The wireless synchronization problem

In this paper, we study the wireless synchronization problem which requires devices activated at different times on a congested single-hop radio network to synchronize their round numbering. We assume a collection of n synchronous devices with access to a shared band of the radio spectrum, divided into F narrowband frequencies. We assume that the communication medium suffers from unpredictable, perhaps even malicious interference, which we model by an adversary that can disrupt up to t frequencies per round. Devices begin executing in different rounds and the exact number of participants is not known in advance. We first prove a lower bound, demonstrating that at least Ω (log² n/(F-t) log log n + Ft/F-t log n) rounds are needed to synchronize. We then describe two algorithms. The first algorithm almost matches the lower bound, yielding a running time of Ω (F/F-t log² n + Ft/F-t log n) rounds. The second algorithm is adaptive, terminating in O (t′ log³ n) rounds in good executions, that is, when the devices begin executing at the same time, and there are never more than t′ frequencies disrupted in any given round, for some t′ < t. In all executions, even those that are not good, it terminates in O (F log³ n) rounds.