The convergence time of a random walk on a graph towards its stationary distribution is an important indication of the efficiency of random algorithms based on it. Quantum random walks have been shown to allow quadratically accelerated convergence for large graphs, at least in some cases. The famous Grover search algorithm has been shown to actually fit this framework in an abstracted setting. Yet also with classical dynamics, simple mechanisms have been proposed which allow to quadratically accelerate the convergence with respect to a standard random walk. Some basic principles have been conjectured to cause this acceleration, basically transforming a diffusion-like behavior into a more transport-like behavior, but with remaining trail. We have worked out the equivalence of all these accelerating settings for the simplest example: the cycle graph. Quantum coherences turn out to play no major role and a classical feedback structure can be identified.