Routing permutations on spectral expanders via matchings

We consider the following matching-based routing problem. Initially, each vertex $v$ of a connected graph $G$ is occupied by a pebble which has a unique destination $\pi (v)$. In each round the pebbles across the edges of a selected matching in $G$ are swapped, and the goal is to route each pebble to its destination vertex in as few rounds as possible. We show that if $G$ is a sufficiently strong $d$-regular spectral expander then any permutation $\pi$ can be achieved in $O(\log n)$ rounds. This is optimal for constant $d$ and resolves a problem of Alon, Chung, and Graham [SIAM J. Discrete Math., 7 (1994), pp. 516–530].