Near-Quadratic Bounds for the $L_{1}$ Voronoi Diagram of Moving Points

Abstract

Given a set of $n$ moving points in the plane, how many topological changes occur in the Voronoi diagram of the points? If each point has constant velocity then there is an upper bound of $O(n^{3})$ [Guibas, Mitchell and Roos] and an easy lower bound of $\Omega(n^{2})$. It is widely believed that the true upper bound should be close to $O(n^{2})$. We show this belief to be true for the case of Voronoi diagrams based on the $L_{1}$ (or $L_{\infty}$) metric; the number of changes is shown to be $O(n^{2} \alpha (n))$ where $\alpha(n)$ grows so slowly it is effectively a small constant for all reasonable values of $n$.