On coresets for k-means and k-median clustering

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On coresets for k-means and k-median clustering

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Annual ACM Symposium on Theory of Computing archive
Proceedings of the thirty-sixth annual ACM symposium on Theory of computing table of contents

Chicago, IL, USA

SESSION: Session 7B table of contents

Pages: 291 - 300

Year of Publication: 2004

ISBN:1-58113-852-0

Authors

Sariel Har-Peled	University of Illinois, Urbana-Champaign, Urbana, IL
Soham Mazumdar	University of Illinois, Urbana-Champaign, Urbana, IL

Sponsors

ACM: Association for Computing Machinery
SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

ACM New York, NY, USA

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Downloads (6 Weeks): 23, Downloads (12 Months): 159, Citation Count: 30

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ABSTRACT

In this paper, we show the existence of small coresets for the problems of computing k-median and k-means clustering for points in low dimension. In other words, we show that given a point set P in R^d, one can compute a weighted set S ⊆ P, of size O(k ε^-d log n), such that one can compute the k-median/means clustering on S instead of on P, and get an (1+ε)-approximation. As a result, we improve the fastest known algorithms for (1+ε)-approximate k-means and k-median. Our algorithms have linear running time for a fixed k and ε. In addition, we can maintain the (1+ε)-approximate k-median or k-means clustering of a stream when points are being only inserted, using polylogarithmic space and update time.

REFERENCES

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	1	Pankaj K. Agarwal , Sariel Har-Peled , Kasturi R. Varadarajan, Approximating extent measures of points, Journal of the ACM (JACM), v.51 n.4, p.606-635, July 2004 [doi>10.1145/1008731.1008736]
	2	Pankaj K. Agarwal , Cecilia Magdalena Procopiuc , Kasturi R. Varadarajan, Approximation Algorithms for k-Line Center, Proceedings of the 10th Annual European Symposium on Algorithms, p.54-63, September 17-21, 2002
	3	Arnon Amir , Alon Efrat , Piotr Indyk , Hanan Samet, Efficient Regular Data Structures and Algorithms for Location and Proximity Problems, Proceedings of the 40th Annual Symposium on Foundations of Computer Science, p.160, October 17-18, 1999
	4	Sanjeev Arora, Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, Journal of the ACM (JACM), v.45 n.5, p.753-782, Sept. 1998 [doi>10.1145/290179.290180]
	5	Sanjeev Arora , Prabhakar Raghavan , Satish Rao, Approximation schemes for Euclidean k-medians and related problems, Proceedings of the thirtieth annual ACM symposium on Theory of computing, p.106-113, May 24-26, 1998, Dallas, Texas, United States [doi>10.1145/276698.276718]
	6	Sunil Arya , David M. Mount , Nathan S. Netanyahu , Ruth Silverman , Angela Y. Wu, An optimal algorithm for approximate nearest neighbor searching fixed dimensions, Journal of the ACM (JACM), v.45 n.6, p.891-923, Nov. 1998 [doi>10.1145/293347.293348]
	7	Vijay Arya , Naveen Garg , Rohit Khandekar , Kamesh Munagala , Vinayaka Pandit, Local search heuristic for k-median and facility location problems, Proceedings of the thirty-third annual ACM symposium on Theory of computing, p.21-29, July 2001, Hersonissos, Greece [doi>10.1145/380752.380755]
	8	M. Badoiu and K. L. Clarkson. Optimal core-sets for balls. In Proc. 14th ACM-SIAM Sympos. Discrete Algorithms, pages 801--802, 2003.
	9	Mihai Bādoiu , Sariel Har-Peled , Piotr Indyk, Approximate clustering via core-sets, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada [doi>10.1145/509907.509947]
	10	J. L. Bentley and J. B. Saxe. Decomposable searching problems i: Static-to-dynamic transformation. J. Algorithms, 1(4):301--358, 1980.
	11	Moses Charikar , Sudipto Guha, Improved Combinatorial Algorithms for the Facility Location and k-Median Problems, Proceedings of the 40th Annual Symposium on Foundations of Computer Science, p.378, October 17-18, 1999
	12	Moses Charikar , Sudipto Guha , Éva Tardos , David B. Shmoys, A constant-factor approximation algorithm for the k-median problem (extended abstract), Proceedings of the thirty-first annual ACM symposium on Theory of computing, p.1-10, May 01-04, 1999, Atlanta, Georgia, United States [doi>10.1145/301250.301257]
	13	W. Fernandez de la Vega , Marek Karpinski , Claire Kenyon , Yuval Rabani, Approximation schemes for clustering problems, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 09-11, 2003, San Diego, CA, USA [doi>10.1145/780542.780550]
	14	Tomás Feder , Daniel Greene, Optimal algorithms for approximate clustering, Proceedings of the twentieth annual ACM symposium on Theory of computing, p.434-444, May 02-04, 1988, Chicago, Illinois, United States [doi>10.1145/62212.62255]
	15	Mordecai Golin , Rajeev Raman , Christian Schwarz , Michiel Smid, Simple randomized algorithms for closest pair problems, Nordic Journal of Computing, v.2 n.1, p.3-27, March 1995
	16	T. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci., 38:293--306, 1985.
	17	S. Guha , N. Mishra , R. Motwani , L. O'Callaghan, Clustering data streams, Proceedings of the 41st Annual Symposium on Foundations of Computer Science, p.359, November 12-14, 2000
	18	S. Har-Peled, Clustering Motion, Proceedings of the 42nd IEEE symposium on Foundations of Computer Science, p.84, October 14-17, 2001
	19	S. Har-Peled, A Replacement for Voronoi Diagrams of Near Linear Size, Proceedings of the 42nd IEEE symposium on Foundations of Computer Science, p.94, October 14-17, 2001
	20	Sariel Har-Peled , Kasturi Varadarajan, Projective clustering in high dimensions using core-sets, Proceedings of the eighteenth annual symposium on Computational geometry, p.312-318, June 05-07, 2002, Barcelona, Spain [doi>10.1145/513400.513440]
	21	Mary Inaba , Naoki Katoh , Hiroshi Imai, Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering: (extended abstract), Proceedings of the tenth annual symposium on Computational geometry, p.332-339, June 06-08, 1994, Stony Brook, New York, United States [doi>10.1145/177424.178042]
	22	Piotr Indyk, Sublinear time algorithms for metric space problems, Proceedings of the thirty-first annual ACM symposium on Theory of computing, p.428-434, May 01-04, 1999, Atlanta, Georgia, United States [doi>10.1145/301250.301366]
	23	P. Indyk. (1+ε)-approximate bi-criterion algorithm for k-median on a stream. In Proc. 36th Annu. ACM Sympos. Theory Comput., 2004.
	24	Kamal Jain , Vijay V. Vazirani, Primal-Dual Approximation Algorithms for Metric Facility Location and k-Median Problems, Proceedings of the 40th Annual Symposium on Foundations of Computer Science, p.2, October 17-18, 1999
	25	Tapas Kanungo , David M. Mount , Nathan S. Netanyahu , Christine D. Piatko , Ruth Silverman , Angela Y. Wu, A local search approximation algorithm for k-means clustering, Proceedings of the eighteenth annual symposium on Computational geometry, p.10-18, June 05-07, 2002, Barcelona, Spain [doi>10.1145/513400.513402]
	26	Stavros G. Kolliopoulos , Satish Rao, A Nearly Linear-Time Approximation Scheme for the Euclidean kappa-median Problem, Proceedings of the 7th Annual European Symposium on Algorithms, p.378-389, July 16-18, 1999
	27	Moses Charikar , Liadan O'Callaghan , Rina Panigrahy, Better streaming algorithms for clustering problems, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 09-11, 2003, San Diego, CA, USA [doi>10.1145/780542.780548]
	28	J. Matousek. On approximate geometric k-clustering. Discrete Comput. Geom., 24:61--84, 2000.
	29	R. Mettu and C. G. Plaxton. Optimal time bounds for approximate clustering. In In Proceedings of the 18th Conference on Uncertainty in Artifical Intelligence, pages 344--351, 2002.

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