Multivariate Analysis of Orthogonal Range Searching and Graph Distances.

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 log n) , where k is linear in the treewidth of the graph. For every ϵ > 0 , this bound is n 1 + ϵ exp O (k) , which matches a hardness result of Abboud et al. (in: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 2016. 10.1137/1.9781611974331.ch28) 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 (Comput Geom 42:815–824, 2009. 10.1016/j.comgeo.2009.02.001) 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 Algorithms 1:60–74, 1980. 10.1016/0196-6774(80)90005-X). We also investigate the parameterization by vertex cover number. [ABSTRACT FROM AUTHOR]