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

U2 - 10.4171/JEMS/79

DO - 10.4171/JEMS/79

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AN - SCOPUS:33846646989

SN - 1435-9855

VL - 9

SP - 253

EP - 275

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 2

ER -