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,光球仪的表面工程为光扇提供了各种图形类中的最佳光亮度,如一般图形、欧洲cliidean光谱仪和无微小图形。以前对光球仪的工程的三个缺点是:(一) 这些建筑的运行时间几乎总是次优,通常远非最佳。 (二) 这些建筑在标准意义和粗度上是最佳的,但并不是考虑到更广泛的所涉参数的精细意义上的。 (三) 这些技术是每个图形类的临时性的,因此无法广泛应用。这项工作的目的是通过在各种图形类中提供一个光扇的统一框架来克服这些缺点。非正式地说,这些框架的运行时间从稀薄的光扇向光窗扇转变;由于对光球仪来说,这种最先进的结构比对光球员来说要先进得多。首先,我们应用了我们的框架来设计快速的构造,对几个图形类而言,最优的光度无法广泛应用。第二,我们应用我们最优的架构来解决每个图形类中最优化的等级。 明显地,我们最优化的图表将每个图表都绑定了。</s>