research-article

Nearly optimal vertex fault-tolerant spanners in optimal time: sequential, distributed, and parallel

Author:

Merav ParterAuthors Info & Claims

STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

Pages 1080 - 1092

https://doi.org/10.1145/3519935.3520047

Published: 10 June 2022 Publication History

Abstract

We (nearly) settle the time complexity for computing vertex fault-tolerant (VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT spanners are sparse subgraphs that preserve distance information, up to a small multiplicative stretch, in the presence of vertex failures. These structures were introduced by [Chechik et al., STOC 2009] and have received a lot of attention since then.

Recent work provided algorithms for computing VFT spanners with optimal sparsity but in exponential runtime. The first polynomial time algorithms for these structures have been given by [Bodwin, Dinitz and Robelle, SODA 2021]. Their algorithms, as all other prior algorithms, are greedy and thus inherently sequential. We provide algorithms for computing nearly optimal f-VFT spanners for any n-vertex m-edge graph, with near optimal running time in several computational models:

(i) A randomized sequential algorithm with a runtime of O(m) (i.e., independent in the number of faults f). The state-of-the-art time bound is O(f^1−1/k· n^2+1/k+f² m) by [Bodwin, Dinitz and Robelle, SODA 2021].

(ii) A distributed congest algorithm of O(1) rounds. Improving upon [Dinitz and Robelle, PODC 2020] that obtained FT spanners with near-optimal sparsity in O(f²) rounds.

(iii) A PRAM (CRCW) algorithm with O(m) work and O(1) depth. Prior bounds implied by [Dinitz and Krauthgamer, PODC 2011] obtained sub-optimal FT spanners using O(f³m) work and O(f³) depth.

An immediate corollary provides the first nearly-optimal PRAM algorithm for computing nearly optimal λ-vertex connectivity certificates using polylogarithmic depth and near-linear work. This improves the state-of-the-art parallel bounds of O(1) depth and O(λ m) work, by [Karger and Motwani, STOC’93].

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Cited By

C. S. KParter M(2024)Deterministic Replacement Path CoveringACM Transactions on Algorithms10.1145/367376020:4(1-35)Online publication date: 5-Aug-2024
https://dl.acm.org/doi/10.1145/3673760
Busch CKowalski DRobinson PAgrawal KPetrank E(2024)Sparse Spanners with Small Distance and Congestion StretchesProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659954(383-393)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3626183.3659954
Baswana SBhanja KPandey A(2023)Minimum+1 (s, t)-cuts and Dual-edge Sensitivity OracleACM Transactions on Algorithms10.1145/362327119:4(1-41)Online publication date: 14-Oct-2023
https://dl.acm.org/doi/10.1145/3623271

Index Terms

Nearly optimal vertex fault-tolerant spanners in optimal time: sequential, distributed, and parallel
1. Theory of computation
  1. Design and analysis of algorithms

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cover image ACM Conferences

STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

June 2022

1698 pages

ISBN:9781450392648

DOI:10.1145/3519935

General Chair:
Stefano Leonardi
Sapienza University of Rome, Italy
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Anupam Gupta
Carnegie Mellon University, USA

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Published: 10 June 2022

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STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing

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Cited By

C. S. KParter M(2024)Deterministic Replacement Path CoveringACM Transactions on Algorithms10.1145/367376020:4(1-35)Online publication date: 5-Aug-2024
https://dl.acm.org/doi/10.1145/3673760
Busch CKowalski DRobinson PAgrawal KPetrank E(2024)Sparse Spanners with Small Distance and Congestion StretchesProceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3626183.3659954(383-393)Online publication date: 17-Jun-2024
https://dl.acm.org/doi/10.1145/3626183.3659954
Baswana SBhanja KPandey A(2023)Minimum+1 (s, t)-cuts and Dual-edge Sensitivity OracleACM Transactions on Algorithms10.1145/362327119:4(1-41)Online publication date: 14-Oct-2023
https://dl.acm.org/doi/10.1145/3623271

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