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.2018.45
URN: urn:nbn:de:0030-drops-87581
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8758/
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van der Hoog, Ivor ; Khramtcova, Elena ; Löffler, Maarten

Dynamic Smooth Compressed Quadtrees

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LIPIcs-SoCG-2018-45.pdf (0.6 MB)

Abstract

We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in R^d can be made smooth and dynamic subject to insertions and deletions of points. The second version uses the first but must additionally deal with compression and alignment of quadtree components. In both cases our updates take 2^{O(d log d)} time, except for the point location part in the second version which has a lower bound of Omega(log n); but if a pointer (finger) to the correct quadtree cell is given, the rest of the updates take worst-case constant time. Our result implies that several classic and recent results (ranging from ray tracing to planar point location) in computational geometry which use quadtrees can deal with arbitrary point sets on a real RAM pointer machine.

BibTeX - Entry

@InProceedings{hoogvd_et_al:LIPIcs:2018:8758,
  author =	{Ivor van der Hoog and Elena Khramtcova and Maarten L{\"o}ffler},
  title =	{{Dynamic Smooth Compressed Quadtrees}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{45:1--45:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8758},
  URN =		{urn:nbn:de:0030-drops-87581},
  doi =		{10.4230/LIPIcs.SoCG.2018.45},
  annote =	{Keywords: smooth, dynamic, data structure, quadtree, compression, alignment, Real RAM}
}

Keywords: smooth, dynamic, data structure, quadtree, compression, alignment, Real RAM
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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