SIAM Journal on Computing, Volume 49, Issue 2, Page 429-447, January 2020.
The greedy spanner is arguably the simplest and most well-studied spanner construction. Experimental results demonstrate that it is at least as good as any other spanner construction in terms of both the size and weight parameters. However, a rigorous proof for this statement has remained elusive. In this work we fill in the theoretical gap via a surprisingly simple observation: The greedy spanner is existentially optimal (or existentially near-optimal) for several important graph families in terms of both size and weight. Roughly speaking, the greedy spanner is said to be existentially optimal (or near-optimal) for a graph family $\mathcal G$ if the worst performance of the greedy spanner over all graphs in $\mathcal G$ is just as good (or nearly as good) as the worst performance of an optimal spanner over all graphs in $\mathcal G$. Focusing on the weight parameter, the state-of-the-art spanner constructions for both general graphs (due to Chechik and Wulff-Nilsen [ACM Trans. Algorithms, 14 (2018), 33]) and doubling metrics (due to Gottlieb [Proceedings of the $56$th Annual IEEE Symposium on Foundations of Computer Science, 2015, pp. 759--772]) are complex. Plugging our observation into these results, we conclude that the greedy spanner achieves near-optimal weight guarantees for both general graphs and doubling metrics, thus resolving two longstanding conjectures in the area. Further, we observe that approximate-greedy spanners are existentially near-optimal as well. Consequently, we provide an $O(n \log n)$-time construction of $(1+\epsilon)$-spanners for doubling metrics with constant lightness and degree. Our construction improves Gottlieb's construction, whose runtime is $O(n \log^2 n)$ and whose number of edges and degree are unbounded, and, remarkably, it matches the state-of-the-art Euclidean result (due to Gudmundsson, Levcopoulos, and Narasimhan [SIAM J. Comput., 31 (2002), pp. 1479--1500]) in all of the involved parameters (up to dependencies on $\epsilon$ and the dimension).

中文翻译：

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贪婪扳手是最优的

SIAM计算学报，第49卷，第2期，第429-447页，2020年1月。
贪婪的扳手可以说是最简单，研究最深入的扳手构造。实验结果表明，就尺寸和重量参数而言，它至少与任何其他扳手构造一样好。但是，对于此声明的严格证据仍然难以捉摸。在这项工作中，我们通过一个令人惊讶的简单观察填补了理论上的空白：贪婪的扳手在大小和权重方面对于几个重要的图族是存在最优的（或存在接近最优的）。粗略地说，如果贪婪扳手在$ \ mathcal G $的所有图中表现最差，那么贪婪扳手对于图族$ \ mathcal G $来说是存在最优的（或接近最优）。几乎等于$ \ mathcal G $中所有图的最佳扳手的最差性能。着重于权重参数，这两个通用图（由于Chechik和Wulff-Nilsen [ACM Trans。Algorithms，14（2018），33]）和加倍度量（由于Gottlieb [第56届IEEE年度计算机科学基础年会论文集，2015，第759--772页]的内容很复杂。将我们的观察结果插入这些结果中，我们得出结论，贪婪的扳手对于一般图形和加倍度量均实现了近乎最优的权重保证，从而解决了该地区两个长期存在的猜想。此外，我们观察到近似贪婪的扳手在本质上也接近最优。因此，我们提供了$（1+ \ epsilon）$跨度的$ O（n \ log n）$时间构造，以便在亮度和度数不变的情况下使指标加倍。我们的建筑改进了Gottlieb的建筑，