On the Power of the Semi-Separated Pair Decomposition

Abstract

A Semi-Separated Pair Decomposition (SSPD), with parameter s > 1, of a set $S\subset {\mathbb R}^d$ 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 $d(D_{A_i},D_{B_i}) \geq s \cdot$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 $S\subset {\mathbb R}^d$ of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with ${\mathcal O}(n\log n/(t-1)^d)$ edges which can be computed in ${\mathcal O}(n\log n/(t-1)^d)$ time. If all balls have the same radius, the number of edges reduces to ${\mathcal O}(n/(t-1)^d)$. 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 ${\mathcal O}(n^2\log^2 n)$ time using ${\mathcal O}(n\log n)$ space and answers a query in ${\mathcal O}(n^{1/2+\epsilon})$ time, for any *** > 0. By reducing the preprocessing time to ${\mathcal O}(n^{1+\epsilon})$ and using ${\mathcal O}(n\log^2 n)$ space, the query can be answered in ${\mathcal O}(n^{3/4+\epsilon})$ 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.