An output-sensitive approach for the L1/L∞k-nearest-neighbor Voronoi diagram

Authors:
Chih-Hung Liu;Evanthia Papadopoulou;D. T. Lee
Affiliations:
Research Center for Information Technology Innovation, Academia Sinica, Taiwan;Faculty of Informatics, University of Lugano, Lugano, Switzerland;Research Center for Information Technology Innovation, Academia Sinica, Taiwan and Institute of Information Science, Academia Sinica, Taipei, Taiwan
Venue:
ESA'11 Proceedings of the 19th European conference on Algorithms
Year:
2011

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Abstract

This paper revisits the k-nearest-neighbor (k-NN) Voronoi diagram and presents the first output-sensitive paradigm for its construction. It introduces the k-NN Delaunay graph, which corresponds to the graph theoretic dual of the k-NN Voronoi diagram, and uses it as a base to directly compute the k-NN Voronoi diagram in R2. In the L1, L∞ metrics this results in O((n + m) log n) time algorithm, using segment-dragging queries, where m is the structural complexity (size) of the k-NN Voronoi diagram of n point sites in the plane. The paper also gives a tighter bound on the structural complexity of the k-NN Voronoi diagram in the L∞ (equiv. L1) metric, which is shown to be O(min{k(n - k), (n - k)2}).