In the balanced allocations framework, the goal is to allocate $m$ balls into $n$ bins, so as to minimize the gap (difference of maximum to average load). The One-Choice process allocates each ball to a randomly sampled bin and achieves w.h.p. a $\Theta(\sqrt{(m/n) \cdot \log n})$ gap. The Two-Choice process allocates to the least loaded of two randomly sampled bins, and achieves w.h.p. a $\log_2 \log n + \Theta(1)$ gap. Finally, the $(1+\beta)$ process mixes between these two processes with probability $\beta \in (0, 1)$, and achieves w.h.p. an $\Theta(\log n/\beta)$ gap. We focus on the outdated information setting of [BCEFN12], where balls are allocated in batches of size $b$. For almost the entire range $b \in [1,O(n \log n)]$, it was shown in [LS22a] that Two-Choice achieves w.h.p. the asymptotically optimal gap and for $b = \Omega(n\log n)$ it was shown in [LS22b] that it achieves w.h.p. a $\Theta(b/n)$ gap. In this work, we establish that the $(1+\beta)$ process for appropriately chosen $\beta$, achieves w.h.p. the asymptotically optimal gap of $O(\sqrt{(b/n) \cdot \log n})$ for any $b \in [2n \log n, n^3]$. This not only proves the surprising phenomenon that allocating greedily based on Two-Choice is not the best, but also that mixing two processes (One-Choice and Two-Choice) leads to a process with a gap that is better than both. Furthermore, the upper bound on the gap applies to a larger family of processes and continues to hold in the presence of weights sampled from distributions with bounded MGFs.

翻译：在均衡分配问题中，目标是将$m$个球分配到$n$个槽中，以最小化间隙（最大负载与平均负载之差）。单选择过程将每个球随机分配到一个槽中，并以高概率实现$\Theta(\sqrt{(m/n) \cdot \log n})$的间隙。双选择过程将球分配到两个随机采样槽中负载较轻的一个，并以高概率实现$\log_2 \log n + \Theta(1)$的间隙。最后，$(1+\beta)$过程以概率$\beta \in (0, 1)$混合上述两种过程，并以高概率实现$\Theta(\log n/\beta)$的间隙。我们关注[BCEFN12]中的信息过时设置，其中球以大小为$b$的批次分配。对于几乎整个范围$b \in [1,O(n \log n)]$，[LS22a]证明双选择过程以高概率实现渐近最优间隙；对于$b = \Omega(n\log n)$，[LS22b]证明其以高概率实现$\Theta(b/n)$的间隙。在本工作中，我们证明对于适当选择的$\beta$，$(1+\beta)$过程对于任意$b \in [2n \log n, n^3]$，以高概率实现渐近最优间隙$O(\sqrt{(b/n) \cdot \log n})$。这不仅证明了基于双选择的贪心分配并非最佳这一惊人现象，还表明混合两种过程（单选择和双选择）可产生间隙优于两者的过程。此外，该间隙上界适用于更广泛的过程族，并在存在从具有有界矩生成函数的分布中采样的权重时仍然成立。