Document Open Access Logo

Separating k-Player from t-Player One-Way Communication, with Applications to Data Streams

Authors David P. Woodruff, Guang Yang



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2019.97.pdf
  • Filesize: 0.52 MB
  • 14 pages

Document Identifiers

Author Details

David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, USA
Guang Yang
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
  • Conflux, Beijing, China

Acknowledgements

We would like to thank Yuval Ishai and Eyal Kushilevitz for initiating the problem of separating worst-case partition communication complexity from streaming complexity, which was our starting point. We also thank the ICALP referees for very helpful comments which helped us revise our initial submission. D. Woodruff would also like to thank the Chinese Academy of Sciences, as well as the Simons Institute for the Theory of Computing.

Cite AsGet BibTex

David P. Woodruff and Guang Yang. Separating k-Player from t-Player One-Way Communication, with Applications to Data Streams. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 97:1-97:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.97

Abstract

In a k-party communication problem, the k players with inputs x_1, x_2, ..., x_k, respectively, want to evaluate a function f(x_1, x_2, ..., x_k) using as little communication as possible. We consider the message-passing model, in which the inputs are partitioned in an arbitrary, possibly worst-case manner, among a smaller number t of players (t<k). The t-player communication cost of computing f can only be smaller than the k-player communication cost, since the t players can trivially simulate the k-player protocol. But how much smaller can it be? We study deterministic and randomized protocols in the one-way model, and provide separations for product input distributions, which are optimal for low error probability protocols. We also provide much stronger separations when the input distribution is non-product. A key application of our results is in proving lower bounds for data stream algorithms. In particular, we give an optimal Omega(epsilon^{-2}log(N) log log(mM)) bits of space lower bound for the fundamental problem of (1 +/-{epsilon})-approximating the number |x |_0 of non-zero entries of an n-dimensional vector x after m updates each of magnitude M, and with success probability >= 2/3, in a strict turnstile stream. Our result matches the best known upper bound when epsilon >= 1/polylog(mM). It also improves on the prior Omega({epsilon}^{-2}log(mM)) lower bound and separates the complexity of approximating L_0 from approximating the p-norm L_p for p bounded away from 0, since the latter has an O(epsilon^{-2}log(mM)) bit upper bound.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming models
  • Theory of computation → Complexity classes
  • Theory of computation → Lower bounds and information complexity
Keywords
  • Communication complexity
  • multi-player communication
  • one-way communication
  • streaming complexity

Metrics

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

References

  1. Farid Ablayev. Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theoretical Computer Science, 157(2):139-159, 1996. Google Scholar
  2. Yuqing Ai, Wei Hu, Yi Li, and David P. Woodruff. New Characterizations in Turnstile Streams with Applications. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 20:1-20:22, 2016. Google Scholar
  3. Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. Journal of Computer and System Sciences, 68(4):702-732, June 2004. Google Scholar
  4. Paul Beame, Toniann Pitassi, Nathan Segerlind, and Avi Wigderson. A Direct Sum Theorem for Corruption and the Multiparty NOF Communication Complexity of Set Disjointness. In 20th Annual IEEE Conference on Computational Complexity (CCC 2005), 11-15 June 2005, San Jose, CA, USA, pages 52-66, 2005. Google Scholar
  5. Mark Braverman and Ankit Garg. Public vs Private Coin in Bounded-Round Information. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 502-513, 2014. Google Scholar
  6. Joshua Brody, Harry Buhrman, Michal Koucký, Bruno Loff, Florian Speelman, and Nikolay K. Vereshchagin. Towards a Reverse Newman’s Theorem in Interactive Information Complexity. Algorithmica, 76(3):749-781, 2016 (also CCC 2013). Google Scholar
  7. Graham Cormode, Mayur Datar, Piotr Indyk, and S. Muthukrishnan. Comparing Data Streams Using Hamming Norms (How to Zero In). IEEE Trans. Knowl. Data Eng., 15(3):529-540, 2003. Google Scholar
  8. André Gronemeier. Asymptotically Optimal Lower Bounds on the NIH-Multi-Party Information Complexity of the AND-Function and Disjointness. In 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009, February 26-28, 2009, Freiburg, Germany, Proceedings, pages 505-516, 2009. Google Scholar
  9. Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. ACM, 53(3):307-323, 2006. Google Scholar
  10. T. S. Jayram. Hellinger strikes back: A note on the multi-party information complexity of AND. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX '09 / RANDOM '09, Berkeley, CA, USA, August 21 - 23, 2009, pages 562-573, 2009. Google Scholar
  11. Daniel M. Kane, Jelani Nelson, Ely Porat, and David P. Woodruff. Fast moment estimation in data streams in optimal space. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 745-754, 2011. Google Scholar
  12. Daniel M. Kane, Jelani Nelson, and David P. Woodruff. An optimal algorithm for the distinct elements problem. In Proceedings of the twenty-ninth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, pages 41-52, 2010. Google Scholar
  13. Daniel M. Kane, Jelani Nelson, and David P. Woodruff. On the Exact Space Complexity of Sketching and Streaming Small Norms. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 1161-1178, 2010. Google Scholar
  14. Ilan Kremer, Noam Nisan, and Dana Ron. On randomized one-round communication complexity. Computational Complexity, 8(1):21-49, 1999. Google Scholar
  15. Christos H. Papadimitriou and Michael Sipser. Communication complexity. Journal of Computer and System Sciences, 28(2):260-269, 1984. Google Scholar
  16. Emanuele Viola. The communication complexity of addition. SODA, 2013. Google Scholar
  17. David P. Woodruff and Qin Zhang. Tight bounds for distributed functional monitoring. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 941-960, 2012. 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