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.2017.26
URN: urn:nbn:de:0030-drops-72279
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2017/7227/
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Chan, Timothy M.

Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry

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LIPIcs-SoCG-2017-26.pdf (0.5 MB)

Abstract

We apply the polynomial method - specifically, Chebyshev polynomials - to obtain a number of new results on geometric approximation algorithms in low constant dimensions. For example, we give an algorithm for constructing epsilon-kernels (coresets for approximate width and approximate convex hull) in close to optimal time O(n + (1/epsilon)^{(d-1)/2}), up to a small near-(1/epsilon)^{3/2} factor, for any d-dimensional n-point set. We obtain an improved data structure for Euclidean *approximate nearest neighbor search* with close to O(n log n + (1/epsilon)^{d/4} n) preprocessing time and O((1/epsilon)^{d/4} log n) query time. We obtain improved approximation algorithms for discrete Voronoi diagrams, diameter, and bichromatic closest pair in the L_s-metric for any even integer constant s >= 2. The techniques are general and may have further applications.

BibTeX - Entry

@InProceedings{chan:LIPIcs:2017:7227,
  author =	{Timothy M. Chan},
  title =	{{Applications of Chebyshev Polynomials to Low-Dimensional Computational Geometry}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Boris Aronov and Matthew J. Katz},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2017/7227},
  URN =		{urn:nbn:de:0030-drops-72279},
  doi =		{10.4230/LIPIcs.SoCG.2017.26},
  annote =	{Keywords: diameter, coresets, approximate nearest neighbor search, the polynomial method, streaming}
}

Keywords: diameter, coresets, approximate nearest neighbor search, the polynomial method, streaming
Collection: 33rd International Symposium on Computational Geometry (SoCG 2017)
Issue Date: 2017
Date of publication: 20.06.2017

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