Constructing levels in arrangements and higher order Voronoi diagrams

Abstract

We give a simple lazy randomized incremental algorithm to compute ≤k-levels in arrangements of x-monotone Jordan curves in the plane, and in arrangements of planes in three-dimensional space. If each pair of curves intersects in at most s points, the expected running time of the algorithm is O(k2ls(n/k)+min(ls(n)log2n,k2ls(n/k)logn)). For the three-dimensional case the expected running time is O(nk2+min(nlog3n,nk2logn)). The algorithm also works for computing the ≤k-level in a set of discs, with an expected running time of O(nk+min(nlog2n,nklogn)). Furthermore, we give a simple algorithm for computing the order-k Voronoi diagram of a set of n points in the plane that runs in expected time O(k(n−k)logn+nlog3n).