Constructing levels in arrangements and higher order Voronoi diagrams

Abstract

We give simple randomized incremental algorithms for computing the ≤k-level in an arrangement of n hyperplanes in two- and three-dimensional space. The expected running time of our algorithms is O(nk+nα(n) logn) for the planar case, and O(nk2+n log3 n) for the three-dimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the ≤k-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also be used to compute the k-level; this yields a randomized algorithm for computing the order-k Voronoi diagram of n points in the plane in expected time O(k(n - k) log n + n log3 n).