 27–42 (2013). https://doi.org/10.1007/s10898-012-9932-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-9932-1