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Geometric Optimization Problems Over Sliding Windows

  • Timothy M. Chan
  • Bashir S. Sadjad
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3341)

Abstract

We study the problem of maintaining a (1+ε)-factor approximation of the diameter  of a stream of points under the sliding window   model. In one dimension, we give a simple algorithm that only needs to store \(O(\frac{1}{\epsilon}{\rm log}R)\) points at any time, where the parameter R denotes the “spread” of the point set. This bound is optimal and improves Feigenbaum, Kannan, and Zhang’s recent solution by two logarithmic factors. We then extend our one-dimensional algorithm to higher constant dimensions and, at the same time, correct an error in the previous solution. In high nonconstant dimensions, we also observe a constant-factor approximation algorithm that requires sublinear space. Related optimization problems, such as the width, are also considered in the two-dimensional case.

Keywords

Approximation Algorithm Data Stream Computational Geometry Sliding Window Related Optimization Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agarwal, P.K., Har-Peled, S., Varadarajan, R.: Approximating extent measures of points. In: Journal of the ACM (to appear) Google Scholar
  2. 2.
    Agarwal, P.K., Matoušek, J., Suri, S.: Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Computational Geometry: Theory and Applications 1, 189–201 (1991)Google Scholar
  3. 3.
    Agarwal, P.K., Sharir, M.: Efficient randomized algorithms for some geometric optimization problems. Discrete & Computational Geometry 16, 317–337 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Barequet, G., Har-Peled, S.: Efficiently approximating the minimum-volume bounding box of a point set in three dimensions. Journal of Algorithms 38, 91–109 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Bentley, J.L., Saxe, J.B.: Decomposable searching problems I: Static-to-dynamic transformations. Journal of Algorithms 1(4), 301–358 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chan, T.M.: Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. International Journal on Computational Geometry and Applications 12, 67–85 (2002)zbMATHCrossRefGoogle Scholar
  7. 7.
    Chan, T.M.: Faster core-set constructions and data stream algorithms in fixed dimensions. In: Proceedings of the 20th Annual Symposium on Computational Geometry, pp. 152–159 (2004)Google Scholar
  8. 8.
    Clarkson, K.L., Shor, P.W.: Applications of random sampling in computational geometry, II. Discrete & Computational Geometry 4(1), 387–421 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Duncan, C.A., Goodrich, M.T., Ramos, E.A.: Efficient approximation and optimization algorithms for computational metrology. In: Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 121–130 (1997)Google Scholar
  10. 10.
    Feigenbaum, J., Kannan, S., Zhang, J.: Computing diameter in the streaming and sliding-window models. Algorithmica, to appear; or as Tech. Report DCS/TR-1245, Yale University (2002), http://cs-www.cs.yale.edu/homes/jf/FKZ.ps
  11. 11.
    Goel, A., Indyk, P., Varadarajan, K.: Reductions among high dimensional proximity problems. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 769–778 (2001)Google Scholar
  12. 12.
    Henzinger, M., Raghavan, P., Rajagopalan, S.: Computing on data streams. Technical Report SRC-TN-1998-011, Hewlett Packard Laboratories (1998)Google Scholar
  13. 13.
    Hershberger, J., Suri, S.: Convex hulls and related problems in data streams. In: ACM SIGMOD/PODS Workshop on Management and Processing of Data Streams, pp. 148–168 (2003)Google Scholar
  14. 14.
    Indyk, P.: Better algorithms for high-dimensional proximity problems via asymmetric embeddings. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 539–545 (2003)Google Scholar
  15. 15.
    Muthukrishnan, S.M.: Data streams: Algorithms and applications. Rutgers University Technical Report (2003), http://athos.rutgers.edu/~muthu/stream-1-1.ps
  16. 16.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, Heidelberg (1985)Google Scholar
  17. 17.
    Ramos, E.A.: An optimal deterministic algorithm for computing the diameter of a three-dimensional point set. Discrete & Computational Geometry 26, 233–244 (2001)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Timothy M. Chan
    • 1
  • Bashir S. Sadjad
    • 1
  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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