Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with \(\Omega (n \log n)\) edges, or with a hopbound \(n^{\Omega (1)}\). In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on \(\epsilon \)) \((\log \log n)^{\log \log n + O(1)}\). This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [19] for computing hopsets with a constant (i.e., independent of n) hopbound requires \(n^{\Omega (1)}\) time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [19]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.

Hopset 是一种基本的图论和图算法构造，它们广泛用于各种计算设置中与距离相关的问题。目前现有的 hopsets 结构产生带有\(\Omega (n \log n)\)边的 hopsets，或者带有 hopbound \(n^{\Omega (1)}\)的 hopsets 。在本文中，我们设计了一种具有 hopbound 的线性大小hopsets 的构造（忽略对\(\epsilon \)的依赖）\((\log \log n)^{\log \log n + O(1)}\) . 这几乎成倍地改善了线性大小 hopset 的先前 hopbound. 我们还设计了在 PRAM 和分布式环境中构建的有效实现。唯一现有的 PRAM 算法 [19] 用于计算具有常数（即，独立于n）跳边界的跳集需要\(n^{\Omega (1)}\)时间。我们设计了一种具有多对数运行时间的 PRAM 算法，用于计算具有恒定 hopbound 的 hopset，即，我们的运行时间比前一个算法好得多。此外，这些 hopset 也比 [19] 中的对应物要稀疏得多。我们应用这些 hopsets 在 PRAM 模型中实现以下最短路径的在线变体：在多对数时间内预处理给定的加权图，然后给定任何查询顶点v，报告从v在恒定时间内的所有近似最短路径。所有先前的 hopset 结构都需要每次查询的多对数时间或多项式预处理时间。