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Graph Connectivity and Single Element Recovery via Linear and OR Queries

Authors Sepehr Assadi, Deeparnab Chakrabarty, Sanjeev Khanna

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Author Details

Sepehr Assadi
  • Rutgers University, New Brunswick, NJ, USA
Deeparnab Chakrabarty
  • Dartmouth College, Hanover, NH, USA
Sanjeev Khanna
  • University of Pennsylvania, Philadelphia, PA, USA

Funding

  • Assadi, Sepehr: Supported in part by NSF CAREER award CCF-2047061, and gift from Google Research.
  • Chakrabarty, Deeparnab: Supported in part by the NSF awards CCF-1813053 and CCF-2041920.
  • Khanna, Sanjeev: Supported in part by the NSF awards CCF-1763514, CCF-1617851, CCF-1934876, and CCF-2008305.

Cite AsGet BibTex

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 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 7:1-7:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ESA.2021.7

Abstract

We study the problem of finding a spanning forest in an undirected, n-vertex multi-graph under two basic query models. One are Linear queries which are linear measurements on the incidence vector induced by the edges; the other are the weaker OR queries which only reveal whether a given subset of plausible edges is empty or not. At the heart of our study lies a fundamental problem which we call the single element recovery problem: given a non-negative vector x ∈ ℝ^{N}_{≥ 0}, the objective is to return a single element x_j > 0 from the support. Queries can be made in rounds, and our goals is to understand the trade-offs between the query complexity and the rounds of adaptivity needed to solve these problems, for both deterministic and randomized algorithms. These questions have connections and ramifications to multiple areas such as sketching, streaming, graph reconstruction, and compressed sensing. Our main results are as follows: - For the single element recovery problem, it is easy to obtain a deterministic, r-round algorithm which makes (N^{1/r}-1)-queries per-round. We prove that this is tight: any r-round deterministic algorithm must make ≥ (N^{1/r} - 1) Linear queries in some round. In contrast, a 1-round O(polylog)-query randomized algorithm is known to exist. - We design a deterministic O(r)-round, Õ(n^{1+1/r})-OR query algorithm for graph connectivity. We complement this with an Ω̃(n^{1 + 1/r})-lower bound for any r-round deterministic algorithm in the OR-model. - We design a randomized, 2-round algorithm for the graph connectivity problem which makes Õ(n)-OR queries. In contrast, we prove that any 1-round algorithm (possibly randomized) requires Ω̃(n²)-OR queries. A randomized, 1-round algorithm making Õ(n)-Linear queries is already known. All our algorithms, in fact, work with more natural graph query models which are special cases of the above, and have been extensively studied in the literature. These are Cross queries (cut-queries) and BIS (bipartite independent set) queries.

Subject Classification

ACM Subject Classification
  • Theory of computation
Keywords
  • Query Models
  • Graph Connectivity
  • Group Testing
  • Duality

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References

  1. Hasan Abasi and Nader H. Bshouty. On learning graphs with edge-detecting queries. CoRR, abs/1803.10639, 2018. Google Scholar
  2. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proc., SODA, pages 459-467, 2012. Google Scholar
  3. Noga Alon and Vera Asodi. Learning a hidden subgraph. SIAM Journal on Discrete Mathematics (SIDMA), 18(4):697-712, 2005. Google Scholar
  4. Noga Alon, Richard Beigel, Simon Kasif, Steven Rudich, and Benny Sudakov. Learning a hidden matching. In Proc., FOCS, page 197, 2002. Google Scholar
  5. Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. J. Comput. System Sci., 58(1):137-147, 1999. Google Scholar
  6. Dana Angluin and Jiang Chen. Learning a hidden graph using O(log n) queries per edge. J. Comput. System Sci., 74(4):546-556, 2008. Google Scholar
  7. Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph connectivity and single element recovery via linear and OR queries. arXiv preprint arXiv:2007.06098, 2020. Google Scholar
  8. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Polynomial pass lower bounds for graph streaming algorithms. In Proc., STOC, pages 265-276, 2019. Google Scholar
  9. Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge estimation with independent set oracles. ACM Trans. on Algorithms (TALG), 16(4):1-27, 2020. Preliminary version in Proc. ITCS, 2018. Google Scholar
  10. Omri Ben-Eliezer, Rajesh Jayaram, David P. Woodruff, and Eylon Yogev. A framework for adversarially robust streaming algorithms. In Dan Suciu, Yufei Tao, and Zhewei Wei, editors, Proc., ACM Symposium on Principles of Database Systems (PODS), 2020. Google Scholar
  11. Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Triangle estimation using polylogarithmic queries. CoRR, abs/1808.00691, 2018. Google Scholar
  12. Nader H Bshouty. Optimal algorithms for the coin weighing problem with a spring scale. In Proc., Conf. on Learning Theory, 2009. Google Scholar
  13. Nader H Bshouty. Lower bound for non-adaptive estimation of the number of defective items. In Proc., International Symposium on Algorithms and Computation (ISAAC 2019), pages 2:1-2:9, 2019. Google Scholar
  14. Nader H Bshouty and Hanna Mazzawi. Reconstructing weighted graphs with minimal query complexity. Theoretical Computer Science, 412(19):1782-1790, 2011. Google Scholar
  15. Nader H. Bshouty and Hanna Mazzawi. Toward a deterministic polynomial time algorithm with optimal additive query complexity. Theoretical Computer Science, 417:23-35, 2012. Google Scholar
  16. Sung-Soon Choi. Polynomial time optimal query algorithms for finding graphs with arbitrary real weights. In Proc., Conf. on Learning Theory, pages 797-818, 2013. Google Scholar
  17. Sung-Soon Choi and Jeong Han Kim. Optimal query complexity bounds for finding graphs. In Proc., STOC, pages 749-758, 2008. Google Scholar
  18. Graham Cormode and Donatella Firmani. A unifying framework for 𝓁₀-sampling algorithms. Distributed and Parallel Databases, 32(3):315-335, 2014. Google Scholar
  19. Graham Cormode and S Muthukrishnan. Combinatorial algorithms for compressed sensing. In SIROCCO, pages 280-294. Springer, 2006. Google Scholar
  20. Peter Damaschke and Azam Sheikh Muhammad. Competitive group testing and learning hidden vertex covers with minimum adaptivity. Discrete Mathematics, Algorithms and Applications, 2(03):291-311, 2010. Google Scholar
  21. Holger Dell and John Lapinskas. Fine-grained reductions from approximate counting to decision. In Proc., STOC, pages 281-288. ACM, 2018. Google Scholar
  22. David L Donoho et al. Compressed sensing. IEEE Transactions on information theory, 52(4):1289-1306, 2006. Google Scholar
  23. Robert Dorfman. The detection of defective members of large populations. The Annals of Mathematical Statistics, 14(4):436-440, 1943. Google Scholar
  24. Dingzhu Du and Frank K Hwang. Combinatorial group testing and its applications, volume 12. World Scientific, 2000. Google Scholar
  25. AG D'yachkov, VS Lebedev, PA Vilenkin, and SM Yekhanin. Cover-free families and superimposed codes: constructions, bounds and applications to cryptography and group testing. In Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No. 01CH37252), page 117. IEEE, 2001. Google Scholar
  26. Moein Falahatgar, Ashkan Jafarpour, Alon Orlitsky, Venkatadheeraj Pichapati, and Ananda Theertha Suresh. Estimating the number of defectives with group testing. In Proc., IEEE International Symposium on Information Theory (ISIT), pages 1376-1380, 2016. Google Scholar
  27. Philippe Flajolet and G Nigel Martin. Probabilistic counting algorithms for data base applications. J. Comput. System Sci., 31(2):182-209, 1985. Google Scholar
  28. Gereon Frahling, Piotr Indyk, and Christian Sohler. Sampling in dynamic data streams and applications. International Journal of Computational Geometry & Applications, 18:3-28, 2008. Google Scholar
  29. Sumit Ganguly. Lower bounds on frequency estimation of data streams. In International Computer Science Symposium in Russia (CSR), pages 204-215, 2008. Google Scholar
  30. Vladimir Grebinski and Gregory Kucherov. Optimal reconstruction of graphs under the additive model. Algorithmica, 28(1):104-124, 2000. Google Scholar
  31. Jacob Holm, Valerie King, Mikkel Thorup, Or Zamir, and Uri Zwick. Random k-out subgraph leaves only O(n/k) inter-component edges. In Proc., FOCS, pages 896-909. IEEE, 2019. Google Scholar
  32. FK Hwang and VT Sós. Non-adaptive hypergeometric group testing. Studia Sci. Math. Hungar, 22(1-4):257-263, 1987. Google Scholar
  33. Piotr Indyk. Explicit constructions for compressed sensing of sparse signals. In Proc., SODA, pages 30-33, 2008. Google Scholar
  34. Hossein Jowhari, Mert Sağlam, and Gábor Tardos. Tight bounds for 𝓁_p samplers, finding duplicates in streams, and related problems. In Proc., ACM Symposium on Principles of Database Systems (PODS), pages 49-58, 2011. Google Scholar
  35. John Kallaugher and Eric Price. Separations and equivalences between turnstile streaming and linear sketching. In Proc., STOC, pages 1223-1236, 2020. Google Scholar
  36. Michael Kapralov, Yin Tat Lee, CN Musco, Christopher Paul Musco, and Aaron Sidford. Single pass spectral sparsification in dynamic streams. SIAM Journal on Computing (SICOMP), 46(1):456-477, 2017. Google Scholar
  37. Michael Kapralov, Jelani Nelson, Jakub Pachocki, Zhengyu Wang, David P Woodruff, and Mobin Yahyazadeh. Optimal lower bounds for universal relation, and for samplers and finding duplicates in streams. In Proc., FOCS, pages 475-486, 2017. Google Scholar
  38. W Kautz and Roy Singleton. Nonrandom binary superimposed codes. IEEE Transactions on Information Theory, 10(4):363-377, 1964. Google Scholar
  39. Yi Li, Huy L Nguyen, and David P Woodruff. Turnstile streaming algorithms might as well be linear sketches. In Proc., STOC, pages 174-183, 2014. Google Scholar
  40. Bernt Lindström. On Möbius functions and a problem in combinatorial number theory. Canadian Mathematical Bulletin, 14(4):513-516, 1971. Google Scholar
  41. Hanna Mazzawi. Optimally reconstructing weighted graphs using queries. In Proc., SODA, pages 608-615, 2010. Google Scholar
  42. Jelani Nelson and Huacheng Yu. Optimal lower bounds for distributed and streaming spanning forest computation. In Proc., SODA, pages 1844-1860, 2019. Google Scholar
  43. Hung Q Ngo and Ding-Zhu Du. A survey on combinatorial group testing algorithms with applications to dna library screening. Discrete mathematical problems with medical applications, 55:171-182, 2000. Google Scholar
  44. Noam Nisan. The demand query model for bipartite matching. Proc., SODA, pages 592-599, 2021. Google Scholar
  45. Ely Porat and Amir Rothschild. Explicit non-adaptive combinatorial group testing schemes. In Proc., ICALP, pages 748-759, 2008. Google Scholar
  46. Cyrus Rashtchian, David P. Woodruff, and Hanlin Zhu. Vector-matrix-vector queries for solving linear algebra, statistics, and graph problems. In Proc., International Workshop on Randomization and Computation (RANDOM), pages 26:1-26:20, 2020. Google Scholar
  47. Lev Reyzin and Nikhil Srivastava. Learning and verifying graphs using queries with a focus on edge counting. In Proc., International Conference on Algorithmic Learning Theory (ALT), pages 285-297. Springer, 2007. Google Scholar
  48. 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):15, 2016. Google Scholar
  49. Aviad Rubinstein, Tselil Schramm, and S. Matthew Weinberg. Computing exact minimum cuts without knowing the graph. In Proc., Innovations in Theoretical Computer Science (ITCS), pages 39:1-39:16, 2018. Google Scholar
  50. Miklós Ruszinkó and Peter Vanroose. How an Erdős-Rényi-type search approach gives an explicit code construction of rate 1 for random access with multiplicity feedback. IEEE Transactions on Information Theory, 43(1):368-373, 1997. Google Scholar
  51. Xiaoming Sun, David P. Woodruff, Guang Yang, and Jialin Zhang. Querying a matrix through matrix-vector products. In Proc., ICALP, pages 94:1-94:16, 2019. Google Scholar
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