Seminal works on light spanners over the years provide spanners with optimal or near-optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Two shortcomings of all previous work on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly (e.g., some require large stretch and are thus suitable to general graphs, while others are naturally suitable to stretch $1 + \epsilon$). (2) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. This work aims at initiating a unified theory of light spanners by presenting a single framework that can be used to construct light spanners in a variety of graph classes. This theory is developed in two papers. The current paper is the first of the two -- it lays the foundations of the theory of light spanners and then applies it to design fast constructions with optimal lightness for several graph classes. Our new constructions are significantly faster than the state-of-the-art for every examined graph class; moreover, our runtimes are near-linear and usually optimal. Specifically, this paper includes the following results: (i) An $O(m \alpha(m,n))$ -time construction of $(2k-1)(1+\epsilon)$-spanner with lightness $O(n^{1/k})$ for general graphs; (ii) An $O(n\log n)$-time construction of Euclidean $(1+\epsilon)$-spanners with lightness and degree both bounded by constants in the basic algebraic computation tree (ACT) model. This construction resolves a major problem in the area of geometric spanners, which was open for three decades; (ii) An $O(n\log n)$-time construction of $(1+\epsilon)$-spanners with constant lightness and degree, in the ACT model for unit disk graphs; (iv) a linear-time algorithm for constructing $(1+\epsilon)$-spanners with constant lightness for minor-free graphs.

中文翻译：

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迈向 Light Spanner 的统一理论 I：快速（但最佳）结构

多年来关于轻型扳手的开创性工作为扳手在各种图类中提供了最佳或接近最佳的亮度，例如一般图、欧几里得扳手和无次要图。以前关于轻型扳手的所有工作的两个缺点是：（1）这些技术是针对每个图类的临时技术，因此不能广泛应用（例如，一些需要大的拉伸，因此适用于一般图，而另一些则自然适合拉伸 $1 + \epsilon$）。(2) 这些结构的运行时间几乎总是次优的，通常远非最优。这项工作旨在通过提出一个单一的框架来启动一个统一的光扳手理论，该框架可用于在各种图类中构建光扳手。这个理论是在两篇论文中发展起来的。当前的论文是这两篇论文中的第一篇——它奠定了光扳手理论的基础，然后将其应用于为多个图类设计具有最佳亮度的快速结构。对于每个检查的图类，我们的新结构都比最先进的结构快得多；此外，我们的运行时间接近线性并且通常是最优的。具体来说，本文包括以下结果： (i) 一个 $O(m \alpha(m,n))$ -time 构造 $(2k-1)(1+\epsilon)$-spanner 的亮度为 $O( n^{1/k})$ 用于一般图形；(ii) 欧几里得 $(1+\epsilon)$-spanner 的 $O(n\log n)$-time 构造，其亮度和度均受基本代数计算树 (ACT) 模型中的常数限制。这种结构解决了几何扳手领域的一个重大问题，该领域已经开放了三年；(ii) 在单位圆盘图的 ACT 模型中，具有恒定亮度和度数的 $(1+\epsilon)$-spanner 的 $O(n\log n)$-time 构造；(iv) 一种线性时间算法，用于构造具有恒定亮度的 $(1+\epsilon)$-spanners，用于无次要图。