A pseudo-linear time algorithm for the optimal discrete speed minimizing energy consumption

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.