Shape sensitive geometric monitoring

A fundamental problem in distributed computation is the distributed evaluation of functions. The goal is to determine the value of a function over a set of distributed inputs, in a communication efficient manner. Specifically, we assume that each node holds a time varying input vector, and we are interested in determining, at any given time, whether the value of an arbitrary function on the average of these vectors crosses a predetermined threshold. In this paper, we introduce a new method for monitoring distributed data, which we term shape sensitive geometric monitoring. It is based on a geometric interpretation of the problem, which enables to define local constraints on the data received at the nodes. It is guaranteed that as long as none of these constraints has been violated, the value of the function does not cross the threshold. We generalize previous work on geometric monitoring, and solve two problems which seriously hampered its performance: as opposed to the constraints used so far, which depend only on the current values of the local input vectors, here we incorporate their temporal behavior into the constraints. Also, the new constraints are tailored to the geometric properties of the specific function which is being monitored, while the previous constraints were generic. Experimental results on real world data reveal that using the new geometric constraints reduces communication by up to three orders of magnitude in comparison to existing approaches, and considerably narrows the gap between existing results and a newly defined lower bound on the communication complexity.