Self-Stabilizing Symmetry Breaking in Constant Space

We investigate the problem of self-stabilizing round-robin token management on a bidirectional ring of identical processors. Each processor is an asynchronous probabilistic finite state (i.e., constant space) machine which sends and receives constant-size messages and whose state transition is triggered by the receipt of a message. We also show that this problem is equivalent to symmetry breaking (i.e., leader election). We justify and suggest a two-layer (hardware and software) solution to the token management problem: The subproblem of reducing an arbitrary but nonzero number of tokens (in an otherwise arbitrary initial system state) to exactly one token (and a legal system state) is solved in hardware and takes only small polynomial time. The detection of a complete lack of tokens (communication deadlock) is done by a software clock. In high-speed networks the hardware layer can be implemented using fast universal switches (i.e., finite state machines) independent of the size of the network. We note that randomization is essential, since Dijkstra showed that for arbitrary rings the subproblem does not have a deterministic solution (regardless of the computational power of the identical processors). The use of the software layer (deadlock detection) in our solution is minimized.