Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Abstract

AbstractWe show that the eccentricities, diameter, radius, and Wiener index of an undirected n-vertex graph with nonnegative edge lengths can be computed in time $$O(n\cdot \left( {\begin{array}{c}k+\lceil \log n\rceil \\ k\end{array}}\right) \cdot 2^k \log n)$$ O ( n · k + ⌈ log n ⌉ k · 2 k log n ) , where k is linear in the treewidth of the graph. For every $$\epsilon >0$$ ϵ > 0 , this bound is $$n^{1+\epsilon }\exp O(k)$$ 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 $$\log ^d n$$ log d n to $$\left( {\begin{array}{c}d+\lceil \log n\rceil \\ d\end{array}}\right)$$ 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.