Balls into Bins via Local Search

Abstract

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 (log log n) on expander graphs. In addition, we show that for d-dimensional grids the maximum load is log n log log n 1 d+1 . Finally, for almost regular graphs with minimum degree (log n), we prove that the maximum load is constant and also reveal a fundamental difference between random and arbitrary tie-breaking rules.