Abstract
We unveil an alluring alternative to parametric search that applies
to both the nongeodesic and geodesic Fr{'\e}chet optimization
problems. This randomized approach is based on a variant of
redblue 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 nongeodesic 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 nongeodesic 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 = {193204},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897064},
ISSN = {18688969},
year = {2008},
volume = {1},
editor = {Susanne Albers and Pascal Weil},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1330},
URN = {urn:nbn:de:0030drops13303},
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: 

06.02.2008 