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The Geodesic Farthest-Point Voronoi Diagram in a Simple Polygon

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Abstract

Given a set of point sites in a simple polygon, the geodesic farthest-point Voronoi diagram partitions the polygon into cells, at most one cell per site, such that every point in a cell has the same farthest site with respect to the geodesic metric. We present an \(O(n\log \log n+ m\log m)\)-time algorithm to compute the geodesic farthest-point Voronoi diagram of m point sites in a simple n-gon. This improves the previously best known algorithm by Aronov et al. (Discrete Comput Geom 9(3):217–255, 1993). In the case that all point sites are on the boundary of the simple polygon, we can compute the geodesic farthest-point Voronoi diagram in \(O((n+m) \log \log n)\) time.

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Notes

  1. The paper [8] shows that their running time is \(O(n+m\log (n+m))\). But it is \(O(n+m\log m)\). To see this, observe that it is O(n) for \(m=O(n/\log n)\). Also, it is \(O(n+m\log m)\) for \(m=\Omega (n/\log n)\).

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Acknowledgements

Funding was provided by Ministry of Science and ICT (KR) (Grant No. IITP-2017-0-00905).

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Authors and Affiliations

  1. Pohang University of Science and Technology, Pohang, Korea

    Eunjin Oh & Hee-Kap Ahn

  2. Department of Computer Science, ETH Zürich, Zurich, Switzerland

    Luis Barba

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  1. Eunjin Oh
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  2. Luis Barba
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Correspondence to Hee-Kap Ahn.

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This work was supported by the NRF Grant 2011-0030044 (SRC-GAIA) funded by the government of Korea and the MSIT (Ministry of Science and ICT), Korea, under the SW Starlab support program (IITP-2017-0-00905) supervised by the IITP (Institute for Information & communications Technology Promotion).

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Oh, E., Barba, L. & Ahn, HK. The Geodesic Farthest-Point Voronoi Diagram in a Simple Polygon. Algorithmica 82, 1434–1473 (2020). https://doi.org/10.1007/s00453-019-00651-z

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