Balanced allocation on graphs: a random walk approach

We propose algorithms for allocating n sequential balls into n bins that are interconnected as a d-regular n-vertex graph G, where d ≥ 3 can be any integer. In general, the algorithms proceeds in n succeeding rounds. Let ℓ > 0 be an integer, which is given as an input to the algorithms. In each round, ball 1 ≤ t ≤ n picks a node of G uniformly at random and performs a nonbacktracking random walk of length ℓ from the chosen node and simultaneously collects the load information of a subset of the visited nodes. It then allocates itself to one of them with the minimum load (ties are broken uniformly at random). For graphs with sufficiently large girths, we obtain upper and lower bounds for the maximum number of balls at any bin after allocating all n balls in terms of ℓ, with high probability.