research-article

Tight Analysis of Parallel Randomized Greedy MIS

Authors:

Manuela Fischer,

Andreas NoeverAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 16, Issue 1

Article No.: 6, Pages 1 - 13

https://doi.org/10.1145/3326165

Published: 05 December 2019 Publication History

Abstract

We provide a tight analysis that settles the round complexity of the well-studied parallel randomized greedy MIS algorithm, thus answering the main open question of Blelloch, Fineman, and Shun [SPAA’12].

The parallel/distributed randomized greedy Maximal Independent Set (MIS) algorithm works as follows. An order of the vertices is chosen uniformly at random. Then, in each round, all vertices that appear before their neighbors in the order are added to the independent set and removed from the graph along with their neighbors. The main question of interest is the number of rounds it takes until the graph is empty. This algorithm has been studied since 1987, initiated by Coppersmith, Raghavan, and Tompa [FOCS’87], and the previously best known bounds were O(log n) rounds in expectation for Erdős-Rényi random graphs by Calkin and Frieze [Random Struc. Alg.’90] and O(log² n) rounds with high probability for general graphs by Blelloch, Fineman, and Shun [SPAA’12].

We prove a high probability upper bound of O(log n) on the round complexity of this algorithm in general graphs and that this bound is tight. This also shows that parallel randomized greedy MIS is as fast as the celebrated algorithm of Luby [STOC’85, JALG’86].

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Cited By

Fedorov AHashemi DNadiradze GAlistarh DAgrawal KShun J(2023)Provably-Efficient and Internally-Deterministic Parallel Union-FindProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591082(261-271)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591082
Behnezhad SCharikar MMa WTan L(2022)Almost 3-Approximate Correlation Clustering in Constant Rounds2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00074(720-731)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00074
Behnezhad S(2022)Time-Optimal Sublinear Algorithms for Matching and Vertex Cover2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00089(873-884)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00089
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Index Terms

Tight Analysis of Parallel Randomized Greedy MIS
1. Theory of computation
  1. Design and analysis of algorithms
    1. Parallel algorithms
      1. Shared memory algorithms

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 16, Issue 1

Special Issue on Soda'18 and Regular Papers

January 2020

369 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3372373

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2019 ACM.

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Association for Computing Machinery

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Publication History

Published: 05 December 2019

Accepted: 01 April 2019

Revised: 01 January 2019

Received: 01 March 2018

Published in TALG Volume 16, Issue 1

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Cited By

Fedorov AHashemi DNadiradze GAlistarh DAgrawal KShun J(2023)Provably-Efficient and Internally-Deterministic Parallel Union-FindProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591082(261-271)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591082
Behnezhad SCharikar MMa WTan L(2022)Almost 3-Approximate Correlation Clustering in Constant Rounds2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00074(720-731)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00074
Behnezhad S(2022)Time-Optimal Sublinear Algorithms for Matching and Vertex Cover2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS52979.2021.00089(873-884)Online publication date: Feb-2022
https://doi.org/10.1109/FOCS52979.2021.00089
Harris D(2020)Oblivious Resampling Oracles and Parallel Algorithms for the Lopsided Lovász Local LemmaACM Transactions on Algorithms10.1145/339203517:1(1-32)Online publication date: 31-Dec-2020
https://dl.acm.org/doi/10.1145/3392035

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