research-article

Tight analysis of parallel randomized greedy MIS

Authors:

Manuela Fischer,

Andreas NoeverAuthors Info & Claims

SODA '18: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

Pages 2152 - 2160

Published: 07 January 2018 Publication History

Abstract

We provide a tight analysis which 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].

References

[1]

{ABI86} Noga Alon, László Babai, and Alon Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of algorithms, 7(4):567--583, 1986.

Digital Library

[2]

{BAAS09} Robert Bocchino, Vikram Adve, Sarita Adve, and Marc Snir. Parallel programming must be deterministic by default. In Proceedings of the First USENIX conference on Hot topics in parallelism, pages 4--4, 2009.

Digital Library

[3]

{BEPS12} Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed symmetry breaking. In Foundations of Computer Science (FOCS) 2012, pages 321--330. IEEE, 2012.

Digital Library

[4]

{BEPS16} Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed symmetry breaking. J. ACM, 63(3):20:1--20:45, 2016.

Digital Library

[5]

{BFGS12} Guy E. Blelloch, Jeremy T. Fineman, Phillip B. Gibbons, and Julian Shun. Internally deterministic parallel algorithms can be fast. In ACM SIGPLAN Notices, volume 47, pages 181--192. ACM, 2012.

Digital Library

[6]

{BFS12} Guy E. Blelloch, Jeremy T. Fineman, and Julian Shun. Greedy sequential maximal independent set and matching are parallel on average. In Proceedings of the twenty-fourth annual ACM symposium on Parallelism in algorithms and architectures, pages 308--317. ACM, 2012.

Digital Library

[7]

{CF90} Neil J. Calkin and Alan M. Frieze. Probabilistic analysis of a parallel algorithm for finding maximal independent sets. Random Struct. Algorithms, 1(1):39--B50, 1990.

[8]

{Coo83} Stephen A. Cook. An overview of computational complexity. Communications of the ACM, 26(6):400--408, 1983.

Digital Library

[9]

{CRT87} Don Coppersmith, Prabhakar Raghavan, and Martin Tompa. Parallel graph algorithms that are efficient on average. In 28th Annual Symposium on Foundations of Computer Science, Los Angeles, California, USA, 27--29 October 1987, pages 260--269, 1987.

Digital Library

[10]

{Gha16} Mohsen Ghaffari. An improved distributed algorithm for maximal independent set. In Pro. of ACM-SIAM Symp. on Disc. Alg. (SODA), 2016.

Digital Library

[11]

{Gol86} Mark K. Goldberg. Parallel algorithms for three graph problems. Congressus Numerantium, 54(111--121):4--1, 1986.

[12]

{GPS87} Andrew Goldberg, Serge Plotkin, and Gregory Shannon. Parallel symmetry-breaking in sparse graphs. In Proceedings of the nineteenth annual ACM symposium on Theory of computing, pages 315--324. ACM, 1987.

Digital Library

[13]

{GS89a} Mark K. Goldberg and Thomas Spencer. Constructing a maximal independent set in parallel. SIAM Journal on Discrete Mathematics, 2(3):322--328, 1989.

Digital Library

[14]

{GS89b} Mark K. Goldberg and Thomas Spencer. A new parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 18(2):419--427, 1989.

Digital Library

[15]

{KW85} Richard M. Karp and Avi Wigderson. A fast parallel algorithm for the maximal independent set problem. Journal of the ACM (JACM), 32(4):762--773, 1985.

Digital Library

[16]

{Lin87} Nathan Linial. Distributive graph algorithms global solutions from local data. In Proc. of the Symp. on Found. of Comp. Sci. (FOCS), pages 331--335. IEEE, 1987.

Digital Library

[17]

{Lin92} Nathan Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193--201, 1992.

Digital Library

[18]

{Lub85} Michael Luby. A simple parallel algorithm for the maximal independent set problem. In Proc. of the Symp. on Theory of Comp. (STOC), pages 1--10, 1985.

Digital Library

[19]

{MRSDZ10} Yves Métivier, John Michael Robson, Nasser Saheb-Djahromi, and Akka Zemmari. An optimal bit complexity randomized distributed mis algorithm. In Structural Information and Communication Complexity, pages 323--337. Springer, 2010.

Digital Library

[20]

{Val82} Leslie G. Valiant. Parallel computation. In Proc. 7th IBM Symp. on Math. Found. of Comp. Sci., 1982.

Cited By

Ghaffari MPortmann JAgrawal KLee I(2022)Average Awake Complexity of MIS and MatchingProceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3490148.3538566(45-55)Online publication date: 11-Jul-2022
https://dl.acm.org/doi/10.1145/3490148.3538566
Behnezhad SDhulipala LEsfandiari HLacki JMirrokni VSchudy W(2020)Parallel graph algorithms in constant adaptive roundsProceedings of the VLDB Endowment10.14778/3424573.342457913:13(3588-3602)Online publication date: 1-Sep-2020
https://dl.acm.org/doi/10.14778/3424573.3424579
Harris DChan T(2019)Oblivious resampling oracles and parallel algorithms for the lopsided lovász local lemmaProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310487(841-860)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310487
Show More Cited By

Tight analysis of parallel randomized greedy MIS

Recommendations

Tight Analysis of Parallel Randomized Greedy MIS
Special Issue on Soda'18 and Regular Papers

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 ...
Constant-Time Randomized Parallel String Matching

Given a pattern string of length m for the string-matching problem, we design an algorithm that computes deterministic samples of a sufficiently long substring of the pattern in constant time. This problem used to be the bottleneck in the pattern ...
Tight bounds for parallel randomized load balancing

Given a distributed system of $$n$$n balls and $$n$$n bins, how evenly can we distribute the balls to the bins, minimizing communication__ __ The fastest non-adaptive and symmetric algorithm achieving a constant maximum bin load requires $$\varTheta (\...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

SODA '18: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

January 2018

2859 pages

ISBN:9781611975031

Program Chair:
Artur Czumaj
University of Warwick, United Kingdom

Sponsors

SIAM Activity Group on Discrete Mathematics
SIGACT: ACM Special Interest Group on Algorithms and Computation Theory

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 07 January 2018

Check for updates

Qualifiers

Research-article

Conference

SODA '18

Sponsor:

SIGACT

SODA '18: Symposium on Discrete Algorithms

January 7 - 10, 2018

Louisiana, New Orleans

Acceptance Rates

Overall Acceptance Rate 411 of 1,322 submissions, 31%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

8
Total Citations
View Citations
62
Total Downloads

Downloads (Last 12 months)6
Downloads (Last 6 weeks)2

Reflects downloads up to 17 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Ghaffari MPortmann JAgrawal KLee I(2022)Average Awake Complexity of MIS and MatchingProceedings of the 34th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3490148.3538566(45-55)Online publication date: 11-Jul-2022
https://dl.acm.org/doi/10.1145/3490148.3538566
Behnezhad SDhulipala LEsfandiari HLacki JMirrokni VSchudy W(2020)Parallel graph algorithms in constant adaptive roundsProceedings of the VLDB Endowment10.14778/3424573.342457913:13(3588-3602)Online publication date: 1-Sep-2020
https://dl.acm.org/doi/10.14778/3424573.3424579
Harris DChan T(2019)Oblivious resampling oracles and parallel algorithms for the lopsided lovász local lemmaProceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3310435.3310487(841-860)Online publication date: 6-Jan-2019
https://dl.acm.org/doi/10.5555/3310435.3310487
Harris D(2019)Derandomized Concentration Bounds for Polynomials, and Hypergraph Maximal Independent SetACM Transactions on Algorithms10.1145/332617115:3(1-29)Online publication date: 16-Jul-2019
https://dl.acm.org/doi/10.1145/3326171
Alistarh DNadiradze GKoval NScheideler CBerenbrink P(2019)Efficiency Guarantees for Parallel Incremental Algorithms under Relaxed SchedulersThe 31st ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3323165.3323201(145-154)Online publication date: 17-Jun-2019
https://dl.acm.org/doi/10.1145/3323165.3323201
Alistarh DBrown TKopinsky JNadiradze GNewport CKeidar I(2018)Relaxed Schedulers Can Efficiently Parallelize Iterative AlgorithmsProceedings of the 2018 ACM Symposium on Principles of Distributed Computing10.1145/3212734.3212756(377-386)Online publication date: 23-Jul-2018
https://dl.acm.org/doi/10.1145/3212734.3212756
Ghaffari MGouleakis TKonrad CMitrović SRubinfeld RNewport CKeidar I(2018)Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex CoverProceedings of the 2018 ACM Symposium on Principles of Distributed Computing10.1145/3212734.3212743(129-138)Online publication date: 23-Jul-2018
https://dl.acm.org/doi/10.1145/3212734.3212743
Dhulipala LBlelloch GShun JScheideler CFineman J(2018)Theoretically Efficient Parallel Graph Algorithms Can Be Fast and ScalableProceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures10.1145/3210377.3210414(393-404)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3210377.3210414

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents