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, 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.

翻译：将度量空间嵌入到结构化空间（特别是层次化良好分离树（HSTs））中，是在线算法设计中的基本工具。在经典在线嵌入设置中，点依次到达，且必须在到达时不可撤销地嵌入，从而导致强失真下界 $Ω(\min(n, \log n\log Δ))$，其中 $n$ 为点的数量，$Δ$ 为其长宽比。我们提出一种新的松弛方法——在线单调度量嵌入，允许目标空间中嵌入点之间的距离随时间单调递减。此类松弛嵌入仍与许多在线算法兼容。此外，该松弛打破了现有下界壁垒，使得嵌入到 HSTs 中的失真度可达 $O(\log^2 n)$。我们还研究了一种动态变体，其中点可同时到达和离开，寻求关于同时存在的最大点数 $l$ 的失真保证。对于传统嵌入而言，此类下界是不可能的，并且这一限制即使在确定性单调嵌入中依然存在。令人惊讶的是，概率性单调嵌入允许 $O(l \log l)$ 的失真度，考虑到 $Ω(l)$ 的下界，这几乎是紧的。