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

Abstract

In a $k$-party communication problem, the $k$ players with inputs $x_1,x_2,\dots ,x_k$, respectively, want to evaluate a function $f(x_1,x_2,\dots ,x_k)$ 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 $\hat{I}\text{c◯}(\hat{I}{\mu }^{-2}\log (N)\log \log (mM))$ bits of space lower bound for the fundamental problem of $(1\pm \hat{I}\mu )$-approximating the number $\Vert x{\Vert }_0$ 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 $\hat{I}\text{c◯}(\hat{I}{\mu }^{-2}\log (N)\log \log (T))$ 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 $\hat{I}\mu \ge 1/\mathrm{polylog}(mM)$ and when $T=2^{\mathrm{poly}(1/\hat{I}\mu )}$ respectively. It also improves on the prior $\hat{I}\text{c◯}(\hat{I}{\mu }^{-2}\log (mM))$ 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 $O(\hat{I}{\mu }^{-2}\log (mM))$ bit upper bound.

Preliminary version appeared in ICALP 2019, submitted to ToC