Seminal works on light spanners over the years have provided spanners with optimal lightness in various graph classes, such as general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (i) The runtimes of these constructions are almost always sub-optimal and usually far from optimal. (ii) These constructions are optimal in the standard and crude sense but not in a refined sense that takes into account a wider range of involved parameters. (iii) The techniques are ad hoc per graph class and thus can't be applied broadly. This work aims at addressing these shortcomings by presenting a unified framework of light spanners in a variety of graph classes. Informally, the framework boils down to a transformation from sparse spanners to light spanners; since the state-of-the-art for sparse spanners is much more advanced than that for light spanners, such a transformation is powerful. First, we apply our framework to design fast constructions with optimal lightness for several graph classes. Second, we apply our framework to achieve more refined optimality bounds for several graph classes, i.e., the bounds remain optimal when taking into account a wider range of involved parameters, most notably $\epsilon$. Our new constructions are significantly better than the state-of-the-art for every examined graph class.

翻译：多年来关于轻量级生成树的经典研究为各种图类（如一般图、欧几里得生成树及无小图）提供了具有最优轻量度的生成树。此前在轻量级生成树研究中有三个不足之处：(i) 这些构造的运行时间几乎总是次优的，且通常远未达到最优；(ii) 这些构造在标准粗粒度意义下最优，但在考虑更广泛参数范围的精细化意义上并非最优；(iii) 这些技术针对不同图类采用特定方法，因此无法广泛应用。本研究旨在通过提出适用于多种图类的统一轻量级生成树框架来解决上述缺陷。非正式地说，该框架可归结为从稀疏生成树到轻量级生成树的变换；由于稀疏生成树的研究现状远优于轻量级生成树，此类变换具有强大效能。首先，我们应用该框架为若干图类设计了具有最优轻量度的快速构造方法。其次，我们应用该框架为若干图类实现了更精细化的最优性边界——即当考虑更广泛参数（尤其是$\epsilon$）时，这些边界仍保持最优性。对于所研究的每个图类，我们的新构造方法均显著优于当前最优方案。