We introduce a new class of balanced allocation processes which bias towards underloaded bins (those with load below the mean load) either by skewing the probability by which a bin is chosen for an allocation (probability bias), or alternatively, by adding more balls to an underloaded bin (weight bias). A prototypical process satisfying the probability bias condition is Mean-Thinning: At each round, we sample one bin and if it is underloaded, we allocate one ball; otherwise, we allocate one ball to a second bin sample. Versions of this process have been in use since at least 1986. An example of a process, introduced by us, which satisfies the weight bias condition is Twinning: At each round, we only sample one bin. If the bin is underloaded, then we allocate two balls; otherwise, we allocate only one ball. Our main result is that for any process with a probability or weight bias, with high probability the gap between maximum and minimum load is logarithmic in the number of bins. This result holds for any number of allocated balls (heavily loaded case), covers many natural processes that relax the Two-Choice process, and we also prove it is tight for many such processes, including Mean-Thinning and Twinning. Our analysis employs a delicate interplay between linear, quadratic and exponential potential functions. It also hinges on a phenomenon we call "mean quantile stabilization", which holds in greater generality than our framework and may be of independent interest.

翻译：我们引入了一类新的均衡分配过程，通过向欠载箱子（负载低于平均负载的箱子）倾斜分配概率（概率偏倚），或向欠载箱子额外添加球数（权重偏倚），来调控负载均衡。满足概率偏倚条件的典型过程是"均值稀释"：在每一轮中，先采样一个箱子，若其欠载则分配一个球；否则将球分配给第二个采样箱子。该过程的变体至少自1986年起便投入使用。由我们提出的满足权重偏倚条件的示例过程是"配对"：每轮仅采样一个箱子，若该箱子欠载则分配两个球，否则仅分配一个球。我们的主要结论是：对于任意具有概率偏倚或权重偏倚的过程，最大负载与最小负载之间的差距高概率为箱子数量的对数级。该结论适用于任意分配的球数（重载情形），涵盖了许多放宽"二选一"过程的自然过程，并且我们证明了该下界对均值稀释和配对等过程的紧性。分析中我们巧妙结合了线性、二次与指数势函数，并依赖于一种称为"均值分位稳定化"的现象——该现象在比我们框架更广的范围内成立，可能具有独立的研究价值。