On the power of the semi-separated pair decomposition

A Semi-Separated Pair Decomposition (SSPD), with parameter s>1, of a set SâŠ"^Rd is a set {(^Ai,^Bi)} of pairs of subsets of S such that for each i, there are balls D_Ai and D_Bi containing ^Ai and ^Bi respectively such that d(D_Ai,D_Bi)≥s×min(radius(D_Ai), radius(D_Bi)), and for any two points p,qâ̂̂S there is a unique index i such that pâ̂̂^Ai and qâ̂̂^Bi or vice versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set SâŠ"^Rd of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with O(nlogn/(^t-1)d) edges that can be computed in O(nlogn/(^t-1)d) time. If all balls have the same radius, the number of edges reduces to O(n/(^t-1)d). Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in O(^{n2 log2}n) time using O(nlogn) space and answers a query in O(n1 ^/2+ε) time, for any ε>0. By reducing the preprocessing time to O(n1^+ε) and using O(n^log2n) space, the query can be answered in O(n3^/4+ε) time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k-partite graphs and low-diameter spanners, and show how to use the SSPD to obtain simple algorithms for these problems.