Revised: January 8, 2023

Published: December 31, 2023

**Keywords:**streaming, space complexity, Hamming norm, approximation, algorithms with predictions, direct sum

**Categories:**complexity, communication complexity, lower bounds, streaming, space complexity, Hamming norm, approximation, algorithms with predictions, direct sum

**ACM Classification:**F.1.3, F.2.3, F.2.1

**AMS Classification:**68Q17, 68W25, 68W20

**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.

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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).