Skip to main content
SpringerLink
Log in

Direct Sum Fails for Zero-Error Average Communication

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We show that in the model of zero-error communication complexity, direct sum fails for average communication complexity as well as for external information complexity. Our example also refutes a version of a conjecture by Braverman et al. that in the zero-error case amortized communication complexity equals external information complexity. In our examples the underlying distributions do not have full support. One interpretation of a distribution of non full support is as a promise given to the players (the players have a guarantee on their inputs). This brings up the issue of promise versus non-promise problems in this context.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. This is because in any non trivial case the communication complexity is at least one.

  2. The limit exists since the corresponding sequence is Cauchy.

  3. Even for fixed (x, y) the transcript T(x, y) is a random variable that depends on the private coins.

  4. \(\mathrm{supp}(\mu )\) is the set of all \((x,y)\in \mathcal{X}\times \mathcal{Y}\) such that \(\mu (x,y)>0\).

  5. By \(\pi (x,y,r) = f(x,y)\) we mean that knowledge of the transcript and the public randomness yields knowledge of f as in Sect. 2.1.

References

  1. Barak, B., Braverman, M., Chen, X., Rao, A.: How to compress interactive communication. SIAM J. Comput. 42(3), 1327–1363 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Braverman, M.: Coding for interactive computation: progress and challenges. In: Allerton (2012)

  3. Braverman, M.: Interactive information complexity. SIAM J. Comput. 44(6), 1698–1739 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Braverman, M., Garg, A., Pankratov, D., Weinstein, O.: From information to exact communication. In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) Symposium on Theory of Computing Conference, STOC’13, Palo Alto, CA, USA, June 1–4, 2013, pp. 151–160. ACM (2013)

  5. Braverman, M., Rao, A.: Information equals amortized communication. IEEE Trans. Inf. Theory 60(10), 6058–6069 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brody, J., Chakrabarti, A., Kondapally, R., Woodruff, D.P., Yaroslavtsev, G.: Certifying equality with limited interaction. In: Jansen, K., Rolim, J. D. P., Devanur, N.R., Moore, C. (eds.) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4–6, 2014, Barcelona, Spain, volume 28 of LIPIcs, pp. 545–581. Schloss Dagstuhl—Leibniz-Zentrum fuer Informatik (2014)

  7. Chakrabarti, A., Shi, Y., Wirth, A., Yao, A.C.-C.: Informational complexity and the direct sum problem for simultaneous message complexity. In: 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14–17 October 2001, Las Vegas, Nevada, USA, pp. 270–278. IEEE Computer Society (2001)

  8. Feder, T., Kushilevitz, E., Naor, M., Nisan, N.: Amortized communication complexity. SIAM J. Comput. 24(4), 736–750 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Harsha, P., Jain, R., McAllester, D.A., Radhakrishnan, J.: The communication complexity of correlation. IEEE Trans. Inf. Theory 56(1), 438–449 (2010)

    Article  MathSciNet  Google Scholar 

  10. Jain, R.: New strong direct product results in communication complexity. JACM 62(3), 20 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jain, R., Radhakrishnan, J., Sen, P.: A direct sum theorem in communication complexity via message compression. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.), Automata, Languages and Programming, 30th International Colloquium, ICALP 2003, Eindhoven, The Netherlands, June 30–July 4, 2003. Proceedings, volume 2719 of Lecture Notes in Computer Science, pp. 300–315. Springer (2003)

  12. Klauck, H.: A strong direct product theorem for disjointness. In: Schulman, L.J. (ed.) Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5–8 June 2010, pp. 77–86. ACM (2010)

  13. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, USA

    Gillat Kol

  2. Department of Computer Science, Technion-IIT, Haifa, Israel

    Shay Moran

  3. Max Planck Institute for Informatics, Saarbrücken, Germany

    Shay Moran

  4. Department of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Amir Shpilka

  5. Department of Mathematics, Technion-IIT, Haifa, Israel

    Amir Yehudayoff

Authors
  1. Gillat Kol
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shay Moran
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amir Shpilka
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Amir Yehudayoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shay Moran.

Additional information

Gillat Kol: Research at the IAS supported by The Fund For Math and the Weizmann Institute of Science National Postdoctoral Award Program for Advancing Women in Science. Part of this work was done while the author was visiting the Simons Institute for the Theory of Computing, Berkeley, CA, and the Technion, Israel.

Shay Moran and Amir Shpilka: The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement number 257575.

Amir Yehudayoff: Horev fellow—supported by the Taub foundation. Research is also supported by ISF and BSF.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kol, G., Moran, S., Shpilka, A. et al. Direct Sum Fails for Zero-Error Average Communication. Algorithmica 76, 782–795 (2016). https://doi.org/10.1007/s00453-016-0144-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-016-0144-9

Keywords

Search

Navigation