Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Wenk, Carola; Cook, Atlas F. https://www.dagstuhl.de/lipics License: Creative Commons Attribution-NoDerivs 3.0 Unported license (CC BY-ND 3.0)
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URN: urn:nbn:de:0030-drops-13303
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Geodesic Fréchet Distance Inside a Simple Polygon

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Abstract

We unveil an alluring alternative to parametric search that applies
to both the non-geodesic and geodesic Fr{'\e}chet optimization
problems. This randomized approach is based on a variant of
red-blue intersections and is appealing due to its elegance and
practical efficiency when compared to parametric search.

We present the first algorithm for the geodesic Fr{'\e}chet distance
between two polygonal curves $A$ and $B$ inside a simple bounding
polygon $P$. The geodesic Fr{'\e}chet decision problem is solved
almost as fast as its non-geodesic sibling and requires $O(N^{2log
k)$ time and $O(k+N)$ space after $O(k)$ preprocessing, where $N$
is the larger of the complexities of $A$ and $B$ and $k$ is the
complexity of $P$. The geodesic Fr{'\e}chet optimization problem is
solved by a randomized approach in $O(k+N^{2log kNlog N)$
expected time and $O(k+N^{2)$ space. This runtime is only a
logarithmic factor larger than the standard non-geodesic Fr{'\e}chet
algorithm (Alt and Godau 1995). Results are also presented for the
geodesic Fr{'\e}chet distance in a polygonal domain with obstacles and
the geodesic Hausdorff distance for sets of points or sets of line
segments inside a simple polygon $P$.


BibTeX - Entry

@InProceedings{wenk_et_al:LIPIcs:2008:1330,
  author =	{Carola Wenk and Atlas F. Cook},
  title =	{{Geodesic Fr{\'e}chet Distance Inside a Simple Polygon}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{193--204},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Susanne Albers and Pascal Weil},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2008/1330},
  URN =		{urn:nbn:de:0030-drops-13303},
  doi =		{10.4230/LIPIcs.STACS.2008.1330},
  annote =	{Keywords: Fr{\'e}chet Distance, Geodesic, Parametric Search, Simple Polygon}
}

Keywords: Fréchet Distance, Geodesic, Parametric Search, Simple Polygon
Seminar: 25th International Symposium on Theoretical Aspects of Computer Science
Issue date: 2008
Date of publication: 06.02.2008


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