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URN: urn:nbn:de:0030-drops-138259
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Approximating the (Continuous) Fréchet Distance

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Abstract

We describe the first strongly subquadratic time algorithm with subexponential approximation ratio for approximately computing the Fréchet distance between two polygonal chains. Specifically, let P and Q be two polygonal chains with n vertices in d-dimensional Euclidean space, and let α ∈ [√n, n]. Our algorithm deterministically finds an O(α)-approximate Fréchet correspondence in time O((n³ / α²) log n). In particular, we get an O(n)-approximation in near-linear O(n log n) time, a vast improvement over the previously best know result, a linear time 2^O(n)-approximation. As part of our algorithm, we also describe how to turn any approximate decision procedure for the Fréchet distance into an approximate optimization algorithm whose approximation ratio is the same up to arbitrarily small constant factors. The transformation into an approximate optimization algorithm increases the running time of the decision procedure by only an O(log n) factor.

BibTeX - Entry

@InProceedings{colombe_et_al:LIPIcs.SoCG.2021.26,
  author =	{Colombe, Connor and Fox, Kyle},
  title =	{{Approximating the (Continuous) Fr\'{e}chet Distance}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{26:1--26:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/13825},
  URN =		{urn:nbn:de:0030-drops-138259},
  doi =		{10.4230/LIPIcs.SoCG.2021.26},
  annote =	{Keywords: Fr\'{e}chet distance, approximation algorithm, approximate decision procedure}
}

Keywords: Fréchet distance, approximation algorithm, approximate decision procedure
Seminar: 37th International Symposium on Computational Geometry (SoCG 2021)
Issue date: 2021
Date of publication: 02.06.2021


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