Worst-case asymmetric distributed function computation

Abstract

We consider a distributed function computation problem where an information sink communicates with N correlated information sources to compute a given deterministic function of source data. A data-vector is drawn from a discrete and finite probability distribution and component x i is revealed to ith, , source. We address this problem in asymmetric communication scenarios where only the sink knows the distribution P. We are interested in computing the minimum number of source bits required, in the worst-case, to solve this problem. We propose the notion of functional ambiguity to carry out the worst-case information-theoretic analysis of distributed function computation problems. We establish its various characteristics and prove that it leads to a valid information measure. Then, we provide a constructive solution for the distributed function computation problem in terms of an interactive communication protocol and prove its optimality. Finally, we establish two equivalence classes of compressible and incompressible functions to classify the set of all computable multivariate functions based on the minimum number of source bits needed in the worst-case to compute a function in the distributed function computation set-up.