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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.26
URN: urn:nbn:de:0030-drops-186796
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18679/
Buchin, Kevin ;
Gudmundsson, Joachim ;
Kalb, Antonia ;
Popov, Aleksandr ;
Rehs, Carolin ;
van Renssen, André ;
Wong, Sampson
Oriented Spanners
Abstract
Given a point set P in the Euclidean plane and a parameter t, we define an oriented t-spanner as an oriented subgraph of the complete bi-directed graph such that for every pair of points, the shortest cycle in G through those points is at most a factor t longer than the shortest oriented cycle in the complete bi-directed graph. We investigate the problem of computing sparse graphs with small oriented dilation.
As we can show that minimising oriented dilation for a given number of edges is NP-hard in the plane, we first consider one-dimensional point sets. While obtaining a 1-spanner in this setting is straightforward, already for five points such a spanner has no plane embedding with the leftmost and rightmost point on the outer face. This leads to restricting to oriented graphs with a one-page book embedding on the one-dimensional point set. For this case we present a dynamic program to compute the graph of minimum oriented dilation that runs in ?(n⁸) time for n points, and a greedy algorithm that computes a 5-spanner in ?(nlog n) time.
Expanding these results finally gives us a result for two-dimensional point sets: we prove that for convex point sets the greedy triangulation results in an oriented ?(1)-spanner.
BibTeX - Entry
@InProceedings{buchin_et_al:LIPIcs.ESA.2023.26,
author = {Buchin, Kevin and Gudmundsson, Joachim and Kalb, Antonia and Popov, Aleksandr and Rehs, Carolin and van Renssen, Andr\'{e} and Wong, Sampson},
title = {{Oriented Spanners}},
booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)},
pages = {26:1--26:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-295-2},
ISSN = {1868-8969},
year = {2023},
volume = {274},
editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18679},
URN = {urn:nbn:de:0030-drops-186796},
doi = {10.4230/LIPIcs.ESA.2023.26},
annote = {Keywords: computational geometry, spanner, oriented graph, greedy triangulation}
}
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Keywords: |
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computational geometry, spanner, oriented graph, greedy triangulation |
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Collection: |
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31st Annual European Symposium on Algorithms (ESA 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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30.08.2023 |
