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的一项突破性成果[SODA '16]证明：每个$n$节点图$G$存在轻量度为$O_{\varepsilon}(n^{1/k})$的$(1+\varepsilon)(2k-1)$-生成子。近期Le与Solomon的后续工作[STOC '23]将该证明策略进行了推广，并改进了对$\varepsilon$的依赖关系。本文给出了该结论的新证明（采用改进后的$\varepsilon$依赖关系）。我们的证明直接分析了常用于研究的贪心生成子，可视为分析生成子稀疏性的经典Moore界的推广。