Sparse euclidean spanners with tiny diameter

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Abstract

In STOC'95, Arya et al. [1995] showed that for any set of n points in ℝd, a (1 + ε)-spanner with diameter at most 2 (respectively, 3) and O(n log n) edges (respectively, O(n log log n) edges) can be built in O(n log n) time. Moreover, it was shown in Arya et al. [1995] and Narasimhan and Smid [2007] that for any k ≥ 4, one can build in O(n(log n)2kαk(n)) time a (1+ε)-spanner with diameter at most 2k and O(nkαk (n)) edges. The function αk is the inverse of a certain function at the [k/2] th level of the primitive recursive hierarchy, where α0(n) = [n/2], α1 (n) = [√n], α2 (n) = [log n], α3 (n) = [log log n], α4 (n) = log* n, α5 (n) = [1/2 log* n], ⋯, etc. It is also known [Narasimhan and Smid 2007] that if one allows quadratic time, then these bounds can be improved. Specifically, for any k ≥ 4, a (1+ε)-spanner with diameter at most k and O(nkαk(n)) edges can be constructed in O(n2) time [Narasimhan and Smid 2007]. A major open question in this area is whether one can construct within time O(n log n+nkαk(n) ) a (1+ε)- spanner with diameter at most k and O(nkαk(n)) edges. In this article, we answer this question in the affirmative. Moreover, in fact, we provide a stronger result. Specifically, we show that for any k ≥ 4, a (1 + ε)-spanner with diameter at most k and O(nαk(n)) edges can be built in optimal time O(n log n).

Original languageEnglish
Article number28
Number of pages33
JournalACM Transactions on Algorithms
Volume9
Issue number3
DOIs
StatePublished - Jun 2013

Keywords

  • Diameter
  • Euclidean metrics
  • Euclidean spanners
  • Graph spanners

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)

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