Efficient threshold detection in a distributed environment

Consider a distributed network in which events occur at arbitrary nodes and at unpredicted times. An event occurring at node u is sensed only by u which in turn may invoke a communication protocol that allows nodes to exchange messages with their neighbors. We are interested in the following threshold detection (TD) problem inherent to distributed computing: Given some threshold k, the goal of a TD protocol is to broadcast a termination signal when at least k events have occurred (throughout the network). In this paper we develop a randomized TD protocol that may fail with negligible probability but which significantly improves previous results in terms of the message complexity, namely, the total number of messages sent by all participating nodes. With the right choice of parameters our randomized protocol turns into a deterministic one that guarantees low communication burden for any node. This is a principal complexity measure in many applications of wireless networks and which, to the best of our knowledge, has not been bounded before in the context of such problems.