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：我们研究了经典的完工时间最小化调度问题，其中$n$个作业必须在$m$台相同机器上调度。利用加权随机采样，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。两种算法不仅给出最优完工时间的$(1+3ε)$近似解，还生成草图调度。我们的第一种算法针对已知$n$的情况，在加权随机采样下单轮抽取样本，其运行时间为$\tilde{O}(\tfrac{m^5}{ε^4} \sqrt{n}+A(\ceiling{m\over ε}, ε ))$，其中$A(\mathcal{N}, α)$是任意针对$\mathcal{N}$个作业的完工时间最小化问题的$(1+α)$近似方案的时间复杂度。第二种算法处理未知$n$的情况，采用自适应加权随机采样，即通过多轮抽取样本并在每轮后调整样本数量，其亚线性运行时间为$\tilde{O}\left( \tfrac{m^5} {ε^4} \sqrt{n} + A(\ceiling{m\over ε}, ε )\right)$。我们还提供了一种实现方案，该方案使用$O(\log n)$个均匀随机样本生成加权随机样本。