The well-separated pair decomposition for balls
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Given a real number t>1, a geometric t-spanner is a geometric graph for a point set in 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 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 Ω(n^2) edges. Then, we propose an algorithm that computes a Well-Separated Pair Decomposition (WSPD) of size 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 O(n n+n/(t-1)^d) time an imprecise t-spanner with O(n/(t-1)^d) edges for balls.
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