Online embedding of metrics

TL;DR

This paper investigates the limits and possibilities of online metric embeddings into various normed spaces and trees, highlighting fundamental bounds and constructing near-optimal embeddings.

Contribution

It establishes a polynomial lower bound for Euclidean embeddings, a tight exponential bound for line embeddings, and a near-optimal high-dimensional $ ext{l}_ ext{infinity}$ embedding.

Findings

Polynomial lower bound on Euclidean embedding distortion.

Exponential upper bound for embedding into the line.

High-dimensional $(1+\epsilon)$-distortion embedding into $ ext{l}_\infty$.

Abstract

We study deterministic online embeddings of metrics spaces into normed spaces and into trees against an adaptive adversary. Main results include a polynomial lower bound on the (multiplicative) distortion of embedding into Euclidean spaces, a tight exponential upper bound on embedding into the line, and a $(1+\epsilon)$-distortion embedding in $\ell_\infty$ of a suitably high dimension.