License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2023.6
URN: urn:nbn:de:0030-drops-180581
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18058/
Aamand, Anders ;
Karczmarz, Adam ;
Łącki, Jakub ;
Parotsidis, Nikos ;
Rasmussen, Peter M. R. ;
Thorup, Mikkel
Optimal Decremental Connectivity in Non-Sparse Graphs
Abstract
We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in O(m + n poly log n) total time. Interspersed with the deletions, it can answer queries whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental c-edge-connectivity to a variant of fully-dynamic c-edge-connectivity on a sparse graph.
For non-sparse graphs with Ω(n poly log n) edges, our connectivity and 2-edge-connectivity algorithms handle all deletions in optimal linear total time, using existing algorithms for the respective fully-dynamic problems. This improves upon an O(m log (n² / m) + n poly log n)-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with Ω(n²) edges.
Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an Ω(log n/log log n) worst-case time lower bound in the decremental setting [Alstrup, Husfeldt, and Rauhe FOCS'98] as well as an Ω(log n) amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].
BibTeX - Entry
@InProceedings{aamand_et_al:LIPIcs.ICALP.2023.6,
author = {Aamand, Anders and Karczmarz, Adam and {\L}\k{a}cki, Jakub and Parotsidis, Nikos and Rasmussen, Peter M. R. and Thorup, Mikkel},
title = {{Optimal Decremental Connectivity in Non-Sparse Graphs}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {6:1--6:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-278-5},
ISSN = {1868-8969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18058},
URN = {urn:nbn:de:0030-drops-180581},
doi = {10.4230/LIPIcs.ICALP.2023.6},
annote = {Keywords: decremental connectivity, dynamic connectivity}
}
Keywords: |
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decremental connectivity, dynamic connectivity |
Collection: |
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50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) |
Issue Date: |
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2023 |
Date of publication: |
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05.07.2023 |