Over the last fifteen years, a large group of algorithms emerged which compute various predicates from distributed data with a focus on communication efficiency. These algorithms are often called "communication-efficient", "geometric-monitoring", or "local" algorithms. We jointly call them distributed convex thresholding algorithms, for reasons which will be explained in this work. Distributed convex thresholding algorithms have found their applications in domains in which bandwidth is a scarce resource, such as wireless sensor networks and peer-to-peer systems, or in scenarios in which data rapidly streams to the different processors but outcome of the predicate rarely changes. Common to all of these algorithms is the use of a data dependent criteria to determine when further messaging is required.
This work presents two very simple yet exceedingly general theorems from which the correctness of all distributed convex thresholding algorithms can be elicited, and demonstrates that for key examples. Because the theorems are general, they extend the range of predicates which can be computed in a communication efficient manner beyond what is currently known. Unlike the previous correction proofs given to these algorithms, the proofs of the theorems presented here do not depend on the communication infrastructure. So the correctness of any distributed convex thresholding algorithm is immediately extended from broadcast enabled networks or from cycle free networks to general networks. Inspecting existing algorithms in light of the new theorems reveals that they contain redundant requirements, which cause them to send messages when indeed none are needed.