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

Abstract

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. (1) A language is recognized by an algorithm that uses O ( n ) bits if any only if it is regular. 2. (2) Every non-regular language requires at least Ω(n log n) bits for its recognition (clearly, every language requires no more than O ( n 2 ) bits for its recognition). 3. (3) For every function g ( n ), Ω(n log n)≤g(n)≤O(n 2 ) , there is a language that requires Θ ( g ( n )) bits for its recognition.