迈向光spanner统一理论 I：快速（却最优）的构造
Towards a Unified Theory of Light Spanners I: Fast (Yet Optimal) Constructions
AI总结：
本文提出了一种统一框架，用于在多种图类中快速构造具有最优轻量级特性的spanner，解决了以往方法在通用性、运行时间和参数优化上的不足。
摘要：
多年来关于光spanner的研究在各种图类中提供了最优的轻量级特性，例如在一般图、欧几里得spanner和无简图中。先前关于光spanner工作的三个不足之处是：（1）技术是针对每个图类的，因此无法广泛应用。（2）这些构造的运行时间几乎总是次优，通常远非最优。（3）这些构造在标准和粗略的意义上是最佳的，但不考虑更广泛涉及的参数的细化意义。本文旨在通过在各种图类中提出光spanner的统一框架来解决这些不足。非正式地说，该框架归结为从稀疏spanner到轻量spanner的转换；由于稀疏spanner的最新进展比轻量spanner更先进，因此这种转换非常强大。我们的框架是在两篇论文中开发的。当前论文是其中的第一篇——它奠定了统一框架的基础，然后将其应用于设计几种图类的快速构造，具有最优的轻量级特性。在各种应用和影响中，我们特别强调以下几点：_ 在低维欧几里得空间中，我们提出了一个O(n log n)时间的(1+ε)-spanner构造，其轻量级和度数都受常数限制，该构造在代数计算树（ACT）中解决了该领域的一个重大问题，该问题已存在三十年未解。_ 在一般图中，对于任何k≥2，我们构造了一个(2k-1)(1+ε)-spanner，其轻量级为O(n^{1/k})，时间复杂度为O(m α(m,n))。这一结果对于一般加权图中的轻量spanner来说令人惊讶，因为它优于稀疏spanner的相应结果。
英文摘要：
Seminal works on light spanners over the years provide spanners with optimal lightness in various graph classes, such as in general graphs, Euclidean spanners, and minor-free graphs. Three shortcomings of previous works on light spanners are: (1) The techniques are ad hoc per graph class, and thus can't be applied broadly. (2) The runtimes of these constructions are almost always sub-optimal, and usually far from optimal. (3) 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. 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. Our framework is developed in two papers. The current paper is the first of the two -- it lays the basis of the unified framework and then applies it to design fast constructions with optimal lightness for several graph classes. Among various applications and implications of our framework, we highlight here the following: _ In low-dimensional Euclidean spaces, we present an $O(n\log n)$-time construction of $(1+ε)$-spanners with lightness and degree both bounded by constants in the algebraic computation tree (ACT). Our construction resolves a major problem in the area of geometric spanners, which was open for three decades. _ In general graphs, for any $k \geq 2$, we construct a $(2k-1)(1+ε)$-spanner with lightness $O(n^{1/k})$ in $O(m α(m,n))$ time. This result for light spanners in general weighted graphs is surprising, as it outperforms the analog one for sparse spanners.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
