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The k-Nearest-Neighbor Voronoi Diagram Revisited

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Abstract

We revisit the k-nearest-neighbor (k-NN) Voronoi diagram and present a new paradigm for its construction. We introduce the k-NN Delaunay graph, which is the graph-theoretic dual of the k-NN Voronoi diagram, and use it as a base to directly compute this diagram in R 2. We implemented our paradigm in the L 1 and L ∞ metrics, using segment-dragging queries, resulting in the first output-sensitive, O((n+m)logn)-time algorithm to compute the k-NN Voronoi diagram of n points in the plane, where m is the structural complexity (size) of this diagram. We also show that the structural complexity of the k-NN Voronoi diagram in the L ∞ (equiv. L 1) metric is O(min{k(n−k),(n−k)2}). Efficient implementation of our paradigm in the L 2 (resp. L p , 1<p<∞) metric remains an open problem.

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Notes

  1. Note the k-NN Delaunay graph is different from the geometric k-Delaunay graph of [1, 18].

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Authors and Affiliations

  1. Academia Sinica, Taipei, Taiwan

    Chih-Hung Liu & Der-Tsai Lee

  2. Università della Svizzera Italiana, Lugano, Switzerland

    Evanthia Papadopoulou

  3. National Chung Hsing University, Taichung, Taiwan

    Der-Tsai Lee

Authors
  1. Chih-Hung Liu
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  2. Evanthia Papadopoulou
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  3. Der-Tsai Lee
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Corresponding author

Correspondence to Der-Tsai Lee.

Additional information

A preliminary version appeared in Proc. European Symposium on Algorithms (ESA) 2011 [22]. This work was supported in part by the Swiss National Science Foundation under Grants SNF-200021-127137, and SNF-20GG21-134355, the latter under the ESF EUROCORES program EuroGIGA/VORONOI, and by the National Science Council, Taiwan under Grants Nos. NSC-99-2911-I-001-506, NSC-99-2911-I-001-511, NSC-101-2221-E-005-019-MY2, and NSC-101-2221-E-005-026-MY2.

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Liu, CH., Papadopoulou, E. & Lee, DT. The k-Nearest-Neighbor Voronoi Diagram Revisited. Algorithmica 71, 429–449 (2015). https://doi.org/10.1007/s00453-013-9809-9

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