Constructing Levels in Arrangements and Higher Order Voronoi Diagrams
Pankaj K. Agarwal, Mark de Berg, Jiřı́ Matoušek, Otfried Cheong Schwarzkopf · SIAM Journal on Computing · 1998
We give simple randomized incremental algorithms for computing the Amk-level in an arrangement of n lines in the plane or in an arrangement of n planes in $\Reals^3$. The expected running time of our algorithms is $O(nk+n\alpha(n)\log n)$ for the planarcase and O(nk 2 + n log 3 n) for the three-dimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the Amk-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also 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 log 3 n).