TY - GEN
T1 - Algorithms and hardness for diameter in dynamic graphs
AU - Ancona, Bertie
AU - Henzinger, Monika
AU - Roditty, Liam
AU - Williams, Virginia Vassilevska
AU - Wein, Nicole
N1 - Publisher Copyright: © Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein; licensed under Creative Commons License CC-BY
PY - 2019/7/1
Y1 - 2019/7/1
N2 - 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.
AB - 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.
KW - Dynamic algorithms
KW - Fine-grained complexity
KW - Graph algorithms
UR - http://www.scopus.com/inward/record.url?scp=85069213653&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.ICALP.2019.13
DO - https://doi.org/10.4230/LIPIcs.ICALP.2019.13
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
A2 - Baier, Christel
A2 - Chatzigiannakis, Ioannis
A2 - Flocchini, Paola
A2 - Leonardi, Stefano
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
Y2 - 9 July 2019 through 12 July 2019
ER -