Near-optimal lower bounds on the multi-party communication complexity of set disjointness

We study the communication complexity of the set disjointness problem in the general multi-party model. For t players, each holding a subset of a universe of size n, we establish a near-optimal lower bound of Ω(n/(t log t)) on the communication complexity of the problem of determining whether their sets are disjoint. In the more restrictive one-way communication model, in which the players are required to speak in a predetermined order, we improve our bound to an optimal Ω(n/t). These results improve upon the earlier bounds of Ω(n/t²) in the general model, and Ω(ε ²n/t^1+ε) in the one-way model, due to Bar-Yossef, Jayram, Kumar, and Sivakumar [5]. As in the case of earlier results, our bounds apply to the unique intersection promise problem. This communication problem is known to have connections with the space complexity of approximating frequency moments in the data stream model. Our results lead to an improved space complexity lower bound of Ω(n^1-2/k/log n) for approximating the k^th frequency moment with a constant number of passes over the input, and a technical improvement to Ω(n^1-2/k) if only one pass over the input is permitted. Our proofs rely on the information theoretic direct sum decomposition paradigm of Bar-Yossef et al [5]. Our improvements stem from novel analytical techniques, as opposed to earlier techniques based on Hellinger and related distances, for estimating the information cost of protocols for one-bit functions.