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 S1, ldots, Sk, 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: beginitemize item for ℓ2-Heavy Hitters, any O(1)-pass streaming algorithm in the insertion-only model for detecting if an eps-ℓ2-heavy hitter exists requires min(frac1eps2log fraceps2nδ, frac1epsn1/2) bits of memory, which is optimal up to a log n factor. For deterministic algorithms and constant eps, this gives an Omega(n1/2) lower bound, improving the prior Omega(log n) lower bound. We also obtain lower bounds for Zipfian distributions. item for ℓp-Estimation, p > 2, we show an O(1)-pass Omega(n1-2/p log(1/δ)) bit lower bound for outputting an O(1)-approximation with probability 1-δ, in the insertion-only model. This is optimal, and the best previous lower bound was Omega(n1-2/p + log(1/δ)). enditemize