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Shortest Path Problems on a Polyhedral Surface

Authors: Carola Wenk and Atlas F. Cook

Published in: Dagstuhl Seminar Proceedings, Volume 9111, Computational Geometry (2009)

Abstract

We develop algorithms to compute edge sequences, Voronoi diagrams, shortest path maps, the Fréchet distance, and the diameter for a polyhedral surface. Distances on the surface are measured either by the length of a Euclidean shortest path or by link distance. Our main result is a linear-factor speedup for computing all shortest path edge sequences on a convex polyhedral surface.

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Carola Wenk and Atlas F. Cook. Shortest Path Problems on a Polyhedral Surface. In Computational Geometry. Dagstuhl Seminar Proceedings, Volume 9111, pp. 1-30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{wenk_et_al:DagSemProc.09111.5,
  author =	{Wenk, Carola and Cook, Atlas F.},
  title =	{{Shortest Path Problems on a Polyhedral Surface}},
  booktitle =	{Computational Geometry},
  pages =	{1--30},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9111},
  editor =	{Pankaj Kumar Agarwal and Helmut Alt and Monique Teillaud},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.09111.5},
  URN =		{urn:nbn:de:0030-drops-20332},
  doi =		{10.4230/DagSemProc.09111.5},
  annote =	{Keywords: Shortest paths, edge sequences}
}

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DOI: 10.4230/LIPIcs.STACS.2008.1330

Geodesic Fréchet Distance Inside a Simple Polygon

Authors: Carola Wenk and Atlas F. Cook

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

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 $P$.

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Carola Wenk and Atlas F. Cook. Geodesic Fréchet Distance Inside a Simple Polygon. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 193-204, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{wenk_et_al:LIPIcs.STACS.2008.1330,
  author =	{Wenk, Carola and Cook, Atlas F.},
  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 =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.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}
}

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Documents authored by Cook, Atlas F.

Shortest Path Problems on a Polyhedral Surface

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Geodesic Fréchet Distance Inside a Simple Polygon

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