Routing permutations on spectral expanders via matchings
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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 π(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 π 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].
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