On the bit complexity of distributed computations in a ring with a leader

We study the bit complexity of pattern recognition in a distributed ring with a leader. Each processor gets as input a letter from some alphabet, and these concatenated letters, starting at the leader, form the pattern of the ring. The leader initiates an algorithm that accepts or rejects this pattern. Thus each algorithm recognizes a language over a given alphabet. We prove the following (n is the size of the ring, not known a priori to any of the processors) : (1) A language is recognized by an algorithm that uses 0 (n) bits if and only if it is regular. (2) Every non-regular language requires at least g2(n logn) bits for its recognition (clearly, every language requkes no more than O (n 2) bits for its recognition). (3) For every function g (n), ~(n logn ) _< g (n) _< 0 (n 2), there is a language that requires ®(g (n)) bits for its recognition. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specfic permission.