arXiv 2602.040592026-02-03cs.DS

通过加权随机采样在子线性时间内最小化作业跨度

Minimizing Makespan in Sublinear Time via Weighted Random Sampling

Bin Fu, Yumei Huo, Hairong Zhao

AI总结：

本文提出两种基于加权随机采样的子线性时间算法，用于在未知或已知作业数情况下近似最小化作业跨度。

摘要：

我们考虑经典的作业跨度最小化调度问题，其中n个作业必须在m台相同机器上进行调度。通过加权随机采样，我们开发了两种子线性时间近似方案：一种是在n已知的情况下，另一种是在n未知的情况下。这两种算法不仅为最优作业跨度提供了(1+3ε)的近似解，还生成了一个草图调度。我们的第一个算法针对n已知的情况，在加权随机采样中进行单轮采样，其运行时间为~O(m^5/ε^4√n + A(ceil(m/ε), ε))，其中A(N, α)是任何(1+α)近似方案的作业跨度最小化的时间复杂度。第二个算法针对n未知的情况。它使用自适应加权随机采样，即在多轮中进行采样，在每轮后调整样本数量，并在子线性时间内运行~O(m^5/ε^4√n + A(ceil(m/ε), ε))。我们还提供了一种生成加权随机样本的实现，使用O(log n)个均匀随机样本。

英文摘要：

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.

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