Document Open Access Logo

Ruling Sets in Random Order and Adversarial Streams

Authors Sepehr Assadi, Aditi Dudeja

PDF

File

Thumbnail PDF
PDF
LIPIcs.DISC.2021.6.pdf
  • Filesize: 1.02 MB
  • 18 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
  • Theory of computation → Lower bounds and information complexity
  • Theory of computation → Random order and robust communication complexity
Keywords
  • Symmetry breaking
  • Ruling sets
  • Lower bounds
  • Communication Complexity

Metrics

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

Abstract

The goal of this paper is to understand the complexity of a key symmetry breaking problem, namely the (α,β)-ruling set problem in the graph streaming model. Given a graph G = (V,E), an (α, β)-ruling set is a subset I ⊆ V such that the distance between any two vertices in I is at least α and the distance between a vertex in V and the closest vertex in I is at most β. This is a fundamental problem in distributed computing where it finds applications as a useful subroutine for other problems such as maximal matching, distributed colouring, or shortest paths. Additionally, it is a generalization of MIS, which is a (2,1)-ruling set.
Our main results are two algorithms for (2,2)-ruling sets:  
1) In adversarial streams, where the order in which edges arrive is arbitrary, we give an algorithm with Õ(n^{4/3}) space, improving upon the best known algorithm due to Konrad et al. [DISC 2019], with space Õ(n^{3/2}). 
2) In random-order streams, where the edges arrive in a random order, we give a semi-streaming algorithm, that is an algorithm that takes Õ(n) space.  Finally, we present new algorithms and lower bounds for (α,β)-ruling sets for other values of α and β. Our algorithms improve and generalize the previous work of Konrad et al. [DISC 2019] for (2,β)-ruling sets, while our lower bound establishes the impossibility of obtaining any non-trivial streaming algorithm for (α,α-1)-ruling sets for all even α > 2.

Cite As Get BibTex

Sepehr Assadi and Aditi Dudeja. Ruling Sets in Random Order and Adversarial Streams. In 35th International Symposium on Distributed Computing (DISC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 209, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.DISC.2021.6

Author Details

Sepehr Assadi
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Aditi Dudeja
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA

Funding

  • Assadi, Sepehr: Research supported in part by a NSF CAREER Grant CCF-2047061, and a gift from Google Research.
  • Dudeja, Aditi: This work was done while funded by NSF Award 1942010.

Acknowledgements

We thank anonymous reviewers for their insightful feedback.

References

  1. Amir Abboud, Keren Censor-Hillel, Seri Khoury, and Ami Paz. Smaller cuts, higher lower bounds. arXiv preprint, 2019. URL: http://arxiv.org/abs/1901.01630.
  2. Kook Jin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, and Anthony Wirth. Correlation clustering in data streams. In Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37, ICML'15, page 2237–2246. JMLR.org, 2015. Google Scholar
  3. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing Graph Structure via Linear Measurements, pages 459-467. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.40.
  4. Sepehr Assadi, MohammadHossein Bateni, Aaron Bernstein, Vahab S. Mirrokni, and Cliff Stein. Coresets meet EDCS: algorithms for matching and vertex cover on massive graphs. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 1616-1635. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.98.
  5. Sepehr Assadi and Soheil Behnezhad. Beating two-thirds for random-order streaming matching. CoRR, abs/2102.07011, 2021. URL: http://arxiv.org/abs/2102.07011.
  6. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Sublinear algorithms for (Δ + 1) vertex coloring. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019, pages 767-786. SIAM, 2019. URL: https://doi.org/10.1137/1.9781611975482.48.
  7. B. Awerbuch, M. Luby, A.V. Goldberg, and S.A. Plotkin. Network decomposition and locality in distributed computation. In 30th Annual Symposium on Foundations of Computer Science, pages 364-369, 1989. URL: https://doi.org/10.1109/SFCS.1989.63504.
  8. Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed symmetry breaking. J. ACM, 63(3), 2016. URL: https://doi.org/10.1145/2903137.
  9. Aaron Bernstein. Improved bounds for matching in random-order streams. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 12:1-12:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.12.
  10. Tushar Bisht, Kishore Kothapalli, and Sriram Pemmaraju. Brief announcement: Super-fast t-ruling sets. In PODC, New York, NY, USA, 2014. Association for Computing Machinery. URL: https://doi.org/10.1145/2611462.2611512.
  11. Amit Chakrabarti, Graham Cormode, and Andrew McGregor. Robust lower bounds for communication and stream computation. In Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC '08, page 641–650, New York, NY, USA, 2008. Association for Computing Machinery. URL: https://doi.org/10.1145/1374376.1374470.
  12. Yi-Jun Chang, Wenzheng Li, and Seth Pettie. An optimal distributed (δ+1)-coloring algorithm? In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, page 445–456, New York, NY, USA, 2018. Association for Computing Machinery. URL: https://doi.org/10.1145/3188745.3188964.
  13. Graham Cormode, Jacques Dark, and Christian Konrad. Independent sets in vertex-arrival streams. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 45:1-45:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.45.
  14. Devdatt P. Dubhashi and Desh Ranjan. Balls and bins: A study in negative dependence. Random Struct. Algorithms, 13(2):99-124, 1998. URL: https://doi.org/10.1002/(SICI)1098-2418(199809)13:2%3C99::AID-RSA1%3E3.0.CO;2-M.
  15. Alireza Farhadi, MohammadTaghi Hajiaghayi, Tung Mai, Anup Rao, and Ryan A. Rossi. Approximate maximum matching in random streams. In Proceedings of the Thirty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '20, page 1773–1785, USA, 2020. Society for Industrial and Applied Mathematics. Google Scholar
  16. Buddhima Gamlath, Sagar Kale, Slobodan Mitrovic, and Ola Svensson. Weighted matchings via unweighted augmentations. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC '19, page 491–500, New York, NY, USA, 2019. Association for Computing Machinery. URL: https://doi.org/10.1145/3293611.3331603.
  17. Beat Gfeller and Elias Vicari. A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs. In Proceedings of the Twenty-Sixth Annual ACM Symposium on Principles of Distributed Computing, PODC '07, page 53–60, New York, NY, USA, 2007. Association for Computing Machinery. URL: https://doi.org/10.1145/1281100.1281111.
  18. 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
  19. Monika Henzinger, Sebastian Krinninger, and Danupon Nanongkai. A deterministic almost-tight distributed algorithm for approximating single-source shortest paths. In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, STOC '16, page 489–498, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2897518.2897638.
  20. H. Jowhari, M. Saglam, and G. Tardos. Tight bounds for lp samplers, finding duplicates in streams, and related problems. arXiv, abs/1012.4889, 2011. URL: http://arxiv.org/abs/1012.4889.
  21. Tomasz Jurdziński and Krzysztof Nowicki. Mst in o(1) rounds of congested clique. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, page 2620–2632, USA, 2018. Society for Industrial and Applied Mathematics. Google Scholar
  22. Christian Konrad. MIS in the congested clique model in o(log log Δ) rounds. CoRR, abs/1802.07647, 2018. URL: http://arxiv.org/abs/1802.07647.
  23. Christian Konrad, Frédéric Magniez, and Claire Mathieu. Maximum matching in semi-streaming with few passes. In Anupam Gupta, Klaus Jansen, José D. P. Rolim, and Rocco A. Servedio, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Cambridge, MA, USA, August 15-17, 2012. Proceedings, volume 7408 of Lecture Notes in Computer Science, pages 231-242. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32512-0_20.
  24. Christian Konrad, Sriram V. Pemmaraju, Talal Riaz, and Peter Robinson. The Complexity of Symmetry Breaking in Massive Graphs. In Jukka Suomela, editor, 33rd International Symposium on Distributed Computing (DISC 2019), volume 146 of Leibniz International Proceedings in Informatics (LIPIcs), pages 26:1-26:18, Dagstuhl, Germany, 2019. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.DISC.2019.26.
  25. 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.
  26. Ilan Kremer, Noam Nisan, and Dana Ron. On randomized one-round communication complexity. In Frank Thomson Leighton and Allan Borodin, editors, Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, 29 May-1 June 1995, Las Vegas, Nevada, USA, pages 596-605. ACM, 1995. URL: https://doi.org/10.1145/225058.225277.
  27. Fabian Kuhn, Thomas Moscibroda, and Roger Wattenhofer. Local computation: Lower and upper bounds. J. ACM, 63(2):17:1-17:44, 2016. URL: https://doi.org/10.1145/2742012.
  28. Andrew McGregor. Graph stream algorithms: a survey. SIGMOD Rec., 43(1):9-20, 2014. URL: https://doi.org/10.1145/2627692.2627694.
  29. Morteza Monemizadeh, S. Muthukrishnan, Pan Peng, and Christian Sohler. Testable bounded degree graph properties are random order streamable. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 131:1-131:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.131.
  30. Gopal Pandurangan, Peter Robinson, and Michele Scquizzato. Fast distributed algorithms for connectivity and mst in large graphs. In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '16, page 429–438, New York, NY, USA, 2016. Association for Computing Machinery. URL: https://doi.org/10.1145/2935764.2935785.
  31. Johannes Schneider and Roger Wattenhofer. A new technique for distributed symmetry breaking. In Proceedings of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC '10, page 257–266, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1835698.1835760.
  32. David Wajc. Negative association-definition, properties, and applications, 2017. URL: http://www.cs.cmu.edu/~dwajc/notes/Negative%20Association.pdf.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact