The well-separated pair decomposition for balls

Given a real number $t>1$, a geometric $t$-spanner is a geometric graph for a point set in $\mathbb{R}^d$ with straight lines between vertices such that the ratio of the shortest-path distance between every pair of vertices in the graph (with Euclidean edge lengths) to their actual Euclidean distance is at most $t$. An imprecise point set is modeled by a set $R$ of regions in $\mathbb{R}^d$. If one chooses a point in each region of $R$, then the resulting point set is called a precise instance of~$R$. An imprecise $t$-spanner for an imprecise point set $R$ is a graph $G=(R,E)$ such that for each precise instance $S$ of $R$, graph $G_S=(S,E_S)$, where $E_S$ is the set of edges corresponding to $E$, is a $t$-spanner. In this paper, we show that, given a real number $t>1$, there is an imprecise point set $R$ of $n$ straight-line segments in the plane such that any imprecise $t$-spanner for $R$ has $\Omega(n^2)$ edges. Then, we propose an algorithm that computes a Well-Separated Pair Decomposition (WSPD) of size ${\cal O}(n)$ for a set of $n$ pairwise disjoint $d$-dimensional balls with arbitrary sizes. Given a real number $t>1$ and given a set of $n$ pairwise disjoint $d$-balls with arbitrary sizes, we use this WSPD to compute in ${\cal O}(n\log n+n/(t-1)^d)$ time an imprecise $t$-spanner with ${\cal O}(n/(t-1)^d)$ edges for balls.