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Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Authors Karl Bringmann, Thore Husfeldt , Måns Magnusson



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Author Details

Karl Bringmann
  • Max-Planck-Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany
Thore Husfeldt
  • BARC, IT University of Copenhagen, Denmark, and Lund University, Sweden
Måns Magnusson
  • Department of Computer Science, Lund University, Sweden

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Karl Bringmann, Thore Husfeldt, and Måns Magnusson. Multivariate Analysis of Orthogonal Range Searching and Graph Distances. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 4:1-4:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.IPEC.2018.4

Abstract

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 * binom{k+ceil[log n]}{k} * 2^k k^2 log n), where k is the treewidth of the graph. For every epsilon>0, this bound is 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 log^d n to binom{d+ceil[log n]}{d}, as originally observed by Monier (J. Alg. 1980). We also investigate the parameterization by vertex cover number.

Subject Classification

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
Keywords
  • Diameter
  • radius
  • Wiener index
  • orthogonal range searching
  • treewidth
  • vertex cover number

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References

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