Algorithms and hardness for diameter in dynamic graphs

Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, Nicole Wein

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

18 Scopus citations

Abstract

The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported. This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include: Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP. Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+ε)-approximation to Diameter in directed or undirected n-vertex, medge graphs can be maintained decrementally in total time m1+o(1)√n/ε2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal.

Original languageEnglish
Title of host publication46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
EditorsChristel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771092
DOIs
StatePublished - 1 Jul 2019
Event46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 - Patras, Greece
Duration: 9 Jul 201912 Jul 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume132
ISSN (Print)1868-8969

Conference

Conference46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
Country/TerritoryGreece
CityPatras
Period9/07/1912/07/19

Bibliographical note

Publisher Copyright:
© Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein; licensed under Creative Commons License CC-BY

Keywords

  • Dynamic algorithms
  • Fine-grained complexity
  • Graph algorithms

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