Self-Stabilizing Repeated Balls-into-Bins

We study the following synchronous process that we call repeated balls-into-bins. The process is started by assigning n balls to n bins in an arbitrary way. Then, in every subsequent round, one ball is chosen according to some fixed strategy (random, FIFO, etc) from each non-empty bin, and re-assigned to one of the n bins uniformly at random. This process corresponds to a non-reversible Markov chain and our aim is to study its self-stabilization properties with respect to the maximum(bin) load and some related performance measures. We define a configuration (i.e., a state) legitimate if its maximum load is O(log n). We first prove that, starting from any legitimate configuration, the process will only take on legitimate configurations over a period of length bounded by any polynomial in n, with high probability (w.h.p.). Further we prove that, starting from any configuration, the process converges to a legitimate configuration in linear time, w.h.p. This implies that the process is self-stabilizing w.h.p. and, moreover, that every ball traverses all bins in O(n log2 n) rounds, w.h.p. The latter result can also be interpreted as an almost tight bound on the cover time for the problem of parallel resource assignment in the complete graph.