Let $G = (V, E)$ be an undirected connected simple graph on $n$ vertices. A cut-equivalent tree of $G$ is an edge-weighted tree on the same vertex set $V$, such that for any pair of vertices $s, t\in V$, the minimum $(s, t)$-cut in the tree is also a minimum $(s, t)$-cut in $G$, and these two cuts have the same cut value. In a recent paper [Abboud, Krauthgamer and Trabelsi, 2021], the authors propose the first subcubic time algorithm for constructing a cut-equivalent tree. More specifically, their algorithm has $\widetilde{O}(n^{2.5})$ running time. In this paper, we improve the running time to $\hat{O}(n^2)$ if almost-linear time max-flow algorithms exist. Also, using the currently fastest max-flow algorithm by [van den Brand et al, 2021], our algorithm runs in time $\widetilde{O}(n^{17/8})$.

翻译：让$G = (V, E) 美元 = (V, t) 是一个未方向连接的简单图表 。 切成等值的$G$树是在同一顶端上设定为 $V 的边缘加权树。 更具体地说, 对于树上任何一对顶端的美元, t = 美元, t = 美元 = 美元, 树上最小的美元 = 美元 = (s, t) 折价 $, 而这两个切价是相同的切价 。 在最近的一篇论文中[Abboud, Krauthgamer 和 Trabelsi, 2021], 作者提出了建造一个切成等值树的第一个亚基值时间算法 。 更具体地说, 他们的算法是 $\ 百利特尔德{ O} (n 2.5} 运行时间。 在本文中, 如果存在几乎线上时间最大流算法的话, 我们的运行时间将改进为 $\ 2 。 此外, 我们的算算法是 $17\ = 全时程 = = = yal_ y