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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.SoCG.2023.16
URN: urn:nbn:de:0030-drops-178669
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/17866/
de Berg, Sarita ;
van Kreveld, Marc ;
Staals, Frank
The Complexity of Geodesic Spanners
Abstract
A geometric t-spanner for a set S of n point sites is an edge-weighted graph for which the (weighted) distance between any two sites p,q ∈ S is at most t times the original distance between p and q. We study geometric t-spanners for point sets in a constrained two-dimensional environment P. In such cases, the edges of the spanner may have non-constant complexity. Hence, we introduce a novel spanner property: the spanner complexity, that is, the total complexity of all edges in the spanner. Let S be a set of n point sites in a simple polygon P with m vertices. We present an algorithm to construct, for any constant ε > 0 and fixed integer k ≥ 1, a (2k + ε)-spanner with complexity O(mn^{1/k} + nlog² n) in O(nlog²n + mlog n + K) time, where K denotes the output complexity. When we consider sites in a polygonal domain P with holes, we can construct such a (2k+ε)-spanner of similar complexity in O(n² log m + nmlog m + K) time. Additionally, for any constant ε ∈ (0,1) and integer constant t ≥ 2, we show a lower bound for the complexity of any (t-ε)-spanner of Ω(mn^{1/(t-1)} + n).
BibTeX - Entry
@InProceedings{deberg_et_al:LIPIcs.SoCG.2023.16,
author = {de Berg, Sarita and van Kreveld, Marc and Staals, Frank},
title = {{The Complexity of Geodesic Spanners}},
booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)},
pages = {16:1--16:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-273-0},
ISSN = {1868-8969},
year = {2023},
volume = {258},
editor = {Chambers, Erin W. and Gudmundsson, Joachim},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/17866},
URN = {urn:nbn:de:0030-drops-178669},
doi = {10.4230/LIPIcs.SoCG.2023.16},
annote = {Keywords: spanner, simple polygon, polygonal domain, geodesic distance, complexity}
}
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Keywords: |
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spanner, simple polygon, polygonal domain, geodesic distance, complexity |
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Collection: |
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39th International Symposium on Computational Geometry (SoCG 2023) |
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Issue Date: |
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2023 |
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Date of publication: |
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09.06.2023 |
