## Abstract

Given a metric M = (V, d), a graph G = (V, E) is a t-spanner for M if every pair of nodes in V has a "short" path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every such short path also uses at most D edges. We consider the problem of constructing sparse (1 + ε)-spanners with small hop diameter for metrics of low doubling dimension. In this paper, we show that given any metric with constant doubling dimension k, and any 0 < ε < 1, one can find a (1 + ε)-spanner for the metric with nearly linear number of edges (i.e., only O(n log* n + nε^{-O(k)}) edges) and a constant hop diameter, and also a (1 + ε)-spanner with linear number of edges (i.e., only nε^{-O(k)} edges) which achieves a hop diameter that grows like the functional inverse of the Ackermann's function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal.

Original language | English (US) |
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Pages | 70-78 |

Number of pages | 9 |

DOIs | |

State | Published - 2006 |

Event | Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms - Miami, FL, United States Duration: Jan 22 2006 → Jan 24 2006 |

### Other

Other | Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms |
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Country/Territory | United States |

City | Miami, FL |

Period | 1/22/06 → 1/24/06 |

## ASJC Scopus subject areas

- Software
- General Mathematics