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.ISAAC.2019.50
URN: urn:nbn:de:0030-drops-115462
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11546/
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Alves Akitaya, Hugo ; Buchin, Maike ; Ryvkin, Leonie ; Urhausen, Jérôme

The k-Fréchet Distance: How to Walk Your Dog While Teleporting

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LIPIcs-ISAAC-2019-50.pdf (0.7 MB)

Abstract

We introduce a new distance measure for comparing polygonal chains: the k-Fréchet distance. As the name implies, it is closely related to the well-studied Fréchet distance but detects similarities between curves that resemble each other only piecewise. The parameter k denotes the number of subcurves into which we divide the input curves (thus we allow up to k-1 "teleports" on each input curve). The k-Fréchet distance provides a nice transition between (weak) Fréchet distance and Hausdorff distance. However, we show that deciding this distance measure turns out to be NP-hard, which is interesting since both (weak) Fréchet and Hausdorff distance are computable in polynomial time. Nevertheless, we give several possibilities to deal with the hardness of the k-Fréchet distance: besides a short exponential-time algorithm for the general case, we give a polynomial-time algorithm for k=2, i.e., we ask that we subdivide our input curves into two subcurves each. We can also approximate the optimal k by factor 2. We then present a more intricate FPT algorithm using parameters k (the number of allowed subcurves) and z (the number of segments of one curve that intersect the epsilon-neighborhood of a point on the other curve).

BibTeX - Entry

@InProceedings{alvesakitaya_et_al:LIPIcs:2019:11546,
  author =	{Hugo Alves Akitaya and Maike Buchin and Leonie Ryvkin and J{\'e}r{\^o}me Urhausen},
  title =	{{The k-Fr{\'e}chet Distance: How to Walk Your Dog While Teleporting}},
  booktitle =	{30th International Symposium on Algorithms and Computation (ISAAC 2019)},
  pages =	{50:1--50:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-130-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{149},
  editor =	{Pinyan Lu and Guochuan Zhang},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2019/11546},
  URN =		{urn:nbn:de:0030-drops-115462},
  doi =		{10.4230/LIPIcs.ISAAC.2019.50},
  annote =	{Keywords: Measures, Fr{\'e}chet distance, Hardness, FPT}
}

Keywords: Measures, Fréchet distance, Hardness, FPT
Collection: 30th International Symposium on Algorithms and Computation (ISAAC 2019)
Issue Date: 2019
Date of publication: 28.11.2019

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