Optimal computation of the Voronoi diagram of disjoint clusters

Abstract

Let $\text{d(a,b)}$ denote the Euclidean distance between two points $\text{a}$, $\text{b}$ in the plane. For a cluster $\text{C}$ of sites, and for a point $\text{p}$, define $\text{d(p,C)=}\text{max}\text{{d(p,x)∣x∈C}}$ as the distance between $\text{p}$ and $\text{C}$. The Voronoi diagram of a set $\text{S}$ of clusters $\text{C}{}_1\text{,…,C}{}_{\mathrm{m}}$ is a partition of the plane into domains, one for each cluster, such that a point $\text{p}$ belongs to the domain of $\text{C}{}_{\mathrm{i}}$ if and only if $\text{d(p,C}{}_{\mathrm{i}}\text{)⩽d(p,C}{}_{\mathrm{j}}\text{),}\text{i≠j}$. In this note, we present an optimal time $\text{O}\text{(n}\text{log}\text{n)}$ algorithm for computing the Voronoi diagram of a set of convex-hull disjoint clusters, where $\text{n}$ is the sum of cardinalities of all clusters. This improves upon the previous $\text{O}\text{(n}{}^2\text{α(n))}$ time bound, where $\text{α(n)}$ is the inverse Ackermann function. Our result is obtained by examining a new variant of the Voronoi diagram where each site is associated with a convex and unbounded region in which it is active.