### 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 | |

State | 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 |
---|---|

Country | United States |

City | Berkeley, CA |

Period | 10/21/06 → 10/24/06 |

### Fingerprint

### 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

**Ramsey partitions and proximity data structures.** / Mendel, Manor; Naor, Assaf.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006.*, 4031348, pp. 109-118, 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006, Berkeley, CA, United States, 10/21/06. https://doi.org/10.1109/FOCS.2006.65

}

TY - GEN

T1 - Ramsey partitions and proximity data structures

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2006

Y1 - 2006

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=35448988431&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35448988431&partnerID=8YFLogxK

U2 - 10.1109/FOCS.2006.65

DO - 10.1109/FOCS.2006.65

M3 - Conference contribution

SN - 0769527205

SN - 9780769527208

SP - 109

EP - 118

BT - 47th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2006

ER -