Constructing Levels in Arrangements and Higher Order Voronoi Diagrams

We give simple randomized incremental algorithms for computing the ≤κ-level in an arrangement of n lines in the plane or in an arrangement of n planes in ℝ³. The expected running time of our algorithms is O(nκ + nα(n) logn) for the planar case and O(nκ² + nlog³n) for the three-dimensional case. Both bounds are optimal unless κ is very small. The algorithm generalizes to computing the ≤κ-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also compute the κ-level; this yields a randomized algorithm for computing the order-κ Voronoi diagram of n points in the plane in expected time O(κ(n - κ) log n + n log³ n)