### Abstract

This paper addresses the non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce and construct optimal Ramsey partitions, and use them to show that for every ε ∈ (0,1), any n-point metric space has a subset of size n
^{1-ε} which embeds into Hilbert space with distortion 0(1/ε). This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor [5], in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in [26]. Namely, we show that for any n point metric space X, and k > 1, there exists an O (k)-approximate distance oracle whose storage requirement is O(n
^{1+1/k}), and whose query time is a universal constant. We also discuss applications to various other geometric data structures, and the relation to well separated pair decompositions.

Original language | English (US) |
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Title of host publication | 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 |

Pages | 109-118 |

Number of pages | 10 |

DOIs | |

Publication status | Published - 2006 |

Event | 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 - Berkeley, CA, United States Duration: Oct 21 2006 → Oct 24 2006 |

### Other

Other | 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006 |
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Country | United States |

City | Berkeley, CA |

Period | 10/21/06 → 10/24/06 |

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### ASJC Scopus subject areas

- Engineering(all)

### Cite this

*47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006*(pp. 109-118). [4031348] https://doi.org/10.1109/FOCS.2006.65