Balls into Bins via Local Search

We propose a natural process for allocating n balls into n bins that are organized as the vertices of an undirected graph G. Each ball first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. In our main result, we prove that this process yields a maximum load of only Θ(log logn) on expander graphs. In addition, we show that for d-dimensional grids the maximum load is Θ (( logn log logn ) 1 d+1 ) . Finally, for almost regular graphs with minimum degree Ω(logn), we prove that the maximum load is constant and also reveal a fundamental difference between random and arbitrary tie-breaking rules.