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Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles

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Abstract

We present an algorithm to compute the geodesic \(L_1\) farthest-point Voronoi diagram of m point sites in the presence of n rectangular obstacles in the plane. It takes \(O(nm+n \log n + m\log m)\) construction time using O(nm) space. This is the first optimal algorithm for constructing the farthest-point Voronoi diagram in the presence of obstacles. We can construct a data structure in the same construction time and space that answers a farthest-neighbor query in \(O(\log (n+m))\) time.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Pohang University of Science and Technology, Pohang, Korea

    Mincheol Kim & Taehoon Ahn

  2. Graduate School of Artificial Intelligence, Pohang University of Science and Technology, Pohang, Korea

    Chanyang Seo

  3. Graduate School of Artificial Intelligence, Department of Computer Science and Engineering, Pohang University of Science and Technology, Pohang, Korea

    Hee-Kap Ahn

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  1. Mincheol Kim
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Correspondence to Hee-Kap Ahn.

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This research was partly supported by the Institute of Information & communications Technology Planning & Evaluation(IITP) Grant funded by the Korea government(MSIT) (No. 2017-0-00905, Software Star Lab (Optimal Data Structure and Algorithmic Applications in Dynamic Geometric Environment)) and (No. 2019-0-01906, Artificial Intelligence Graduate School Program(POSTECH)).

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Kim, M., Seo, C., Ahn, T. et al. Farthest-Point Voronoi Diagrams in the Presence of Rectangular Obstacles. Algorithmica 85, 2214–2237 (2023). https://doi.org/10.1007/s00453-022-01094-9

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