License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2019.24
URN: urn:nbn:de:0030-drops-104288
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10428/
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Chan, Timothy M.

Dynamic Geometric Data Structures via Shallow Cuttings

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LIPIcs-SoCG-2019-24.pdf (0.5 MB)

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].

BibTeX - Entry

@InProceedings{chan:LIPIcs:2019:10428,
  author =	{Timothy M. Chan},
  title =	{{Dynamic Geometric Data Structures via Shallow Cuttings}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{24:1--24:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Gill Barequet and Yusu Wang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10428},
  URN =		{urn:nbn:de:0030-drops-104288},
  doi =		{10.4230/LIPIcs.SoCG.2019.24},
  annote =	{Keywords: dynamic data structures, convex hulls, nearest neighbor search, closest pair, shallow cuttings}
}

Keywords: dynamic data structures, convex hulls, nearest neighbor search, closest pair, shallow cuttings
Collection: 35th International Symposium on Computational Geometry (SoCG 2019)
Issue Date: 2019
Date of publication: 11.06.2019

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