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

Let D be a set of n pairwise disjoint unit disks in the plane. We describe how to build a data structure for 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 log n) expected time and has linear size. Given P, we can find its onion decomposition in O(n log k) time, where k is the number of layers. We also provide a lower bound showing that the running time must depend on k. Our solution is based on a recursive space decomposition, combined with a fast algorithm to compute the union of two disjoint onion decompositions.


slides
keywords: Computational Geometry, Convex Hulls, Data Structures, Data Imprecision, UDG

Journal Article (peer-reviewed)

Maarten Löffler, Wolfgang Mulzer
Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition
Journal of Computational Geometry
5, 1, 1–13, 2014
http://jocg.org/v5n1p1

Conference Proceedings (peer-reviewed)

Maarten Löffler, Wolfgang Mulzer
Unions of Onions: Preprocessing Imprecise Points for Fast Onion Layer Decomposition
Proc. 13th Algorithms and Data Structures Symposium
487–498, 2013

Workshop or Poster (weakly reviewed)

Maarten Löffler, Wolfgang Mulzer
Unions of Onions
Proc. 29th European Workshop on Computational Geometry
61–64, 2013

Archived Publication (not reviewed)

Maarten Löffler, Wolfgang Mulzer
Unions of Onions
1302.5328, 2013
http://arXiv.org/abs/1302.5328

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