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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2008.1330
URN: urn:nbn:de:0030-drops-13303
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2008/1330/
Wenk, Carola ;
Cook, Atlas F.
Geodesic Fréchet Distance Inside a Simple Polygon
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: |
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Fréchet Distance, Geodesic, Parametric Search, Simple Polygon |
Collection: |
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25th International Symposium on Theoretical Aspects of Computer Science |
Issue Date: |
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2008 |
Date of publication: |
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06.02.2008 |