SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 1030-1056, March 2024.
Abstract. Given a metric space [math], a weighted graph [math] over [math] is a metric [math]-spanner of [math] if for every [math], [math], where [math] is the shortest path metric in [math]. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points [math], where the points are presented one at a time (i.e., after [math] steps, we see [math]). The algorithm is allowed to add edges to the spanner when a new point arrives; however, it is not allowed to remove any edge from the spanner. The goal is to maintain a [math]-spanner [math] for [math] for all [math], while minimizing the number of edges, and their total weight. We construct online [math]-spanners in the Euclidean [math]-space, [math]-spanners for general metrics, and [math]-spanners for ultrametrics. Most notably, in the Euclidean plane, we construct a [math]-spanner with competitive ratio [math], bypassing the classic lower bound [math] for lightness, which compares the weight of the spanner to that of the minimum spanning tree.

中文翻译：

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度量空间中的在线扳手

SIAM 离散数学杂志，第 38 卷，第 1 期，第 1030-1056 页，2024 年 3 月。
摘要。给定一个度量空间 [math]，如果对于每个 [math]、[math]，则 [math] 上的加权图 [math] 是 [math] 的度量 [math]-spanner，其中 [math] 是最短路径度量在[数学]中。在本文中，我们在在线设置中为度量空间中的有限集构造了扳手。在这里，我们给出了一系列点 [math]，其中这些点一次呈现一个（即，在 [math] 步骤之后，我们看到 [math]）。当新点到达时，算法允许向扳手添加边；但是，不允许去除扳手上的任何边缘。目标是为所有[数学]维护[数学]扳手[数学]，同时最小化边的数量及其总重量。我们在欧几里得[数学]空间中构造在线[数学]扳手，用于一般指标的[数学]扳手，以及用于超计量学的[数学]扳手。最值得注意的是，在欧几里得平面中，我们构造了一个具有竞争比[math]的[math]-spanner，绕过了经典的轻量下限[math]，它将spanner的权重与最小生成树的权重进行了比较。