The metric skip-list is a data structure designed for efficient nearest and $k$-nearest neighbor search in metric spaces. For many real-world datasets with reasonable distributions - specifically, those with a constant expansion rate - it supports $\tilde{O}(n)$ construction time and $O(k\log n)$ query time, where $n$ is the input size and $k$ is the number of nearest neighbors in queries. Notably, unlike alternative approaches, it does not require a bounded aspect ratio, making it more flexible for input data distributions. However, the inherently sequential nature of its original construction has, to our knowledge, precluded any existing parallel algorithm. In this paper, we present highly parallel and work-efficient algorithms for constructing metric skip lists. Under the assumption of a constant expansion rate, our approach achieves an expected work of $O(n \log n)$ and a polylogarithmic span with high probability. Our design is based on novel algorithmic insights that improves the sequential procedure, enabling a divide-and-conquer strategy that facilitates parallelism while maintaining efficiency. With our algorithms, we can also support improved bounds for relevant applications using nearest neighbor as building blocks, including bichromatic closest pair (BCP), density-based clustering, and $k$-NN graph construction, among others. To our knowledge, many of these results represent the first solutions to achieve both work efficiency and polylogarithmic span, relying solely on the assumption of a constant expansion rate.

翻译：度量跳表是一种专为度量空间中高效最近邻和$k$-最近邻搜索而设计的数据结构。对于许多具有合理分布（特别是具有恒定扩展率）的真实世界数据集，它支持$\tilde{O}(n)$构建时间和$O(k\log n)$查询时间，其中$n$为输入规模，$k$为查询中最近邻的数量。值得注意的是，与其他方法不同，它不需要有界纵横比，从而对输入数据分布更具灵活性。然而，其原始构造固有的顺序性质，据我们所知，使得现有并行算法无法实现。本文提出了高度并行且工作高效的度量跳表构造算法。在恒定扩展率的假设下，我们的方法实现了期望工作量为$O(n \log n)$且高概率下为多对数级别的跨度。我们的设计基于新颖的算法洞见，改进了顺序过程，使得能够采用分治策略在保持效率的同时促进并行性。利用我们的算法，我们还能为以最近邻为构建模块的相关应用（包括双色最近点对（BCP）、基于密度的聚类和$k$-最近邻图构建等）提供改进的界限。据我们所知，其中许多结果代表了仅依赖恒定扩展率假设即可同时实现工作高效和多对数跨度的首批解决方案。