An almost linear time algorithm testing whether the Markoff graph modulo $p$ is connected

The Markoff graph modulo $p$ is known to be connected for all but finitely many primes $p$ (see Eddy, Fuchs, Litman, Martin, Tripeny, and Vanyo [arxiv:2308.07579]), and it is conjectured that these graphs are connected for all primes. In this paper, we provide an algorithmic realization of the process introduced by Bourgain, Gamburd, and Sarnak [arxiv:1607.01530] to test whether the Markoff graph modulo $p$ is connected for arbitrary primes. Our algorithm runs in $o(p^{1 + \epsilon})$ time for every $\epsilon > 0$. We demonstrate this algorithm by confirming that the Markoff graph modulo $p$ is connected for all primes less than one million.