Loading...

On the power of the semi-separated pair decomposition

Abam, M. A ; Sharif University of Technology | 2013

545 Viewed
  1. Type of Document: Article
  2. DOI: 10.1016/j.comgeo.2013.02.003
  3. Publisher: 2013
  4. Abstract:
  5. 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
  6. Keywords:
  7. Semi-separated pair decomposition ; Spanners for complete k-partite graphs ; Axis parallel rectangles ; Built in ; Closest-pair query ; Complete k-partite graphs ; Geometric t-spanners ; Half-planes ; Imprecise spanners ; Preprocessing time ; SIMPLE algorithm ; Two-point ; Separation ; Data structures
  8. Source: Computational Geometry: Theory and Applications ; Volume 46, Issue 6 , 2013 , Pages 631-639 ; 09257721 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S092577211300014X