Ramsey partitions and proximity data structures

Mendel, Manor, and Naor, Assaf. "Ramsey partitions and proximity data structures." Journal of the European Mathematical Society 009.2 (2007): 253-275. <http://eudml.org/doc/277787>.

@article{Mendel2007,
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 $\varepsilon \in (0,1)$, every $n$-point metric space has a subset of size $n^\{1−\varepsilon \}$ which embeds into Hilbert space with distortion $O(1/\varepsilon )$. 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 $k\ge 1$, 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 of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.},
author = {Mendel, Manor, Naor, Assaf},
journal = {Journal of the European Mathematical Society},
keywords = {metric Ramsey theorem; approximate distance oracle; proximity data structure},
language = {eng},
number = {2},
pages = {253-275},
publisher = {European Mathematical Society Publishing House},
title = {Ramsey partitions and proximity data structures},
url = {http://eudml.org/doc/277787},
volume = {009},
year = {2007},
}

TY - JOUR
AU - Mendel, Manor
AU - Naor, Assaf
TI - Ramsey partitions and proximity data structures
JO - Journal of the European Mathematical Society
PY - 2007
PB - European Mathematical Society Publishing House
VL - 009
IS - 2
SP - 253
EP - 275
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 $\varepsilon \in (0,1)$, every $n$-point metric space has a subset of size $n^{1−\varepsilon }$ which embeds into Hilbert space with distortion $O(1/\varepsilon )$. 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 $k\ge 1$, 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 of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking.
LA - eng
KW - metric Ramsey theorem; approximate distance oracle; proximity data structure
UR - http://eudml.org/doc/277787
ER -