Document Open Access Logo

A New Approach to Multi-Party Peer-to-Peer Communication Complexity

Authors Adi Rosén, Florent Urrutia

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2019.64.pdf
  • Filesize: 498 kB
  • 19 pages

Document Identifiers

Author Details

Adi Rosén
  • CNRS and Université Paris Diderot
Florent Urrutia
  • Université Paris Diderot

Funding

  • Rosén, Adi: Research supported in part by ANR project RDAM.
  • Urrutia, Florent: Research supported in part by ERC QCC and by ANR project RDAM.

Cite AsGet BibTex

Adi Rosén and Florent Urrutia. A New Approach to Multi-Party Peer-to-Peer Communication Complexity. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 64:1-64:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ITCS.2019.64

Abstract

We introduce new models and new information theoretic measures for the study of communication complexity in the natural peer-to-peer, multi-party, number-in-hand setting. We prove a number of properties of our new models and measures, and then, in order to exemplify their effectiveness, we use them to prove two lower bounds. The more elaborate one is a tight lower bound of Omega(kn) on the multi-party peer-to-peer randomized communication complexity of the k-player, n-bit function Disjointness, Disj_k^n. The other one is a tight lower bound of Omega(kn) on the multi-party peer-to-peer randomized communication complexity of the k-player, n-bit bitwise parity function, Par_k^n. Both lower bounds hold when n=Omega(k). The lower bound for Disj_k^n improves over the lower bound that can be inferred from the result of Braverman et al. (FOCS 2013), which was proved in the coordinator model and can yield a lower bound of Omega(kn/log k) in the peer-to-peer model. To the best of our knowledge, our lower bounds are the first tight (non-trivial) lower bounds on communication complexity in the natural peer-to-peer multi-party setting. In addition to the above results for communication complexity, we also prove, using the same tools, an Omega(n) lower bound on the number of random bits necessary for the (information theoretic) private computation of the function Disj_k^n.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Mathematics of computing → Information theory
  • Theory of computation → Cryptographic protocols
Keywords
  • communication complexity
  • multi-party communication complexity
  • peer-to-peer communication complexity
  • information complexity
  • private computation

Metrics

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

References

  1. Gilad Asharov and Yehuda Lindell. A Full Proof of the BGW Protocol for Perfectly Secure Multiparty Computation. J. Cryptology, 30(1):58-151, 2017. URL: http://dx.doi.org/10.1007/s00145-015-9214-4.
  2. Reuven Bar-Yehuda, Benny Chor, Eyal Kushilevitz, and Alon Orlitsky. Privacy, additional information and communication. IEEE Transactions on Information Theory, 39(6):1930-1943, 1993. URL: http://dx.doi.org/10.1109/18.265501.
  3. Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci., 68(4):702-732, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2003.11.006.
  4. Boaz Barak, Mark Braverman, Xi Chen, and Anup Rao. How to compress interactive communication. In Proceedings of the 42nd ACM symposium on Theory of computing, STOC '10, pages 67-76, New York, NY, USA, 2010. ACM. URL: http://dx.doi.org/10.1145/1806689.1806701.
  5. Michael Ben-Or, Shafi Goldwasser, and Avi Wigderson. Completeness theorems for non-cryptographic fault-tolerant distributed computation. In Proceedings of the twentieth annual ACM symposium on Theory of computing, STOC '88, pages 1-10, New York, NY, USA, 1988. ACM. URL: http://dx.doi.org/10.1145/62212.62213.
  6. C. Blundo, A. De Santis, G. Persiano, and U. Vaccaro. Randomness complexity of private computation. computational complexity, 8(2):145-168, 1999. URL: http://dx.doi.org/10.1007/s000370050025.
  7. Mark Braverman. Interactive Information Complexity. SIAM J. Comput., 44(6):1698-1739, 2015. URL: http://dx.doi.org/10.1137/130938517.
  8. Mark Braverman, Faith Ellen, Rotem Oshman, Toniann Pitassi, and Vinod Vaikuntanathan. A Tight Bound for Set Disjointness in the Message-Passing Model. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 668-677. IEEE Computer Society, 2013. URL: http://dx.doi.org/10.1109/FOCS.2013.77.
  9. Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. From information to exact communication. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC '13, pages 151-160, New York, NY, USA, 2013. ACM. URL: http://dx.doi.org/10.1145/2488608.2488628.
  10. Mark Braverman and Rotem Oshman. On Information Complexity in the Broadcast Model. In Chryssis Georgiou and Paul G. Spirakis, editors, Proceedings of the 2015 ACM Symposium on Principles of Distributed Computing, PODC 2015, Donostia-San Sebastián, Spain, July 21 - 23, 2015, pages 355-364. ACM, 2015. URL: http://dx.doi.org/10.1145/2767386.2767425.
  11. Mark Braverman and Anup Rao. Information Equals Amortized Communication. IEEE Trans. Information Theory, 60(10):6058-6069, 2014. URL: http://dx.doi.org/10.1109/TIT.2014.2347282.
  12. Amit Chakrabarti and Sagar Kale. Strong Fooling Sets for Multi-player Communication with Applications to Deterministic Estimation of Stream Statistics. In Irit Dinur, editor, IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, New Brunswick, New Jersey, USA, pages 41-50. IEEE Computer Society, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.14.
  13. Amit Chakrabarti, Subhash Khot, and Xiaodong Sun. Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In In IEEE Conference on Computational Complexity, pages 107-117, 2003. Google Scholar
  14. Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Chi-Chih Yao. Informational Complexity and the Direct Sum Problem for Simultaneous Message Complexity. In FOCS, pages 270-278, 2001. URL: http://dx.doi.org/10.1109/SFCS.2001.959901.
  15. Arkadev Chattopadhyay and Sagnik Mukhopadhyay. Tribes Is Hard in the Message Passing Model. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 224-237. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.224.
  16. Arkadev Chattopadhyay and Toniann Pitassi. The Story of Set Disjointness. SIGACT News, 41(3):59-85, September 2010. URL: http://dx.doi.org/10.1145/1855118.1855133.
  17. Arkadev Chattopadhyay, Jaikumar Radhakrishnan, and Atri Rudra. Topology Matters in Communication. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 631-640, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.73.
  18. Arkadev Chattopadhyay and Atri Rudra. The Range of Topological Effects on Communication. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part II, volume 9135 of Lecture Notes in Computer Science, pages 540-551. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47666-6_43.
  19. David Chaum, Claude Crépeau, and Ivan Damgard. Multiparty unconditionally secure protocols. In Proceedings of the twentieth annual ACM symposium on Theory of computing, STOC '88, pages 11-19, New York, NY, USA, 1988. ACM. URL: http://dx.doi.org/10.1145/62212.62214.
  20. Danny Dolev and Tomás Feder. Multiparty Communication Complexity. In 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, North Carolina, USA, 30 October - 1 November 1989, pages 428-433. IEEE Computer Society, 1989. URL: http://dx.doi.org/10.1109/SFCS.1989.63514.
  21. Faith Ellen, Rotem Oshman, Toniann Pitassi, and Vinod Vaikuntanathan. Brief Announcement: Private Channel Models in Multi-party Communication Complexity. In 27th International Symposium on Distributed Computing (DISC), Jerusalem, Israel, pages 575-576, 2013. Google Scholar
  22. Uri Feige, Joe Killian, and Moni Naor. A Minimal Model for Secure Computation (Extended Abstract). In Proceedings of the Twenty-sixth Annual ACM Symposium on Theory of Computing, STOC '94, pages 554-563, New York, NY, USA, 1994. ACM. URL: http://dx.doi.org/10.1145/195058.195408.
  23. Silvio Frischknecht, Stephan Holzer, and Roger Wattenhofer. Networks cannot compute their diameter in sublinear time. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1150-1162. SIAM, 2012. URL: http://dx.doi.org/10.1137/1.9781611973099.
  24. Anna Gál and Parikshit Gopalan. Lower Bounds on Streaming Algorithms for Approximating the Length of the Longest Increasing Subsequence. SIAM J. Comput., 39(8):3463-3479, 2010. URL: http://dx.doi.org/10.1137/090770801.
  25. Anna Gál and Adi Rosén. Omega(log n) Lower Bounds on the Amount of Randomness in 2-Private Computation. SIAM J. Comput., 34(4):946-959, 2005. URL: http://dx.doi.org/10.1137/S0097539703432785.
  26. Andre Gronemeier. Asymptotically Optimal Lower Bounds on the NIH-Multi-Party Information Complexity of the AND-Function and Disjointness. In Susanne Albers and Jean-Yves Marion, editors, 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009, February 26-28, 2009, Freiburg, Germany, Proceedings, volume 3 of LIPIcs, pages 505-516. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany, 2009. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2009.1846.
  27. Zengfeng Huang, Bozidar Radunovic, Milan Vojnovic, and Qin Zhang. Communication Complexity of Approximate Matching in Distributed Graphs. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 460-473. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.STACS.2015.460.
  28. T. S. Jayram. Hellinger Strikes Back: A Note on the Multi-party Information Complexity of AND. In Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX '09 / RANDOM '09, pages 562-573, Berlin, Heidelberg, 2009. Springer-Verlag. URL: http://dx.doi.org/10.1007/978-3-642-03685-9_42.
  29. Bala Kalyanasundaram and Georg Schintger. The Probabilistic Communication Complexity of Set Intersection. SIAM J. Discret. Math., 5(4):545-557, November 1992. URL: http://dx.doi.org/10.1137/0405044.
  30. Iordanis Kerenidis, Adi Rosén, and Florent Urrutia. Multi-Party Protocols, Information Complexity and Privacy. In Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Kraków, Poland, volume 58 of LIPIcs, pages 57:1-57:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.MFCS.2016.57.
  31. Janne H. Korhonen and Jukka Suomela. Brief Announcement: Towards a Complexity Theory for the Congested Clique. In Andréa W. Richa, editor, 31st International Symposium on Distributed Computing, DISC 2017, October 16-20, 2017, Vienna, Austria, volume 91 of LIPIcs, pages 55:1-55:3. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.DISC.2017.55.
  32. Janne H. Korhonen and Jukka Suomela. Towards a complexity theory for the congested clique. CoRR, abs/1705.03284, 2017. URL: http://arxiv.org/abs/1705.03284.
  33. Eyal Kushilevitz and Yishay Mansour. Randomness in Private Computations. SIAM J. Discrete Math., 10(4):647-661, 1997. URL: http://dx.doi.org/10.1137/S0895480196306130.
  34. Eyal Kushilevitz and Noam Nisan. Communication complexity. Cambridge University Press, 1997. Google Scholar
  35. Eyal Kushilevitz, Rafail Ostrovsky, and Adi Rosén. Characterizing Linear Size Circuits in Terms of Pricacy. J. Comput. Syst. Sci., 58(1):129-136, 1999. URL: http://dx.doi.org/10.1006/jcss.1997.1544.
  36. Peter Bro Miltersen, Noam Nisan, Shmuel Safra, and Avi Wigderson. On Data Structures and Asymmetric Communication Complexity. J. Comput. Syst. Sci., 57(1):37-49, 1998. URL: http://dx.doi.org/10.1006/jcss.1998.1577.
  37. Jeff M. Phillips, Elad Verbin, and Qin Zhang. Lower Bounds for Number-in-Hand Multiparty Communication Complexity, Made Easy. SIAM J. Comput., 45(1):174-196, 2016. URL: http://dx.doi.org/10.1137/15M1007525.
  38. A. A. Razborov. On the Distributional Complexity of Disjointness. Theor. Comput. Sci., 106(2):385-390, December 1992. URL: http://dx.doi.org/10.1016/0304-3975(92)90260-M.
  39. Atish Das Sarma, Stephan Holzer, Liah Kor, Amos Korman, Danupon Nanongkai, Gopal Pandurangan, David Peleg, and Roger Wattenhofer. Distributed Verification and Hardness of Distributed Approximation. CoRR, abs/1011.3049, 2010. URL: http://arxiv.org/abs/1011.3049.
  40. C. E. Shannon. A mathematical theory of communication. Bell system technical journal, 27, 1948. Google Scholar
  41. David P. Woodruff and Qin Zhang. Tight Bounds for Distributed Functional Monitoring. CoRR, abs/1112.5153, 2011. URL: http://arxiv.org/abs/1112.5153.
  42. David P. Woodruff and Qin Zhang. When Distributed Computation does not Help. CoRR, abs/1304.4636, 2013. URL: http://arxiv.org/abs/1304.4636.
  43. David P. Woodruff and Qin Zhang. An Optimal Lower Bound for Distinct Elements in the Message Passing Model. In Proceedings of the Twenty-fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '14, pages 718-733, Philadelphia, PA, USA, 2014. Society for Industrial and Applied Mathematics. URL: http://dl.acm.org/citation.cfm?id=2634074.2634128.
  44. Andrew Chi-Chih Yao. Protocols for Secure Computations (Extended Abstract). In 23rd Annual Symposium on Foundations of Computer Science, Chicago, Illinois, USA, 3-5 November 1982, pages 160-164. IEEE Computer Society, 1982. URL: http://dx.doi.org/10.1109/SFCS.1982.38.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact