A Lower Bound on Wait-Free Counting

A counting protocol (mod m) consists of shared memory bits - referred to as the counter - and of a procedure for incrementing the counter value by 1 (mod m). The procedure may be executed by many processes concurrently. It is required to satisfy a very weak correctness requirement; namely, the counter is required to show a correct value only in quiescent states - states in which no process is incrementing the counter. Special cases of counting protocols are "counting networks" [AHS91] and "concurrent counters" [MTY92]. We consider the problem of implementing a wait-free counting protocol, assuming that the basic atomic operation of a process is a read-modify-write on a single bit. Let flip(Pr) be the maximum number of times a single increment operation changes the counter bits in a counting protocol Pr. Our main result is: In any wait-free counting protocol Pr which counts modulo m, log m = f for some integer f ≤ flip(Pr). Thus, flip(Pr) ≥log m and m is a power of 2. By a result of S. Moran, G. Taubenfeld, and I. Yadin (J. Comput. System Sci. 53 (1996), 61-78), the above lower bound on flip(Pr) is tight. This result provides interesting generalizations of lower bounds and impossibility results for counting and smoothing networks.