Online Monotone Metric Embeddings

TL;DR

This paper introduces online monotone metric embeddings that allow decreasing distances over time, breaking previous lower bounds and enabling better embeddings into hierarchically well-separated trees with near-optimal distortion.

Contribution

It proposes a novel relaxation called online monotone metric embeddings, achieving improved distortion bounds and extending to dynamic settings with point arrivals and departures.

Findings

Breaks existing lower bounds with $O( ext{log}^2 n)$ distortion for HST embeddings.

Probabilistic monotone embeddings achieve $O(l ext{log} l)$ distortion in dynamic settings.

Traditional embeddings cannot guarantee such bounds in dynamic scenarios.

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 $\Omega(\min(n, \log n\log \Delta))$, where $n$ is the number of points and $\Delta$ 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 $O(\log^2 n)$. 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 $\Omega(l)$ lower bound.