Self-stabilizing clock synchronization with 1-bit messages

Abstract

We study the fundamental problem of distributed clock synchronization in a basic probabilistic communication setting. We consider a synchronous fully-connected network of n agents, where each agent has a local clock, that is, a counter increasing by one modulo T in each round. The clocks have arbitrary values initially, and they must all indicate the same time eventually. We assume a pull communication model, where in every round each agent receives an l-bit message from a random agent. We devise several fast synchronization algorithms that use small messages and are self-stabilizing, that is, the complete initial state of each agent (not just its clock value) can be arbitrary. We first provide a surprising algorithm for synchronizing a binary clock (T = 2) using 1-bit messages (l = 1). This is a variant of the voter model and converges in O(log n) rounds w.h.p., unlike the voter model which needs polynomial time. Next we present an elegant extension of our algorithm that synchronizes a modulo T = 4 clock, with l = 1, in O(log n) rounds. Using these two algorithms, we refine an algorithm of Boczkowski et al. (SODA'17), that synchronizes a modulo T clock in polylogarithmic time (in n and T). The original algorithm uses l = 3 bit messages, and each agent receives messages from two agents per round. Our algorithm reduces the message size to l = 2, and the number of messages received to one per round, without increasing the running time. Finally, we present two algorithms that simulate our last algorithm achieving l < 2, without hurting the asymptotic running time. The first algorithm uses a message space of size 3, i.e., l = log2(3). The second requires a rough upper bound on log n, and uses just 1-bit messages. More generally, our constructions can simulate any self-stabilizing algorithm that requires a shared clock, without increasing the message size and by only increasing the running time by a constant factor and a polylogarithmic term.