Tight Bounds for Set Disjointness in the Message Passing Model

In a multiparty message-passing model of communication, there are k players. Each player has a private input, and they communicate by sending messages to one another over private channels. While this model has been used extensively in distributed computing and in multiparty computation, lower bounds on communication complexity in this model and related models have been somewhat scarce. In recent work [40, 46, 47], strong lower bounds of the form Ω(n ·k) were obtained for several functions in the message-passing model; however, a lower bound on the classical Set Disjointness problem remained elusive. In this paper, we prove tight lower bounds of the form Ω(n · k) for the Set Disjointness problem in the message passing model. Our bounds are obtained by developing information complexity tools in the message-passing model, and then proving an information complexity lower bound for Set Disjointness. As a corollary, we show a tight lower bound for the task allocation problem [19] via a reduction from Set Disjointness. Department of Computer Science, Princeton University, [email protected],Research supported in part by an Alfred P. Sloan Fellowship, an NSF CAREER award (CCF-1149888), and a Turing Centenary Fellowship. Department of Computer Science, University of Toronto, [email protected], Research supported in part by NSERC. Department of Computer Science, University of Toronto, [email protected], Research supported in part by NSERC. Department of Computer Science, University of Toronto, [email protected],Research supported in part by NSERC. Department of Computer Science, University of Toronto, [email protected], Research supported in part by NSERC, DARPA award FA8750-11-2-0225 and an Alfred P. Sloan Fellowship.