Constructing Levels in Arrangements and Higher Order Voronoi Diagrams

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 R3. The expected running time of our algorithms is O(nk + n(n) log n) 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 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). Key words. arrangements, random sampling, Voronoi diagrams AMS subject classifications. 65Y25, 68Q25, 68U05 PII. S0097539795281840

• Type: Conference
• Year: 1994
• Where: COMPGEOM
• Authors: Pankaj K. Agarwal, Mark de Berg, Jirí Matousek, Otfried Schwarzkopf
• Keywords: Algorithm, COMPGEOM 1994, Discrete Geometry, Incremental Algorithms, Voronoi Diagrams