Constructing Levels in Arrangements and Higher Order Voronoi Diagrams

Pankaj K. Agarwal, Mark de Berg, Jirí Matousek, Otfried Schwarzkopf
1998 SIAM journal on computing (Print)  
We give simple randomized incremental algorithms for computing the ≤k-level in an arrangement of n lines in the plane or in an arrangement of n planes in R 3 . The expected running time of our algorithms is O(nk + nα(n) log n) for the planar case 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 ≤k-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also
more » ... e 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).
doi:10.1137/s0097539795281840 fatcat:wgncoxmqcrfqvaesiwhdcqllkq