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箱中的关键组成部分。在线度量嵌入中，点按序到达，目标是以不可撤销的方式将其嵌入到简单空间中，同时最小化失真度。我们的第一个结果是将一般度量确定性地在线嵌入到欧几里得空间中，失真度为 $O(\log n)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$（若度量具有倍增维度 $d$，则为 $O(d)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$），这解决了 Newman 和 Rabinovich（2020）的一个猜想，并将 Indyk 等人（2010）结果中对纵横比 $\Phi$ 的依赖从线性改进为平方根级别。我们的第二个结果是度量空间到树结构的随机嵌入，其期望失真度为 $O(d\cdot \log\Phi)$，推广了先前的结果（Indyk 等人（2010），Bartal 等人（2020））。接着，我们研究\emph{在线最小权完美匹配}问题，其中 $2n$ 个度量点成对到达，算法需始终保持一个完美匹配。我们允许调整操作（否则到达顺序将完全决定匹配）。目标是在所有时刻返回一个近似\emph{最小权}完美匹配的匹配，同时最小化调整次数。我们的第三个结果是针对迟钝对手的随机算法，其竞争比为 $O(d\cdot \log \Phi)$，调整次数为 $O(\log \Phi)$，该结果通过我们提出的新型随机在线嵌入获得。第四个结果是针对自适应对手的确定性算法，使用 $O(\log^2 n)$ 次调整，始终保持一个权重至多为最小生成树权重 $O(\log n)$ 倍的匹配，即具有 $O(\log n)$ 轻量性的匹配。我们通过为迟钝对手设计一个策略来补充上界，该策略在调整次数 $r$ 下，为竞争比和轻量性同时建立了 $\Omega(\frac{\log n}{r \log r})$ 的下界。