Computing on an anonymous ring

Abstract

The computational capabilities of a system of n indistinguishable (anonymous) processors arranged on a ring in the synchronous and asynchronous models of distributed computation are analyzed. A precise characterization of the functions that can be computed in this setting is given. It is shown that any of these functions can be computed in O ( n 2 ) messages in the asynchronous model. This is also proved to be a lower bound for such elementary functions as AND, SUM, and Orientation. In the synchronous model any computable function can be computed in O ( n log n ) messages. A ring can be oriented and start synchronized within the same bounds. The main contribution of this paper is a new technique for proving lower bounds in the synchronous model. With this technique tight lower bounds of θ( n log n ) (for particular n ) are proved for XOR, SUM, Orientation, and Start Synchronization. The technique is based on a string-producing mechanism from formal language theory, first introduced by Thue to study square-free words. Two methods for generalizing the synchronous lower bounds to arbitrary ring sizes are presented.