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Dynamic Distribution-Sensitive Point Location

Authors Siu-Wing Cheng , Man-Kit Lau



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Siu-Wing Cheng
  • Department of Computer Science and Engineering, HKUST, Hong Kong, China
Man-Kit Lau
  • Department of Computer Science and Engineering, HKUST, Hong Kong, China

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Siu-Wing Cheng and Man-Kit Lau. Dynamic Distribution-Sensitive Point Location. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 30:1-30:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.SoCG.2020.30

Abstract

We propose a dynamic data structure for the distribution-sensitive point location problem. Suppose that there is a fixed query distribution in ℝ², and we are given an oracle that can return in O(1) time the probability of a query point falling into a polygonal region of constant complexity. We can maintain a convex subdivision S with n vertices such that each query is answered in O(OPT) expected time, where OPT is the minimum expected time of the best linear decision tree for point location in S. The space and construction time are O(n log² n). An update of S as a mixed sequence of k edge insertions and deletions takes O(k log⁵ n) amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of n sites can be performed in O(n log⁵ n) expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • dynamic planar point location
  • convex subdivision
  • linear decision tree

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References

  1. U. Adamy and R. Seidel. On the exact worst case query complexity of planar point location. Journal of Algorithms, 27(1):189-217, 2000. Google Scholar
  2. P. Afshani, J. Barbay, and T. Chan. Instance-optimal geometric algorithms. Journal of the ACM, 64(1):3:1-3:38, 2017. Google Scholar
  3. L. Arge, G.S. Brodal, and L Georgiadis. Improved dynamic planar point location. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 305-314, 2006. Google Scholar
  4. S. Arya, T. Malamatos, D. Mount, and K. Wong. Optimal expected-case planar point location. SIAM Journal on Computing, 37(2):584-610, 2007. Google Scholar
  5. H. Baumgarten, H. Jung, and K. Mehlhorn. Dynamic point location in general subdivisions. Journal of Algorithms, 17(3):342-380, 1994. Google Scholar
  6. S.W. Bent, D.D. Sleator, and R.E. Tarjan. Biased search trees. SIAM Journal on Computing, 14(3):545-568, 1985. Google Scholar
  7. Prosenjit Bose, Luc Devroye, Karim Douieb, Vida Dujmovic, James King, and Pat Morin. Odds-on trees, 2010. URL: http://arxiv.org/abs/1002.1092.
  8. T. Chan and Y. Nekrich. Towards an optimal method for dynamic planar point location. SIAM Journal on Computing, 47(6):2337-2361, 2018. Google Scholar
  9. S.-W. Cheng and R. Janardan. New results on dynamic planar point location. SIAM Journal on Computing, 21(5):972-999, 1992. Google Scholar
  10. S.-W. Cheng and M.-K. Lau. Adaptive planar point location. In Proceedings of the 33rd International Symposium of Computational Geometry, pages 30:1-30:15, 2017. Google Scholar
  11. S.-W. Cheng and M.-K. Lau. Adaptive point location in planar convex subdivisions. International Journal of Computational Geometry and Applications, 27(1-2):3-12, 2017. Google Scholar
  12. S.-W. Cheng and M.-K. Lau. Adaptive planar point location, 2018. URL: http://arxiv.org/abs/1810.00715.
  13. S.-W. Cheng and M.-K. Lau. Dynamic distribution-sensitive point location, 2020. URL: http://arxiv.org/abs/2003.08288.
  14. Y.-J. Chiang, F.P. Preparata, and R. Tamassia. A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps. SIAM Journal on Computing, 25(1):207-233, 1996. Google Scholar
  15. Y.-J. Chiang and R. Tamassia. Dynamization of the trapezoid method for planar point location in monotone subdivisions. Internatational Journal of Computational Geometry and Applications, 2(3):311-333, 1992. Google Scholar
  16. S. Collette, V. Dujmović, J. Iacono, S. Langerman, and P. Morin. Entropy, triangulation, and point location in planar subdivisions. ACM Transactions on Algorithms, 8(3):29:1-29:18, 2012. Google Scholar
  17. D.P. Dobkin and D.G. Kirkpatrick. Determining the separation of preprocessed polyhedra - a unified approach. In Proceedings of the 17th International Colloquium on Automata, Languages and Programming, pages 400-413, 1990. Google Scholar
  18. J.R. Driscoll, N. Sarnak, D.D. Sleator, and R.E. Tarjan. Making data structures persistent. Journal of Computer and System Sciences, 38(1):86-124, 1989. Google Scholar
  19. H. Edelsbrunner, L. J Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM Journal on Computing, 15(2):317-340, 1986. Google Scholar
  20. M.T. Goodrich and R. Tamassia. Dynamic ray shooting and shortest paths in planar subdivisions via balanced geodesic triangulations. Journal of Algorithms, 23(1):51-73, 1997. Google Scholar
  21. M.T. Goodrich and R. Tamassia. Dynamic trees and dynamic point location. SIAM Journal on Computing, 28(2):612-636, 1998. Google Scholar
  22. J. Hershberger and S. Suri. A pedestrian approach to ray shooting: Shoot a ray, take a walk. Journal of Algorithms, 18(3):403-431, 1995. Google Scholar
  23. J. Iacono. Expected asymptotically optimal planar point location. Computational Geometry: Theory and Applications, 29(1):19-22, 2004. Google Scholar
  24. J. Iacono and W. Mulzer. A static optimality transformation with applications to planar point location. International Journal of Computational Geometry and Applications, 22(4):327-340, 2012. Google Scholar
  25. D. G. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12(1):28-35, 1983. Google Scholar
  26. E. Oh. Point location in incremental planar subdivisions. In Proceedings of the 29th International Symposium on Algorithms and Computation, pages 51:1-51:12, 2018. Google Scholar
  27. E. Oh and H.-K. Ahn. Point location in dynamic planar subdivision. In Proceedings of the 34th International Symposium on Computational Geometry, pages 63:1-53:14, 2018. Google Scholar
  28. F.P. Preparata and R. Tamassia. Fully dynamic point location in a monotone subdivision. SIAM Journal on Computing, 18(4):811-830, 1989. Google Scholar
  29. N. Sarnak and R. E. Tarjan. Planar point location using persistent search trees. Communications of ACM, 29(7):669-679, 1986. Google Scholar
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