License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2018.4
URN: urn:nbn:de:0030-drops-102050
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10205/
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Bringmann, Karl ; Husfeldt, Thore ; Magnusson, Måns

Multivariate Analysis of Orthogonal Range Searching and Graph Distances

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LIPIcs-IPEC-2018-4.pdf (0.5 MB)

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.

BibTeX - Entry

@InProceedings{bringmann_et_al:LIPIcs:2019:10205,
  author =	{Karl Bringmann and Thore Husfeldt and M{\aa}ns Magnusson},
  title =	{{Multivariate Analysis of Orthogonal Range Searching and Graph Distances}},
  booktitle =	{13th International Symposium on Parameterized and Exact  Computation (IPEC 2018)},
  pages =	{4:1--4:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-084-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{115},
  editor =	{Christophe Paul and Michal Pilipczuk},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10205},
  URN =		{urn:nbn:de:0030-drops-102050},
  doi =		{10.4230/LIPIcs.IPEC.2018.4},
  annote =	{Keywords: Diameter, radius, Wiener index, orthogonal range searching, treewidth, vertex cover number}
}

Keywords: Diameter, radius, Wiener index, orthogonal range searching, treewidth, vertex cover number
Collection: 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)
Issue Date: 2019
Date of publication: 05.02.2019

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