A major open problem in the field of metric embedding is the existence of dimension reduction for @math -point subsets of Euclidean space, such that both distortion and dimension depend only on the doubling constant of the pointset, and not on its cardinality. In this paper, we negate this possibility for @math spaces with @math . In particular, we introduce an @math -point subset of @math with doubling constant O(1), and demonstrate that any embedding of the set into @math with distortion @math must have @math .
There are several results on embedding metric spaces with low intrinsic dimension into low dimensional normed space with low distortion: Assouad @cite_12 showed that the snowflakes of doubling metrics For @math , an @math -snowflake of a metric @math is the metric @math , that is, all distances are taken to power @math . embed with constant distortion into constant dimensional Euclidean space. In particular, this is a positive answer to question:d for this special case.
@cite_12
100489996
This paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of @math , with the guarantee that for each @math simultaneously, the distortion of a fraction @math of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion @math . For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion @math . These bounds are tight even for embedding into arbitrary trees. These results imply that the average distortion of the embedding is constant and that the @math distortion is @math . For probabilistic embedding into spanning trees we prove a scaling distortion of @math , which implies constant @math -distortion for every fixed @math . The doubling constant of a metric space (X, d) is the smallest value spl lambda such that every ball in X can be covered by spl lambda balls of half the radius. The doubling dimension of X is then defined as dim (X) = log sub 2 spl lambda . A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaces which contains many families of metrics that occur in applied settings. We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane according to T. J. Laakso (2002). Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L sub 2 . The doubling dimension of a metric is the smallest k such that any ball of radius 2r can be covered using 2k balls of radius r. This concept for abstract metrics has been proposed as a natural analog to the dimension of a Euclidean space. If we could embed metrics with low doubling dimension into low dimensional Euclidean spaces, they would inherit several algorithmic and structural properties of the Euclidean spaces. Unfortunately however, such a restriction on dimension does not suffice to guarantee embeddibility in a normed space.In this paper we explore the option of bypassing the embedding. In particular we show the following for low dimensional metrics: Quasi-polynomial time (1+e)-approximation algorithm for various optimization problems such as TSP, k-median and facility location. (1+e)-approximate distance labeling scheme with optimal label length. (1+e)-stretch polylogarithmic storage routing scheme. 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 BFM86 ). We then proceed to construct optimal Ramsey partitions, and use them to show that for every @math , every @math -point metric space has a subset of size @math which embeds into Hilbert space with distortion @math . This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor BLMN05 , 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 TZ05 . Namely, we show that for every @math point metric space @math , and @math , there exists an @math -approximate distance oracle whose storage requirement is @math , 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.