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⋅(kk+⌈logn⌉)⋅2kk2logn), where
k is the treewidth of the graph. For every ϵ>0, this bound is
n1+ϵexpO(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 logdn to
(dd+⌈logn⌉), as originally observed by Monier (J. Alg.
1980).
We also investigate the parameterization by vertex cover number