Document Open Access Logo

Multivariate Analysis of Orthogonal Range Searching and Graph Distances

Authors Karl Bringmann, Thore Husfeldt , Måns Magnusson

PDF
Thumbnail PDF

File

LIPIcs.IPEC.2018.4.pdf
  • Filesize: 0.51 MB
  • 13 pages

Document Identifiers

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

Funding

  • Husfeldt, Thore: Swedish Research Council grant VR-2016-03855 and Villum Foundation grant 16582.

Cite As Get BibTex

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, Va, USA, January 10-12, 2016, pages 377-391. SIAM, 2016. URL: http://dx.doi.org/10.1137/1.9781611974331.ch28.
  2. Matthias Bentert and André Nichterlein. Parameterized Complexity of Diameter. CoRR, abs/1802.10048, 2018. URL: http://arxiv.org/abs/1802.10048.
  3. Jon Louis Bentley. Multidimensional Divide-and-Conquer. Commun. ACM, 23(4):214-229, 1980. URL: http://dx.doi.org/10.1145/358841.358850.
  4. Manuel Blum, Robert W. Floyd, Vaughan R. Pratt, Ronald L. Rivest, and Robert Endre Tarjan. Time Bounds for Selection. J. Comput. Syst. Sci., 7(4):448-461, 1973. URL: http://dx.doi.org/10.1016/S0022-0000(73)80033-9.
  5. Hans L. Bodlaender, Pål Grønås Drange, Markus S. Dregi, Fedor V. Fomin, Daniel Lokshtanov, and Michał Pilipczuk. An O(c^k n) 5-approximation algorithm for treewidth. SIAM J. Comput., 45(2):317-378, 2016. URL: http://dx.doi.org/10.1137/130947374.
  6. Sergio Cabello and Christian Knauer. Algorithms for bounded treewidth with orthogonal range searching. Comput. Geom., 42(9):815-824, 2009. URL: http://dx.doi.org/10.1016/j.comgeo.2009.02.001.
  7. Timothy M. Chan. All-Pairs Shortest Paths with Real Weights in O(n³/log n) Time. Algorithmica, 50(2):236-243, 2008. URL: http://dx.doi.org/10.1007/s00453-007-9062-1.
  8. Bernard Chazelle. Lower Bounds for Orthogonal Range Searching: I. The Reporting Case. J. Assoc. Comput. Mach., 37(2):200-212, 1990. URL: http://dx.doi.org/10.1145/77600.77614.
  9. Thore Husfeldt. Computing Graph Distances Parameterized by Treewidth and Diameter. In 11th International Symposium on Parameterized and Exact Computation (IPEC 2016), volume 63 of Leibniz International Proceedings in Informatics (LIPIcs), pages 16:1-16:11, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2016.16.
  10. Russel Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: http://dx.doi.org/10.1006/jcss.2001.1774.
  11. Der-Tsai Lee and Chakkuen K. Wong. Quintary Trees: A File Structure for Multidimensional Database Systems. ACM Trans. Database Syst., 5(3):339-353, 1980. URL: http://dx.doi.org/10.1145/320613.320618.
  12. George S. Lueker. A Data Structure for Orthogonal Range Queries. In 19th Annual Symposium on Foundations of Computer Science, Ann Arbor, Michigan, USA, 16-18 October 1978, pages 28-34. IEEE Computer Society, 1978. URL: http://dx.doi.org/10.1109/SFCS.1978.1.
  13. Louis Monier. Combinatorial Solutions of Multidimensional Divide-and-Conquer Recurrences. J. Algorithms, 1(1):60-74, 1980. URL: http://dx.doi.org/10.1016/0196-6774(80)90005-X.
  14. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 515-524, 2013. URL: http://dx.doi.org/10.1145/2488608.2488673.
  15. Qiaosheng Shi. Efficient algorithms for network center/covering location optimization problems. PhD thesis, School of Computing Science, Simon Fraser University, 2008. Google Scholar
  16. Dan E. Willard. New data structures for orthogonal range queries. SIAM J. Comput., 14(1):232-253, 1985. URL: http://dx.doi.org/10.1137/0214019.
  17. Virginia Vassilevska Williams. Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk). In 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, September 16-18, 2015, Patras, Greece, pages 17-29, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.17.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact