Given a pointed metric space $(X,\mathsf{dist}, w)$ on $n$ points, its Gromov's approximating tree is a 0-hyperbolic pseudo-metric space $(X,\mathsf{dist}_T)$ such that $\mathsf{dist}(x,w)=\mathsf{dist}_T(x,w)$ and $\mathsf{dist}(x, y)-2 \delta \log_2n \leq \mathsf{dist}_T (x, y) \leq \mathsf{dist}(x, y)$ for all $x, y \in X$ where $\delta$ is the Gromov hyperbolicity of $X$. On the other hand, the all pairs bottleneck paths (APBP) problem asks, given an undirected graph with some capacities on its edges, to find the maximal path capacity between each pair of vertices. In this note, we prove: $\bullet$ Computing Gromov's approximating tree for a metric space with $n+1$ points from its matrix of distances reduces to solving the APBP problem on an connected graph with $n$ vertices. $\bullet$ There is an explicit algorithm that computes Gromov's approximating tree for a graph from its adjacency matrix in quadratic time. $\bullet$ Solving the APBP problem on a weighted graph with $n$ vertices reduces to finding Gromov's approximating tree for a metric space with $n+1$ points from its distance matrix.

翻译：给定一个包含$n$个点的带基点度量空间$(X,\mathsf{dist}, w)$，其Gromov近似树是一个0-双曲伪度量空间$(X,\mathsf{dist}_T)$，满足对于所有$x, y \in X$有$\mathsf{dist}(x,w)=\mathsf{dist}_T(x,w)$且$\mathsf{dist}(x, y)-2 \delta \log_2n \leq \mathsf{dist}_T (x, y) \leq \mathsf{dist}(x, y)$，其中$\delta$为$X$的Gromov双曲度。另一方面，全对瓶颈路径（APBP）问题要求：给定一个边带容量的无向图，找出每对顶点间的最大路径容量。本文证明：$\bullet$ 从距离矩阵计算具有$n+1$个点的度量空间的Gromov近似树，可归约为在具有$n$个顶点的连通图上求解APBP问题。$\bullet$ 存在一个显式算法，可从图的邻接矩阵出发，在二次时间内计算其Gromov近似树。$\bullet$ 在具有$n$个顶点的加权图上求解APBP问题，可归约为从距离矩阵计算具有$n+1$个点的度量空间的Gromov近似树。