## 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.

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

### 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

1302.5328, 2013

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