A spanner is a sparse subgraph of a given graph $G$ which preserves distances, measured w.r.t.\ some distance metric, up to a multiplicative stretch factor. This paper addresses the problem of constructing graph spanners w.r.t.\ the group Steiner metric, which generalizes the recently introduced beer distance metric. In such a metric we are given a collection of groups of required vertices, and we measure the distance between two vertices as the length of the shortest path between them that traverses at least one required vertex from each group. We discuss the relation between group Steiner spanners and classic spanners and we show that they exhibit strong ties with sourcewise spanners w.r.t.\ the shortest path metric. Nevertheless, group Steiner spanners capture several interesting scenarios that are not encompassed by existing spanners. This happens, e.g., for the singleton case, in which each group consists of a single required vertex, thus modeling the setting in which routes need to traverse certain points of interests (in any order). We provide several constructions of group Steiner spanners for both the all-pairs and single-source case, which exhibit various size-stretch trade-offs. Notably, we provide spanners with almost-optimal trade-offs for the singleton case. Moreover, some of our spanners also yield novel trade-offs for classical sourcewise spanners. Finally, we also investigate the query times that can be achieved when our spanners are turned into group Steiner distance oracles with the same size, stretch, and building time.

翻译：扩张器是给定图$G$的一个稀疏子图，它能够保持距离（相对于某种距离度量）至多一个乘法拉伸因子。本文研究了针对组Steiner度量构造图扩张器的问题，该度量推广了最近提出的啤酒距离度量。在此度量下，给定一组必需顶点集合，我们测量两个顶点间的距离为它们之间至少经过每个集合中一个必需顶点的最短路径长度。我们探讨了组Steiner扩张器与经典扩张器的关系，并证明它们与基于最短路径度量的源感知扩张器存在紧密联系。尽管如此，组Steiner扩张器捕捉了现有扩张器未能涵盖的若干有趣场景。例如在单元素情形中，每个集合仅包含一个必需顶点，从而建模了路径需要经过特定兴趣点（顺序任意）的场景。我们针对全点对和单源情形提供了多种组Steiner扩张器构造方案，展示了不同的规模-拉伸权衡。值得注意的是，我们为单元素情形提供了具有近乎最优权衡的扩张器。此外，我们提出的部分扩张器也为经典源感知扩张器带来了新的权衡结果。最后，我们还研究了将所提出的扩张器转换为具有相同规模、拉伸和构建时间的组Steiner距离预言机时所能达到的查询时间。