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 (a.k.a. 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\geq 1
, there exists an
O(k)
-approximate distance oracle whose storage requirement is
O\left( n^{1+1/k}\right)
, 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.