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An Output-Sensitive Approach for the L 1/L  ∞  k-Nearest-Neighbor Voronoi Diagram

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Algorithms – ESA 2011 (ESA 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6942))

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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 R 2. In the L 1, L  ∞  metrics this results in O((n + m)logn) 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. L 1) metric, which is shown to be O(min{k(n − k), (n − k)2}).

This work was performed while the first and third authors visited University of Lugano in September/October 2010. It was supported in part by the University of Lugano during the visit, by the Swiss National Science Foundation under grant SNF-200021-127137, and by the National Science Council, Taiwan under grants No. NSC-98-2221-E-001-007-MY3, No. NSC-98-2221-E-001-008-MY3, and No. NSC-99-2918-I-001-009.

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Author information

Authors and Affiliations

  1. Research Center for Information Technology Innovation, Academia Sinica, Taiwan

    Chih-Hung Liu & D. T. Lee

  2. Faculty of Informatics, University of Lugano, Lugano, Switzerland

    Evanthia Papadopoulou

  3. Institute of Information Science, Academia Sinica, Taipei, Taiwan

    D. T. Lee

Authors
  1. Chih-Hung Liu
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  2. Evanthia Papadopoulou
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  3. D. T. Lee
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Editor information

Editors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, University of Rome " La Sapienza", via Ariosto 25, 00185, Roma, Italy

    Camil Demetrescu

  2. School of Computer Science, Reykjavik University, Menntavegur 1, 101, Reykjavik, Iceland

    Magnús M. Halldórsson

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Liu, CH., Papadopoulou, E., Lee, D.T. (2011). An Output-Sensitive Approach for the L 1/L  ∞  k-Nearest-Neighbor Voronoi Diagram. In: Demetrescu, C., Halldórsson, M.M. (eds) Algorithms – ESA 2011. ESA 2011. Lecture Notes in Computer Science, vol 6942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23719-5_7

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  • DOI: https://doi.org/10.1007/978-3-642-23719-5_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23718-8

  • Online ISBN: 978-3-642-23719-5

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