### Abstract

This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for every ε ∈ (0, 1), every n-point metric space has a subset of size n^{1-ε} which embeds into Hubert space with distortion O(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 [32]. Namely, we show that for every n-point metric space X, and k ≥ 1, there exists an O(k)-approximate distance oracle whose storage requirement is 0(n^{1+1/k}), and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.

Original language | English (US) |
---|---|

Pages (from-to) | 253-275 |

Number of pages | 23 |

Journal | Journal of the European Mathematical Society |

Volume | 9 |

Issue number | 2 |

State | Published - 2007 |

### Fingerprint

### Keywords

- Approximate distance oracle
- Metric Ramsey theorem
- Proximity data structure

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the European Mathematical Society*,

*9*(2), 253-275.

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

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 9, no. 2, pp. 253-275.

}

TY - JOUR

T1 - Ramsey partitions and proximity data structures

AU - Mendel, Manor

AU - Naor, Assaf

PY - 2007

Y1 - 2007

N2 - This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for every ε ∈ (0, 1), every n-point metric space has a subset of size n1-ε which embeds into Hubert space with distortion O(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 [32]. Namely, we show that for every n-point metric space X, and k ≥ 1, there exists an O(k)-approximate distance oracle whose storage requirement is 0(n1+1/k), and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.

AB - This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in [8]). We then proceed to construct optimal Ramsey partitions, and use them to show that for every ε ∈ (0, 1), every n-point metric space has a subset of size n1-ε which embeds into Hubert space with distortion O(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 [32]. Namely, we show that for every n-point metric space X, and k ≥ 1, there exists an O(k)-approximate distance oracle whose storage requirement is 0(n1+1/k), and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.

KW - Approximate distance oracle

KW - Metric Ramsey theorem

KW - Proximity data structure

UR - http://www.scopus.com/inward/record.url?scp=33846646989&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33846646989&partnerID=8YFLogxK

M3 - Article

VL - 9

SP - 253

EP - 275

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 2

ER -