Multivariate Analysis of Orthogonal Range Searching and Graph Distances Parameterized by Treewidth

We show that the eccentricities, diameter, radius, and Wiener index of an undirected n-vertex graph with nonnegative edge lengths can be computed in time O(n·k+⌈log n⌉k· 2^k k^2 log n), where k is the treewidth of the graph. For every ϵ>0, this bound is n^1+ϵexp O(k), which matches a hardness result of Abboud, Vassilevska Williams, and Wang (SODA 2015) and closes an open problem in the multivariate analysis of polynomial-time computation. To this end, we show that the analysis of an algorithm of Cabello and Knauer (Comp. Geom., 2009) in the regime of non-constant treewidth can be improved by revisiting the analysis of orthogonal range searching, improving bounds of the form log^d n to d+⌈log n⌉d, as originally observed by Monier (J. Alg. 1980). We also investigate the parameterization by vertex cover number.