Tree embedding has been a fundamental method in algorithm design with wide applications. We focus on the efficiency of building tree embedding in various computational settings under high-dimensional Euclidean $\mathbb{R}^d$. We devise a new tree embedding construction framework that operates on an arbitrary metric decomposition with bounded diameter, offering a tradeoff between distortion and the locality of its algorithmic steps. This framework works for general metric spaces and may be of independent interest beyond the Euclidean setting. Using this framework, we obtain a dynamic algorithm that maintains an $O_ε(\log n)$-distortion tree embedding with update time $\tilde O(n^ε+ d)$ subject to point insertions/deletions, and a massively parallel algorithm that achieves $O_ε(\log n)$-distortion in $O(1)$ rounds and total space $\tilde O(n^{1 + ε})$ (for constant $ε\in (0, 1)$). These new tree embedding results allow for a wide range of applications. Notably, under a similar performance guarantee as in our tree embedding algorithms, i.e., $\tilde O(n^ε+ d)$ update time and $O(1)$ rounds, we obtain $O_ε(\log n)$-approximate dynamic and MPC algorithms for $k$-median and earth-mover distance in $\mathbb{R}^d$.

翻译：树嵌入一直是算法设计中的基本方法，具有广泛的应用。我们关注在高维欧几里得空间 $\mathbb{R}^d$ 中，不同计算环境下构建树嵌入的效率问题。我们设计了一种新的树嵌入构建框架，该框架可在直径有界的任意度量分解上运行，并在失真度与算法步骤的局部性之间提供权衡。此框架适用于一般度量空间，其意义可能超越欧几里得环境本身。利用该框架，我们获得了一种动态算法，可在点插入/删除操作下以 $\tilde O(n^ε+ d)$ 的更新时间维护 $O_ε(\log n)$ 失真度的树嵌入；以及一种大规模并行算法，可在 $O(1)$ 轮内实现 $O_ε(\log n)$ 失真度，总空间开销为 $\tilde O(n^{1 + ε})$（其中 $ε\in (0, 1)$ 为常数）。这些新的树嵌入结果为众多应用提供了可能。值得注意的是，在与我们树嵌入算法类似的性能保证下（即 $\tilde O(n^ε+ d)$ 更新时间和 $O(1)$ 轮数），我们得到了 $\mathbb{R}^d$ 中 $k$-中位数和推土机距离的 $O_ε(\log n)$ 近似动态算法与MPC算法。