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.