On the power of the semi-separated pair decomposition

A Semi-Separated Pair Decomposition (SSPD), with parameter s > 1, of a set is a set {(A _i ,B _i )} of pairs of subsets of S such that for each i, there are balls and containing A _i and B _i respectively such that min ( radius ) , radius ), and for any two points p, q S there is a unique index i such that p A _i and q B _i 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 of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with edges which can be computed in time. If all balls have the same radius, the number of edges reduces to . 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 time using space and answers a query in time, for any ε> 0. By reducing the preprocessing time to and using space, the query can be answered in 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.