2024/08/09 by Cornect, Anders, Martínez-Pedroza, Eduardo
#Combinatorics (math.CO) #Computational Complexity (cs.CC) #Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2408.05338
Given a pointed metric space (X,dist, w) on n points, its Gromov's approximating tree is a 0-hyperbolic pseudo-metric space (X,distT) such that dist(x,w)=distT(x,w) and dist(x, y)-2 δlog2n ≤ distT (x, y) ≤ dist(x, y) for all x, y ∈ X where δ 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.