
Revised: January 8, 2023
Published: December 31, 2023
Abstract: [Plain Text Version]
In a k-party communication problem, the k players with inputs x_1, x_2, \ldots, x_k want to evaluate a function f(x_1, x_2, \ldots, 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(\gap^{-2}\log(N) \log \log(mM)) bits of space lower bound for the fundamental problem of (1\pm\gap)-approximating the number \|x\|_0 of non-zero entries of an n-dimensional vector x after m integer updates each of magnitude at most M, and with success probability \ge 2/3, in a strict turnstile stream. We additionally prove the matching \Omega(\gap^{-2}\log(N) \log \log(T)) space lower bound for the problem when we have access to a heavy hitters oracle with threshold T. Our results match the best known upper bounds when \gap\ge 1/\polylog(mM) and when T = 2^{\poly(1/\epsilon)}, respectively. It also improves on the prior \Omega(\gap^{-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(\gap^{-2}\log (mM)) bit upper bound.
--------------------------
A preliminary version of this paper, by a subset of the authors, appeared in the Proceedings of the 46th International Colloquium on Automata, Languages and Programming, 2019 (ICALP'19).