Unions of Onions: Preprocessing Imprecise Points for Fast Onion 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 ) 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 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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