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Near-Optimal Distributed 2-Ruling Sets on Graphs with Low Arboricity

Authors Malte Baumecker , Rustam Latypov , Yannic Maus , Jara Uitto

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ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Probabilistic algorithms
  • Theory of computation → Massively parallel algorithms
Keywords
  • 2-ruling set
  • LOCAL model
  • CONGEST model
  • MPC model

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Abstract

Given a graph G = (V,E), a β-ruling set is a subset of nodes S ⊆ V that is independent, and each node in V is at distance at most β from some node in S. In this paper, we present almost optimal distributed algorithms for finding 2-ruling sets in the classical LOCAL model. Our main contribution is a randomized algorithm that w.h.p. computes a 2-ruling set on any n-node graph with bounded arboricity in O(log log n) rounds. In fact, the algorithm works up to arboricity O(log log n), improves exponentially over the prior state of the art that can be achieved by combining [Barenboim, Elkin, Pettie, Schneider; JACM'16], [Ghaffari; SODA'16], and [Bisht, Kothapalli and Pemmaraju; PODC'14], and nearly matches the lower bound of Ω(log log n / log log log n) [Balliu, Brandt, Kuhn, Olivetti; FOCS'20]. The domination parameter β = 2 is optimal for algorithms with runtime log^{o(1)}n: on graphs with arboricity 2, there is a lower bound of Ω(√{log n}) rounds for MIS (i.e., β = 1) [Khoury, Schild; FOCS'25].
Additionally, we obtain improved algorithms for larger arboricity. For general graphs with arboricity α, we present a randomized algorithm that computes a 2-ruling set in Õ(log^{5/8} α +log^{5/3} log n) rounds. This improves exponentially over the state of the art for a large range of non-constant arboricity.
Our techniques extend beyond distributed computing. We present an O(log log log n)-round algorithm in the low-space Massively Parallel Computation (MPC) model that w.h.p. computes a 2-ruling set on any graph with arboricity up to 2^poly(log log n), improving exponentially over the state of the art from [Kothapalli, Pai, Pemmaraju; FSTTCS'20] combined with [Fischer, Giliberti, Grunau; SPAA'23].

Cite As Get BibTex

Malte Baumecker, Rustam Latypov, Yannic Maus, and Jara Uitto. Near-Optimal Distributed 2-Ruling Sets on Graphs with Low Arboricity. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 15:1-15:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.15

Author Details

Malte Baumecker
  • Graz University of Technology, Austria
Rustam Latypov
  • Aalto University, Finland
Yannic Maus
  • Graz University of Technology, Austria
Jara Uitto
  • Aalto University, Finland

Funding

  • Baumecker, Malte: Supported by the Austrian Science Fund (FWF) https://doi.org/10.55776/I6915.
  • Latypov, Rustam: Supported by the Research Council of Finland (RCF), Grant 371062.
  • Maus, Yannic: Supported by the Austrian Science Fund (FWF) https://doi.org/10.55776/I6915.

Acknowledgements

We thank Lasse Leskelä for helpful discussions on probabilistic arguments for handling dependencies in an earlier version of this work.

References

  1. Noga Alon, Lásló 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. URL: https://doi.org/10.1016/0196-6774(86)90019-2.
  2. Sepehr Assadi and Aditi Dudeja. Ruling Sets in Random Order and Adversarial Streams. In the International Symposium on Distributed Computing (DISC), pages 6:1-6:18, 2021. URL: https://doi.org/10.4230/LIPIcs.DISC.2021.6.
  3. Baruch Awerbuch, Andrew V. Goldberg, Michael Luby, and Serge A. Plotkin. Network Decomposition and Locality in Distributed Computation. In the Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 364-369, 1989. URL: https://doi.org/10.1109/SFCS.1989.63504.
  4. Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti, Mikaël Rabie, and Jukka Suomela. Lower Bounds for Maximal Matchings and Maximal Independent Sets. In the Proceedings of the Symposium on Foundations of Computer Science (FOCS), 2019. Google Scholar
  5. Alkida Balliu, Sebastian Brandt, Fabian Kuhn, and Dennis Olivetti. Distributed Δ-coloring plays hide-and-seek. In Proc. 54th ACM Symp. on Theory of Computing (STOC), 2022. Google Scholar
  6. Alkida Balliu, Sebastian Brandt, and Dennis Olivetti. Distributed Lower Bounds for Ruling Sets. In the Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 365-376, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00042.
  7. Alkida Balliu, Sebastian Brandt, and Dennis Olivetti. Distributed lower bounds for ruling sets. SIAM J. Comput., 51(1):70-115, 2022. URL: https://doi.org/10.1137/20M1381770.
  8. Alkida Balliu, Mohsen Ghaffari, Fabian Kuhn, and Dennis Olivetti. Node and edge averaged complexities of local graph problems. In Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing, PODC'22, pages 4-14, New York, NY, USA, 2022. Association for Computing Machinery. URL: https://doi.org/10.1145/3519270.3538419.
  9. L. Barenboim and M. Elkin. Sublogarithmic distributed mis algorithm for sparse graphs using nash-williams decomposition. Distributed Computing, 22(5):363-379, 2010. URL: https://doi.org/10.1007/s00446-009-0088-2.
  10. Leonid Barenboim, Michael Elkin, and Fabian Kuhn. Distributed (δ+1)-coloring in linear (in δ) time. SIAM Journal on Computing, 43(1):72-95, 2014. URL: https://doi.org/10.1137/12088848X.
  11. 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.
  12. Malte Baumecker, Rustam Latypov, Yannic Maus, and Jara Uitto. Near-optimal distributed 2-ruling sets on graphs with low arboricity, 2026. URL: https://doi.org/10.48550/arXiv.2606.11974.
  13. Malte Baumecker, Yannic Maus, and Jara Uitto. Nearly-optimal distributed ruling sets for trees and high-girth graphs. In Proceedings of the ACM Symposium on Principles of Distributed Computing, PODC '25, pages 88-98, New York, NY, USA, 2025. Association for Computing Machinery. URL: https://doi.org/10.1145/3732772.3733547.
  14. Paul Beame, Paraschos Koutris, and Dan Suciu. Communication Steps for Parallel Query Processing. the Journal of the ACM, 64(6), 2017. URL: https://doi.org/10.1145/3125644.
  15. Tushar Bisht, Kishore Kothapalli, and Sriram Pemmaraju. Brief announcement: Super-fast t-ruling sets. In the Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 379-381, 2014. URL: https://doi.org/10.1145/2611462.2611512.
  16. Mélanie Cambus, Davin Choo, Havu Miikonen, and Jara Uitto. Massively Parallel Correlation Clustering in Bounded Arboricity Graphs. In Seth Gilbert, editor, 35th International Symposium on Distributed Computing (DISC 2021), volume 209 of Leibniz International Proceedings in Informatics (LIPIcs), pages 15:1-15:18, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2021.15.
  17. Mélanie Cambus, Fabian Kuhn, Shreyas Pai, and Jara Uitto. Time and Space Optimal Massively Parallel Algorithm for the 2-Ruling Set Problem. In Rotem Oshman, editor, 37th International Symposium on Distributed Computing, (DISC 2023), volume 281 of LIPIcs, pages 11:1-11:12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.DISC.2023.11.
  18. Chandra Chekuri, Aleksander Bjørn Christiansen, Jacob Holm, Ivor van der Hoog, Kent Quanrud, Eva Rotenberg, and Chris Schwiegelshohn. Adaptive Out-Orientations with Applications, pages 3062-3088. Society for Industrial and Applied Mathematics, 2024. URL: https://doi.org/10.1137/1.9781611977912.110.
  19. Aleksander Bjørn Grodt Christiansen, Krzysztof Nowicki, and Eva Rotenberg. Improved dynamic colouring of sparse graphs. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 1201-1214, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585111.
  20. Corinna Coupette and Christoph Lenzen. A Breezing Proof of the KMW Bound, pages 184-195. Society for Industrial and Applied Mathematics, 2021. URL: https://doi.org/10.1137/1.9781611976496.21.
  21. 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 '20, pages 175-185, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3350755.3400282.
  22. Jeffrey Dean and Sanjay Ghemawat. MapReduce: Simplified Data Processing on Large Clusters. Communications of the ACM, pages 107-113, 2008. URL: https://doi.org/10.1145/1327452.1327492.
  23. Michal Dory, Mohsen Ghaffari, and Saeed Ilchi. Near-optimal distributed dominating set in bounded arboricity graphs. Distributed Computing, 37(4):387-398, May 2023. URL: https://doi.org/10.1007/s00446-023-00447-z.
  24. Talya Eden, Saleet Mossel, and Dana Ron. Approximating the Arboricity in Sublinear Time, pages 2404-2425. Society for Industrial and Applied Mathematics, 2022. URL: https://doi.org/10.1137/1.9781611977073.96.
  25. Talya Eden, Dana Ron, and C. Seshadhri. Faster sublinear approximation of the number of k-cliques in low-arboricity graphs, pages 1467-1478. Society for Industrial and Applied Mathematics, 2020. URL: https://doi.org/10.1137/1.9781611975994.89.
  26. James D. Esary, Frank Proschan, and David W. Walkup. Association of random variables, with applications. The Annals of Mathematical Statistics, 38(5):1466-1474, 1967. URL: https://doi.org/10.1214/aoms/1177698701.
  27. Salwa Faour and Fabian Kuhn. Brief Announcement: Faster CONGEST Approximation Algorithms for Maximum Weighted Independent Set in Sparse Graphs. In Dariusz R. Kowalski, editor, 39th International Symposium on Distributed Computing (DISC 2025), volume 356 of Leibniz International Proceedings in Informatics (LIPIcs), pages 54:1-54:7, Dagstuhl, Germany, 2025. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2025.54.
  28. Manuela Fischer, Jeff Giliberti, and Christoph Grunau. Deterministic massively parallel symmetry breaking for sparse graphs. In Proceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '23, pages 89-100, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3558481.3591081.
  29. Dmitry Gavinsky, Shachar Lovett, Michael Saks, and Srikanth Srinivasan. A tail bound for read-k families of functions. Random Structures & Algorithms, 47(1):99-108, 2015. URL: https://doi.org/10.1002/rsa.20532.
  30. B. Gfeller and E. Vicari. A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs. In the Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 53-60, 2007. Google Scholar
  31. Mohsen Ghaffari. An Improved Distributed Algorithm for Maximal Independent Set. In the Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 270-277, 2016. URL: https://doi.org/10.1137/1.9781611974331.CH20.
  32. Mohsen Ghaffari and Christoph Grunau. Dynamic o(arboricity) coloring in polylogarithmic worst-case time. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, STOC 2024, pages 1184-1191, New York, NY, USA, 2024. Association for Computing Machinery. URL: https://doi.org/10.1145/3618260.3649782.
  33. Mohsen Ghaffari and Christoph Grunau. Near-optimal deterministic network decomposition and ruling set, and improved MIS. In the Proceedings of the Symposium on Foundations of Computer Science (FOCS), pages 2148-2179, 2024. URL: https://doi.org/10.1109/FOCS61266.2024.00007.
  34. Mohsen Ghaffari, Christoph Grunau, Bernhard Haeupler, Saeed Ilchi, and Václav Rozhon. Improved distributed network decomposition, hitting sets, and spanners, via derandomization. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 2532-2566. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH97.
  35. Mohsen Ghaffari, Christoph Grunau, and Ce Jin. Improved MPC Algorithms for MIS, Matching, and Coloring on Trees and Beyond. In Hagit Attiya, editor, 34th International Symposium on Distributed Computing (DISC 2020), volume 179 of Leibniz International Proceedings in Informatics (LIPIcs), pages 34:1-34:18, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2020.34.
  36. Mohsen Ghaffari, Juho Hirvonen, Fabian Kuhn, and Yannic Maus. Improved distributed delta-coloring. In the Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 427-436, 2018. URL: https://dl.acm.org/citation.cfm?id=3212764.
  37. Mohsen Ghaffari and Jara Uitto. Sparsifying Distributed Algorithms with Ramifications in Massively Parallel Computation and Centralized Local Computation, pages 1636-1653. Society for Industrial and Applied Mathematics, 2019. URL: https://doi.org/10.1137/1.9781611975482.99.
  38. Jeff Giliberti and Zahra Parsaeian. Massively Parallel Ruling Set Made Deterministic. In Dan Alistarh, editor, 38th International Symposium on Distributed Computing (DISC 2024), volume 319 of Leibniz International Proceedings in Informatics (LIPIcs), pages 29:1-29:21, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2024.29.
  39. M. Henzinger, S. Krinninger, and D. Nanongkai. A deterministic almost-tight dis- tributed algorithm for approximating single-source shortest paths. In Proc. 48th ACM Symp. on Theory of Computing (STOC), pages 489-498, 2016. Google Scholar
  40. G. Iliopoulos and N. Balakrishnan. Conditional independence of blocked ordered data. Statistics and Probability Letters, 79(8):1008-1015, 2009. URL: https://doi.org/10.1016/j.spl.2008.12.005.
  41. Michael Isard, Mihai Budiu, Yuan Yu, Andrew Birrell, and Dennis Fetterly. Dryad: Distributed Data-Parallel Programs from Sequential Building Blocks. ACM SIGOPS Operating Systems Review, pages 59-72, 2007. URL: https://doi.org/10.1145/1272996.1273005.
  42. Hongyan Ji, Kishore Kothapalli, Sriram V Pemmaraju, and Ajitanshu Singh. Fast deterministic massively parallel ruling sets algorithms. In Proceedings of the 26th International Conference on Distributed Computing and Networking, ICDCN '25, pages 152-160, New York, NY, USA, 2025. Association for Computing Machinery. URL: https://doi.org/10.1145/3700838.3700872.
  43. Howard J. Karloff, Siddharth Suri, and Sergei Vassilvitskii. A Model of Computation for MapReduce. In the Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 938-948, 2010. URL: https://doi.org/10.1137/1.9781611973075.76.
  44. Seri Khoury and Aaron Schild. Breaking barriers for distributed mis by faster degree reduction, 2025. URL: https://doi.org/10.48550/arXiv.2505.15652.
  45. Seri Khoury and Aaron Schild. Round elimination via self-reduction: Closing gaps for distributed maximal matching, 2025. URL: https://doi.org/10.48550/arXiv.2505.15654.
  46. Kishore Kothapalli, Shreyas Pai, and Sriram V. Pemmaraju. Sample-And-Gather: Fast Ruling Set Algorithms in the Low-Memory MPC Model. In Nitin Saxena and Sunil Simon, editors, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020), volume 182 of Leibniz International Proceedings in Informatics (LIPIcs), pages 28:1-28:18, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2020.28.
  47. Kishore Kothapalli and Sriram Pemmaraju. Super-Fast 3-Ruling Sets. In FSTTCS, pages 136-147, 2012. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2012.136.
  48. F. Kuhn, T. Moscibroda, and R. Wattenhofer. Local computation: Lower and upper bounds. J. of the ACM, 63(2), 2016. URL: https://doi.org/10.1145/2742012.
  49. Fabian Kuhn, Yannic Maus, and Simon Weidner. Deterministic distributed ruling sets of line graphs. In Zvi Lotker and Boaz Patt-Shamir, editors, Structural Information and Communication Complexity - 25th International Colloquium, SIROCCO 2018, Ma'ale HaHamisha, Israel, June 18-21, 2018, Revised Selected Papers, volume 11085 of Lecture Notes in Computer Science, pages 193-208. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-01325-7_19.
  50. Christoph Lenzen and Roger Wattenhofer. Brief Announcement: Exponential Speed-up of Local Algorithms Using Non-Local Communication. In the Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 295-296, 2010. URL: https://doi.org/10.1145/1835698.1835772.
  51. Christoph Lenzen and Roger Wattenhofer. Minimum dominating set approximation in graphs of bounded arboricity. In Nancy A. Lynch and Alexander A. Shvartsman, editors, Distributed Computing, pages 510-524, Berlin, Heidelberg, 2010. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/978-3-642-15763-9_48.
  52. Nati Linial. Locality in Distributed Graph Algorithms. SIAM Journal on Computing, 21(1):193-201, 1992. URL: https://doi.org/10.1137/0221015.
  53. M. Luby. A Simple Parallel Algorithm for the Maximal Independent Set Problem. SIAM Journal on Computing, 15:1036-1053, 1986. URL: https://doi.org/10.1137/0215074.
  54. Yannic Maus. Distributed graph coloring made easy. In Kunal Agrawal and Yossi Azar, editors, SPAA '21: 33rd ACM Symposium on Parallelism in Algorithms and Architectures, Virtual Event, USA, 6-8 July, 2021, pages 362-372. ACM, 2021. URL: https://doi.org/10.1145/3409964.3461804.
  55. Yannic Maus, Saku Peltonen, and Jara Uitto. Distributed symmetry breaking on power graphs via sparsification. In Proceedings of the 2023 ACM Symposium on Principles of Distributed Computing, PODC '23, pages 157-167, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3583668.3594579.
  56. Adir Morgan, Shay Solomon, and Nicole Wein. Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial. In Seth Gilbert, editor, 35th International Symposium on Distributed Computing (DISC 2021), volume 209 of Leibniz International Proceedings in Informatics (LIPIcs), pages 33:1-33:19, Dagstuhl, Germany, 2021. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2021.33.
  57. Shreyas Pai, Gopal Pandurangan, Sriram V. Pemmaraju, Talal Riaz, and Peter Robinson. Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets. In Andréa Richa, editor, 31st International Symposium on Distributed Computing (DISC 2017), volume 91 of Leibniz International Proceedings in Informatics (LIPIcs), pages 38:1-38:16, Dagstuhl, Germany, 2017. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2017.38.
  58. Shreyas Pai and Sriram V. Pemmaraju. Brief Announcement: Deterministic Massively Parallel Algorithms for Ruling Sets. In the Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 366-368, 2022. URL: https://doi.org/10.1145/3519270.3538472.
  59. David Peleg. Distributed computing: A locality-sensitive approach. In Discrete Mathematics and Applications Distributed Computing: A Locality-Sensitive Approach, 1987. URL: https://api.semanticscholar.org/CorpusID:58843423.
  60. Sriram Pemmaraju and Talal Riaz. Using Read-k Inequalities to Analyze a Distributed MIS Algorithm. In Panagiota Fatourou, Ernesto Jiménez, and Fernando Pedone, editors, 20th International Conference on Principles of Distributed Systems (OPODIS 2016), volume 70 of Leibniz International Proceedings in Informatics (LIPIcs), pages 9:1-9:17, Dagstuhl, Germany, 2017. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.OPODIS.2016.9.
  61. Václav Rozhoň and Mohsen Ghaffari. Polylogarithmic-time Deterministic Network Decomposition and Distributed Derandomization. In the Proceedings of the Annual ACM Symposium on Theory of Computing (STOC), pages 350-363, 2020. Google Scholar
  62. Johannes Schneider, Michael Elkin, and Roger Wattenhofer. Symmetry breaking depending on the chromatic number or the neighborhood growth. Theoretical Computer Science, 509:40-50, 2013. Google Scholar
  63. Johannes Schneider, Michael Elkin, and Roger Wattenhofer. Symmetry breaking depending on the chromatic number or the neighborhood growth. Theor. Comput. Sci., 509:40-50, 2013. URL: https://doi.org/10.1016/J.TCS.2012.09.004.
  64. Johannes Schneider and Roger Wattenhofer. A new technique for distributed symmetry breaking. In the Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 257-266, 2010. URL: https://doi.org/10.1145/1835698.1835760.
  65. Tom White. Hadoop: The Definitive Guide. O'Reilly Media, Inc., 2009. Google Scholar
  66. Matei Zaharia, Mosharaf Chowdhury, Michael J. Franklin, Scott Shenker, and Ion Stoica. Spark: Cluster Computing with Working Sets. In the Proceedings of the USENIX Conference on Hot Topics in Cloud Computing (HotCloud), page 10, 2010. URL: https://doi.org/10.5555/1863103.1863113.
  67. Yufei Zhao. Probabilistic methods in combinatorics: Lecture 15, correlation inequalities. MIT OpenCourseWare, 18.226 Probabilistic Methods in Combinatorics, 2022. Accessed: 2026-01-06. URL: https://ocw.mit.edu/courses/18-226-probabilistic-methods-in-combinatorics-fall-2022/resources/mit18_226_f22_lec15_pdf/.
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