Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Cardinal, Jean; Dallant, Justin; Iacono, John https://www.dagstuhl.de/lipics License: Creative Commons Attribution 4.0 license (CC BY 4.0)
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URN: urn:nbn:de:0030-drops-146057
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An Instance-Optimal Algorithm for Bichromatic Rectangular Visibility

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Abstract

Afshani, Barbay and Chan (2017) introduced the notion of instance-optimal algorithm in the order-oblivious setting. An algorithm A is instance-optimal in the order-oblivious setting for a certain class of algorithms 𝒜 if the following hold:
- A takes as input a sequence of objects from some domain;
- for any instance σ and any algorithm A' ∈ 𝒜, the runtime of A on σ is at most a constant factor removed from the runtime of A' on the worst possible permutation of σ. If we identify permutations of a sequence as representing the same instance, this essentially states that A is optimal on every possible input (and not only in the worst case).
We design instance-optimal algorithms for the problem of reporting, given a bichromatic set of points in the plane S, all pairs consisting of points of different color which span an empty axis-aligned rectangle (or reporting all points which appear in such a pair). This problem has applications for training-set reduction in nearest-neighbour classifiers. It is also related to the problem consisting of finding the decision boundaries of a euclidean nearest-neighbour classifier, for which Bremner et al. (2005) gave an optimal output-sensitive algorithm.
By showing the existence of an instance-optimal algorithm in the order-oblivious setting for this problem we push the methods of Afshani et al. closer to their limits by adapting and extending them to a setting which exhibits highly non-local features. Previous problems for which instance-optimal algorithms were proven to exist were based solely on local relationships between points in a set.

BibTeX - Entry

@InProceedings{cardinal_et_al:LIPIcs.ESA.2021.24,
  author =	{Cardinal, Jean and Dallant, Justin and Iacono, John},
  title =	{{An Instance-Optimal Algorithm for Bichromatic Rectangular Visibility}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{24:1--24:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14605},
  URN =		{urn:nbn:de:0030-drops-146057},
  doi =		{10.4230/LIPIcs.ESA.2021.24},
  annote =	{Keywords: computational geometry, instance-optimality, colored point sets, empty rectangles, visibility}
}

Keywords: computational geometry, instance-optimality, colored point sets, empty rectangles, visibility
Seminar: 29th Annual European Symposium on Algorithms (ESA 2021)
Issue date: 2021
Date of publication: 31.08.2021


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