On the Communication Complexity of Distributed Algebraic Computation

We consider a situation where two processors F'l and Pz are to evaluate a collection of functions f],.… f, of two-vector variables x, v, under the assumption that processor P] (respectively, Pz ) has access only to the value of the variable x (respectively, y) and the functional form of ~1,.… f,. We provide some new bounds on the communication complexity (the amount of information that has to be exchanged between the processors) for this problem. An almost optimal bound is derived for the case of one-way communication when the functions are polynomials. We also derive some new lower bounds for the case of two-way communication that improve on earlier bounds by Abelson [2]. As an application, we consider the case where x and y are n X t~ matrices and f(x, y) is a particular entry of the inverse of.r + y. Under a certain restriction on the class of allowed communication protocols, we obtain an fl(n2) lower bound, in contrast to the Q(n) lower bound obtained by applying Abelson's results. Our results are based on certain tools from classical algebraic geomet~ and field extension theory.