We consider the classical problem of minimizing off-line the total energy consumption required to execute a set of n real-time jobs on a single processor with a finite number of available speeds. Each real-time job is defined by its release time, size, and deadline (all bounded integers). The goal is to find a processor speed schedule, such that no job misses its deadline and the energy consumption is minimal. We propose a pseudo-linear time algorithm that checks the schedulability of the given set of n jobs and computes an optimal speed schedule. The time complexity of our algorithm is in ${\mathcal {O}}(n)$ , to be compared with ${\mathcal {O}}(n\log (n))$ for the best known solution. Besides the complexity gain, the main interest of our algorithm is that it is based on a completely different idea: instead of computing the critical intervals, it sweeps the set of jobs and uses a dynamic programming approach to compute an optimal speed schedule. Our linear time algorithm is still valid (with some changes) with arbitrary (non-convex) power functions and when switching costs are taken into account.

中文翻译：

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最优离散速度最小化能耗的伪线性时间算法

我们考虑了在具有有限数量的可用速度的单个处理器上执行一组 n 个实时作业所需的离线总能耗最小化的经典问题。每个实时作业都由其发布时间、大小和截止日期（所有有界整数）定义。目标是找到处理器速度计划，以便没有作业错过其最后期限并且能耗最小。我们提出了一种伪线性时间算法，用于检查给定的 n 个作业集的可调度性并计算最佳速度调度。我们算法的时间复杂度在 ${\mathcal {O}}(n)$ 中，与 ${\mathcal {O}}(n\log (n))$ 进行比较以获得最佳解决方案。除了复杂度增益之外，我们算法的主要兴趣在于它基于一个完全不同的想法：而不是计算临界区间，它扫描作业集并使用动态规划方法来计算最佳速度计划。我们的线性时间算法对于任意（非凸）幂函数和考虑切换成本时仍然有效（有一些变化）。