Multi-Embedding of Metric Spaces

TL;DR

This paper introduces multi-embeddings of metric spaces that reduce distortion bounds, enabling near-optimal embeddings into trees for expanders and improving algorithmic solutions for optimization problems.

Contribution

It proposes multi-embeddings that bypass traditional distortion lower bounds, achieving better embeddings into ultrametrics and trees, with applications to optimization problems.

Findings

Multi-embeddings achieve O(log Delta loglog Delta) distortion.

Constant distortion embeddings for expanders into trees.

Improved algorithms for group Steiner tree and metrical task systems.

Abstract

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion as a function of its size. Using probabilistic metric embeddings, the bound on the distortion reduces to logarithmic in the size. We make a step in the direction of bypassing the lower bound on the distortion in terms of the size of the metric. We define "multi-embeddings" of metric spaces in which a point is mapped onto a set of points, while keeping the target metric of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In particular, for expander graphs, we are able to obtain constant distortion embeddings into trees in contrast with the Omega(log n) lower bound for all previous notions of embeddings. We demonstrate the algorithmic application of the new embeddings for two optimization problems: group Steiner tree and metrical task systems.