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Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition

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Algorithms and Data Structures (WADS 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8037))

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Abstract

Let \(\mathcal{D}\) be a set of n pairwise disjoint unit disks in the plane. We describe how to build a data structure for \(\mathcal{D}\) so that for any point set P containing exactly one point from each disk, we can quickly find the onion decomposition (convex layers) of P.

Our data structure can be built in O(n logn) time and has linear size. Given P, we can find its onion decomposition in O(n logk) time, where k is the number of layers. We also provide a matching lower bound.

Our solution is based on a recursive space decomposition, combined with a fast algorithm to compute the union of two disjoint onion decompositions.

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Author information

Authors and Affiliations

  1. Department of Information and Computing Sciences, Universiteit Utrecht, The Netherlands

    Maarten Löffler

  2. Institut für Informatik, Freie Universität Berlin, Germany

    Wolfgang Mulzer

Authors
  1. Maarten Löffler
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  2. Wolfgang Mulzer
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Editor information

Editors and Affiliations

  1. School of Computer Science, Parallel Computing and Bioinformatics Laboratory, Carleton University, VISM Building, Room 6210, 1125 Colonel By Drive, K1S 5B6, Ottawa, ON, Canada

    Frank Dehne

  2. Department of Computer Science, The University of Western Ontario, N6A 5B7, London, ON, Canada

    Roberto Solis-Oba

  3. School of Computer Science, Carleton University, K1S 5B6, Ottawa, ON, Canada

    Jörg-Rüdiger Sack

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Löffler, M., Mulzer, W. (2013). Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition. In: Dehne, F., Solis-Oba, R., Sack, JR. (eds) Algorithms and Data Structures. WADS 2013. Lecture Notes in Computer Science, vol 8037. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40104-6_42

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  • DOI: https://doi.org/10.1007/978-3-642-40104-6_42

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40103-9

  • Online ISBN: 978-3-642-40104-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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