On the power of the semi-separated pair decomposition

Mohammad Ali Abam, Paz Carmi, Mohammad Farshi, Michiel Smid

Research output: Contribution to journalArticlepeer-review

13 Scopus citations

Abstract

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) 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.

Original languageEnglish
Pages (from-to)631-639
Number of pages9
JournalComputational Geometry: Theory and Applications
Volume46
Issue number6
DOIs
StatePublished - 12 Mar 2013
Externally publishedYes

Keywords

  • Closest-pair query
  • Imprecise spanners
  • Semi-separated pair decomposition
  • Spanners for complete k-partite graphs

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics

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