A Simple Proof of a New Set Disjointness with Applications to Data Streams

The multiplayer promise set disjointness is one of the most widely used problems from communication complexity in applications. In this problem there are $k$ players with subsets $S^1, \ldots, S^k$, each drawn from $\{1, 2, \ldots, n\}$, and we are promised that either the sets are (1) pairwise disjoint, or (2) there is a unique element $j$ occurring in all the sets, which are otherwise pairwise disjoint. The total communication of solving this problem with constant probability in the blackboard model is $\Omega(n/k)$. We observe for most applications, it instead suffices to look at what we call the ``mostly'' set disjointness problem, which changes case (2) to say there is a unique element $j$ occurring in at least half of the sets, and the sets are otherwise disjoint. This change gives us a much simpler proof of an $\Omega(n/k)$ randomized total communication lower bound, avoiding Hellinger distance and Poincare inequalities. Using this we show several new results for data streams: \begin{itemize} \item for $\ell_2$-Heavy Hitters, any $O(1)$-pass streaming algorithm in the insertion-only model for detecting if an $\eps$-$\ell_2$-heavy hitter exists requires $\min(\frac{1}{\eps^2}\log \frac{\eps^2n}{\delta}, \frac{1}{\eps}n^{1/2})$ bits of memory, which is optimal up to a $\log n$ factor. For deterministic algorithms and constant $\eps$, this gives an $\Omega(n^{1/2})$ lower bound, improving the prior $\Omega(\log n)$ lower bound. We also obtain lower bounds for Zipfian distributions. \item for $\ell_p$-Estimation, $p > 2$, we show an $O(1)$-pass $\Omega(n^{1-2/p} \log(1/\delta))$ bit lower bound for outputting an $O(1)$-approximation with probability $1-\delta$, in the insertion-only model. This is optimal, and the best previous lower bound was $\Omega(n^{1-2/p} + \log(1/\delta))$. \end{itemize}