A Randomized Divide and Conquer Algorithm for Higher-Order Abstract Voronoi Diagrams

Given a set of sites in the plane, their order-\(k\) Voronoi diagram partitions the plane into regions such that all points within one region have the same \(k\) nearest sites. The order-\(k\) abstract Voronoi diagram is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance, and it represents a wide class of order-\(k\) concrete Voronoi diagrams. In this paper we develop a randomized divide-and-conquer algorithm to compute the order-\(k\) abstract Voronoi diagram in expected \(O(kn^{1+\varepsilon })\) operations. For solving small sub-instances in the divide-and-conquer process, we also give two sub-algorithms with expected \(O(k^2n\log n)\) and \(O(n^22^{\alpha (n)}\log n)\) time, respectively. This directly implies an \(O(kn^{1+\varepsilon })\)-time algorithm for several concrete order-\(k\) instances such as points in any convex distance, disjoint line segments and convex polygons of constant size in the \(L_p\) norm, and others.