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Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Bringmann, Karl ; Husfeldt, Thore LU and Magnusson, Måns (2019) 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). In LIPIcs 115. p.1-13
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
host publication
13th International Symposium on Parameterized and Exact Computation, IPEC 2018 : August 20-24, 2018, Helsinki, Finland - August 20-24, 2018, Helsinki, Finland
series title
LIPIcs
editor
Paul, Christophe ; Philipczuk, Michał ; and
volume
115
article number
4
pages
13 pages
publisher
Schloss Dagstuhl - Leibniz-Zentrum für Informatik
conference name
13th International Symposium on Parameterized and Exact Computation (IPEC 2018).
conference location
Helsinki, Finland
conference dates
2018-08-20 - 2018-08-24
ISBN
978-3-95977-084-2
DOI
10.4230/LIPIcs.IPEC.2018.4
project
Algebraic Graph Algorithms
language
English
LU publication?
yes
id
b966bcba-fa40-4a84-8fa7-7179f00d8073
date added to LUP
2019-02-27 09:35:17
date last changed
2019-02-27 11:31:13
@inproceedings{b966bcba-fa40-4a84-8fa7-7179f00d8073,
  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.},
  author       = {Bringmann, Karl and Husfeldt, Thore and Magnusson, Måns},
  booktitle    = {13th International Symposium on Parameterized and Exact Computation, IPEC 2018 : August 20-24, 2018, Helsinki, Finland},
  editor       = {Paul, Christophe and Philipczuk, Michał},
  isbn         = {978-3-95977-084-2},
  language     = {eng},
  month        = {02},
  pages        = {1--13},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  series       = {LIPIcs},
  title        = {Multivariate Analysis of Orthogonal Range Searching and Graph Distances},
  url          = {http://dx.doi.org/10.4230/LIPIcs.IPEC.2018.4},
  doi          = {10.4230/LIPIcs.IPEC.2018.4},
  volume       = {115},
  year         = {2019},
}