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

  • Maarten Löffler
  • Wolfgang Mulzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


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.


Convex Hull Delaunay Triangulation Simplicial Partition Onion Peeling Convex Chain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Maarten Löffler
    • 1
  • Wolfgang Mulzer
    • 2
  1. 1.Department of Information and Computing SciencesUniversiteit UtrechtThe Netherlands
  2. 2.Institut für InformatikFreie Universität BerlinGermany

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