License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2023.4
URN: urn:nbn:de:0030-drops-178544
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17854/
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Agarwal, Pankaj K. ; Har-Peled, Sariel

Computing Instance-Optimal Kernels in Two Dimensions

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LIPIcs-SoCG-2023-4.pdf (1 MB)

Abstract

Let P be a set of n points in ℝ². For a parameter ε ∈ (0,1), a subset C ⊆ P is an ε-kernel of P if the projection of the convex hull of C approximates that of P within (1-ε)-factor in every direction. The set C is a weak ε-kernel of P if its directional width approximates that of P in every direction. Let ?_ε(P) (resp. ?^?_ε(P)) denote the minimum-size of an ε-kernel (resp. weak ε-kernel) of P. We present an O(n ?_ε(P)log n)-time algorithm for computing an ε-kernel of P of size ?_ε(P), and an O(n²log n)-time algorithm for computing a weak ε-kernel of P of size ?^?_ε(P). We also present a fast algorithm for the Hausdorff variant of this problem.
In addition, we introduce the notion of ε-core, a convex polygon lying inside ch(P), prove that it is a good approximation of the optimal ε-kernel, present an efficient algorithm for computing it, and use it to compute an ε-kernel of small size.

BibTeX - Entry

@InProceedings{agarwal_et_al:LIPIcs.SoCG.2023.4,
  author =	{Agarwal, Pankaj K. and Har-Peled, Sariel},
  title =	{{Computing Instance-Optimal Kernels in Two Dimensions}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{4:1--4:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/17854},
  URN =		{urn:nbn:de:0030-drops-178544},
  doi =		{10.4230/LIPIcs.SoCG.2023.4},
  annote =	{Keywords: Coreset, approximation, kernel}
}

Keywords: Coreset, approximation, kernel
Collection: 39th International Symposium on Computational Geometry (SoCG 2023)
Issue Date: 2023
Date of publication: 09.06.2023

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