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
We assume d>8.
References
Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Approximating extent measures of points. J. ACM 51, 606–635 (2004)
Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Geometric approximation via coresets. In: Goodman, J., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry, pp. 1–30. Cambridge University Press, Cambridge (2005)
Agarwal, P.K., Matoušek, J., Suri, S.: Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Comput. Geom. 1, 189–201 (1992)
Agarwal, P.K., Yu, H.: A space-optimal data-stream algorithm for coresets in the plane. In: Proceedings of 23rd Annual Symposium on Computational Geometry, pp. 1–10 (2007)
Aggarwal, C.: Data Streams: Models and Algorithms. Springer, Berlin (2007)
Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. J. Comput. Syst. Sci. 58, 137–147 (1999)
Andoni, A., Croitoru, D., Patrascu, M.: Hardness of nearest neighbor under L-infinity. In: Proceedings of 49th Annual IEEE Symposium on Foundations of Computer Science, pp. 424–433 (2008)
Ball, K.: An elementary introduction to modern convex geometry. In: Flavors of Geometry, vol. 31, pp. 1–58 (1997)
Bădoiu, M., Clarkson, K.L.: Optimal core-sets for balls. Comput. Geom. 40, 14–22 (2008)
Bădoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proceedings of 34th Annual ACM Symposium on Theory of Computing, pp. 250–257 (2002)
Chan, T.M.: Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. Int. J. Comput. Geom. Appl. 12, 67–85 (2002)
Chan, T.M.: Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput. Geom. 35, 20–35 (2006)
Chan, T.M., Pathak, V.: Streaming and dynamic algorithms for minimum enclosing balls in high dimensions. In: Proceedings of 12th International Symposium on Algorithms on Data Structures, pp. 195–206 (2011)
Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L., Sharir, M., Welzl, E.: Improved bounds on weak ε-nets for convex sets. Discrete Comput. Geom. 13, 1–15 (1995)
Chazelle, B., Matoušek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimension. J. Algorithms 21, 579–597 (1996)
Clarkson, K.L.: Las Vegas algorithms for linear and integer programming when the dimension is small. J. ACM 42, 488–499 (1995)
Clarkson, K.L.: Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm. In: Proceedings of 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 922–931 (2008)
Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry, II. Discrete Comput. Geom. 2, 195–222 (1987)
Clarkson, K.L., Woodruff, D.P.: Numerical linear algebra in the streaming model. In: Proceedings of 41st Annual ACM Symposium on Theory of Computing, pp. 205–214 (2009)
Gärtner, B., Jaggi, M.: Coresets for polytope distance. In: Proceedings of 25th Annual Symposium on Computational Geometry, pp. 33–42 (2009)
Goel, A., Indyk, P., Varadarajan, K.: Reductions among high dimensional proximity problems. In: Proceedings of 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 769–778 (2001)
Har-Peled, S., Varadarajan, K.R.: Projective clustering in high dimensions using core-sets. In: Proceedings of 18th Annual Symposium on Computational Geometry, pp. 312–318 (2003)
Indyk, P.: Better algorithms for high-dimensional proximity problems via asymmetric embeddings. In: Proceedings of 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 539–545 (2003)
Kumar, P., Mitchell, J.S.B., Yildirim, E.A.: Approximate minimum enclosing balls in high dimensions using core-sets. ACM J. Exp. Algorithmics 8, 1.1 (2003)
Kumar, P., Yildirim, E.A.: Minimum-volume enclosing ellipsoids and core sets. J. Optim. Theory Appl. 126, 1–21 (2005)
Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)
Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. Algorithmica 16, 498–516 (1996)
Muthukrishnan, S.: Data Streams: Algorithms and Applications. Now Publishers, Hanover (2005)
Ramos, E.A.: An optimal deterministic algorithm for computing the diameter of a three-dimensional point set. Discrete Comput. Geom. 26, 233–244 (2001)
Zarrabi-Zadeh, H., Chan, T.: A simple streaming algorithm for minimum enclosing balls. In: Proceedings of 18th Annual Canadian Conference on Computational Geometry, pp. 139–142 (2006)
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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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DOI: https://doi.org/10.1007/s00453-013-9846-4