Ramsey partitions and proximity data structures
Manor Mendel, Assaf Naor · Journal of the European Mathematical Society · 2007
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.