## 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 D_{Ai} and D_{Bi} containing ^{Ai} and ^{Bi} respectively such that d(D_{Ai},D_{Bi})≥s×min(radius(D_{Ai}), radius(D_{Bi})), and for any two points p,qâ̂̂S there is a unique index i such that pâ̂̂^{Ai} and qâ̂̂^{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 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} ^{log2}n) 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(n^{log2}n) 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 language | English |
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Pages (from-to) | 631-639 |

Number of pages | 9 |

Journal | Computational Geometry: Theory and Applications |

Volume | 46 |

Issue number | 6 |

DOIs | |

State | Published - 12 Mar 2013 |

Externally published | Yes |

## 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