Self-Stabilizing Repeated Balls-into-Bins

TL;DR

This paper proves that a repeated balls-into-bins process is self-stabilizing, converging quickly to a balanced state with logarithmic maximum load, and maintains this stability over polynomially bounded periods with high probability.

Contribution

It establishes that the repeated balls-into-bins process is self-stabilizing, converging in linear time to a balanced configuration from any initial state.

Findings

Converges to a legitimate configuration in linear time

Maintains legitimate configurations over polynomial time with high probability

Each ball visits all bins in O(n log^2 n) rounds

Abstract

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. In every subsequent round, from each non-empty bin one ball is chosen according to some fixed strategy (random, FIFO, etc), and re-assigned to one of the $n$ bins uniformly at random. We define a configuration "legitimate" if its maximum load is $\mathcal{O}(\log n)$. We prove that, starting from any configuration, the process will converge to a legitimate configuration in linear time and then it will only take on legitimate configurations over a period of length bounded by any polynomial in $n$, with high probability (w.h.p.). This implies that the process is self-stabilizing and that every ball traverses all bins in $\mathcal{O}(n \log^2 n)$ rounds, w.h.p.