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Dynamic Geometric Data Structures via Shallow Cuttings

Author Timothy M. Chan

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LIPIcs.SoCG.2019.24.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
Keywords
  • dynamic data structures
  • convex hulls
  • nearest neighbor search
  • closest pair
  • shallow cuttings

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Abstract

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].

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.SoCG.2019.24

Author Details

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

Funding

  • Chan, Timothy M.: Work supported in part by NSF Grant CCF-1814026.

Acknowledgements

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.

References

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