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(n log n/(t − 1)d ) edges which can be computed in O(n log n/(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 log2 n) time usi...

• Type: Conference
• Year: 2009
• Where: WADS
• Authors: Mohammad Ali Abam, Paz Carmi, Mohammad Farshi, Michiel H. M. Smid
• Keywords: Algorithms, Dbi Containing Ai, Imprecise Points, Query Data Structure, WADS 2009