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DOI: 10.4230/LIPIcs.SOCG.2015.719
URN: urn:nbn:de:0030-drops-51353
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5135/
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Chan, Timothy M. ; Tsakalidis, Konstantinos

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

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Abstract

We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (1992) and the optimal but randomized algorithm of Ramos (1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, (<= k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, epsilon-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (1991) and Chazelle (1993).

BibTeX - Entry

@InProceedings{chan_et_al:LIPIcs:2015:5135,
  author =	{Timothy M. Chan and Konstantinos Tsakalidis},
  title =	{{Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{719--732},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5135},
  URN =		{urn:nbn:de:0030-drops-51353},
  doi =		{10.4230/LIPIcs.SOCG.2015.719},
  annote =	{Keywords: shallow cuttings, derandomization, halfspace range reporting, geometric data structures}
}

Keywords: shallow cuttings, derandomization, halfspace range reporting, geometric data structures
Seminar: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 11.06.2015


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