Given a metric space (X, d), a set of terminals \(K\subseteq X\), and a parameter \(0<\epsilon <1\), we consider metric structures (e.g., spanners, distance oracles, embedding into normed spaces) that preserve distances for all pairs in \(K\times X\) up to a factor of \(1+\epsilon\), and have small size (e.g. number of edges for spanners, dimension for embeddings). While such terminal (aka source-wise) metric structures are known to exist in several settings, no terminal spanner or embedding with distortion close to 1, is currently known. Here we devise such terminal metric structures for doubling metrics, and show that essentially any metric structure with distortion \(1+\epsilon\) and space s(|X|) has its terminal counterpart, with distortion \(1+O(\epsilon )\) and space \(s(|K|)+n\). In particular, for any doubling metric on n points, a set of k terminals, and constant \(0<\epsilon <1\), there exists

• A spanner with stretch \(1+\epsilon\) for pairs in \(K\times X\), with \(n+O(k)\) edges.

• A labeling scheme with stretch \(1+\epsilon\) for pairs in \(K\times X\), with label size \(\approx \log k\).

• An embedding into \(\ell _\infty ^d\) with distortion \(1+\epsilon\) for pairs in \(K\times X\), where \(d=O(\log k)\).

Moreover, surprisingly, the last two results apply if only the metric space on K is doubling, while the metric on X can be arbitrary.

给定一个度量空间 ( X ,  d )、一组终端\(K\subseteq X\)和一个参数\(0<\epsilon <1\)，我们考虑度量结构（例如，扳手、距离预言机、嵌入到赋范空间），保留\(K\times X\) 中所有对的距离，最大为\(1+\epsilon\)，并且具有小尺寸（例如扳手的边数，嵌入的尺寸）。虽然已知这种终端（又名源级）度量结构存在于多种设置中，但目前还没有终端生成器或失真接近 1 的嵌入是已知的。在这里，我们设计了这样的终端度量结构来加倍度量，并表明基本上任何具有失真\(1+\epsilon\)和空间s (| X |) 的度量结构都有其终端对应物，具有失真\(1+O(\epsilon )\)和空间\(s (|K|)+n\)。特别是，对于n个点、一组k 个终端和常数\(0<\epsilon <1\)上的任何加倍度量，存在

• 一个带有\(1+\epsilon\)的扳手，用于\(K\times X\) 中的对，具有\(n+O(k)\)边。

• 对\(K\times X\) 中的对进行拉伸\(1+\epsilon\)的标记方案，标签大小为\(\approx \log k\)。

• 对于\(K\times X\) 中的对，嵌入\(\ell _\infty ^d\)并带有失真\(1+\epsilon \)，其中\(d=O(\log k)\)。

此外，令人惊讶的是，如果仅K上的度量空间加倍，则最后两个结果适用，而X上的度量可以是任意的。