Separating k-Player from t-Player One-Way Communication, with Applications to Data Streams

Authors David P. Woodruff, Guang Yang

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Author Details

David P. Woodruff
  • Carnegie Mellon University, Pittsburgh, PA, USA
Guang Yang
  • Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
  • Conflux, Beijing, China


We would like to thank Yuval Ishai and Eyal Kushilevitz for initiating the problem of separating worst-case partition communication complexity from streaming complexity, which was our starting point. We also thank the ICALP referees for very helpful comments which helped us revise our initial submission. D. Woodruff would also like to thank the Chinese Academy of Sciences, as well as the Simons Institute for the Theory of Computing.

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David P. Woodruff and Guang Yang. Separating k-Player from t-Player One-Way Communication, with Applications to Data Streams. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 97:1-97:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


In a k-party communication problem, the k players with inputs x_1, x_2, ..., x_k, respectively, want to evaluate a function f(x_1, x_2, ..., x_k) using as little communication as possible. We consider the message-passing model, in which the inputs are partitioned in an arbitrary, possibly worst-case manner, among a smaller number t of players (t<k). The t-player communication cost of computing f can only be smaller than the k-player communication cost, since the t players can trivially simulate the k-player protocol. But how much smaller can it be? We study deterministic and randomized protocols in the one-way model, and provide separations for product input distributions, which are optimal for low error probability protocols. We also provide much stronger separations when the input distribution is non-product. A key application of our results is in proving lower bounds for data stream algorithms. In particular, we give an optimal Omega(epsilon^{-2}log(N) log log(mM)) bits of space lower bound for the fundamental problem of (1 +/-{epsilon})-approximating the number |x |_0 of non-zero entries of an n-dimensional vector x after m updates each of magnitude M, and with success probability >= 2/3, in a strict turnstile stream. Our result matches the best known upper bound when epsilon >= 1/polylog(mM). It also improves on the prior Omega({epsilon}^{-2}log(mM)) lower bound and separates the complexity of approximating L_0 from approximating the p-norm L_p for p bounded away from 0, since the latter has an O(epsilon^{-2}log(mM)) bit upper bound.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming models
  • Theory of computation → Complexity classes
  • Theory of computation → Lower bounds and information complexity
  • Communication complexity
  • multi-player communication
  • one-way communication
  • streaming complexity


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