On the power of the semi-separated pair decomposition

A Semi-Separated Pair Decomposition (SSPD), with parameter s>1, of a set S@?R^d is a set {(A"i,B"i)} of pairs of subsets of S such that for each i, there are balls D"A"""i and D"B"""i containing A"i and B"i respectively such that d(D"A"""i,D"B"""i)>=s@?min(radius(D"A"""i),radius(D"B"""i)), 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 S@?R^d 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(n^2log^2n) time using O(nlogn) space and answers a query in O(n^1^/^2^+^@e) time, for any @e>0. By reducing the preprocessing time to O(n^1^+^@e) and using O(nlog^2n) space, the query can be answered in O(n^3^/^4^+^@e) 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.