, Volume 76, Issue 3, pp 782–795 | Cite as

Direct Sum Fails for Zero-Error Average Communication

  • Gillat Kol
  • Shay Moran
  • Amir Shpilka
  • Amir Yehudayoff


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.


Communication complexity Information complexity External information Amortized communication complexity Promise problems 


  1. 1.
    Barak, B., Braverman, M., Chen, X., Rao, A.: How to compress interactive communication. SIAM J. Comput. 42(3), 1327–1363 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Braverman, M.: Coding for interactive computation: progress and challenges. In: Allerton (2012)Google Scholar
  3. 3.
    Braverman, M.: Interactive information complexity. SIAM J. Comput. 44(6), 1698–1739 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 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)Google Scholar
  5. 5.
    Braverman, M., Rao, A.: Information equals amortized communication. IEEE Trans. Inf. Theory 60(10), 6058–6069 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 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)Google Scholar
  7. 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)Google Scholar
  8. 8.
    Feder, T., Kushilevitz, E., Naor, M., Nisan, N.: Amortized communication complexity. SIAM J. Comput. 24(4), 736–750 (1995)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Harsha, P., Jain, R., McAllester, D.A., Radhakrishnan, J.: The communication complexity of correlation. IEEE Trans. Inf. Theory 56(1), 438–449 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jain, R.: New strong direct product results in communication complexity. JACM 62(3), 20 (2015)MathSciNetCrossRefMATHGoogle Scholar
  11. 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)Google Scholar
  12. 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)Google Scholar
  13. 13.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Department of Computer ScienceTechnion-IITHaifaIsrael
  3. 3.Max Planck Institute for InformaticsSaarbrückenGermany
  4. 4.Department of Computer ScienceTel Aviv UniversityTel AvivIsrael
  5. 5.Department of MathematicsTechnion-IITHaifaIsrael

Personalised recommendations