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Diameter and k-Center in Sliding Windows

Authors Vincent Cohen-Addad, Chris Schwiegelshohn, Christian Sohler

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Vincent Cohen-Addad
Chris Schwiegelshohn
Christian Sohler

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Vincent Cohen-Addad, Chris Schwiegelshohn, and Christian Sohler. Diameter and k-Center in Sliding Windows. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 19:1-19:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.ICALP.2016.19

Abstract

In this paper we develop streaming algorithms for the diameter problem and the k-center clustering problem in the sliding window model. In this model we are interested in maintaining a solution for the N most recent points of the stream. In the diameter problem we would like to maintain two points whose distance approximates the diameter of the point set in the window. Our algorithm computes a (3 + epsilon)-approximation and uses O(1/epsilon*ln(alpha)) memory cells, where alpha is the ratio of the largest and smallest distance and is assumed to be known in advance. We also prove that under reasonable assumptions obtaining a (3 - epsilon)-approximation requires Omega(N1/3) space.

For the k-center problem, where the goal is to find k centers that minimize the maximum distance of a point to its nearest center, we obtain a (6 + epsilon)-approximation using O(k/epsilon*ln(alpha)) memory cells and a (4 + epsilon)-approximation for the special case k = 2. We also prove that any algorithm for the 2-center problem that achieves an approximation ratio of less than 4 requires Omega(N^{1/3}) space.

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Keywords
  • Streaming
  • k-Center
  • Diameter
  • Sliding Windows

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References

  1. Pankaj K. Agarwal, Jirí Matousek, and Subhash Suri. Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Comput. Geom., 1:189-201, 1991. URL: http://dx.doi.org/10.1016/0925-7721(92)90001-9.
  2. Pankaj K. Agarwal and R. Sharathkumar. Streaming algorithms for extent problems in high dimensions. Algorithmica, 72(1):83-98, 2015. URL: http://dx.doi.org/10.1007/s00453-013-9846-4.
  3. Brian Babcock, Mayur Datar, Rajeev Motwani, and Liadan O'Callaghan. Maintaining variance and k-medians over data stream windows. In Proceedings of the Twenty-Second ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, June 9-12, 2003, San Diego, CA, USA, pages 234-243, 2003. URL: http://dx.doi.org/10.1145/773153.773176.
  4. Vladimir Braverman, Harry Lang, Keith Levin, and Morteza Monemizadeh. Clustering on sliding windows in polylogarithmic space. In 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2015, December 16-18, 2015, Bangalore, India, pages 350-364, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.350.
  5. Vladimir Braverman, Harry Lang, Keith Levin, and Morteza Monemizadeh. Clustering problems on sliding windows. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1374-1390, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch95.
  6. Vladimir Braverman and Rafail Ostrovsky. Effective computations on sliding windows. SIAM J. Comput., 39(6):2113-2131, 2010. URL: http://dx.doi.org/10.1137/090749281.
  7. Timothy M. Chan and Vinayak Pathak. Streaming and dynamic algorithms for minimum enclosing balls in high dimensions. Comput. Geom., 47(2):240-247, 2014. URL: http://dx.doi.org/10.1016/j.comgeo.2013.05.007.
  8. Timothy M. Chan and Bashir S. Sadjad. Geometric optimization problems over sliding windows. Int. J. Comput. Geometry Appl., 16(2-3):145-158, 2006. URL: http://dx.doi.org/10.1142/S0218195906001975.
  9. Moses Charikar, Chandra Chekuri, Tomás Feder, and Rajeev Motwani. Incremental clustering and dynamic information retrieval. In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 626-635, 1997. URL: http://dx.doi.org/10.1145/258533.258657.
  10. Moses Charikar, Liadan O'Callaghan, and Rina Panigrahy. Better streaming algorithms for clustering problems. In Proc. of the 35th Annual ACM Symp. on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 30-39, 2003. URL: http://dx.doi.org/10.1145/780542.780548.
  11. Michael S. Crouch, Andrew McGregor, and Daniel Stubbs. Dynamic graphs in the sliding-window model. In Algorithms - ESA 2013 - 21st Annual European Symposium, Sophia Antipolis, France, September 2-4, 2013. Proceedings, pages 337-348, 2013. URL: http://dx.doi.org/10.1007/978-3-642-40450-4_29.
  12. Mayur Datar, Aristides Gionis, Piotr Indyk, and Rajeev Motwani. Maintaining stream statistics over sliding windows. SIAM J. Comput., 31(6):1794-1813, 2002. URL: http://dx.doi.org/10.1137/S0097539701398363.
  13. Joan Feigenbaum, Sampath Kannan, and Jian Zhang. Computing diameter in the streaming and sliding-window models. Algorithmica, 41(1):25-41, 2004. URL: http://dx.doi.org/10.1007/s00453-004-1105-2.
  14. Teofilo F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci., 38:293-306, 1985. URL: http://dx.doi.org/10.1016/0304-3975(85)90224-5.
  15. Sudipto Guha. Tight results for clustering and summarizing data streams. In Database Theory - ICDT 2009, 12th International Conference, St. Petersburg, Russia, March 23-25, 2009, Proceedings, pages 268-275, 2009. URL: http://dx.doi.org/10.1145/1514894.1514926.
  16. Dorit S. Hochbaum and David B. Shmoys. A unified approach to approximation algorithms for bottleneck problems. J. ACM, 33(3):533-550, 1986. URL: http://dx.doi.org/10.1145/5925.5933.
  17. Piotr Indyk. Better algorithms for high-dimensional proximity problems via asymmetric embeddings. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, January 12-14, 2003, Baltimore, Maryland, USA., pages 539-545, 2003. Google Scholar
  18. Sang-Sub Kim and Hee-Kap Ahn. An improved data stream algorithm for clustering. Comput. Geom., 48(9):635-645, 2015. URL: http://dx.doi.org/10.1016/j.comgeo.2015.06.003.
  19. Richard Matthew McCutchen and Samir Khuller. Streaming algorithms for k-center clustering with outliers and with anonymity. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, 11th International Workshop, APPROX 2008, and 12th International Workshop, RANDOM 2008, Boston, MA, USA, August 25-27, 2008. Proceedings, pages 165-178, 2008. URL: http://dx.doi.org/10.1007/978-3-540-85363-3_14.
  20. Hamid Zarrabi-Zadeh. Core-preserving algorithms. In Proc. of the 20th Annual Canadian Conf. on Computational Geometry, Montréal, Canada, August 13-15, 2008, 2008. Google Scholar
  21. Hamid Zarrabi-Zadeh. An almost space-optimal streaming algorithm for coresets in fixed dimensions. Algorithmica, 60(1):46-59, 2011. URL: http://dx.doi.org/10.1007/s00453-010-9392-2.
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