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 cite{phillips12,woodruff12,woodruff13}, strong lower bounds of the form $Omega(n cdot k)$ 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 $Omega(n cdot k)$ 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 cite{DruckerKuhnOshman} via a reduction from Set Disjointness.

Gradient Clock Synchronization using Reference Broadcasts

In this paper we suggest a method by which reference broadcast synchronization (RBS), and other methods of estimating clock values, can be incorporated in standard clock synchronization algorithms to improve synchronization quality. We advocate a logical separation of the task of estimating the clock values of other nodes in the network from the task of using these estimates to output a logical clock value. The separation is achieved by means of a virtual estimate graph, overlaid on top of the real network graph, which represents the information various nodes can obtain about each other. RBS estimates are represented in the estimate graph as edges between nodes at distance 2 from each other in the original network graph. A clock synchronization algorithm then operates on the estimate graph as though it were the original network. To illustrate the merits of this approach, we modify a recent optimal gradient clock synchronization algorithm to work in this setting. The modified algorithm transparently takes advantage of RBS estimates and any other means by which nodes can estimate each others clock values.