1998/06/01 by Pankaj K. Agarwal, Mark de Berg, Jiřı́ Matoušek +1 · 1 citation
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Point processes and geometric inequalities #Data Management and Algorithms #Voronoi diagram #Monotone polygon #Mathematics #Combinatorics #Plane (geometry) #Computational geometry #Binary logarithm #Order (exchange) #Randomized algorithm #Diagram #Discrete mathematics #Algorithm #Geometry
paper · doi:10.1137/s0097539795281840
openalex publication_date 1998/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
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 \Reals3. The expected running time of our algorithms is O(nk+nα(n)log n) for the planarcase and O(nk2 + n log3n) 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 log3n).