论文标题
轻度跨越的统一框架
A Unified Framework for Light Spanners
论文作者
论文摘要
多年来,开创性的作品在各种图形类别中为跨度提供了最佳的轻度,例如通用图,欧几里得跨越和次要图形。以前在轻度跨越的作品的三个缺点是:(i)这些构造的运行时间几乎总是优化的,通常远非最佳。 (ii)这些结构在标准和粗略的意义上是最佳的,但在精致的意义上却不考虑到涉及的参数范围更大。 (iii)这些技术是每个图类别的临时,因此不能广泛应用。 这项工作旨在通过在各种图形类中展示统一的光跨度框架来解决这些缺点。非正式地,该框架归结为从稀疏的跨度转变为轻跨度;由于稀疏跨度的最先进的稀疏器要比轻跨度的先进得多,因此这种转换非常有力。首先,我们将框架应用于设计快速构造,以最佳的轻度为几个图形类。其次,我们将框架应用于几个图形类别的更精致的最优界限,即,考虑到更广泛的涉及参数,最著名的是$ε$,范围仍然是最佳的。对于每个检查的图形类别,我们的新结构都比最先进的构造要好得多。
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 $ε$. Our new constructions are significantly better than the state-of-the-art for every examined graph class.