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在[SODA '16]中取得突破性成果，证明任意$n$节点图$G$均存在轻量度为$O_{\varepsilon}(n^{1/k})$的$(1+\varepsilon)(2k-1)$-稀疏支撑子。随后Le与Solomon在[STOC '23]中的工作推广了该证明策略并优化了其对$\varepsilon$的依赖关系。本文给出该结果的新证明，同时保持了改进后的$\varepsilon$依赖关系。我们的证明直接分析了常被研究的贪心稀疏支撑子，可视为用于分析稀疏支撑子稀疏性的经典Moore界的延伸。