We consider the classical makespan minimization scheduling problem where $n$ jobs must be scheduled on $m$ identical machines. Using weighted random sampling, we developed two sublinear time approximation schemes: one for the case where $n$ is known and the other for the case where $n$ is unknown. Both algorithms not only give a $(1+3ε)$-approximation to the optimal makespan but also generate a sketch schedule. Our first algorithm, which targets the case where $n$ is known and draws samples in a single round under weighted random sampling, has a running time of $\tilde{O}(\tfrac{m^5}{ε^4} \sqrt{n}+A(\ceiling{m\over ε}, ε ))$, where $A(\mathcal{N}, α)$ is the time complexity of any $(1+α)$-approximation scheme for the makespan minimization of $\mathcal{N}$ jobs. The second algorithm addresses the case where $n$ is unknown. It uses adaptive weighted random sampling, %\textit{that is}, it draws samples in several rounds, adjusting the number of samples after each round, and runs in sublinear time $\tilde{O}\left( \tfrac{m^5} {ε^4} \sqrt{n} + A(\ceiling{m\over ε}, ε )\right)$. We also provide an implementation that generates a weighted random sample using $O(\log n)$ uniform random samples.

翻译：我们研究经典的最小化最大完工时间调度问题，该问题需将$n$个作业分配到$m$台同构机器上。通过加权随机采样，我们开发了两种亚线性时间近似方案：一种针对已知$n$的情况，另一种针对未知$n$的情况。两种算法不仅能给出最优最大完工时间的$(1+3ε)$-近似解，还能生成草图调度方案。第一种算法针对已知$n$的情况，在加权随机采样下通过单轮采样实现，运行时间为$\tilde{O}(\tfrac{m^5}{ε^4} \sqrt{n}+A(\ceiling{m\over ε}, ε))$，其中$A(\mathcal{N}, α)$表示任意$(1+α)$-近似方案对$\mathcal{N}$个作业进行最小化最大完工时间调度的时间复杂度。第二种算法解决未知$n$的情况，采用自适应加权随机采样（即多轮采样，每轮后调整采样数量），运行时间同样为$\tilde{O}\left( \tfrac{m^5}{ε^4} \sqrt{n} + A(\ceiling{m\over ε}, ε) \right)$。我们还提供了利用$O(\log n)$个均匀随机样本生成加权随机采样的实现方案。