On the power of the semi-separated pair decomposition

Abstract

A Semi-Separated Pair Decomposition (SSPD), with parameter s>1, of a set S⊂Rd is a set {(Ai,Bi)} of pairs of subsets of S such that for each i, there are balls DAi and DBi containing Ai and Bi respectively such that d(DAi,DBi)⩾s⋅min(radius(DAi),radius(DBi)), and for any two points p,q∈S there is a unique index i such that p∈Ai and q∈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 S⊂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(n2log2n) 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(nlog2n) 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.