research-article

Connectivity Oracles for Graphs Subject to Vertex Failures

Authors:

Seth PettieAuthors Info & Claims

SIAM Journal on Computing, Volume 49, Issue 6

Pages 1363 - 1396

https://doi.org/10.1137/17M1146610

Published: 01 January 2020 Publication History

Abstract

We introduce new data structures for answering connectivity queries in graphs subject to batched vertex failures. A deterministic structure processes a batch of $d\leq d_\star$ failed vertices in $\tilde{O}(d^3)$ time and thereafter answers connectivity queries in $O(d)$ time. It occupies space $O(d_\star m\log n)$. We develop a randomized Monte Carlo version of our data structure with update time $\tilde{O}(d^2)$, query time $O(d)$, and space $\tilde{O}(m)$ for any failure bound $d\le n$. This is the first connectivity oracle for general graphs that can efficiently deal with an unbounded number of vertex failures. We also develop a more efficient Monte Carlo edge failure connectivity oracle. Using space $O(n\log^2 n)$, $d$ edge failures are processed in $O(d\log d\log\log n)$ time, and thereafter, connectivity queries are answered in $O(\log\log n)$ time, which are correct with high probability. Our data structures are based on a new decomposition theorem for an undirected graph $G=(V,E)$, which is of independent interest. It states that for any terminal set $U\subseteq V$ we can remove a set $B$ of $|U|/(s-2)$ vertices such that the remaining graph contains a Steiner forest for $U-B$ with maximum degree $s$.

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Cited By

Parter MPetruschka APettie SMohar BShinkar IO'Donnell R(2024)Connectivity Labeling and Routing with Multiple Vertex FailuresProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649729(823-834)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649729
Izumi TEmek YWadayama TMasuzawa TOshman RNolin AHalldorsson MBalliu A(2023)Deterministic Fault-Tolerant Connectivity Labeling SchemeProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594584(190-199)Online publication date: 19-Jun-2023
https://dl.acm.org/doi/10.1145/3583668.3594584
Duan RRen HLeonardi SGupta A(2022)Maintaining exact distances under multiple edge failuresProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520002(1093-1101)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520002

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cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 49, Issue 6

ISSN:0097-5397

DOI:10.1137/smjcat.49.6

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© 2020, Society for Industrial and Applied Mathematics.

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Published: 01 January 2020

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Parter MPetruschka APettie SMohar BShinkar IO'Donnell R(2024)Connectivity Labeling and Routing with Multiple Vertex FailuresProceedings of the 56th Annual ACM Symposium on Theory of Computing10.1145/3618260.3649729(823-834)Online publication date: 10-Jun-2024
https://dl.acm.org/doi/10.1145/3618260.3649729
Izumi TEmek YWadayama TMasuzawa TOshman RNolin AHalldorsson MBalliu A(2023)Deterministic Fault-Tolerant Connectivity Labeling SchemeProceedings of the 2023 ACM Symposium on Principles of Distributed Computing10.1145/3583668.3594584(190-199)Online publication date: 19-Jun-2023
https://dl.acm.org/doi/10.1145/3583668.3594584
Duan RRen HLeonardi SGupta A(2022)Maintaining exact distances under multiple edge failuresProceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing10.1145/3519935.3520002(1093-1101)Online publication date: 9-Jun-2022
https://dl.acm.org/doi/10.1145/3519935.3520002

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