Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion $O(\log n)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$ (or, $O(d)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$ if the metric has doubling dimension $d$), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio $\Phi$ from Indyk et al.\ (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion $O(d\cdot \log\Phi)$, generalizing previous results (Indyk et al.\ (2010), Bartal et al.\ (2020)). Next, we study the \emph{online minimum-weight perfect matching} problem, where a sequence of $2n$ metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the \emph{minimum-weight} perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio $O(d\cdot \log \Phi)$ and recourse $O(\log \Phi)$ against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using $O(\log^2 n)$ recourse, that maintains a matching of weight at most $O(\log n)$ times the weight of the MST, i.e., a matching of lightness $O(\log n)$. We complement our upper bounds with a strategy for an oblivious adversary that, with recourse $r$, establishes a lower bound of $\Omega(\frac{\log n}{r \log r})$ for both competitive ratio and lightness.

翻译：低失真度量嵌入是现代算法工具包中的关键组成部分。在在线度量嵌入问题中，点序列依次到达，目标是以不可逆的方式将其嵌入简单空间，同时最小化失真。我们的第一个结果是：将一般度量空间确定性在线嵌入欧几里得空间，失真为 $O(\log n)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$（若度量具有倍增维度 $d$，则失真为 $O(d)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$），这解决了Newman和Rabinovich（2020）的一个猜想，并将Indyk等人（2010）结果中对长宽比 $\Phi$ 的依赖二次改进。第二个结果是：将度量空间随机嵌入树结构，期望失真为 $O(d\cdot \log\Phi)$，推广了先前结果（Indyk等（2010），Bartal等（2020））。接下来，我们研究*在线最小权重完美匹配*问题：$2n$ 个度量点成对序列到达，需始终保持完美匹配。我们允许反悔（否则到达顺序决定匹配），目标是在所有时刻返回近似*最小权重*完美匹配的匹配，同时最小化反悔次数。第三个结果是：针对 oblivious 对手的随机算法，其竞争比为 $O(d\cdot \log \Phi)$，反悔次数为 $O(\log \Phi)$，该结果基于我们新的随机在线嵌入。第四个结果是：针对自适应对手的确定性算法，使用 $O(\log^2 n)$ 次反悔，维持的匹配权重不超过MST权重的 $O(\log n)$ 倍，即匹配的轻量性为 $O(\log n)$。我们为这些上界补充了 oblivious 对手的策略：当反悔次数为 $r$ 时，该策略对竞争比和轻量性均建立 $\Omega(\frac{\log n}{r \log r})$ 的下界。