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

M. Feldmann, A. Khazraei, C. Scheideler, in: Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), ACM, 2020.

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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 \cite{firefly_sync} (where $T$ is required to be at least $4n$, so the bound is no better than $O(nD)$). 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.
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Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)
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Feldmann M, Khazraei A, Scheideler C. Time- and Space-Optimal Discrete Clock Synchronization in the Beeping Model. In: Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). ACM; 2020. doi:10.1145/3350755.3400246
Feldmann, M., Khazraei, A., & Scheideler, C. (2020). Time- and Space-Optimal Discrete Clock Synchronization in the Beeping Model. In Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). ACM. https://doi.org/10.1145/3350755.3400246
@inproceedings{Feldmann_Khazraei_Scheideler_2020, title={Time- and Space-Optimal Discrete Clock Synchronization in the Beeping Model}, DOI={10.1145/3350755.3400246}, booktitle={Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)}, publisher={ACM}, author={Feldmann, Michael and Khazraei, Ardalan and Scheideler, Christian}, year={2020} }
Feldmann, Michael, Ardalan Khazraei, and Christian Scheideler. “Time- and Space-Optimal Discrete Clock Synchronization in the Beeping Model.” In Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). ACM, 2020. https://doi.org/10.1145/3350755.3400246.
M. Feldmann, A. Khazraei, and C. Scheideler, “Time- and Space-Optimal Discrete Clock Synchronization in the Beeping Model,” in Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2020.
Feldmann, Michael, et al. “Time- and Space-Optimal Discrete Clock Synchronization in the Beeping Model.” Proceedings of the 32nd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), ACM, 2020, doi:10.1145/3350755.3400246.

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arXiv 2005.07388