Document Open Access Logo

Improved MPC Algorithms for MIS, Matching, and Coloring on Trees and Beyond

Authors Mohsen Ghaffari, Christoph Grunau, Ce Jin

PDF
Thumbnail PDF

File

LIPIcs.DISC.2020.34.pdf
  • Filesize: 0.57 MB
  • 18 pages

Document Identifiers

Author Details

Mohsen Ghaffari
  • ETH Zürich, Switzerland
Christoph Grunau
  • ETH Zürich, Switzerland
Ce Jin
  • Tsinghua University, Beijing, China

Funding

This work was supported in part by funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 853109), and the Swiss National Foundation (project grant 200021-184735).

Cite AsGet BibTex

Mohsen Ghaffari, Christoph Grunau, and Ce Jin. Improved MPC Algorithms for MIS, Matching, and Coloring on Trees and Beyond. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 34:1-34:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.DISC.2020.34

Abstract

We present O(log log n) round scalable Massively Parallel Computation algorithms for maximal independent set and maximal matching, in trees and more generally graphs of bounded arboricity, as well as for coloring trees with a constant number of colors. Following the standards, by a scalable MPC algorithm, we mean that these algorithms can work on machines that have capacity/memory as small as n^{δ} for any positive constant δ < 1. Our results improve over the O(log²log n) round algorithms of Behnezhad et al. [PODC'19]. Moreover, our matching algorithm is presumably optimal as its bound matches an Ω(log log n) conditional lower bound of Ghaffari, Kuhn, and Uitto [FOCS'19].

Subject Classification

ACM Subject Classification
  • Theory of computation → Massively parallel algorithms
Keywords
  • Massively Parallel Computation
  • MIS
  • Matching
  • Coloring

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Noga Alon, László Babai, and Alon Itai. A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem. J. Algorithms, 7(4):567-583, 1986. URL: https://doi.org/10.1016/0196-6774(86)90019-2.
  2. Alexandr Andoni, Zhao Song, Clifford Stein, Zhengyu Wang, and Peilin Zhong. Parallel graph connectivity in log diameter rounds. In Proceedings of the 59th IEEE Symposium on Foundations of Computer Science (FOCS), pages 674-685, 2018. URL: https://doi.org/10.1109/FOCS.2018.00070.
  3. Sepehr Assadi, MohammadHossein Bateni, Aaron Bernstein, Vahab Mirrokni, and Cliff Stein. Coresets meet EDCS: Algorithms for matching and vertex cover on massive graphs. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1616-1635, 2019. URL: https://doi.org/10.1137/1.9781611975482.98.
  4. Leonid Barenboim and Michael Elkin. Sublogarithmic Distributed MIS Algorithm for Sparse Graphs Using Nash-Williams Decomposition. Distributed Computing, 22(5-6):363-379, 2010. URL: https://doi.org/10.1007/s00446-009-0088-2.
  5. Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed symmetry breaking. Journal of the ACM, 63(3):20:1-20:45, 2016. URL: https://doi.org/10.1145/2903137.
  6. MohammadHossein Bateni, Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, and Vahab S. Mirrokni. Brief announcement: Mapreduce algorithms for massive trees. In Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP), pages 162:1-162:4, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.162.
  7. MohammadHossein Bateni, Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, and Vahab S. Mirrokni. Massively parallel dynamic programming on trees. arXiv preprint, 2018. URL: http://arxiv.org/abs/1809.03685.
  8. Paul Beame, Paraschos Koutris, and Dan Suciu. Skew in Parallel Query Processing. In Proceedings of the 33rd ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems (PODS), pages 212-223, 2014. URL: https://doi.org/10.1145/2594538.2594558.
  9. Soheil Behnezhad, Sebastian Brandt, Mahsa Derakhshan, Manuela Fischer, MohammadTaghi Hajiaghayi, Richard M. Karp, and Jara Uitto. Massively parallel computation of matching and MIS in sparse graphs. In Proceedings of the 38th ACM Symposium on Principles of Distributed Computing (PODC), pages 481-490, 2019. URL: https://doi.org/10.1145/3293611.3331609.
  10. Soheil Behnezhad, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, and Richard M. Karp. Massively Parallel Symmetry Breaking on Sparse Graphs: MIS and Maximal Matching. arXiv preprint, 2018. URL: http://arxiv.org/abs/1807.06701.
  11. Soheil Behnezhad, Laxman Dhulipala, Hossein Esfandiari, Jakub Łącki, and Vahab Mirrokni. Near-optimal massively parallel graph connectivity. In Proceedings of the 60th IEEE Symposium on Foundations of Computer Science (FOCS), pages 1615-1636, 2019. URL: https://doi.org/10.1109/FOCS.2019.00095.
  12. Soheil Behnezhad, MohammadTaghi Hajiaghayi, and David G. Harris. Exponentially faster massively parallel maximal matching. In Proceedings of the 60th IEEE Symposium on Foundations of Computer Science (FOCS), pages 1637-1649, 2019. URL: https://doi.org/10.1109/FOCS.2019.00096.
  13. Sebastian Brandt, Manuela Fischer, and Jara Uitto. Matching and MIS for uniformly sparse graphs in the low-memory MPC model. arXiv preprint, 2018. URL: http://arxiv.org/abs/1807.05374.
  14. Sebastian Brandt, Manuela Fischer, and Jara Uitto. Breaking the linear-memory barrier in MPC: Fast MIS on trees with strongly sublinear memory. In Proceedings of the 26th International Colloquium on Structural Information and Communication Complexity (SIROCCO), pages 124-138, 2019. URL: https://doi.org/10.1007/978-3-030-24922-9_9.
  15. Richard Cole and Uzi Vishkin. Deterministic coin tossing and accelerating cascades: Micro and macro techniques for designing parallel algorithms. In Proceedings of the 18th ACM Symposium on Theory of Computing, pages 206-219, 1986. Google Scholar
  16. Artur Czumaj, Peter Davies, and Merav Parter. Graph sparsification for derandomizing massively parallel computation with low space. In Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 175-185, 2020. URL: https://doi.org/10.1145/3350755.3400282.
  17. Artur Czumaj, Jakub Łącki, Aleksander Mądry, Slobodan Mitrović, Krzysztof Onak, and Piotr Sankowski. Round compression for parallel matching algorithms. SIAM Journal on Computing, pages STOC18-1-STOC18-44, 2019. URL: https://doi.org/10.1137/18M1197655.
  18. Jeffrey Dean and Sanjay Ghemawat. MapReduce: Simplified Data Processing on Large Clusters. Communications of the ACM, 51(1):107-113, 2008. URL: https://doi.org/10.1145/1327452.1327492.
  19. Mohsen Ghaffari. Distributed MIS via all-to-all communication. In Proceedings of the 36th ACM Symposium on Principles of Distributed Computing (PODC), pages 141-149, 2017. URL: https://doi.org/10.1145/3087801.3087830.
  20. Mohsen Ghaffari, Themis Gouleakis, Christian Konrad, Slobodan Mitrović, and Ronitt Rubinfeld. Improved massively parallel computation algorithms for MIS, matching, and vertex cover. In Proceedings of the 37th ACM Symposium on Principles of Distributed Computing (PODC), pages 129-138, 2018. URL: https://doi.org/10.1145/3212734.3212743.
  21. Mohsen Ghaffari, Fabian Kuhn, and Jara Uitto. Conditional hardness results for massively parallel computation from distributed lower bounds. In Proceedings of the 60th IEEE Symposium on Foundations of Computer Science (FOCS), pages 1650-1663, 2019. URL: https://doi.org/10.1109/FOCS.2019.00097.
  22. Mohsen Ghaffari and Jara Uitto. Sparsifying distributed algorithms with ramifications in massively parallel computation and centralized local computation. In Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1636-1653, 2019. URL: https://doi.org/10.1137/1.9781611975482.99.
  23. Michael T. Goodrich, Nodari Sitchinava, and Qin Zhang. Sorting, Searching, and Simulation in the MapReduce Framework. In Proceedings of the 22nd International Symposium on Algorithms and Computation, pages 374-383, 2011. URL: https://doi.org/10.1007/978-3-642-25591-5_39.
  24. Michael Isard, Mihai Budiu, Yuan Yu, Andrew Birrell, and Dennis Fetterly. Dryad: Distributed Data-Parallel Programs from Sequential Building Blocks. In Proceedings of the 2nd ACM SIGOPS/EuroSys European Conference on Computer Systems, pages 59-72, 2007. URL: https://doi.org/10.1145/1272996.1273005.
  25. Howard Karloff, Siddharth Suri, and Sergei Vassilvitskii. A Model of Computation for MapReduce. In Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 938-948, 2010. URL: https://doi.org/10.1137/1.9781611973075.76.
  26. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. Local computation: Lower and upper bounds. Journal of the ACM, 63(2):17:1-17:44, 2016. URL: https://doi.org/10.1145/2742012.
  27. Silvio Lattanzi, Benjamin Moseley, Siddharth Suri, and Sergei Vassilvitskii. Filtering: A method for solving graph problems in MapReduce. In Proceedings of the 23rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 85-94, 2011. URL: https://doi.org/10.1145/1989493.1989505.
  28. Christoph Lenzen and Roger Wattenhofer. Brief Announcement: Exponential Speed-Up of Local Algorithms Using Non-Local Communication. In Proceedings of the 29th ACM Symposium on Principles of Distributed Computing (PODC), pages 295-296, 2010. URL: https://doi.org/10.1145/1835698.1835772.
  29. Nathan Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193-201, 1992. URL: https://doi.org/10.1137/0221015.
  30. Michael Luby. A Simple Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing, 15(4):1036-1053, 1986. URL: https://doi.org/10.1137/0215074.
  31. Crispin St. John Alvah Nash-Williams. Decomposition of finite graphs into forests. J. London Math. Soc., 39(1):12, 1964. URL: https://doi.org/10.1112/jlms/s1-39.1.12.
  32. Tim Roughgarden, Sergei Vassilvitskii, and Joshua R. Wang. Shuffles and circuits (on lower bounds for modern parallel computation). Journal of the ACM, 65(6):41:1-41:24, 2018. URL: https://doi.org/10.1145/3232536.
  33. Tom White. Hadoop: The Definitive Guide. O'Reilly, 2012. Google Scholar
  34. Matei Zaharia, Mosharaf Chowdhury, Michael J. Franklin, Scott Shenker, and Ion Stoica. Spark: Cluster Computing with Working Sets. In Proceedings of the 2nd USENIX Conference on Hot Topics in Cloud Computing, 2010. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact