Unions of Onions

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.