Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Wenk, Carola; Cook, Atlas F. http://www.dagstuhl.de/lipics License
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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 =		{http://dx.doi.org/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: 2008


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