AI 中文总结
研究度量空间到低维欧氏空间的在线嵌入，针对固定实心图\(K_5\)及特定树度量，给出多项式失真在线嵌入方法，反驳相关猜想，还表明特定树度量在线与离线嵌入差距不大，能转移概率嵌入结果。
AI 中文摘要
我们研究度量空间到固定维度\(d>1\)的欧几里得空间的在线嵌入，对抗适应性对手。\(d = 1\)的情况已被充分理解，高维情况知之甚少。对于\(d = 2\)，最坏情况失真是否随暴露点数呈指数增长未知。首先研究固定的“实心”图\(K_5\)，证明其可在线嵌入\(\mathbb{R}^2\)且失真为多项式，反驳了之前关于利用\(K_5\)平面拓扑不可嵌入性建立指数下界的猜想。其次研究特定类型树度量的在线嵌入，表明此类度量到\(\mathbb{R}^d\)的最坏情况在线嵌入与离线嵌入相差不大，这一结果可将度量到HST的概率嵌入结果以近乎最优方式转移到低维欧几里得空间。
英文摘要
We study online embeddings of metric spaces into Euclidean spaces of a constant dimension $d>1$, against an adaptive adversary. While the case of $d=1$ is well understood, for higher dimensions little is known. In particular, even for $d=2$ it remains unknown whether the worst-case distortion grows exponentially with the number of exposed points, as it does in the case for the line, or whether it is polynomial, as in the case for unbounded $d$. Our first result is about fixed {\em solid} graphs, i.e., $K_5$, whose edges are solid intervals, equipped with the shortest-path metric. We show that if the input points arrive from such a metric space, they can indeed be online-embedded into ${\mathbb R}^2$ with a polynomial distortion. This refutes the previously believed conjecture that the topological non-embeddability of $K_5$ into the plane could be exploited for establishing exponential lower bounds. The second results is about online embeddings of tree metrics of a certain type, including, e.g., ultrametrics and HST's. Somewhat surprisingly, we show that for metrics from this class the worst-case online embedding into ${\mathbb R}^d$ is not much worse that the offline embedding, both being $n^{Θ(1/d)}$, and this holds even when $d = Θ(\log n)$. This is in a stark contrast to the more common situation where the online-offline gap is typically huge, and even exponential. This result allows us to transfer results about probabilistic embeddings of metrics into HST's to low-dimensional Euclidean spaces, in an almost optimal possible manner.
Comments10 main pages, bibliography and Appendix
