Balanced Allocations: The Heavily Loaded Case with Deletions

Abstract

In the 2-choice allocation problem, m balls are placed into n bins, and each ball must choose between two random bins $i,j\in[n]$ that it has been assigned to. It has been known for more than two decades, that if each ball follows the GREEDY strategy (i.e., always pick the less-full bin), then the maximum load will be $m/n+O(\log\log n)$ with high probability in n (and $m/n+O(\log m)$ with high probability in m). It has remained an open question whether the same bounds hold in the dynamic version of the same game, where balls are inserted/deleted with no more than m balls present at a time.We show that, somewhat surprisingly, these bounds do not hold in the dynamic setting: already on 4 bins, there exists a sequence of insertions/deletions that cause the GREEDY strategy to incur a maximum load of $m/4+\Omega(\sqrt{m})$ with probability $\Omega(1)$—this is the same bound that one gets in the single-choice allocation model where each ball is assigned to a random bin!This raises the question of whether any 2-choice allocation strategy can offer a strong bound in the dynamic setting. Our second result answers this question in the affirmative: we present a new strategy, called MODULATEDGREEDY, that guarantees a maximum load of $m/n+O(\log m)$, at any given moment, with high probability in m. We also show how to generalize ModulatedGreedy to obtain dynamic guarantees for the $(1+\beta)$-choice setting, and for the setting of balls-and-bins on a graph.Finally, we consider an extension of the dynamic setting in which balls can be reinserted after they are deleted, and where the pair i, j that a given ball uses is consistent across insertions. This seemingly small modification renders tight load balancing impossible: on 4 bins, any balls-and-bins strategy that is oblivious to the specific identities of balls being inserted/deleted must allow for a maximum load of $m/4+\mathrm{poly}(m)$ at some point in the first poly (m) insertions/deletions, with high probability in m. This is a remarkable departure from the m=n case where the maximum load of O(loglogn) holds independently of whether reinsertions are allowed or not.