Multivariate Analysis of Orthogonal Range Searching and Graph Distances
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
... 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 ). 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  that a lower bound of the form (2) holds for this parameter as well.