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 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,qâ̂̂S there is a unique index i such that pâ̂̂Ai and qâ̂̂Bi 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 log2n) 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.

Additional Metadata
Keywords Closest-pair query, Imprecise spanners, Semi-separated pair decomposition, Spanners for complete k-partite graphs
Persistent URL dx.doi.org/10.1016/j.comgeo.2013.02.003
Journal Computational Geometry
Abam, M.A. (Mohammad Ali), Carmi, P. (Paz), Farshi, M. (Mohammad), & Smid, M. (2013). On the power of the semi-separated pair decomposition. Computational Geometry, 46(6), 631–639. doi:10.1016/j.comgeo.2013.02.003