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Online Monotone Metric Embeddings

Authors Christian Coester , Yichen Huang

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LIPIcs.ICALP.2026.66.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online Algorithms
  • Metric Embeddings
  • k-Taxi

Metrics

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 nlog Δ)), where n is the number of points and Δ 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.

Cite As Get BibTex

Christian Coester and Yichen Huang. Online Monotone Metric Embeddings. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 66:1-66:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.66

Author Details

Christian Coester
  • Department of Computer Science, University of Oxford, UK
Yichen Huang
  • John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Funding

  • Coester, Christian: Funded by the European Union (ERC, CCOO, 101165139). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
  • Huang, Yichen: Supported by NSF grant CNS-2107078.

Acknowledgements

The authors thank Michael Mitzenmacher for insightful comments on earlier versions of this paper, and the anonymous reviewers for their constructive feedback.

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