Graph Connectivity and Single Element Recovery via Linear and OR Queries
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.
Query Models
Graph Connectivity
Group Testing
Duality
Theory of computation
7:1-7:19
Regular Paper
https://arxiv.org/abs/2007.06098
Sepehr
Assadi
Sepehr Assadi
Rutgers University, New Brunswick, NJ, USA
Supported in part by NSF CAREER award CCF-2047061, and gift from Google Research.
Deeparnab
Chakrabarty
Deeparnab Chakrabarty
Dartmouth College, Hanover, NH, USA
Supported in part by the NSF awards CCF-1813053 and CCF-2041920.
Sanjeev
Khanna
Sanjeev Khanna
University of Pennsylvania, Philadelphia, PA, USA
Supported in part by the NSF awards CCF-1763514, CCF-1617851, CCF-1934876, and CCF-2008305.
10.4230/LIPIcs.ESA.2021.7
Hasan Abasi and Nader H. Bshouty. On learning graphs with edge-detecting queries. CoRR, abs/1803.10639, 2018.
Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing graph structure via linear measurements. In Proc., SODA, pages 459-467, 2012.
Noga Alon and Vera Asodi. Learning a hidden subgraph. SIAM Journal on Discrete Mathematics (SIDMA), 18(4):697-712, 2005.
Noga Alon, Richard Beigel, Simon Kasif, Steven Rudich, and Benny Sudakov. Learning a hidden matching. In Proc., FOCS, page 197, 2002.
Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. J. Comput. System Sci., 58(1):137-147, 1999.
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.
Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna. Graph connectivity and single element recovery via linear and OR queries. arXiv preprint arXiv:2007.06098, 2020.
Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Polynomial pass lower bounds for graph streaming algorithms. In Proc., STOC, pages 265-276, 2019.
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.
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.
Anup Bhattacharya, Arijit Bishnu, Arijit Ghosh, and Gopinath Mishra. Triangle estimation using polylogarithmic queries. CoRR, abs/1808.00691, 2018.
Nader H Bshouty. Optimal algorithms for the coin weighing problem with a spring scale. In Proc., Conf. on Learning Theory, 2009.
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.
Nader H Bshouty and Hanna Mazzawi. Reconstructing weighted graphs with minimal query complexity. Theoretical Computer Science, 412(19):1782-1790, 2011.
Nader H. Bshouty and Hanna Mazzawi. Toward a deterministic polynomial time algorithm with optimal additive query complexity. Theoretical Computer Science, 417:23-35, 2012.
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.
Sung-Soon Choi and Jeong Han Kim. Optimal query complexity bounds for finding graphs. In Proc., STOC, pages 749-758, 2008.
Graham Cormode and Donatella Firmani. A unifying framework for 𝓁₀-sampling algorithms. Distributed and Parallel Databases, 32(3):315-335, 2014.
Graham Cormode and S Muthukrishnan. Combinatorial algorithms for compressed sensing. In SIROCCO, pages 280-294. Springer, 2006.
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.
Holger Dell and John Lapinskas. Fine-grained reductions from approximate counting to decision. In Proc., STOC, pages 281-288. ACM, 2018.
David L Donoho et al. Compressed sensing. IEEE Transactions on information theory, 52(4):1289-1306, 2006.
Robert Dorfman. The detection of defective members of large populations. The Annals of Mathematical Statistics, 14(4):436-440, 1943.
Dingzhu Du and Frank K Hwang. Combinatorial group testing and its applications, volume 12. World Scientific, 2000.
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.
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.
Philippe Flajolet and G Nigel Martin. Probabilistic counting algorithms for data base applications. J. Comput. System Sci., 31(2):182-209, 1985.
Gereon Frahling, Piotr Indyk, and Christian Sohler. Sampling in dynamic data streams and applications. International Journal of Computational Geometry & Applications, 18:3-28, 2008.
Sumit Ganguly. Lower bounds on frequency estimation of data streams. In International Computer Science Symposium in Russia (CSR), pages 204-215, 2008.
Vladimir Grebinski and Gregory Kucherov. Optimal reconstruction of graphs under the additive model. Algorithmica, 28(1):104-124, 2000.
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.
FK Hwang and VT Sós. Non-adaptive hypergeometric group testing. Studia Sci. Math. Hungar, 22(1-4):257-263, 1987.
Piotr Indyk. Explicit constructions for compressed sensing of sparse signals. In Proc., SODA, pages 30-33, 2008.
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.
John Kallaugher and Eric Price. Separations and equivalences between turnstile streaming and linear sketching. In Proc., STOC, pages 1223-1236, 2020.
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.
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.
W Kautz and Roy Singleton. Nonrandom binary superimposed codes. IEEE Transactions on Information Theory, 10(4):363-377, 1964.
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.
Bernt Lindström. On Möbius functions and a problem in combinatorial number theory. Canadian Mathematical Bulletin, 14(4):513-516, 1971.
Hanna Mazzawi. Optimally reconstructing weighted graphs using queries. In Proc., SODA, pages 608-615, 2010.
Jelani Nelson and Huacheng Yu. Optimal lower bounds for distributed and streaming spanning forest computation. In Proc., SODA, pages 1844-1860, 2019.
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.
Noam Nisan. The demand query model for bipartite matching. Proc., SODA, pages 592-599, 2021.
Ely Porat and Amir Rothschild. Explicit non-adaptive combinatorial group testing schemes. In Proc., ICALP, pages 748-759, 2008.
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.
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.
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.
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.
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.
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.
Sepehr Assadi, Deeparnab Chakrabarty, and Sanjeev Khanna
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