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Sample-And-Gather: Fast Ruling Set Algorithms in the Low-Memory MPC Model

Authors Kishore Kothapalli, Shreyas Pai , Sriram V. Pemmaraju

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Author Details

Kishore Kothapalli
  • IIIT Hyderabad, India
Shreyas Pai
  • The University of Iowa, Iowa City, IA, USA
Sriram V. Pemmaraju
  • The University of Iowa, Iowa City, IA, USA

Funding

This work is funded in part by the Department of Science and Technology, Government of India, via Grant number MTR/2017/000849.

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Kishore Kothapalli, Shreyas Pai, and Sriram V. Pemmaraju. Sample-And-Gather: Fast Ruling Set Algorithms in the Low-Memory MPC Model. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.FSTTCS.2020.28

Abstract

Motivated by recent progress on symmetry breaking problems such as maximal independent set (MIS) and maximal matching in the low-memory Massively Parallel Computation (MPC) model (e.g., Behnezhad et al. PODC 2019; Ghaffari-Uitto SODA 2019), we investigate the complexity of ruling set problems in this model. The MPC model has become very popular as a model for large-scale distributed computing and it comes with the constraint that the memory-per-machine is strongly sublinear in the input size. For graph problems, extremely fast MPC algorithms have been designed assuming Ω̃(n) memory-per-machine, where n is the number of nodes in the graph (e.g., the O(log log n) MIS algorithm of Ghaffari et al., PODC 2018). However, it has proven much more difficult to design fast MPC algorithms for graph problems in the low-memory MPC model, where the memory-per-machine is restricted to being strongly sublinear in the number of nodes, i.e., O(n^ε) for constant 0 < ε < 1. In this paper, we present an algorithm for the 2-ruling set problem, running in Õ(log^{1/6} Δ) rounds whp, in the low-memory MPC model. Here Δ is the maximum degree of the graph. We then extend this result to β-ruling sets for any integer β > 1. Specifically, we show that a β-ruling set can be computed in the low-memory MPC model with O(n^ε) memory-per-machine in Õ(β ⋅ log^{1/(2^{β+1}-2)} Δ) rounds, whp. From this it immediately follows that a β-ruling set for β = Ω(log log log Δ)-ruling set can be computed in in just O(β log log n) rounds whp. The above results assume a total memory of Õ(m + n^{1+ε}). We also present algorithms for β-ruling sets in the low-memory MPC model assuming that the total memory over all machines is restricted to Õ(m). For β > 1, these algorithms are all substantially faster than the Ghaffari-Uitto Õ(√{log Δ})-round MIS algorithm in the low-memory MPC model. All our results follow from a Sample-and-Gather Simulation Theorem that shows how random-sampling-based Congest algorithms can be efficiently simulated in the low-memory MPC model. We expect this simulation theorem to be of independent interest beyond the ruling set algorithms derived here.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Massively Parallel Computation
  • Ruling Set
  • Simulation Theorems

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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, December 1986. URL: https://doi.org/10.1016/0196-6774(86)90019-2.
  2. Sepehr Assadi. Simple round compression for parallel vertex cover. CoRR, abs/1709.04599, 2017. URL: http://arxiv.org/abs/1709.04599.
  3. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Sublinear algorithms for (Δ + 1) vertex coloring. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, pages 767-786, 2019. URL: https://doi.org/10.1137/1.9781611975482.48.
  4. Sepehr Assadi, Xiaorui Sun, and Omri Weinstein. Massively parallel algorithms for finding well-connected components in sparse graphs. Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, pages 461-470, 2019. URL: https://doi.org/10.1145/3293611.3331596.
  5. Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed symmetry breaking. J. ACM, 63(3):20:1-20:45, 2016. URL: https://doi.org/10.1145/2903137.
  6. 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 2019 ACM Symposium on Principles of Distributed Computing, PODC ’19, page 481–490, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3293611.3331609.
  7. Soheil Behnezhad, Mohammadtaghi Hajiaghayi, and David G. Harris. Exponentially Faster Massively Parallel Maximal Matching. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, volume 2019-November, pages 1637-1649, 2019. URL: https://doi.org/10.1109/FOCS.2019.00096.
  8. Tushar Bisht, Kishore Kothapalli, and Sriram V. Pemmaraju. Brief announcement: Super-fast t-ruling sets. In Magnús M. Halldórsson and Shlomi Dolev, editors, ACM Symposium on Principles of Distributed Computing, PODC '14, Paris, France, July 15-18, 2014, pages 379-381. ACM, 2014. URL: https://doi.org/10.1145/2611462.2611512.
  9. Yi-Jun Chang, Manuela Fischer, Mohsen Ghaffari, Jara Uitto, and Yufan Zheng. The complexity of (δ+1) coloring in congested clique, massively parallel computation, and centralized local computation. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC ’19, page 471–480, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3293611.3331607.
  10. Moses Charikar, Weiyun Ma, and Li-Yang Tan. Unconditional lower bounds for adaptive massively parallel computation. In Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’20, page 141–151, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3350755.3400230.
  11. Avery Ching, Sergey Edunov, Maja Kabiljo, Dionysios Logothetis, and Sambavi Muthukrishnan. One trillion edges: Graph processing at facebook-scale. Proc. VLDB Endow., 8(12):1804–1815, August 2015. URL: https://doi.org/10.14778/2824032.2824077.
  12. Artur Czumaj, Peter Davies, and Merav Parter. Graph Sparsification for Derandomizing Massively Parallel Computation with Low Space, 2019. URL: http://arxiv.org/abs/1912.05390.
  13. Artur Czumaj, Slobodan Mitrovic, Jakub Łacki, Krzysztof Onak, Aleksander Madry, and Piotr Sankowski. Round compression for parallel matching algorithms. Proceedings of the Annual ACM Symposium on Theory of Computing, pages 471-484, 2018. URL: https://doi.org/10.1145/3188745.3188764.
  14. Jeffrey Dean and Sanjay Ghemawat. Mapreduce: Simplified data processing on large clusters. Commun. ACM, 51(1):107–113, January 2008. URL: https://doi.org/10.1145/1327452.1327492.
  15. Mohsen Ghaffari. An improved distributed algorithm for maximal independent set. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’16, page 270–277, USA, 2016. Society for Industrial and Applied Mathematics. Google Scholar
  16. Mohsen Ghaffari. Distributed MIS via all-to-all communication. Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, Part F129314:141-150, 2017. URL: https://doi.org/10.1145/3087801.3087830.
  17. Mohsen Ghaffari, Themis Gouleakis, Christian Konrad, Slobodan Mitrović, and Ronitt Rubinfeld. Improved massively parallel computation algorithms for MIS, matching, and vertex cover. Proceedings of the Annual ACM Symposium on Principles of Distributed Computing, pages 129-138, 2018. URL: https://doi.org/10.1145/3212734.3212743.
  18. Mohsen Ghaffari, Christoph Grunau, and Ce Jin. Improved MPC Algorithms for MIS, Matching, and Coloring on Trees and Beyond. CoRR, 2020. URL: http://arxiv.org/abs/2002.09610.
  19. Mohsen Ghaffari, Ce Jin, and Daan Nilis. A massively parallel algorithm for minimum weight vertex cover. In Christian Scheideler and Michael Spear, editors, SPAA '20: 32nd ACM Symposium on Parallelism in Algorithms and Architectures, Virtual Event, USA, July 15-17, 2020, pages 259-268. ACM, 2020. URL: https://doi.org/10.1145/3350755.3400260.
  20. Mohsen Ghaffari, Fabian Kuhn, and Jara Uitto. Conditional hardness results for massively parallel computation from distributed lower bounds. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, volume 2019-November, pages 1650-1663, 2019. URL: https://doi.org/10.1109/FOCS.2019.00097.
  21. Mohsen Ghaffari and Jara Uitto. Sparsifying distributed algorithms with ramifications in massively parallel computation and centralized local computation. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1636-1653, July 2019. URL: https://doi.org/10.1137/1.9781611975482.99.
  22. Nicholas J. A. Harvey, Christopher Liaw, and Paul Liu. Greedy and local ratio algorithms in the mapreduce model. In Proceedings of the 30th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA ’18, page 43–52, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3210377.3210386.
  23. James W. Hegeman, Sriram V. Pemmaraju, and Vivek Sardeshmukh. Near-constant-time distributed algorithms on a congested clique. CoRR, abs/1408.2071, 2014. URL: http://arxiv.org/abs/1408.2071.
  24. Howard Karloff, Siddharth Suri, and Sergei Vassilvitskii. A Model of Computation for MapReduce. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 938-948, Philadelphia, PA, January 2010. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.1137/1.9781611973075.76.
  25. Kishore Kothapalli, Shreyas Pai, and Sriram V. Pemmaraju. Sample-and-gather: Fast ruling set algorithms in the low-memory MPC model, 2020. URL: http://arxiv.org/abs/2009.12477.
  26. Kishore Kothapalli and Sriram V. Pemmaraju. Super-fast 3-ruling sets. In Deepak D'Souza, Telikepalli Kavitha, and Jaikumar Radhakrishnan, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, December 15-17, 2012, Hyderabad, India, volume 18 of LIPIcs, pages 136-147. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2012.136.
  27. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. What cannot be computed locally! In Proceedings of the Twenty-Third Annual ACM Symposium on Principles of Distributed Computing, PODC ’04, page 300–309, New York, NY, USA, 2004. Association for Computing Machinery. URL: https://doi.org/10.1145/1011767.1011811.
  28. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. Local computation: Lower and upper bounds. J. ACM, 63(2), March 2016. URL: https://doi.org/10.1145/2742012.
  29. Nathan Linial. Locality in distributed graph algorithms. SIAM J. Comput., 21(1):193–201, February 1992. URL: https://doi.org/10.1137/0221015.
  30. Zvi Lotker, Elan Pavlov, Boaz Patt-Shamir, and David Peleg. MST construction in o(log log n) communication rounds. In SPAA, pages 94-100, 2003. URL: https://doi.org/10.1145/777412.777428.
  31. 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.
  32. Grzegorz Malewicz, Matthew H. Austern, Aart J.C Bik, James C. Dehnert, Ilan Horn, Naty Leiser, and Grzegorz Czajkowski. Pregel: A system for large-scale graph processing. In Proceedings of the 2010 ACM SIGMOD International Conference on Management of Data, SIGMOD ’10, page 135–146, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1807167.1807184.
  33. Merav Parter and Eylon Yogev. Congested clique algorithms for graph spanners. In Ulrich Schmid and Josef Widder, editors, 32nd International Symposium on Distributed Computing, DISC 2018, New Orleans, LA, USA, October 15-19, 2018, volume 121 of LIPIcs, pages 40:1-40:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.DISC.2018.40.
  34. D. Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, 2000. Google Scholar
  35. Tim Roughgarden, Sergei Vassilvitskii, and Joshua R. Wang. Shuffles and circuits (on lower bounds for modern parallel computation). J. ACM, 65(6), November 2018. URL: https://doi.org/10.1145/3232536.
  36. Václav Rozhon and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 350-363, 2020. URL: https://doi.org/10.1145/3357713.3384298.
  37. Grigory Yaroslavtsev and Adithya Vadapalli. Massively parallel algorithms and hardness for single-linkage clustering under 𝓁_p distances. In Jennifer G. Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018, volume 80 of Proceedings of Machine Learning Research, pages 5596-5605. PMLR, 2018. URL: http://proceedings.mlr.press/v80/yaroslavtsev18a.html.
  38. Matei Zaharia, Reynold S. Xin, Patrick Wendell, Tathagata Das, Michael Armbrust, Ankur Dave, Xiangrui Meng, Josh Rosen, Shivaram Venkataraman, Michael J. Franklin, Ali Ghodsi, Joseph Gonzalez, Scott Shenker, and Ion Stoica. Apache spark: A unified engine for big data processing. Commun. ACM, 59(11):56–65, October 2016. URL: https://doi.org/10.1145/2934664.
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