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# Bounded Geometries, Fractals, and Low-Distortion Embeddings

FOCS, pp.534-534, (2003)

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Abstract

The doubling constant of a metric space (X, d) is the smallest value \lambda such that every ball in X can be covered by \lambda balls of half the radius. The doubling dimension of X is then defined as \dim (X) = \log _2 \lambda. A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is...More

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Introduction

- A basic goal in the study of finite metric spaces is to approximate some class of metric spaces by another more simple or tractable class.
- The authors consider embeddings of finite metric spaces into Lp spaces.
- Given a metric (X, d), the goal is to find a map f : X → Lp such that ||f (x)−f (y)||p is close to d(x, y) for all x, y ∈ X.
- The authors' goal is to find bounds on the achievable

Highlights

- A basic goal in the study of finite metric spaces is to approximate some class of metric spaces by another more simple or tractable class
- Apart from being beautiful objects of study lying at the intersection of analysis, combinatorics, and geometry, the ideas and techniques generated in this field have led to a number of powerful algorithmic applications
- 1We are concerned mostly with finite metrics, but most of our results extend to arbitrary metric spaces via standard compactness arguments
- We prove Assouad’s conjecture for trees, showing that every doubling tree can be embedded with constant distortion into constant dimensional lp space for every p ∈ [1, ∞], where both constants depend only on the doubling constant of the tree

Results

**Results and techniques**

The authors are concerned with the broad roles of “volume” and “structure” in determining the embeddability of a metric.- In Section 2, the authors show that every doubling tree metric admits a constant distortion embedding into lOp (1) for any p ∈ [1, ∞]
- This exhibits a very natural class of metric spaces which embed into l2 with O(1) distortion, but not isometrical√ly.
- Some tree metrics require Ω distortion to embed into l2 [5], while w√e prove that some doubling metrics require distortion Ω
- It is precisely the synthesis of these two properties that yields an enormous improvement in embeddability.
- The authors do this by defining a notion of adjacency between paths, and arguing that the resulting graph has bounded chromatic number

Conclusion

- Using the decomposition theorems i√n Section 3, the authors can show that doubling metrics admit (k, log n)-volume respecting embeddings for all k; doubling metrics can be embedded into distributions of doubling trees (HSTs of bounded degree) with distortion O, and they embed into the line with constant average distortion
- These results can be used to improve approximation algorithms and online algorithms for doubling metrics.
- The authors' tree coloring in conjunction with a modification of [11] show that the bandwidth of a graph with local density β is at most O(β1.5 log2 n)

Reference

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