On the power of the semi-separated pair decomposition
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...
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Published in | Computational geometry : theory and applications Vol. 46; no. 6; pp. 631 - 639 |
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Main Authors | , , , |
Format | Journal Article |
Language | English |
Published |
Elsevier B.V
01.08.2013
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Subjects | |
Online Access | Get full text |
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Summary: | 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(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(n2log2n) 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. |
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Bibliography: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
ISSN: | 0925-7721 |
DOI: | 10.1016/j.comgeo.2013.02.003 |