A Tight Lower Bound for Clock Synchronization in Odd-Ary M-Toroids

Synchronizing clocks in a distributed system in which processes communicate through messages with uncertain delays is subject to inherent errors. Prior work has shown upper and lower bounds on the best synchronization achievable in a variety of network topologies and assumptions about the uncertainty on the message delays. However, until now there has not been a tight closed-form expression for the optimal synchronization in $k$-ary $m$-cubes with wraparound, where $k$ is odd. In this paper, we prove a lower bound of $\frac{1}{4}um\left(k-\frac{1}{k}\right)$, where $k$ is the (odd) number of processes in the each of the $m$ dimensions, and $u$ is the uncertainty in delay on every link. Our lower bound matches the previously known upper bound.