The Power of Shared Randomness in Uncertain Communication

Authors Badih Ghazi, Madhu Sudan

Thumbnail PDF


  • Filesize: 0.55 MB
  • 14 pages

Document Identifiers

Author Details

Badih Ghazi
Madhu Sudan

Cite AsGet BibTex

Badih Ghazi and Madhu Sudan. The Power of Shared Randomness in Uncertain Communication. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 49:1-49:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


In a recent work (Ghazi et al., SODA 2016), the authors with Komargodski and Kothari initiated the study of communication with contextual uncertainty, a setup aiming to understand how efficient communication is possible when the communicating parties imperfectly share a huge context. In this setting, Alice is given a function f and an input string x, and Bob is given a function g and an input string y. The pair (x,y) comes from a known distribution mu and f and g are guaranteed to be close under this distribution. Alice and Bob wish to compute g(x,y) with high probability. The lack of agreement between Alice and Bob on the function that is being computed captures the uncertainty in the context. The previous work showed that any problem with one-way communication complexity k in the standard model (i.e., without uncertainty, in other words, under the promise that f=g) has public-coin communication at most O(k(1+I)) bits in the uncertain case, where I is the mutual information between x and y. Moreover, a lower bound of Omega(sqrt{I}) bits on the public-coin uncertain communication was also shown. However, an important question that was left open is related to the power that public randomness brings to uncertain communication. Can Alice and Bob achieve efficient communication amid uncertainty without using public randomness? And how powerful are public-coin protocols in overcoming uncertainty? Motivated by these two questions: - We prove the first separation between private-coin uncertain communication and public-coin uncertain communication. Namely, we exhibit a function class for which the communication in the standard model and the public-coin uncertain communication are O(1) while the private-coin uncertain communication is a growing function of n (the length of the inputs). This lower bound (proved with respect to the uniform distribution) is in sharp contrast with the case of public-coin uncertain communication which was shown by the previous work to be within a constant factor from the certain communication. This lower bound also implies the first separation between public-coin uncertain communication and deterministic uncertain communication. Interestingly, we also show that if Alice and Bob imperfectly share a sequence of random bits (a setup weaker than public randomness), then achieving a constant blow-up in communication is still possible. - We improve the lower-bound of the previous work on public-coin uncertain communication. Namely, we exhibit a function class and a distribution (with mutual information I approx n) for which the one-way certain communication is k bits but the one-way public-coin uncertain communication is at least Omega(sqrt{k}*sqrt{I}) bits. Our proofs introduce new problems in the standard communication complexity model and prove lower bounds for these problems. Both the problems and the lower bound techniques may be of general interest.
  • randomness
  • uncertainty
  • communication
  • imperfectly shared randomness
  • lower bounds


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


  1. Andris Ambainis and Ronald De Wolf. Average-case quantum query complexity. Journal of Physics A: Mathematical and General, 34(35):6741, 2001. Google Scholar
  2. Mohammad Bavarian, Dmitry Gavinsky, and Tsuyoshi Ito. On the role of shared randomness in simultaneous communication. In Automata, Languages, and Programming, pages 150-162. Springer, 2014. Google Scholar
  3. Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David P. Woodruff, and Grigory Yaroslavtsev. Beyond set disjointness: the communication complexity of finding the intersection. In Proceedings of the 2014 ACM symposium on Principles of distributed computing, pages 106-113. ACM, 2014. Google Scholar
  4. Clément Louis Canonne, Venkatesan Guruswami, Raghu Meka, and Madhu Sudan. Communication with imperfectly shared randomness. In Innovations in Theoretical Computer Science, ITCS, pages 257-262, 2015. Google Scholar
  5. Badih Ghazi, Pritish Kamath, and Madhu Sudan. Communication complexity of permutation-invariant functions. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 1902-1921, 2016. URL:
  6. Badih Ghazi, Ilan Komargodski, Pravesh Kothari, and Madhu Sudan. Communication with contextual uncertainty. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 2072-2085, 2016. URL:
  7. Badih Ghazi and Madhu Sudan. The power of shared randomness in uncertain communication. arXiv preprint arXiv:1705.01082, 2017. URL:
  8. Oded Goldreich, Brendan Juba, and Madhu Sudan. A theory of goal-oriented communication. J. ACM, 59(2):8, 2012. Google Scholar
  9. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles are nonnegative juntas. In Proceedings of the 47th Symposium on Theory of Computing (STOC). ACM, 2015. Google Scholar
  10. Mika Göös, Toniann Pitassi, and Thomas Watson. The landscape of communication complexity classes. In Electronic Colloquium on Computational Complexity (ECCC), volume 22, page 49, 2015. Google Scholar
  11. Elad Haramaty and Madhu Sudan. Deterministic compression with uncertain priors. In Innovations in Theoretical Computer Science, ITCS, pages 377-386, 2014. Google Scholar
  12. Brendan Juba, Adam Tauman Kalai, Sanjeev Khanna, and Madhu Sudan. Compression without a common prior: an information-theoretic justification for ambiguity in language. In Innovations in Computer Science, ICS, pages 79-86, 2011. Google Scholar
  13. Brendan Juba and Madhu Sudan. Universal semantic communication I. In 40th Annual ACM Symposium on Theory of Computing, pages 123-132, 2008. Google Scholar
  14. Brendan Juba and Madhu Sudan. Efficient semantic communication via compatible beliefs. In Innovations in Computer Science, ICS, pages 22-31, 2011. Google Scholar
  15. Brendan Juba and Ryan Williams. Massive online teaching to bounded learners. In Innovations in Theoretical Computer Science, ITCS, pages 1-10, 2013. Google Scholar
  16. Hartmut Klauck. Lower bounds for quantum communication complexity. In Foundations of Computer Science, 2001. Proceedings. 42nd IEEE Symposium on, pages 288-297. IEEE, 2001. Google Scholar
  17. Troy Lee, Adi Shraibman, and Robert Spalek. A direct product theorem for discrepancy. In Computational Complexity, 2008. CCC'08. 23rd Annual IEEE Conference on, pages 71-80. IEEE, 2008. Google Scholar
  18. Troy Lee and Shengyu Zhang. Composition theorems in communication complexity. In Automata, Languages and Programming, pages 475-489. Springer, 2010. Google Scholar
  19. Nathan Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193-201, 1992. Google Scholar
  20. Marco Molinaro, David P. Woodruff, and Grigory Yaroslavtsev. Amplification of one-way information complexity via codes and noise sensitivity. In Automata, Languages, and Programming, pages 960-972. Springer, 2015. Google Scholar
  21. Ilan Newman. Private vs. common random bits in communication complexity. Information processing letters, 39(2):67-71, 1991. Google Scholar
  22. Alexander A. Sherstov. The pattern matrix method for lower bounds on quantum communication. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 85-94. ACM, 2008. Google Scholar
  23. Yaoyun Shi and Yufan Zhu. Quantum communication complexity of block-composed functions. arXiv preprint arXiv:0710.0095, 2007. Google Scholar
  24. Hans S. Witsenhausen. On sequences of pairs of dependent random variables. SIAM Journal on Applied Mathematics, 28(1):100-113, 1975. Google Scholar
  25. David P. Woodruff. Efficient and private distance approximation in the communication and streaming models. PhD thesis, Citeseer, 2007. Google Scholar
  26. Andrew Chi-Chih Yao. Some complexity questions related to distributive computing (preliminary report). In 11h Annual ACM Symposium on Theory of Computing, pages 209-213, 1979. Google Scholar