In 2016, a breakthrough result of Chechik and Wulff-Nilsen [SODA '16] established that every $n$-node graph $G$ has a $(1+\varepsilon)(2k-1)$-spanner of lightness $O_{\varepsilon}(n^{1/k})$, and recent followup work by Le and Solomon [STOC '23] generalized the proof strategy and improved the dependence on $\varepsilon$. We give a new proof of this result, with the improved $\varepsilon$-dependence. Our proof is a direct analysis of the often-studied greedy spanner, and can be viewed as an extension of the folklore Moore bounds used to analyze spanner sparsity.

翻译：2016年，Chechik与Wulff-Nilsen[SODA '16]取得突破性成果，证明任意n节点图G均存在轻度为$O_{\varepsilon}(n^{1/k})$的$(1+\varepsilon)(2k-1)$-展形；近期Le与Solomon[STOC '23]的后续工作推广了其证明策略，并改进了对$\varepsilon$的依赖性。本文给出该结果的新证明，其中$\varepsilon$的依赖关系得到优化。我们的证明直接分析了常被研究的贪心展形，可视为分析展形稀疏性时常用的Moore界的推广。