Online monotone metric embeddings

Conference item

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 ∆)), where n is the number of points and ∆ their aspect ratio.

We propose a novel relaxation, 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(log2 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.

Authors: Coester, C; Huang, Y

• Publication status: Accepted
• Peer review status: Peer reviewed
• Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
• Acceptance date: 2026-04-20
• Event title: 53rd EATCS International Colloquium on Automata, Languages, and Programming (ICALP 2026)
• Event location: Egham, UK
• Event website: https://icalppodcspaa2026.cs.rhul.ac.uk/icalp/
• Event start date: 2026-07-07
• Event end date: 2026-07-10
• Language: English
• Keywords: online algorithms; metric embeddings; k-taxi