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

Abstract

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.

This work was supported by the European Science Foundation (ESF) in the EUROCORES collaborative research project EuroGIGA/VORONOI, projects DFG Kl 655/17-1 and SNF 20GG21-134355. The work of the last two authors was also supported by the Swiss National Science Foundation, project 200020-149658.