A $k$-additive spanner of a graph is a subgraph that preserves the distance between any two nodes up to a total additive error of $+k$. Efficient algorithms have been devised for constructing 2 [Aingworth et al. SIAM '99], 6 [Baswana et al. ACM '10, Woodruff ICALP '13], and 8-additive spanners [Knudsen '17], but efficiency hasn't been studied for 4-additive spanner constructions. In this paper we present a modification of Chechik's 4-additive spanner construction [Chechik SODA '13] that produces a 4-additive spanner on $\widetilde{O}(n^{7/5})$ edges, with an improved runtime of $\widetilde{O}(mn^{3/5})$ from $O(mn)$.

中文翻译：

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4-Additive 扳手的快速构建

图的 $k$-additive spanner 是一个子图，它保留任意两个节点之间的距离，直到总加性误差为 $+k$。已经设计了有效的算法来构造 2 [Aingworth 等人。SIAM '99], 6 [Baswana 等人。ACM '10、Woodruff ICALP '13] 和 8 添加剂扳手 [Knudsen '17]，但尚未研究 4 添加剂扳手结构的效率。在本文中，我们提出了 Chechik 的 4-additive spanner 构造 [Chechik SODA '13] 的修改，它在 $\widetilde{O}(n^{7/5})$ 边上产生一个 4-additive spanner，并具有改进的运行时间$\widetilde{O}(mn^{3/5})$ 来自 $O(mn)$。