In the balanced allocations framework, there are $m$ jobs (balls) to be allocated to $n$ servers (bins). The goal is to minimize the gap, the difference between the maximum and the average load. Peres, Talwar and Wieder (RSA 2015) used the hyperbolic cosine potential function to analyze a large family of allocation processes including the $(1+\beta)$-process and graphical balanced allocations. The key ingredient was to prove that the potential drops in every step, i.e., a drift inequality. In this work we improve the drift inequality so that (i) it is asymptotically tighter, (ii) it assumes weaker preconditions, (iii) it applies not only to processes allocating to more than one bin in a single step and (iv) to processes allocating a varying number of balls depending on the sampled bin. Our applications include the processes of (RSA 2015), but also several new processes, and we believe that our techniques may lead to further results in future work.

翻译：在平衡分配框架中，有 $m$ 个任务（球）需要分配给 $n$ 个服务器（箱），目标是最大化减小最大负载与平均负载之间的差值（即间隙）。Peres、Talwar 和 Wieder（RSA 2015）利用双曲余弦势函数分析了包括 $(1+\beta)$-过程和图形化平衡分配在内的大类分配过程，其关键步骤是证明每一步中势函数都会下降（即漂移不等式）。本文改进了该漂移不等式，使其 (i) 在渐近意义上更紧致，(ii) 所需前提假设更弱，(iii) 不仅适用于单步分配至多个箱体的过程，还适用于 (iv) 根据采样箱体分配可变数量球的过程。我们的应用涵盖（RSA 2015）中的过程及若干新过程，并相信相关技术可推动未来研究取得进一步成果。