We introduce a new class of balanced allocation processes which are primarily characterized by ``filling'' underloaded bins. A prototypical example is the Packing process: At each round we only take one bin sample, if the load is below the average load, then we place as many balls until the average load is reached; otherwise, we place only one ball. We prove that for any process in this class the gap between the maximum and average load is $\mathcal{O}(\log n)$ w.h.p. for any number of balls $m\geq 1$. For the Packing process, we also provide a matching lower bound. Additionally, we prove that the Packing process is sample-efficient in the sense that the expected number of balls allocated per sample is strictly greater than one. Finally, we also demonstrate that the upper bound of $\mathcal{O}(\log n)$ on the gap can be extended to the Memory process studied by Mitzenmacher, Prabhakar and Shah (2002).

翻译：我们引入了一类新的平衡分配过程，其主要特征是对欠载箱子进行“填充”。一个典型示例是打包过程：在每一轮中，我们仅抽取一个箱子样本，若其负载低于平均负载，则向其放置球直至达到平均负载；否则只放置一个球。我们证明，对于此类过程中的任意过程，对于任意数量的球 $m\geq 1$，最大负载与平均负载之间的差距以高概率为 $\mathcal{O}(\log n)$。对于打包过程，我们还给出了匹配的下界。此外，我们证明打包过程在样本效率上是高效的，即每次样本分配的期望球数严格大于1。最后，我们还证明，该 $\mathcal{O}(\log n)$ 的上界可扩展至 Mitzenmacher、Prabhakar 和 Shah（2002）研究的记忆过程。