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Dynamic Geometric Data Structures via Shallow Cuttings

Author Timothy M. Chan

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


I thank Sariel Har-Peled for discussions on other problems that indirectly led to the results of this paper. Thanks also to Mitchell Jones for discussions on range searching for points in convex position.

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Timothy M. Chan. Dynamic Geometric Data Structures via Shallow Cuttings. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 24:1-24:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


We present new results on a number of fundamental problems about dynamic geometric data structures: 1) We describe the first fully dynamic data structures with sublinear amortized update time for maintaining (i) the number of vertices or the volume of the convex hull of a 3D point set, (ii) the largest empty circle for a 2D point set, (iii) the Hausdorff distance between two 2D point sets, (iv) the discrete 1-center of a 2D point set, (v) the number of maximal (i.e., skyline) points in a 3D point set. The update times are near n^{11/12} for (i) and (ii), n^{7/8} for (iii) and (iv), and n^{2/3} for (v). Previously, sublinear bounds were known only for restricted "semi-online" settings [Chan, SODA 2002]. 2) We slightly improve previous fully dynamic data structures for answering extreme point queries for the convex hull of a 3D point set and nearest neighbor search for a 2D point set. The query time is O(log^2n), and the amortized update time is O(log^4n) instead of O(log^5n) [Chan, SODA 2006; Kaplan et al., SODA 2017]. 3) We also improve previous fully dynamic data structures for maintaining the bichromatic closest pair between two 2D point sets and the diameter of a 2D point set. The amortized update time is O(log^4n) instead of O(log^7n) [Eppstein 1995; Chan, SODA 2006; Kaplan et al., SODA 2017].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
  • dynamic data structures
  • convex hulls
  • nearest neighbor search
  • closest pair
  • shallow cuttings


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  1. Peyman Afshani and Timothy M. Chan. Optimal halfspace range reporting in three dimensions. In Proc. 20th ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 180-186, 2009. URL:
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  7. Timothy M. Chan. Dynamic planar convex hull operations in near-logarithmic amortized time. J. ACM, 48(1):1-12, 2001. Preliminary version in FOCS 1999. URL:
  8. Timothy M. Chan. A fully dynamic algorithm for planar width. Discrete Comput. Geom., 30(1):17-24, 2003. Preliminary version in SoCG 2001. URL:
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  10. Timothy M. Chan. A dynamic data structure for 3-d convex hulls and 2-d nearest neighbor queries. J. ACM, 57(3):16:1-16:15, 2010. Preliminary version in SODA 2006. URL:
  11. Timothy M. Chan. Optimal partition trees. Discrete Comput. Geom., 47(4):661-690, 2012. Preliminary version in SoCG 2010. URL:
  12. Timothy M. Chan. Three problems about dynamic convex hulls. Int. J. Comput. Geom. Appl., 22(4):341-364, 2012. Preliminary version in SoCG 2011. URL:
  13. Timothy M. Chan, Kasper Green Larsen, and Mihai Pătraşcu. Orthogonal range searching on the RAM, revisited. In Proc. 27th ACM Sympos. Comput. Geom. (SoCG), pages 1-10, 2011. URL:
  14. Timothy M. Chan, Mihai Pătraşcu, and Liam Roditty. Dynamic connectivity: Connecting to networks and geometry. SIAM J. Comput., 40(2):333-349, 2011. Preliminary version in FOCS 2008. URL:
  15. Timothy M. Chan and Konstantinos Tsakalidis. Optimal deterministic algorithms for 2-d and 3-d shallow cuttings. Discrete Comput. Geom., 56(4):866-881, 2016. Preliminary version in SoCG 2015. URL:
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