Distributed symmetric function computation in noisy wireless sensor networks with binary data

We consider a wireless sensor network consisting of n sensors, each having a recorded bit, the sensor's measurement, which has been set to either "0" or "1". The network has a special node called the fusion center whose goal is to compute a symmetric function of these bits; i.e., a function that depends only on the number of sensors that have a "1" The sensors convey information to the fusion center in a multi-hop fashion to enable the function computation. The problem studied is to minimize the total transmission energy used by the network when computing this function, subject to the constraint that this computation is correct with high probability. We assume the wireless channels are binary symmetric channels with a probability of error p, and that each sensor uses units of energy to transmit each bit, where r is the transmission range of the sensor. The main result in this paper is an algorithm whose energy usage is Theta ( n (log log n) ( root log n/n) (alpha)) and we also show that any algorithm satisfying the performance constraints must necessarily have energy usage Omega ( n ( root log n/n (alpha)). Then, we n consider the case where the sensor network observes N events, and each node records one bit per event, thus having N bits to convey. The fusion center now wants to compute N symmetric functions, one for each of the events. In this case, we demonstrate a network algorithm which has energy usage Theta ( n ( root log n/n) (alpha)) per event if the number of events satisfies N = Omega(log n), and we furthermore show that any other feasible algorithm must have energy usage Omega ( n ( root log n/n) (alpha))