ON THE COMMUNICATION COMPLEXITY OF DISTRIBUTED ALGEBRAIC COMPUTATION

We consider a situation where two processors P1 and P2 arc to evaluate a collection of functions f1,..., f(s) of two-vector variables x, y, under the assumption that processor P1 (respectively, P2) has access only to the value of the variable x (respectively, y) and the functional form of f1,..., f(s). 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 f1,..., f(s) are polynomials. We also derive some new lower hounds 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 n matrices and f(x, y) is a particular entry of the inverse of x + y. Under a certain restriction on the class of allowed communication protocols, we obtain an OMEGA(n2) lower bound, in contrast to the OMEGA(n) lower bound obtained by applying Abelson's results. Our results are based on certain tools from classical algebraic geometry and field extension theory.