An Efficient, Practical Algorithm and Implementation for Computing Multiplicatively Weighted Voronoi Diagrams

TL;DR

This paper introduces a simple and efficient algorithm for computing multiplicatively weighted Voronoi diagrams, achieving improved expected runtime and practical performance, with extensions to additive weights and one-dimensional cases.

Contribution

It presents a novel wavefront-like approach that simplifies implementation and improves efficiency for weighted Voronoi diagrams, including experimental validation and extensions.

Findings

Expected runtime of O(n log^4 n) for random weighted points

Practical implementation suggests O(n log^2 n) runtime

Extension to additive weights and 1D case with O(n log n) complexity

Abstract

We present a simple wavefront-like approach for computing multiplicatively weighted Voronoi diagrams of points and straight-line segments in the Euclidean plane. If the input sites may be assumed to be randomly weighted points then the use of a so-called overlay arrangement [Har-Peled&Raichel, Discrete Comput. Geom. 53:547-568, 2015] allows to achieve an expected runtime complexity of $O(n\log^4 n)$, while still maintaining the simplicity of our approach. We implemented the full algorithm for weighted points as input sites, based on CGAL. The results of an experimental evaluation of our implementation suggest $O(n\log^2 n)$ as a practical bound on the runtime. Our algorithm can be extended to handle also additive weights in addition to multiplicative weights, and it yields a truly simple $O(n\log n)$ solution for solving the one-dimensional version of this problem.