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

Abstract

We show that the eccentricities, diameter, radius, and Wiener index of an undirected nn-vertex graph with nonnegative edge lengths can be computed in time O(n(k+lognk)2kk2logn)O(n\cdot \binom{k+\lceil\log n\rceil}{k} \cdot 2^k k^2 \log n), where kk is the treewidth of the graph. For every ϵ>0\epsilon>0, this bound is n1+ϵexpO(k)n^{1+\epsilon}\exp 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 logdn\log^d n to (d+lognd)\binom{d+\lceil\log n\rceil}{d}, as originally observed by Monier (J. Alg. 1980). We also investigate the parameterization by vertex cover number

    Similar works