Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries

Authors Raghavendra Addanki , Andrew McGregor, Cameron Musco



PDF
Thumbnail PDF

File

LIPIcs.ESA.2022.2.pdf
  • Filesize: 0.85 MB
  • 16 pages

Document Identifiers

Author Details

Raghavendra Addanki
  • Adobe Research, San Jose, CA, USA
Andrew McGregor
  • Manning College of Information and Computer Sciences, University of Massachusetts Amherst, MA, USA
Cameron Musco
  • Manning College of Information and Computer Sciences, University of Massachusetts Amherst, MA, USA

Acknowledgements

Most of this work was done while R. Addanki was a student at UMass Amherst. Part of this work was done while R. Addanki was a visiting student at the Simons Institute for the Theory of Computing. We thank the anonymous reviewers for their helpful suggestions.

Cite AsGet BibTex

Raghavendra Addanki, Andrew McGregor, and Cameron Musco. Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ESA.2022.2

Abstract

We study the problem of estimating the number of edges in an n-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (TALG '20). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a (1± ε) relative error approximation to the number of edges, with query complexity Õ(ε^{-5}log⁵ n), where Õ(⋅) hides poly(log log n) dependencies. This is the first non-adaptive algorithm in this setting achieving poly(1/ε,log n) query complexity. Prior work requires Ω(log² n) rounds of adaptivity. We avoid this by taking a fundamentally different approach, inspired by work on single-pass streaming algorithms. Moreover, for constant ε, our query complexity significantly improves on the best known adaptive algorithm due to Bhattacharya et al. (STACS '22), which requires O(ε^{-2} log^{11} n) queries. Building on our edge estimation result, we give the first {non-adaptive} algorithm for outputting a nearly uniformly sampled edge with query complexity Õ(ε^{-6} log⁶ n), improving on the works of Dell et al. (SODA '20) and Bhattacharya et al. (STACS '22), which require Ω(log³ n) rounds of adaptivity. Finally, as a consequence of our edge sampling algorithm, we obtain a Õ(n log^8 n) query algorithm for connectivity, using two rounds of adaptivity. This improves on a three-round algorithm of Assadi et al. (ESA '21) and is tight; there is no non-adaptive algorithm for connectivity making o(n²) queries.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • sublinear graph algorithms
  • bipartite independent set queries
  • edge sampling and counting
  • graph connectivity
  • query adaptivity

Metrics

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

References

  1. Hasan Abasi and Bshouty Nader. On learning graphs with edge-detecting queries. In Algorithmic Learning Theory (ALT), pages 3-30, 2019. Google Scholar
  2. Raghavendra Addanki, Andrew McGregor, and Cameron Musco. Non-adaptive edge counting and sampling via bipartite independent set queries. arXiv, 2022. URL: http://arxiv.org/abs/2207.02817.
  3. Arpit Agarwal, Shivani Agarwal, Sepehr Assadi, and Sanjeev Khanna. Learning with limited rounds of adaptivity: Coin tossing, multi-armed bandits, and ranking from pairwise comparisons. In Proceedings of the 30th Annual Conference on Computational Learning Theory (COLT), pages 39-75, 2017. Google Scholar
  4. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 459-467, 2012. Google Scholar
  5. Maryam Aliakbarpour, Amartya Shankha Biswas, Themis Gouleakis, John Peebles, Ronitt Rubinfeld, and Anak Yodpinyanee. Sublinear-time algorithms for counting star subgraphs via edge sampling. Algorithmica, 80(2):668-697, 2018. Google Scholar
  6. Dana Angluin and Jiang Chen. Learning a hidden graph using o (log n) queries per edge. Journal of Computer and System Sciences, 74(4):546-556, 2008. Google Scholar
  7. Boris Aronov and Sariel Har-Peled. On approximating the depth and related problems. SIAM Journal on Computing, 38(3):899-921, 2008. Google Scholar
  8. Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph connectivity and single element recovery via linear and OR queries. In 29th Annual European Symposium on Algorithms, (ESA), pages 7:1-7:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  9. Eric Balkanski and Yaron Singer. The adaptive complexity of maximizing a submodular function. In Proceedings of the 50th Annual ACM Symposium on Theory of Computing (STOC), pages 1138-1151, 2018. Google Scholar
  10. Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge estimation with independent set oracles. ACM Transactions on Algorithms (TALG), 16(4):1-27, 2020. Google Scholar
  11. Soheil Behnezhad. Time-optimal sublinear algorithms for matching and vertex cover. In Proceedings of the 63rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 873-884, 2022. Google Scholar
  12. Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Hyperedge estimation using polylogarithmic subset queries. arXiv, 2019. URL: http://arxiv.org/abs/1908.04196.
  13. Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. On triangle estimation using tripartite independent set queries. Theory of Computing Systems, pages 1-28, 2021. Google Scholar
  14. Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Faster counting and sampling algorithms using colorful decision oracle. In Proceedings of the 39th International Symposium on Theoretical Aspects of Computer Science (STACS), 2022. Google Scholar
  15. Arijit Bishnu, Arijit Ghosh, Sudeshna Kolay, Gopinath Mishra, and Saket Saurabh. Parameterized query complexity of hitting set using stability of sunflowers. In 29th International Symposium on Algorithms and Computation, 2018. Google Scholar
  16. Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Efficiently sampling and estimating from substructures using linear algebraic queries. arXiv, 2019. URL: http://arxiv.org/abs/1906.07398.
  17. Amartya Shankha Biswas, Talya Eden, and Ronitt Rubinfeld. Towards a decomposition-optimal algorithm for counting and sampling arbitrary motifs in sublinear time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), 2021. Google Scholar
  18. Sergio Cabello and Miha Jejčič. Shortest paths in intersection graphs of unit disks. Computational Geometry, 48(4):360-367, 2015. Google Scholar
  19. Clément L Canonne and Tom Gur. An adaptivity hierarchy theorem for property testing. Computational Complexity, 27(4):671-716, 2018. Google Scholar
  20. Amit Chakrabarti and Manuel Stoeckl. The element extraction problem and the cost of determinism and limited adaptivity in linear queries. arXiv, 2021. URL: http://arxiv.org/abs/2107.05810.
  21. Chandra Chekuri and Kent Quanrud. Parallelizing greedy for submodular set function maximization in matroids and beyond. In Proceedings of the 51st Annual ACM Symposium on Theory of Computing (STOC), pages 78-89, 2019. Google Scholar
  22. Chao L Chen and William H Swallow. Using group testing to estimate a proportion, and to test the binomial model. Biometrics, pages 1035-1046, 1990. Google Scholar
  23. Xi Chen, Amit Levi, and Erik Waingarten. Nearly optimal edge estimation with independent set queries. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2916-2935, 2020. Google Scholar
  24. Graham Cormode and Shan Muthukrishnan. An improved data stream summary: the count-min sketch and its applications. Journal of Algorithms, 55(1):58-75, 2005. Google Scholar
  25. Holger Dell and John Lapinskas. Fine-grained reductions from approximate counting to decision. ACM Transactions on Computation Theory (TOCT), 13(2):1-24, 2021. Google Scholar
  26. Holger Dell, John Lapinskas, and Kitty Meeks. Approximately counting and sampling small witnesses using a colourful decision oracle. In Proceedings of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2201-2211, 2020. Google Scholar
  27. Robert Dorfman. The detection of defective members of large populations. The Annals of Mathematical Statistics, 14(4):436-440, 1943. Google Scholar
  28. Dingzhu Du, Frank K Hwang, and Frank Hwang. Combinatorial group testing and its applications, volume 12. World Scientific, 2000. Google Scholar
  29. Talya Eden, Dana Ron, and C Seshadhri. On approximating the number of k-cliques in sublinear time. SIAM Journal on Computing, 49(4):747-771, 2020. Google Scholar
  30. Talya Eden and Will Rosenbaum. On sampling edges almost uniformly. In 1st Symposium on Simplicity in Algorithms (SOSA), 2018. Google Scholar
  31. Uriel Feige. On sums of independent random variables with unbounded variance and estimating the average degree in a graph. SIAM Journal on Computing, 35(4):964-984, 2006. Google Scholar
  32. Aleksei V Fishkin. Disk graphs: A short survey. In International Workshop on Approximation and Online Algorithms, pages 260-264. Springer, 2003. Google Scholar
  33. Oded Goldreich and Dana Ron. Approximating average parameters of graphs. Random Structures & Algorithms, 32(4):473-493, 2008. Google Scholar
  34. Piotr Indyk, Hung Q Ngo, and Atri Rudra. Efficiently decodable non-adaptive group testing. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1126-1142, 2010. Google Scholar
  35. Piotr Indyk, Eric Price, and David P Woodruff. On the power of adaptivity in sparse recovery. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 285-294, 2011. Google Scholar
  36. Akshay Kamath and Eric Price. Adaptive sparse recovery with limited adaptivity. In Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2729-2744, 2019. Google Scholar
  37. Howard Karloff, Siddharth Suri, and Sergei Vassilvitskii. A model of computation for mapreduce. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 938-948, 2010. Google Scholar
  38. Lidiya Khalidah binti Khalil and Christian Konrad. Constructing large matchings via query access to a maximal matching oracle. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), 2020. Google Scholar
  39. Andrew McGregor. Graph stream algorithms: a survey. ACM SIGMOD Record, 43(1):9-20, 2014. Google Scholar
  40. Ashley Montanaro and Changpeng Shao. Quantum algorithms for learning a hidden graph. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2022. Google Scholar
  41. Vasileios Nakos, Xiaofei Shi, David P Woodruff, and Hongyang Zhang. Improved algorithms for adaptive compressed sensing. In Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP), 2018. Google Scholar
  42. Noam Nisan. The demand query model for bipartite matching. In Proceedings of the 32nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 592-599, 2021. Google Scholar
  43. Krzysztof Onak, Dana Ron, Michal Rosen, and Ronitt Rubinfeld. A near-optimal sublinear-time algorithm for approximating the minimum vertex cover size. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1123-1131, 2012. Google Scholar
  44. Cyrus Rashtchian, David P Woodruff, and Hanlin Zhu. Vector-matrix-vector queries for solving linear algebra, statistics, and graph problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), 2020. Google Scholar
  45. Dana Ron. Sublinear-time algorithms for approximating graph parameters. In Computing and Software Science, pages 105-122. Springer, 2019. Google Scholar
  46. Dana Ron and Gilad Tsur. The power of an example: Hidden set size approximation using group queries and conditional sampling. ACM Transactions on Computation Theory (TOCT), 8(4):1-19, 2016. Google Scholar
  47. Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing Exact Minimum Cuts Without Knowing the Graph. In Proceedings of the 9th Conference on Innovations in Theoretical Computer Science (ITCS), 2018. Google Scholar
  48. C Seshadhri. A simpler sublinear algorithm for approximating the triangle count. arXiv, 2015. URL: http://arxiv.org/abs/1505.01927.
  49. Larry Stockmeyer. The complexity of approximate counting. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC), pages 118-126, 1983. Google Scholar
  50. Larry Stockmeyer. On approximation algorithms for # p. SIAM Journal on Computing, 14(4):849-861, 1985. Google Scholar
  51. Jakub Tětek and Mikkel Thorup. Edge sampling and graph parameter estimation via vertex neighborhood accesses. In Proceedings of the 54th Annual ACM Symposium on Theory of Computing (STOC), pages 1116-1129, 2022. 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