## Abstract

In STOC'95, Arya et al. [1995] showed that for any set of n points in ℝ^{d}, 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)2^{k}α_{k}(n)) time a (1+ε)-spanner with diameter at most 2k and O(n^{k}α_{k} (n)) edges. The function α_{k} is the inverse of a certain function at the [k/2] th level of the primitive recursive hierarchy, where α_{0}(n) = [n/2], α_{1} (n) = [√n], α_{2} (n) = [log n], α_{3} (n) = [log log n], α_{4} (n) = log* n, α_{5} (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(nkα_{k}(n)) edges can be constructed in O(n^{2}) time [Narasimhan and Smid 2007]. A major open question in this area is whether one can construct within time O(n log n+nkα_{k}(n) ) a (1+ε)- spanner with diameter at most k and O(nkα_{k}(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(nα_{k}(n)) edges can be built in optimal time O(n log n).

Original language | English |
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Article number | 28 |

Journal | ACM Transactions on Algorithms |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2013 |

Externally published | Yes |

## Keywords

- Diameter
- Euclidean metrics
- Euclidean spanners
- Graph spanners