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Synchronizing clocks in a distributed system in which processes communicatethrough messages with uncertain delays is subject to inherent errors. Priorwork has shown upper and lower bounds on the best synchronization achievable ina variety of network topologies and assumptions about the uncertainty on themessage delays. However, until now there has not been a tight closed-formexpression for the optimal synchronization in $k$-ary $m$-cubes withwraparound, 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 ofprocesses in the each of the $m$ dimensions, and $u$ is the uncertainty indelay on every link. Our lower bound matches the previously known upper bound.

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