Separating k-Player from t-Player One-Way Communication, with Applications to Data Streams
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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 Ω(ε^-2log(N) loglog(mM)) bits of space lower bound for the fundamental problem of (1±ε)-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 ε> 1/polylog(mM). It also improves on the prior Ω(ε^-2log(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(ε^-2log (mM)) bit upper bound.
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