Time- and Space-Optimal Clock Synchronization in the Beeping Model

Abstract: We consider the clock synchronization problem in the (discrete) beeping model: Given a network of $n$ nodes with each node having a clock value $\delta(v) \in {0,\ldots T-1}$, the goal is to synchronize the clock values of all nodes such that they have the same value in any round. As is standard in clock synchronization, we assume \emph{arbitrary activations} for all nodes, i.e., the nodes start their protocol at an arbitrary round (not limited to ${0,\ldots,T-1}$). We give an asymptotically optimal algorithm that runs in $4D + \Bigl\lfloor \frac{D}{\lfloor T/4 \rfloor} \Bigr \rfloor \cdot (T \mod 4) = O(D)$ rounds, where $D$ is the diameter of the network. Once all nodes are in sync, they beep at the same round every $T$ rounds. The algorithm drastically improves on the $O(T D)$-bound of ACGL'13. Our algorithm is very simple as nodes only have to maintain $3$ bits in addition to the $\lceil \log T \rceil$ bits needed to maintain the clock. Furthermore we investigate the complexity of \emph{self-stabilizing} solutions for the clock synchronization problem: We first show lower bounds of $\Omega(\max{T,n})$ rounds on the runtime and $\Omega(\log(\max{T,n}))$ bits of memory required for any such protocol. Afterwards we present a protocol that runs in $O(\max{T,n})$ rounds using at most $O(\log(\max{T,n}))$ bits at each node, which is asymptotically optimal with regards to both, runtime and memory requirements.