Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition

4 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 5 for any point set P containing exactly one point from each disk, we can quickly find the onion decomposition (convex 6 layers) of P . 7 Our data structure can be built in O(n logn) time and has linear size. Given P , we can find its onion decomposition 8 in O(n log k) time, where k is the number of layers. We also provide a matching lower bound. 9 Our solution is based on a recursive space decomposition, combined with a fast algorithm to compute the union 10 of two disjoint onion decompositions. 11