research-article

Lower Bounds for Distributed Sketching of Maximal Matchings and Maximal Independent Sets

Authors:

Rotem OshmanAuthors Info & Claims

PODC '20: Proceedings of the 39th Symposium on Principles of Distributed Computing

Pages 79 - 88

https://doi.org/10.1145/3382734.3405732

Published: 31 July 2020 Publication History

Abstract

Consider the following distributed graph sketching model: There is a referee and n vertices in an undirected graph G sharing public randomness. Each vertex v only knows its neighborhood in G and the referee receives no input initially. The vertices simultaneously each sends a message, called a sketch, to the referee who then based on the received sketches outputs a solution to some combinatorial problem on G, say, the minimum spanning tree problem.

Previous work on graph sketching have shown that numerous problems, including connectivity, minimum spanning tree, edge or vertex connectivity, cut or spectral sparsifiers, and (Δ + 1)-vertex coloring, all admit efficient algorithms in this model that only require sketches of size polylog(n) per vertex. In contrast, we prove that the two fundamental problems of maximal matching and maximal independent set do not admit such efficient solutions: Any algorithm for either problem that errs with a small constant probability requires sketches of size Ω(n^1/2--ε) for any constant ε > 0.

We prove our results by analyzing communication complexity of these problems in a communication model that allows sharing of inputs between limited number of players, and hence lies between the standard number-in-hand and number-on-forehead multi-party communication models. Our proofs are based on a family of hard instances using Ruzsa-Szemerédi graphs and information-theoretic arguments to establish the communication lower bounds.

References

[1]

Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. 2012. Analyzing Graph Structure via Linear Measurements. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '12). SIAM, 459--467. http://dl.acm.org/citation.cfm?id=2095116.2095156 1, 2, 3

Digital Library

[2]

Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. 2012. Graph sketches: sparsification, spanners, and subgraphs. In Proceedings of the 31st ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS 2012, Scottsdale, AZ, USA, May 20-24, 2012. 5--14. 1, 3

Digital Library

[3]

Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. 2013. Spectral Sparsification in Dynamic Graph Streams. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings. 1--10. 1

[4]

Noga Alon. 2002. Testing subgraphs in large graphs. Random Struct. Algorithms 21, 3-4 (2002), 359--370. 3

Digital Library

[5]

Noga Alon, Ankur Moitra, and Benny Sudakov. 2012. Nearly complete graphs decomposable into large induced matchings and their applications. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012. 1079--1090. 3

Digital Library

[6]

Noga Alon, Noam Nisan, Ran Raz, and Omri Weinstein. 2015. Welfare Maximization with Limited Interaction. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015. 1499--1512. 3

[7]

Noga Alon and Asaf Shapira. 2006. A Characterization of Easily Testable Induced Subgraphs. Combinatorics, Probability & Computing 15, 6 (2006), 791--805. 3

Digital Library

[8]

Alexandr Andoni, Assaf Goldberger, Andrew McGregor, and Ely Porat. 2013. Homomorphic fingerprints under misalignments: sketching edit and shift distances. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013. 931--940. 2

Digital Library

[9]

Sepehr Assadi. 2017. Combinatorial Auctions Do Need Modest Interaction. In Proceedings of the 2017 ACM Conference on Economics and Computation, EC '17, Cambridge, MA, USA, June 26-30, 2017. 145--162. 3

Digital Library

[10]

Sepehr Assadi and Aaron Bernstein. 2019. Towards a Unified Theory of Sparsification for Matching Problems. In 2nd Symposium on Simplicity in Algorithms, SOSA@SODA 2019, January 8-9, 2019 - San Diego, CA, USA. 11:1--11:20. 3

[11]

Sepehr Assadi, Yu Chen, and Sanjeev Khanna. 2019. Sublinear Algorithms for (Δ + 1) Vertex Coloring. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019. 767--786. 1, 2

[12]

Sepehr Assadi, Sanjeev Khanna, and Yang Li. 2016. Tight bounds for single-pass streaming complexity of the set cover problem. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016. 698--711. 3

Digital Library

[13]

Sepehr Assadi, Sanjeev Khanna, and Yang Li. 2017. On Estimating Maximum Matching Size in Graph Streams. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19. 1723--1742. 3

[14]

Sepehr Assadi, Sanjeev Khanna, Yang Li, and Grigory Yaroslavtsev. 2016. Maximum Matchings in Dynamic Graph Streams and the Simultaneous Communication Model. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016. 1345--1364. 2, 3

Digital Library

[15]

László Babai, Anna Gál, Peter G. Kimmel, and Satyanarayana V. Lokam. 2003. Communication Complexity of Simultaneous Messages. SIAM J. Comput. 33, 1 (2003), 137--166. 3

Digital Library

[16]

Florent Becker, Adrian Kosowski, Martín Matamala, Nicolas Nisse, Ivan Rapaport, Karol Suchan, and Ioan Todinca. 2015. Allowing each node to communicate only once in a distributed system: shared whiteboard models. Distributed Comput. 28, 3 (2015), 189--200. 2

Digital Library

[17]

Florent Becker, Martín Matamala, Nicolas Nisse, Ivan Rapaport, Karol Suchan, and Ioan Todinca. 2011. Adding a Referee to an Interconnection Network: What Can(not) Be Computed in One Round. In 25th IEEE International Symposium on Parallel and Distributed Processing, IPDPS 2011, Anchorage, Alaska, USA, 16-20 May, 2011 - Conference Proceedings. 508--514. 2, 3

Digital Library

[18]

Florent Becker, Pedro Montealegre, Ivan Rapaport, and Ioan Todinca. 2014. The Simultaneous Number-in-Hand Communication Model for Networks: Private Coins, Public Coins and Determinism. In Structural Information and Communication Complexity - 21st International Colloquium, SIROCCO 2014, Takayama, Japan, July 23-25, 2014. Proceedings. 83--95. 2

[19]

Florent Becker, Pedro Montealegre, Ivan Rapaport, and Ioan Todinca. 2018. The Impact of Locality on the Detection of Cycles in the Broadcast Congested Clique Model. In LATIN 2018: Theoretical Informatics - 13th Latin American Symposium, Buenos Aires, Argentina, April 16-19, 2018, Proceedings. 134--145. 2

[20]

Felix A Behrend. 1946. On sets of integers which contain no three terms in arithmetical progression. Proceedings of the National Academy of Sciences of the United States of America 32, 12 (1946), 331. 3

[21]

Bertinoro. Bertinoro workshop 2014, problem 65. https://sublinear.info/65. (????). Accessed: 2020-2-14. 2

[22]

Sayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, and Charalampos E. Tsourakakis. 2015. Space- and Time-Efficient Algorithm for Maintaining Dense Subgraphs on One-Pass Dynamic Streams. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015, Portland, OR, USA, June 14-17, 2015. 173--182. 1

Digital Library

[23]

Yitzhak Birk, Nathan Linial, and Roy Meshulam. 1993. On the uniform-traffic capacity of single-hop interconnections employing shared directional multichannels. IEEE Transactions on Information Theory 39, 1 (1993), 186--191. 3

Digital Library

[24]

Mark Braverman and Rotem Oshman. 2017. A Rounds vs. Communication Tradeoff for Multi-Party Set Disjointness. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017. 144--155. 3

[25]

Ashok K. Chandra, Merrick L. Furst, and Richard J. Lipton. 1983. Multi-Party Protocols. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA. 94--99. 3

[26]

Graham Cormode, Jacques Dark, and Christian Konrad. 2019. Independent Sets in Vertex-Arrival Streams. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece. 45:1--45:14. 2, 3

[27]

Thomas M. Cover and Joy A. Thomas. 2006. Elements of information theory (2. ed.). Wiley. 3

Digital Library

[28]

Debarati Das, Michal Koucký, and Michael E. Saks. 2018. Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication. In 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France. 23:1--23:14. 3

[29]

Shahar Dobzinski, Noam Nisan, and Sigal Oren. 2014. Economic efficiency requires interaction. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014. 233--242. 3

Digital Library

[30]

Andrew Drucker, Fabian Kuhn, and Rotem Oshman. 2014. On the power of the congested clique model. In ACM Symposium on Principles of Distributed Computing, PODC '14, Paris, France, July 15-18, 2014. 367--376. 2, 3

Digital Library

[31]

Martin Farach-Colton and Meng-Tsung Tsai. 2016. Tight Approximations of Degeneracy in Large Graphs. In LATIN 2016: Theoretical Informatics - 12th Latin American Symposium, Ensenada, Mexico, April 11-15, 2016, Proceedings. 429--440. 1

[32]

Eldar Fischer, Eric Lehman, Ilan Newman, Sofya Raskhodnikova, Ronitt Rubinfeld, and Alex Samorodnitsky. 2002. Monotonicity testing over general poset domains. In Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada. 474--483. 3

Digital Library

[33]

Orr Fischer, Rotem Oshman, and Uri Zwick. 2016. Public vs. Private Randomness in Simultaneous Multi-party Communication Complexity. In Structural Information and Communication Complexity - 23rd International Colloquium, SIROCCO 2016, Helsinki, Finland, July 19-21, 2016, Revised Selected Papers. 60--74. 3

[34]

Jacob Fox, Hao Huang, and Benny Sudakov. 2015. On graphs decomposable into induced matchings of linear sizes. arXiv preprint arXiv:1512.07852 (2015). 3

[35]

Mohsen Ghaffari, Themis Gouleakis, Christian Konrad, Slobodan Mitrovic, and Ronitt Rubinfeld. 2018. Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover. In Proceedings of the 2018 ACM Symposium on Principles of Distributed Computing, PODC 2018, July 23-27, 2018. 129--138. 2

Digital Library

[36]

Ashish Goel, Michael Kapralov, and Sanjeev Khanna. 2012. On the Communication and Streaming Complexity of Maximum Bipartite Matching. In Proceedings of the Twenty-third Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '12). SIAM, 468--485. http://dl.acm.org/citation.cfm?id=2095116.2095157 3

Digital Library

[37]

Sudipto Guha, Andrew McGregor, and David Tench. 2015. Vertex and Hyperedge Connectivity in Dynamic Graph Streams. In Proceedings of the 34th ACM Symposium on Principles of Database Systems, PODS 2015, Melbourne, Victoria, Australia, May 31 - June 4, 2015. 241--247. 1

Digital Library

[38]

Johan Håstad and Avi Wigderson. 2003. Simple analysis of graph tests for linearity and PCP. Random Struct. Algorithms 22, 2 (2003), 139--160. 3

Digital Library

[39]

Tomasz Jurdzinski, Krzysztof Lorys, and Krzysztof Nowicki. 2018. Communication Complexity in Vertex Partition Whiteboard Model. In Structural Information and Communication Complexity - 25th International Colloquium, SIROCCO 2018, Ma'ale HaHamisha, Israel, June 18-21, 2018, Revised Selected Papers. 264--279. 2, 3

[40]

Tomasz Jurdzinski and Krzysztof Nowicki. 2017. Brief Announcement: On Connectivity in the Broadcast Congested Clique. In 31st International Symposium on Distributed Computing, DISC 2017, October 16-20, 2017, Vienna, Austria. 54:1--54:4. 2

[41]

Tomasz Jurdzinski and Krzysztof Nowicki. 2018. Connectivity and Minimum Cut Approximation in the Broadcast Congested Clique. In Structural Information and Communication Complexity - 25th International Colloquium, SIROCCO 2018, Ma'ale HaHamisha, Israel, June 18-21, 2018, Revised Selected Papers. 331--344. 2

[42]

Michael Kapralov. 2013. Better bounds for matchings in the streaming model. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013. 1679--1697. 3

[43]

Michael Kapralov, Yin Tat Lee, Cameron Musco, Christopher Musco, and Aaron Sidford. 2014. Single Pass Spectral Sparsification in Dynamic Streams. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014. 561--570. 1

Digital Library

[44]

Michael Kapralov, Jelani Nelson, Jakub Pachocki, Zhengyu Wang, David P. Woodruff, and Mobin Yahyazadeh. 2017. Optimal Lower Bounds for Universal Relation, and for Samplers and Finding Duplicates in Streams. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017. 475--486. 2

[45]

Christian Konrad. 2015. Maximum Matching in Turnstile Streams. In Algorithms - ESA 2015 - 23rd Annual European Symposium, September 14-16, 2015, Proceedings. 840--852. 3

[46]

Silvio Lattanzi, Benjamin Moseley, Siddharth Suri, and Sergei Vassilvitskii. 2011. Filtering: a method for solving graph problems in MapReduce. In SPAA 2011: Proceedings of the 23rd Annual ACM Symposium on Parallelism in Algorithms and Architectures, San Jose, CA, USA, June 4-6, 2011 (Co-located with FCRC 2011). 85--94. 2

Digital Library

[47]

Nati Linial, Toniann Pitassi, and Adi Shraibman. 2019. On the Communication Complexity of High-Dimensional Permutations. In 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA. 54:1--54:20. 3

[48]

Andrew McGregor, David Tench, Sofya Vorotnikova, and Hoa T. Vu. 2015. Densest Subgraph in Dynamic Graph Streams. In Mathematical Foundations of Computer Science 2015 - 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part II. 472--482. 1

[49]

Pedro Montealegre, Sebastian Perez-Salazar, Ivan Rapaport, and Ioan Todinca. 2018. Two Rounds Are Enough for Reconstructing Any Graph (Class) in the Congested Clique Model. In Structural Information and Communication Complexity - 25th International Colloquium, SIROCCO 2018, Ma'ale HaHamisha, Israel, June 18-21, 2018, Revised Selected Papers. 134--148. 3

[50]

Jelani Nelson and Huacheng Yu. 2019. Optimal Lower Bounds for Distributed and Streaming Spanning Forest Computation. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6-9, 2019. 1844--1860. 2, 3, 4

[51]

Imre Z Ruzsa and Endre Szemerédi. 1978. Triple systems with no six points carrying three triangles. Combinatorics (Keszthely, 1976), Coll. Math. Soc. J. Bolyai 18 (1978), 939--945. 3

[52]

Terence Tao and Van H Vu. 2006. Additive combinatorics. Vol. 105. Cambridge University Press. 3

[53]

Andrew Chi-Chih Yao. 1977. Probabilistic Computations: Toward a Unified Measure of Complexity (Extended Abstract). In 18th Annual Symposium on Foundations of Computer Science, Providence, Rhode Island, USA, 31 October - 1 November 1977. 222--227. 5

Cited By

Flin MGhaffari MHalldórsson MKuhn FNolin AAgrawal KShun J(2023)Coloring Fast with BroadcastsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591095(455-465)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591095
Assadi SKol GZhang Z(2022)Rounds vs Communication Tradeoffs for Maximal Independent Sets2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00115(1193-1204)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00115
Montealegre PRamírez-Romero DRapaport I(2021)Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary GraphsStabilization, Safety, and Security of Distributed Systems10.1007/978-3-030-91081-5_26(395-409)Online publication date: 9-Nov-2021
https://doi.org/10.1007/978-3-030-91081-5_26

Index Terms

Lower Bounds for Distributed Sketching of Maximal Matchings and Maximal Independent Sets
1. Theory of computation
  1. Design and analysis of algorithms
    1. Distributed algorithms

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cover image ACM Conferences

PODC '20: Proceedings of the 39th Symposium on Principles of Distributed Computing

July 2020

539 pages

ISBN:9781450375825

DOI:10.1145/3382734

General Chair:
Yuval Emek
Technion, Israel
,
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Christian Cachin
University of Bern, Switzerland

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Published: 31 July 2020

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Cited By

Flin MGhaffari MHalldórsson MKuhn FNolin AAgrawal KShun J(2023)Coloring Fast with BroadcastsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591095(455-465)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591095
Assadi SKol GZhang Z(2022)Rounds vs Communication Tradeoffs for Maximal Independent Sets2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS54457.2022.00115(1193-1204)Online publication date: Oct-2022
https://doi.org/10.1109/FOCS54457.2022.00115
Montealegre PRamírez-Romero DRapaport I(2021)Compact Distributed Interactive Proofs for the Recognition of Cographs and Distance-Hereditary GraphsStabilization, Safety, and Security of Distributed Systems10.1007/978-3-030-91081-5_26(395-409)Online publication date: 9-Nov-2021
https://doi.org/10.1007/978-3-030-91081-5_26

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