A Tight Bound for Set Disjointness in the Message-Passing Model

Abstract

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 One of the most natural application domains for communication complexity is distributed computing: When we wish to study the cost of computing in a network spanning multiple cores or physical machines, it is very useful to understand how much communication is necessary, since communication between machines often dominates the cost of the computation. Accordingly, lower bounds in communication complexity have been used to obtain many negative results in distributed computing, from the round complexity of finding a minimum-weight spanning tree [42] to computing functions of distributed data [37,29] and distributed computation and verification of network graph structures and properties [42,23].To the best of our knowledge, however, all applications of communication complexity lower bounds in distributed computing to date have used only two-player lower bounds. The reason for this appears to be twofold: First, the models of multi-party communication favored by the communication complexity community, the number-on-forehead model and the number-in-hand broadcast model, do not correspond to most natural models of distributed computing. Second, two-party lower bounds are surprisingly powerful, even for networks with many players. A typical reduction from a two-player communication complexity problem to a distributed problem T finds a sparse cut in the network, and shows that, to solve T , the two sides of the cut must implicitly solve, say, Set Disjointness [30]. However, there are problems that cannot be addressed by reduction from a two-player problem, because such reductions must reveal almost the entire structure of the network to one of the two players. (One such example is described in [29].)In this paper, we study communication complexity in message-passing models, where each party has a private input, and the parties communicate by sending messages to each other over private channels. These models have been used extensively in distributed computing, for example, to study gossiping protocols [27], to compute various functions...

An Optimal Lower Bound for Distinct Elements in the Message Passing Model

“…The reason this bound cannot be shown using existing techniques is that they either require proving a distributional communication lower bound [24], or they prove a lower bound on the information cost [6]. For the 2-player 2-NEQ problem of deciding whether two strings are not equal, for any distribution µ there is an upper bound of O(log 1/δ), where δ is the error probability over inputs drawn from µ [21].…”

“…The communication is point-to-point, meaning that each pair of players has a private communication channel and messages sent between the two players are not seen by other players. This is also referred to as the message-passing model [12,24,25,27,6,18]. One of the main goals is to minimize the total number of bits exchanged between the players, i.e., the communication complexity of accomplishing this task.…”

Zero-Information Protocols and Unambiguity in Arthur-Merlin Communication

“…One very successful approach for proving communication lower bounds against randomized protocols is the information complexity methodology [16,6,33,15,25,32,18,11,10,9,24]. In this approach one argues that the transcripts of correct protocols must necessarily "leak" information about the input; the amount of information leaked automatically lower bounds the communication.…”

Lower Bounds for Number-in-Hand Multiparty Communication Complexity, Made Easy

“…We notice that after the conference version of this paper, Braverman et al [8] and Huang et al [23] independently developed two new (and different) definitions for icost in the coordinator model, and used them to prove some tight lower bounds in the coordinator model.…”

“…As will be discussed in Section 7.2, due to a limitation of the symmetrization technique, it cannot be used to prove a tight lower bound for the k-player disjointness problem. This was listed as an open problem in the conference version of this paper, and was later settled by Braverman et al [8], using a different technique based on information complexity. Recently Huang et al [23] applied symmetrization together with information complexity to prove tight lower bounds for approximate maximum matchings; Li et al [29] used the symmetrization technique to prove lower bounds for numerical linear algebra problems.…”