Document Open Access Logo

Two-Party Direct-Sum Questions Through the Lens of Multiparty Communication Complexity

Authors Itay Hazan, Eyal Kushilevitz



PDF
Thumbnail PDF

File

LIPIcs.DISC.2017.26.pdf
  • Filesize: 475 kB
  • 15 pages

Document Identifiers

Author Details

Itay Hazan
Eyal Kushilevitz

Cite AsGet BibTex

Itay Hazan and Eyal Kushilevitz. Two-Party Direct-Sum Questions Through the Lens of Multiparty Communication Complexity. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 26:1-26:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.DISC.2017.26

Abstract

Direct-sum questions in (two-party) communication complexity ask whether two parties, Alice and Bob, can compute the value of a function f on l inputs (x_1,y_1),...,(x_l,y_l) more efficiently than by applying the best protocol for f, independently on each input (x_i,y_i). In spite of significant efforts to understand these questions (under various communication-complexity measures), the general question is still far from being well understood. In this paper, we offer a multiparty view of these questions: The direct-sum setting is just a two-player system with Alice having inputs x_1,...,x_l, Bob having inputs y_1,...,y_l and the desired output is f(x_1,y_1),...,f(x_l,y_l). The naive solution of solving the l problems independently, is modeled by a network with l (disconnected) pairs of players Alice i and Bob i, with inputs x_i,y_i respectively, and communication only within each pair. Then, we consider an intermediate ("star") model, where there is one Alice having l inputs x_1,...,x_l and l players Bob_1,...,Bob_l holding y_1,...,y_l, respectively (in fact, we consider few variants of this intermediate model, depending on whether communication between each Bob i and Alice is point-to-point or whether we allow broadcast). Our goal is to get a better understanding of the relation between the two extreme models (i.e., of the two-party direct-sum question). If, for instance, Alice and Bob can do better (for some complexity measure) than solving the l problems independently, we wish to understand what intermediate model already allows to do so (hereby understanding the "source" of such savings). If, on the other hand, we wish to prove that there is no better solution than solving the l problems independently, then our approach gives a way of breaking the task of proving such a statement into few (hopefully, easier) steps. We present several results of both types. Namely, for certain complexity measures, communication problems f and certain pairs of models, we can show gaps between the complexity of solving f on l instances in the two models in question; while, for certain other complexity measures and pairs of models, we can show that such gaps do not exist (for any communication problem f). For example, we prove that if only point-to-point communication is allowed in the intermediate "star" model, then significant savings are impossible in the public-coin randomized setting. On the other hand, in the private-coin randomized setting, if Alice is allowed to broadcast messages to all Bobs in the "star" network, then some savings are possible. While this approach does not lead yet to new results on the original two-party direct-sum question, we believe that our work gives new insights on the already-known direct-sum results, and may potentially lead to more such results in the future.
Keywords
  • Communication Complexity
  • Direct Sum
  • Multiparty Communication

Metrics

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

References

  1. Noga Alon, Klim Efremenko, and Benny Sudakov. Testing equality in communication graphs. arXiv preprint arXiv:1605.01658, 2016. Google Scholar
  2. Ziv Bar-Yossef, Thathachar S Jayram, Ravi Kumar, and D Sivakumar. An information statistics approach to data stream and communication complexity. In Foundations of Computer Science, 2002. Proceedings. The 43rd Annual IEEE Symposium on, pages 209-218. IEEE, 2002. Google Scholar
  3. Mark Braverman. Interactive information complexity. SIAM Journal on Computing, 44(6):1698-1739, 2015. Google Scholar
  4. Mark Braverman, Faith Ellen, Rotem Oshman, Toniann Pitassi, and Vinod Vaikuntanathan. A tight bound for set disjointness in the message-passing model. In Foundations of Computer Science (FOCS), 2013 IEEE 54th Annual Symposium on, pages 668-677. IEEE, 2013. Google Scholar
  5. Mark Braverman and Rotem Oshman. The communication complexity of number-in-hand set disjointness with no promise. In Electronic Colloquium on Computational Complexity (ECCC), volume 22, 2015. Google Scholar
  6. Mark Braverman and Anup Rao. Information equals amortized communication. IEEE Transactions on Information Theory, 60(10):6058-6069, 2014. Google Scholar
  7. Harry Buhrman, Matthias Christandl, and Jeroen Zuiddam. Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication. arXiv preprint arXiv:1603.03757, 2016. Google Scholar
  8. Amit Chakrabarti, Subhash Khot, and Xiaodong Sun. Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In Computational Complexity, 2003. Proceedings. 18th IEEE Annual Conference on, pages 107-117. IEEE, 2003. Google Scholar
  9. Amit Chakrabarti, Yaoyun Shi, Anthony Wirth, and Andrew Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Foundations of Computer Science, 2001. Proceedings. 42nd IEEE Symposium on, pages 270-278. IEEE, 2001. Google Scholar
  10. Tomas Feder, Eyal Kushilevitz, Moni Naor, and Noam Nisan. Amortized communication complexity. SIAM Journal on computing, 24(4):736-750, 1995. Google Scholar
  11. Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication. In Foundations of Computer Science (FOCS), 2014 IEEE 55th Annual Symposium on, pages 176-185. IEEE, 2014. Google Scholar
  12. Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of communication and external information. In STOC, pages 977-986, 2016. Google Scholar
  13. Anat Ganor, Gillat Kol, and Ran Raz. Exponential separation of information and communication for boolean functions. Journal of the ACM (JACM), 63(5):46, 2016. Google Scholar
  14. Bala Kalyanasundaram and Georg Schintger. The probabilistic communication complexity of set intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. Google Scholar
  15. Mauricio Karchmer, Eyal Kushilevitz, and Noam Nisan. Fractional covers and communication complexity. In Structure in Complexity Theory Conference, 1992., Proceedings of the Seventh Annual, pages 262-274. IEEE, 1992. Google Scholar
  16. Mauricio Karchmer, Ran Raz, and Avi Wigderson. Super-logarithmic depth lower bounds via the direct sum in communication complexity. Computational Complexity, 5(3-4):191-204, 1995. Early version in CCC 1991. Google Scholar
  17. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, Cambridge, 1997. Google Scholar
  18. Ilan Newman. Private vs. common random bits in communication complexity. Information processing letters, 39(2):67-71, 1991. Google Scholar
  19. Alon Orlitsky. Worst-case interactive communication. i. two messages are almost optimal. IEEE Transactions on Information Theory, 36(5):1111-1126, 1990. Google Scholar
  20. Alon Orlitsky. Worst-case interactive communication. ii. two messages are not optimal. IEEE Transactions on Information Theory, 37(4):995-1005, 1991. Google Scholar
  21. David Peleg. Distributed computing: a locality-sensitive approach. SIAM, 2000. Google Scholar
  22. Jeff M Phillips, Elad Verbin, and Qin Zhang. Lower bounds for number-in-hand multiparty communication complexity, made easy. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 486-501. SIAM, 2012. Google Scholar
  23. Alexander A. Razborov. On the distributional complexity of disjointness. Theoretical Computer Science, 106(2):385-390, 1992. Google Scholar
  24. David P Woodruff and Qin Zhang. Tight bounds for distributed functional monitoring. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 941-960. ACM, 2012. Google Scholar
  25. David P Woodruff and Qin Zhang. When distributed computation is communication expensive. In International Symposium on Distributed Computing, pages 16-30. Springer, 2013. Google Scholar
  26. 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, pages 718-733. Society for Industrial and Applied Mathematics, 2014. Google Scholar
  27. Andrew Yao. Some complexity questions related to distributed computing. In Proc. 11th STOC, pages 209-213, 1979. Google Scholar
  28. Andrew C Yao. Lower bounds by probabilistic arguments. In Foundations of Computer Science, 1983., 24th Annual Symposium on, pages 420-428. IEEE, 1983. 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