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Simple Multi-Pass Streaming Algorithms for Skyline Points and Extreme Points

Authors Timothy M. Chan, Saladi Rahul

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Author Details

Timothy M. Chan
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Saladi Rahul
  • Dept. of Computer Science and Automation, Indian Institute of Science Bangalore, India

Funding

  • Chan, Timothy M.: Work supported in part by NSF Grant CCF-1814026.
  • Rahul, Saladi: Work supported by IISc start-up research grant.

Cite AsGet BibTex

Timothy M. Chan and Saladi Rahul. Simple Multi-Pass Streaming Algorithms for Skyline Points and Extreme Points. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.22

Abstract

In this paper, we present simple randomized multi-pass streaming algorithms for fundamental computational geometry problems of finding the skyline (maximal) points and the extreme points of the convex hull. For the skyline problem, one of our algorithm occupies O(h) space and performs O(log n) passes, where h is the number of skyline points. This improves the space bound of the currently best known result by Das Sarma, Lall, Nanongkai, and Xu [VLDB'09] by a logarithmic factor. For the extreme points problem, we present the first non-trivial result for any constant dimension greater than two: an O(h log^{O(1)}n) space and O(log^dn) pass algorithm, where h is the number of extreme points. Finally, we argue why randomization seems unavoidable for these problems, by proving lower bounds on the performance of deterministic algorithms for a related problem of finding maximal elements in a poset.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • multi-pass streaming algorithms
  • skyline
  • convex hull
  • extreme points
  • randomized algorithms

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References

  1. Peyman Afshani. Fast computation of output-sensitive maxima in a word RAM. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1414-1423, 2014. Google Scholar
  2. Peyman Afshani, Jérémy Barbay, and Timothy M. Chan. Instance-optimal geometric algorithms. Journal of the ACM, 64(1):3:1-3:38, 2017. Google Scholar
  3. Timothy M. Chan. Output-sensitive results on convex hulls, extreme points, and related problems. Discrete & Computational Geometry, 16(4):369-387, 1996. Google Scholar
  4. Timothy M. Chan. Improved deterministic algorithms for linear programming in low dimensions. ACM Trans. Algorithms, 14(3):30:1-30:10, 2018. URL: https://doi.org/10.1145/3155312.
  5. Timothy M. Chan and Eric Y. Chen. Multi-pass geometric algorithms. Discrete & Computational Geometry, 37(1):79-102, 2007. URL: https://doi.org/10.1007/s00454-006-1275-6.
  6. Timothy M. Chan, Kasper Green Larsen, and Mihai Pǎtraşcu. Orthogonal range searching on the RAM, revisited. In Proceedings of Symposium on Computational Geometry (SoCG), pages 1-10, 2011. Google Scholar
  7. Timothy M. Chan, Jack Snoeyink, and Chee-Keng Yap. Primal dividing and dual pruning: Output-sensitive construction of four-dimensional polytopes and three-dimensional Voronoi diagrams. Discrete & Computational Geometry, 18(4):433-454, 1997. Google Scholar
  8. Kenneth L. Clarkson. More output-sensitive geometric algorithms. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 695-702, 1994. Google Scholar
  9. Kenneth L. Clarkson. Las Vegas algorithms for linear and integer programming when the dimension is small. Journal of the ACM, 42(2):488-499, 1995. URL: https://doi.org/10.1145/201019.201036.
  10. Kenneth L. Clarkson and Peter W. Shor. Application of random sampling in computational geometry, II. Discret. Comput. Geom., 4:387-421, 1989. URL: https://doi.org/10.1007/BF02187740.
  11. Atish Das Sarma, Ashwin Lall, Danupon Nanongkai, and Jun (Jim) Xu. Randomized multi-pass streaming skyline algorithms. PVLDB, 2(1):85-96, 2009. URL: https://doi.org/10.14778/1687627.1687638.
  12. Martin Farach-Colton, Meng Li, and Meng-Tsung Tsai. Streaming algorithms for planar convex hulls. In International Symposium on Algorithms and Computation (ISAAC), pages 47:1-47:13, 2018. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2018.47.
  13. Michael Greenwald and Sanjeev Khanna. Space-efficient online computation of quantile summaries. In Proceedings of ACM Management of Data (SIGMOD), pages 58-66, 2001. Google Scholar
  14. Xiaocheng Hu, Cheng Sheng, Yufei Tao, Yi Yang, and Shuigeng Zhou. Output-sensitive skyline algorithms in external memory. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 887-900, 2013. Google Scholar
  15. J.H. Dula J and R.V. Helgason. A new procedure for identifying the frame of the convex hull of a finite collection of points in multidimensional space. European Journal of Operational Research, 92(2):352-367, 1996. Google Scholar
  16. David G. Kirkpatrick and Raimund Seidel. Output-size sensitive algorithms for finding maximal vectors. In Proceedings of Symposium on Computational Geometry (SoCG), pages 89-96, 1985. URL: https://doi.org/10.1145/323233.323246.
  17. David G. Kirkpatrick and Raimund Seidel. The ultimate planar convex hull algorithm? SIAM Journal of Computing, 15(1):287-299, 1986. URL: https://doi.org/10.1137/0215021.
  18. Ketan Mulmuley. Computational Geometry: An Introduction through Randomized Algorithms. Prentice Hall, 1994. Google Scholar
  19. Thomas Ottmann, Sven Schuierer, and Subbiah Soundaralakshmi. Enumerating extreme points in higher dimensions. Nordic Journal of Computing, 8(2):179-192, 2001. Google Scholar
  20. F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985. Google Scholar
  21. Raimund Seidel. Small-dimensional linear programming and convex hulls made easy. Discrete & Computational Geometry, 6:423-434, 1991. URL: https://doi.org/10.1007/BF02574699.
  22. Cheng Sheng and Yufei Tao. On finding skylines in external memory. In Proceedings of ACM Symposium on Principles of Database Systems (PODS), pages 107-116, 2011. URL: https://doi.org/10.1145/1989284.1989298.
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