Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Karl Bringmann, Thore Husfeldt, Måns Magnusson, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation  
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 > 0, this bound is n 1+ 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
more » ... 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. ACM Subject Classification Theory of computation → Shortest paths, Theory of computation → Parameterized complexity and exact algorithms, Theory of computation → Computational geometry, Mathematics of computing → Paths and connectivity problems Theorem 3 (Implicit in [1]). A data structure for the orthogonal range query problem for the monoid (Z, max) with construction time n · q (n, d) and query time q (n, d), where for some > 0, refutes the Strong Exponential Time hypothesis. We also investigate the same problems parameterized by vertex cover number: Theorem 4. The eccentricities, diameter, and radius of a given undirected, unweighted n-vertex graph G with vertex cover number k can be computed in time O(nk + 2 k k 2 ). The Wiener index can be computed in time O(nk2 k ). Both of these bounds are n exp O(k). It follows from [1] that a lower bound of the form (2) holds for this parameter as well.
doi:10.4230/lipics.ipec.2018.4 dblp:conf/iwpec/BringmannHM18 fatcat:xx3ermqaqfho7d7k2ndpmcvydm