How to synchronize a 2d cellular automaton in $\Theta(\sqrt{n} \log n)$ steps

Question

Let's assume we have given a two dimensional cellular automaton with an initial configuration where alls cells are in an quiescent state, expect for one square of cells. Let $n$ be the number of cells in that square. We want to synchronize all cells in the square, so we basically want to solve the firing squad synchronization problem.

Synchronizing all cells in $\Theta(\sqrt{n})$ is rather easy: All cells at the left and right border of the square serve as generals and we synchronize each row separately with any standard 1-dimensional FSSP algorithm. A row of $k$ cells with generals at both ends can be synchronized in $\Theta(k)$ steps (in our case: $k=\sqrt{n}$).

What I want to do, is to synchronize the whole square in $\Theta(\sqrt{n} \log n)$, so I need to slow down the synchronization process by a factor of $\log n$. But I have no idea how to do so.

Answer 1

A naive solution could be that each general first launch a binary counter in its row and counts the current time until $k\log(k)$ steps and then run the FSSP. To do this you just need to detect the value of $k$ and compute $k\log(k)$ in less than $k\log(k)$ steps: sending a signal back and forth toward the other end while incrementing a binary counter gives you k written in binary in roughly $2k+\log(k)$ steps, so you also know $\log(k)$ (length of the counter). Within the $\Theta(k\log(k))$ steps left you have much more time than necessary to compute the product $k\times \log(k)$. And you're done.

I'm pretty sure that (1) there are more elegant solutions, (2) there is a general acceleration/slowdown theorem that could also be used here.