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An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams

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Abstract

Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in \(O(k(n-k)\log ^2 n +n\log ^3 n)\) steps, where \(O(k(n-k))\) is the number of faces in the worst case. This result applies to disjoint line segments in the \(L_p\) norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, a running time with a polylog factor to the number of faces was only achieved for point sites in the \(L_1\) or Euclidean metric before.

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Notes

  1. Actually, the number of times does not matter. It is sufficient that the size of the sequence of sub-faces is proportional to the size of the boundary of F.

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Acknowledgements

We deeply appreciate all valuable comments from the anonymous reviewers for SoCG 2016 and Algorithmica.

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Authors and Affiliations

  1. Department of Computer Science, University of Bonn, Bonn, Germany

    Cecilia Bohler & Rolf Klein

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

    Chih-Hung Liu

Authors
  1. Cecilia Bohler
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  2. Rolf Klein
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  3. Chih-Hung Liu
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Correspondence to Chih-Hung Liu.

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A preliminary version appeared in Proceedings of International Symposium on Computational Geometry (SoCG) 2016 [5]. This work was supported by Deutsche Forschungsgemeinschaft (DFG Kl 655/19) in a DACH Project.

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Bohler, C., Klein, R. & Liu, CH. An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams. Algorithmica 81, 2317–2345 (2019). https://doi.org/10.1007/s00453-018-00536-7

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