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More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time

Authors Timothy M. Chan , Qizheng He



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Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Qizheng He
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA

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Timothy M. Chan and Qizheng He. More Dynamic Data Structures for Geometric Set Cover with Sublinear Update Time. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 25:1-25:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.25

Abstract

We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an O(1)-approximation in sublinear update time for set cover for axis-aligned squares in 2D . More precisely, we obtain randomized update time O(n^{2/3+δ}) for an arbitrarily small constant δ > 0. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D . As a byproduct, our techniques for dynamic set cover also yield an optimal randomized O(nlog n)-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier O(nlog n(log log n)^{O(1)}) result [SoCG 2020].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Geometric set cover
  • approximation algorithms
  • dynamic data structures
  • sublinear algorithms
  • random sampling

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References

  1. Anna Adamaszek, Sariel Har-Peled, and Andreas Wiese. Approximation schemes for independent set and sparse subsets of polygons. Journal of the ACM, 66(4):29:1-29:40, 2019. URL: https://doi.org/10.1145/3326122.
  2. Pankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dynamic geometric set cover and hitting set. In Proceedings of the 36th Symposium on Computational Geometry (SoCG), volume 164, pages 2:1-2:15, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.2.
  3. Pankaj K. Agarwal and Jeff Erickson. Geometric range searching and its relatives. In B. Chazelle, J. E. Goodman, and R. Pollack, editors, Advances in Discrete and Computational Geometry, pages 1-56. AMS Press, 1999. URL: http://jeffe.cs.illinois.edu/pubs/survey.html.
  4. Pankaj K. Agarwal and Jiangwei Pan. Near-linear algorithms for geometric hitting sets and set covers. Discrete & Computational Geometry, 63(2):460-482, 2020. Preliminary version in SoCG'14. URL: https://doi.org/10.1007/s00454-019-00099-6.
  5. Sunil Arya, David M. Mount, Nathan S. Netanyahu, Ruth Silverman, and Angela Y. Wu. An optimal algorithm for approximate nearest neighbor searching in fixed dimensions. Journal of the ACM, 45(6):891-923, 1998. URL: https://doi.org/10.1145/293347.293348.
  6. Hervé Brönnimann and Michael T. Goodrich. Almost optimal set covers in finite VC-dimension. Discrete & Computational Geometry, 14(4):463-479, 1995. Google Scholar
  7. Timothy M. Chan. Random sampling, halfspace range reporting, and construction of (≤ k)-levels in three dimensions. SIAM Journal on Computing, 30(2):561-575, 2000. URL: https://doi.org/10.1137/S0097539798349188.
  8. Timothy M. Chan. Semi-online maintenance of geometric optima and measures. SIAM Journal on Computing, 32(3):700-716, 2003. URL: https://doi.org/10.1137/S0097539702404389.
  9. Timothy M. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. Journal of the ACM, 57(3):16:1-16:15, 2010. URL: https://doi.org/10.1145/1706591.1706596.
  10. Timothy M. Chan. Optimal partition trees. Discrete & Computational Geometry, 47(4):661-690, 2012. URL: https://doi.org/10.1007/s00454-012-9410-z.
  11. Timothy M. Chan and Qizheng He. Faster approximation algorithms for geometric set cover. In Proceedings of the 36th Symposium on Computational Geometry (SoCG), volume 164, pages 27:1-27:14, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.27.
  12. Timothy M. Chan and Konstantinos Tsakalidis. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. Discrete & Computational Geometry, 56(4):866-881, 2016. Google Scholar
  13. Kenneth L. Clarkson. New applications of random sampling in computational geometry. Discrete & Computational Geometry, 2:195-222, 1987. URL: https://doi.org/10.1007/BF02187879.
  14. Kenneth L. Clarkson. Algorithms for polytope covering and approximation. In Workshop on Algorithms and Data Structures, pages 246-252, 1993. Google Scholar
  15. Kenneth L. Clarkson and Kasturi Varadarajan. Improved approximation algorithms for geometric set cover. Discrete & Computational Geometry, 37(1):43-58, 2007. Google Scholar
  16. Artur Czumaj, Funda Ergün, Lance Fortnow, Avner Magen, Ilan Newman, Ronitt Rubinfeld, and Christian Sohler. Approximating the weight of the Euclidean minimum spanning tree in sublinear time. SIAM Journal on Computing, 35(1):91-109, 2005. URL: https://doi.org/10.1137/S0097539703435297.
  17. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications. Springer, 3rd edition, 2008. URL: https://www.worldcat.org/oclc/227584184.
  18. Greg N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications. SIAM Journal on Computing, 16(6):1004-1022, 1987. Google Scholar
  19. Monika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic approximate maximum independent set of intervals, hypercubes and hyperrectangles. In Proceedings of the 36th Symposium on Computational Geometry (SoCG), volume 164, pages 51:1-51:14, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.51.
  20. Richard J. Lipton and Robert Endre Tarjan. Applications of a planar separator theorem. SIAM Journal on Computing, 9(3):615-627, 1980. URL: https://doi.org/10.1137/0209046.
  21. Chih-Hung Liu, Evanthia Papadopoulou, and D. T. Lee. An output-sensitive approach for the L₁/L_∞ k-nearest-neighbor Voronoi diagram. In Proceedings of the 19th Annual European Symposium on Algorithms (ESA), volume 6942 of Lecture Notes in Computer Science, pages 70-81. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-23719-5_7.
  22. Jiří Matoušek. Efficient partition trees. Discrete & Computational Geometry, 8(3):315-334, 1992. Google Scholar
  23. Jiří Matoušek. Reporting points in halfspaces. Computational Geometry, 2(3):169-186, 1992. Google Scholar
  24. Jiří Matoušek. Range searching with efficient hierarchical cutting. Discrete & Computational Geometry, 10:157-182, 1993. URL: https://doi.org/10.1007/BF02573972.
  25. Nabil H. Mustafa, Rajiv Raman, and Saurabh Ray. Quasi-polynomial time approximation scheme for weighted geometric set cover on pseudodisks and halfspaces. SIAM Journal on Computing, 44(6):1650-1669, 2015. URL: https://doi.org/10.1137/14099317X.
  26. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete & Computational Geometry, 44(4):883-895, 2010. URL: https://doi.org/10.1007/s00454-010-9285-9.
  27. Micha Sharir. On k-sets in arrangement of curves and surfaces. Discrete & Computational Geometry, 6:593-613, 1991. URL: https://doi.org/10.1007/BF02574706.
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