Separating k-Player from t-Player One-Way Communication, with Applications to Data Streams
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.
Communication complexity
multi-player communication
one-way communication
streaming complexity
Theory of computation~Streaming models
Theory of computation~Complexity classes
Theory of computation~Lower bounds and information complexity
97:1-97:14
Track A: Algorithms, Complexity and Games
This work was supported in part by the National Natural Science Foundation of China Grants No. 61433014, 61602440, 61761136014, 61872334, 61502449, and the 973 Program of China Grant No. 2016YFB1000201.
The full version hosted on arXiv https://arxiv.org/abs/1905.07135..
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.
David P.
Woodruff
David P. Woodruff
Carnegie Mellon University, Pittsburgh, PA, USA
Guang
Yang
Guang Yang
Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
Conflux, Beijing, China
10.4230/LIPIcs.ICALP.2019.97
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David P. Woodruff and Guang Yang
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