Distributed function computation in networks: A joint delay-energy perspective

This paper considers the following network computation problem: n nodes are placed on a √n×√n grid, each node is connected to every other node within distance r(n) from itself, and is given an arbitrary input bit. Nodes communicate with each other so that finally a designated sink node can compute a target function f of the input bits. We focus on computing the identity function and the class of symmetric functions under two different communication models. We first consider a noiseless model where links are independent and noise-free, suitable for modeling wired networks. Next, we study a noisy broadcast model in which when a node transmits a bit, each of its neighbors receives a noisy copy of the bit. This is a simple model for wireless communications, originally proposed by El Gamal (1987). We use the protocol model for interference and nodes which do not share neighbors are allowed to transmit simultaneously. For every connection radius r(n), we present lower bounds on the minimum number of transmissions and the minimum number of time slots required to compute f. We then describe efficient protocols which can match both these lower bounds up to a constant factor.