Ramsey分区和邻近数据结构
Ramsey partitions and proximity data structures
摘要：
本文讨论了几何分析与理论计算机科学交叉的两个问题：非线性同构Dvoretzky定理和大变形下良好近似距离预言机的设计。我们引入了有限度量空间的Ramsey划分的概念，并证明了好的Ramsey划分的存在意味着度量Ramsey问题的大失真（a.k.a.）的解决方案。同构Dvoretzky定理的非线性版本，由Bourgain，Figiel和Milman引入）。然后，我们继续构造最优Ramsey划分，并使用它们来证明，对于每个e \in（0，1），任何n点度量空间都有一个大小为n^{1-e}的子集，该子集以失真O（1/e）嵌入希尔伯特空间。这个结果是最好的可能性，并改进了Bartal，Linial，Mendel和Naor的度量拉姆齐定理的一部分，除了大大简化其证明。我们使用新的Ramsey分区来设计最知名的近似距离预言机，当失真很大时，关闭了Thorup和Zwick留下的空白。也就是说，我们证明了对任何$n$点度量空间X，且k>1，存在一个O（k）-近似距离预言机，其存储需求为O（n^{1+1/k}），其查询时间为一个普适常数.我们还讨论了应用拉姆齐分区的各种其他几何数据结构的问题，如设计有效的数据结构的近似排名。
英文摘要：
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). We then proceed to construct optimal Ramsey partitions, and use them to show that for every e\in (0,1), any n-point metric space has a subset of size n^{1-e} which embeds into Hilbert space with distortion O(1/e). This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor, in addition to considerably simplifying its proof. We use our new Ramsey partitions to design the best known approximate distance oracles when the distortion is large, closing a gap left open by Thorup and Zwick. Namely, we show that for any $n$ point metric space X, and k>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.
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