Document Open Access Logo

New Algorithms and Lower Bounds for Streaming Tournaments

Authors Prantar Ghosh , Sahil Kuchlous

PDF
Thumbnail PDF

File

LIPIcs.ESA.2024.60.pdf
  • Filesize: 0.84 MB
  • 19 pages

Document Identifiers

Author Details

Prantar Ghosh
  • Department of Computer Science, Georgetown University, Washington, DC, USA
Sahil Kuchlous
  • Harvard University, Cambridge, MA, USA

Funding

  • Ghosh, Prantar: Supported in part by NSF under award 1918989. Part of this work was done while the author was at DIMACS, Rutgers University, supported in part by a grant (820931) to DIMACS from the Simons Foundation.
  • Kuchlous, Sahil: Research done in part as a participant in the DIMACS REU program 2023 at Rutgers University, supported by NSF grant CCF-2150186.

Acknowledgements

We are extremely grateful to Srijon Mukherjee for letting us know about the IOITC question on determining strong connectivity of a tournament from its set of indegrees, solving and building on which we obtained our results for SCC decomposition. Later we came to know that the question (and its solution) was designed by Sreejata Kishor Bhattacharya and Archit Karandikar at IOITC 2017. We also thank Michael Saks for the observation of simpler analysis for tournaments (mentioned in Remark 12) and Sepehr Assadi, Amit Chakrabarti, and Madhu Sudan for helpful discussions. Finally, we thank the organizers of the DIMACS REU program for making this collaboration possible.

Cite AsGet BibTex

Prantar Ghosh and Sahil Kuchlous. New Algorithms and Lower Bounds for Streaming Tournaments. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 60:1-60:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.60

Abstract

We study fundamental directed graph (digraph) problems in the streaming model. An initial investigation by Chakrabarti, Ghosh, McGregor, and Vorotnikova [SODA'20] on streaming digraphs showed that while most of these problems are provably hard in general, some of them become tractable when restricted to the well-studied class of tournament graphs where every pair of nodes shares exactly one directed edge. Thus, we focus on tournaments and improve the state of the art for multiple problems in terms of both upper and lower bounds. Our primary upper bound is a deterministic single-pass semi-streaming algorithm (using Õ(n) space for n-node graphs, where Õ(.) hides polylog(n) factors) for decomposing a tournament into strongly connected components (SCC). It improves upon the previously best-known algorithm by Baweja, Jia, and Woodruff [ITCS'22] in terms of both space and passes: for p ⩾ 1, they used (p+1) passes and Õ(n^{1+1/p}) space. We further extend our algorithm to digraphs that are close to tournaments and establish tight bounds demonstrating that the problem’s complexity grows smoothly with the "distance" from tournaments. Applying our SCC-decomposition framework, we obtain improved - and in some cases, optimal - tournament algorithms for s,t-reachability, strong connectivity, Hamiltonian paths and cycles, and feedback arc set. On the other hand, we prove lower bounds exhibiting that some well-studied problems - such as (exact) feedback arc set and s,t-distance - remain hard (require Ω(n²) space) on tournaments. Moreover, we generalize the former problem’s lower bound to establish space-approximation tradeoffs: any single-pass (1± ε)-approximation algorithm requires Ω(n/√{ε}) space. Finally, we settle the streaming complexities of two basic digraph problems studied by prior work: acyclicity testing of tournaments and sink finding in DAGs. As a whole, our collection of results contributes significantly to the growing literature on streaming digraphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • tournaments
  • streaming algorithms
  • graph algorithms
  • communication complexity
  • strongly connected components
  • reachability
  • feedback arc set

Metrics

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

Related Versions

References

  1. Farid Ablayev. Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theoretical Computer Science, 175(2):139-159, 1996. Google Scholar
  2. KookJin Ahn, Graham Cormode, Sudipto Guha, Andrew McGregor, and Anthony Wirth. Correlation clustering in data streams. In Proceedings of the 32nd International Conference on Machine Learning, volume 37 of Proceedings of Machine Learning Research, pages 2237-2246. PMLR, 2015. Google Scholar
  3. Sepehr Assadi. Recent advances in multi-pass graph streaming lower bounds. ACM SIGACT News, 54(3):48-75, September 2023. URL: https://doi.org/10.1145/3623800.3623808.
  4. Sepehr Assadi, Arun Jambulapati, Yujia Jin, Aaron Sidford, and Kevin Tian. Semi-streaming bipartite matching in fewer passes and optimal space. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 627-669. SIAM, 2022. URL: https://doi.org/10.1137/1.9781611977073.29.
  5. Sepehr Assadi, Gillat Kol, Raghuvansh R. Saxena, and Huacheng Yu. Multi-pass graph streaming lower bounds for cycle counting, max-cut, matching size, and other problems. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 354-364. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00041.
  6. Sepehr Assadi and Ran Raz. Near-quadratic lower bounds for two-pass graph streaming algorithms. In 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 342-353. IEEE, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00040.
  7. Reuven Bar-Yehuda, Dan Geiger, Joseph (Seffi) Naor, and Ron M. Roth. Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and bayesian inference. SIAM Journal on Computing, 27(4):942-959, 1998. URL: https://doi.org/10.1137/S0097539796305109.
  8. Anubhav Baweja, Justin Jia, and David P. Woodruff. An efficient semi-streaming PTAS for tournament feedback arc set with few passes. In 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 16:1-16:23. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.16.
  9. Soheil Behnezhad, Moses Charikar, Weiyun Ma, and Li-Yang Tan. Single-pass streaming algorithms for correlation clustering. 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 819-849. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH33.
  10. L.W. Beineke and R.J. Wilson. Selected Topics in Graph Theory. Number v. 1-3 in Selected Topics in Graph Theory. Academic Press, 1978. Google Scholar
  11. Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. Certifying equality with limited interaction. In Proc. 18th International Workshop on Randomization and Approximation Techniques in Computer Science, pages 545-581, 2014. Google Scholar
  12. Paul Camion. Chemins et circuits hamiltoniens des graphes complets. COMPTES RENDUS HEBDOMADAIRES DES SEANCES DE L ACADEMIE DES SCIENCES, 249(21):2151-2152, 1959. Google Scholar
  13. Amit Chakrabarti, Graham Cormode, Navin Goyal, and Justin Thaler. Annotations for sparse data streams. In Proc. 25th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 687-706, 2014. Google Scholar
  14. Amit Chakrabarti, Prantar Ghosh, Andrew McGregor, and Sofya Vorotnikova. Vertex ordering problems in directed graph streams. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 1786-1802. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.109.
  15. Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew C. Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Proc. 42nd Annual IEEE Symposium on Foundations of Computer Science, pages 270-278, 2001. Google Scholar
  16. Pierre Charbit, Stéphan Thomassé, and Anders Yeo. The minimum feedback arc set problem is np-hard for tournaments. Combinatorics, Probability & Computing, 16:1-4, January 2007. URL: https://doi.org/10.1017/S0963548306007887.
  17. Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu. Almost optimal super-constant-pass streaming lower bounds for reachability. In STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 570-583. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451038.
  18. Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, and Huacheng Yu. Near-optimal two-pass streaming algorithm for sampling random walks over directed graphs. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 52:1-52:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.52.
  19. Don Coppersmith, Lisa Fleischer, and Atri Rudra. Ordering by weighted number of wins gives a good ranking for weighted tournaments. ACM Trans. Algorithms, 6(3):55:1-55:13, 2010. URL: https://doi.org/10.1145/1798596.1798608.
  20. Abhik Kumar Das and Sriram Vishwanath. On finite alphabet compressive sensing. In 2013 IEEE International Conference on Acoustics, Speech and Signal Processing, pages 5890-5894, 2013. URL: https://doi.org/10.1109/ICASSP.2013.6638794.
  21. Michael Elkin. Distributed exact shortest paths in sublinear time. In Proc. 49th Annual ACM Symposium on the Theory of Computing, pages 757-770, 2017. Google Scholar
  22. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Theor. Comput. Sci., 348(2-3):207-216, 2005. Preliminary version in Proc. 31st International Colloquium on Automata, Languages and Programming , pages 531-543, 2004. Google Scholar
  23. S I Gass. Tournaments, transitivity and pairwise comparison matrices. Journal of the Operational Research Society, 49(6):616-624, 1998. URL: https://doi.org/10.1057/palgrave.jors.2600572.
  24. Xiubo Geng, Tie-Yan Liu, Tao Qin, Andrew Arnold, Hang Li, and Heung-Yeung Shum. Query dependent ranking using k-nearest neighbor. In Proceedings of the 31st Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR '08, pages 115-122. Association for Computing Machinery, 2008. URL: https://doi.org/10.1145/1390334.1390356.
  25. Prantar Ghosh and Sahil Kuchlous. New algorithms and lower bounds for streaming tournaments. CoRR, abs/2405.05952, 2024. URL: https://doi.org/10.48550/arXiv.2405.05952.
  26. Anna C. Gilbert and Piotr Indyk. Sparse recovery using sparse matrices. Proceedings of the IEEE, 98(6):937-947, 2010. Google Scholar
  27. Venkatesan Guruswami and Krzysztof Onak. Superlinear lower bounds for multipass graph processing. In Proceedings of the 28th Conference on Computational Complexity, CCC 2013, K.lo Alto, California, USA, 5-7 June, 2013, pages 287-298. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/CCC.2013.37.
  28. Venkatesan Guruswami, Ameya Velingker, and Santhoshini Velusamy. Streaming complexity of approximating max 2csp and max acyclic subgraph. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2017, August 16-18, 2017, Berkeley, CA, USA, volume 81 of LIPIcs, pages 8:1-8:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.APPROX-RANDOM.2017.8.
  29. Monika R. Henzinger, Prabhakar Raghavan, and Sridhar Rajagopalan. Computing on data streams. External memory algorithms, pages 107-118, 1999. Google Scholar
  30. Ce Jin. Simulating random walks on graphs in the streaming model. In 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, pages 46:1-46:15, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.46.
  31. Richard Karp. Reducibility among combinatorial problems. Complexity of Computer Computations, 40:85-103, 1972. URL: https://doi.org/10.1007/978-3-540-68279-0_8.
  32. Claire Kenyon-Mathieu and Warren Schudy. How to rank with few errors. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC '07, pages 95-103. Association for Computing Machinery, 2007. URL: https://doi.org/10.1145/1250790.1250806.
  33. Luigi Laura and Federico Santaroni. Computing strongly connected components in the streaming model. In Theory and Practice of Algorithms in (Computer) Systems, pages 193-205. Springer Berlin Heidelberg, 2011. Google Scholar
  34. Nikhil S. Mande, Manaswi Paraashar, Swagato Sanyal, and Nitin Saurabh. On the communication complexity of finding a king in a tournament. ArXiv preprint, abs/2402.14751, 2024. URL: https://arxiv.org/abs/2402.14751.
  35. Andrew McGregor. Graph stream algorithms: a survey. SIGMOD Record, 43(1):9-20, 2014. URL: https://doi.org/10.1145/2627692.2627694.
  36. Andrew McGregor and Hoa T. Vu. Better streaming algorithms for the maximum coverage problem. In 20th International Conference on Database Theory, ICDT 2017, March 21-24, 2017, Venice, Italy, volume 68 of LIPIcs, pages 22:1-22:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.ICDT.2017.22.
  37. Alexander Razborov. On the distributional complexity of disjointness. Theor. Comput. Sci., 106(2):385-390, 1992. Preliminary version in Proc. 17th International Colloquium on Automata, Languages and Programming , pages 249-253, 1990. Google Scholar
  38. Atish Das Sarma, Sreenivas Gollapudi, and Rina Panigrahy. Estimating pagerank on graph streams. J. ACM, 58(3):13, 2011. URL: https://doi.org/10.1145/1970392.1970397.
  39. Michael Simpson, Venkatesh Srinivasan, and Alex Thomo. Efficient computation of feedback arc set at web-scale. Proc. VLDB Endow., 10(3):133-144, 2016. URL: https://doi.org/10.14778/3021924.3021930.
  40. Noah Singer, Madhu Sudan, and Santhoshini Velusamy. Streaming approximation resistance of every ordering CSP. In Mary Wootters and Laura Sanità, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021, August 16-18, 2021, University of Washington, Seattle, Washington, USA (Virtual Conference), volume 207 of LIPIcs, pages 17:1-17:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2021.17.
  41. Danny Soroker. Fast parallel algorithms for finding hamiltonian paths and cycles in a tournament. J. Algorithms, 9(2):276-286, June 1988. URL: https://doi.org/10.1016/0196-6774(88)90042-9.
  42. Andrew C. Yao. Some complexity questions related to distributive computing. In Proc. 11th Annual ACM Symposium on the Theory of Computing, pages 209-213, 1979. 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