Online Monotone Metric Embeddings

2026-04-29

AI Summary

This work addresses the limitations of traditional online metric embedding, where points, once embedded, must remain fixed—a constraint that incurs a distortion lower bound of Ω(min(n, log n log Δ)) and hinders adaptability in dynamic settings. The paper introduces a novel paradigm termed “online single-schedule metric embedding,” which permits distances between already-embedded points to monotonically decrease over time, thereby incorporating a relaxation mechanism for inter-point distances. This innovation breaks the classical distortion barrier. Leveraging hierarchically well-separated trees (HSTs) and combining deterministic with randomized monotonic embedding strategies, the proposed method achieves O(log²n) distortion in static scenarios and O(l log l) distortion under dynamic insertions and deletions, where l denotes the maximum number of concurrently present points—closely approaching the theoretical lower bound of Ω(l).

Abstract

Metric embeddings into structured spaces, particularly hierarchically well-separated trees (HSTs), are a fundamental tool in the design of online algorithms. In the classical online embedding setting, points arrive sequentially and must be embedded irrevocably upon arrival, resulting in strong distortion lower bounds of $Ω(\min(n, \log n\log Δ))$, where $n$ is the number of points and $Δ$ their aspect ratio. We propose a novel relaxation, \emph{online monotone metric embeddings}, which allows distances between embedded points in the target space to decrease monotonically over time. Such relaxed embeddings remain compatible with many online algorithms. Moreover, this relaxation breaks existing lower bound barriers, enabling embeddings into HSTs with distortion $O(\log^2 n)$. We also study a dynamic variant, where points may both arrive and depart, seeking distortion guarantees in terms of the maximum number $l$ of simultaneously present points. For traditional embeddings, such bounds are impossible, and this limitation persists even for deterministic monotone embeddings. Surprisingly, probabilistic monotone embeddings allow for $O(l \log l)$ distortion, which is nearly optimal given an $Ω(l)$ lower bound.

Problem

• online metric embeddings
• monotone embeddings
• hierarchically well-separated trees
• distortion bounds
• dynamic embeddings

Innovation

• online monotone metric embeddings
• hierarchically well-separated trees
• distortion bounds
• dynamic metric embedding
• probabilistic embeddings