Deterministic Online Embedding of Metric Spaces into Low Dimensional Spaces

2026-07-12

AI Summary

This study addresses the worst-case distortion incurred when embedding metric spaces into a fixed low-dimensional Euclidean space (\(d > 1\)) under adversarial online settings, with a focus on whether distortion necessarily grows exponentially. The work proposes deterministic online embedding strategies for solid graphs containing a \(K_5\) minor and tree-like metrics. It establishes, for the first time, that \(K_5\) metrics admit online embeddings into \(\mathbb{R}^2\) with only polynomial distortion, thereby refuting a conjectured exponential lower bound. For ultrametrics and other tree-like structures, the approach achieves \(n^{\Theta(1/d)}\) distortion in \(\mathbb{R}^d\), matching the offline optimum up to constant factors in the exponent and substantially narrowing the performance gap between online and offline settings. The methodology integrates metric embedding theory, graph structural analysis, and hierarchical well-separated tree (HST) techniques, enabling efficient derandomization of probabilistic embedding results into low-dimensional Euclidean space.

Abstract

We study online embeddings of metric spaces into Euclidean spaces of a constant dimension $d>1$, against an adaptive adversary. While the case of $d=1$ is well understood, for higher dimensions little is known. In particular, even for $d=2$ it remains unknown whether the worst-case distortion grows exponentially with the number of exposed points, as it does in the case for the line, or whether it is polynomial, as in the case for unbounded $d$. Our first result is about fixed {\em solid} graphs, i.e., $K_5$, whose edges are solid intervals, equipped with the shortest-path metric. We show that if the input points arrive from such a metric space, they can indeed be online-embedded into ${\mathbb R}^2$ with a polynomial distortion. This refutes the previously believed conjecture that the topological non-embeddability of $K_5$ into the plane could be exploited for establishing exponential lower bounds. The second results is about online embeddings of tree metrics of a certain type, including, e.g., ultrametrics and HST's. Somewhat surprisingly, we show that for metrics from this class the worst-case online embedding into ${\mathbb R}^d$ is not much worse that the offline embedding, both being $n^{Θ(1/d)}$, and this holds even when $d = Θ(\log n)$. This is in a stark contrast to the more common situation where the online-offline gap is typically huge, and even exponential. This result allows us to transfer results about probabilistic embeddings of metrics into HST's to low-dimensional Euclidean spaces, in an almost optimal possible manner.

Problem

• online embedding
• metric spaces
• low-dimensional Euclidean space
• distortion
• adaptive adversary

Innovation

• online embedding
• metric spaces
• low-dimensional Euclidean space
• polynomial distortion
• tree metrics