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, …, S^k, each drawn from {1, 2, …, 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 Ω(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 Ω(n/k) randomized total communication lower bound, avoiding Hellinger distance and Poincare inequalities. Our proof also gives strong lower bounds for high probability protocols, which are much larger than what is possible for the set disjointness problem. Using this we show several new results for data streams: 1) for 𝓁₂-Heavy Hitters, any O(1)-pass streaming algorithm in the insertion-only model for detecting if an ε-𝓁₂-heavy hitter exists requires min(1/(ε²)log((ε²n)/δ), 1/(ε)n^{1/2}) bits of memory, which is optimal up to a log n factor. For deterministic algorithms and constant ε, this gives an Ω(n^{1/2}) lower bound, improving the prior Ω(log n) lower bound. We also obtain lower bounds for Zipfian distributions. 2) for 𝓁_p-Estimation, p > 2, we show an O(1)-pass Ω(n^{1-2/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 Ω(n^{1-2/p} + log(1/δ)). 3) for low rank approximation of a sparse matrix in ℝ^{d× n}, if we see the rows of a matrix one at a time in the row-order model, each row having O(1) non-zero entries, any deterministic algorithm requires Ω(√d) memory to output an O(1)-approximate rank-1 approximation. Finally, we consider strict and general turnstile streaming models, and show separations between sketching lower bounds and non-sketching upper bounds for the heavy hitters problem.

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