Abstract
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.
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Notes
The critical interval is the time interval with the highest load per time unit, to be precisely defined later.
The arithmetic complexity of an algorithm is the number of elementary operations it requires, regardless of the size of their arguments.
Increasing paths over a 2D integer lattice are staircases.
To be more precise, \(|{\mathcal {P}}({w})|\) is bounded by \(|{\mathcal {S}}|\), and since \({\mathcal {S}} = \{0, 1, {\ldots } m-1\}\), we have \(|{\mathcal {S}}| = {s_{\max \limits }}+1\).
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This article belongs to the Topical Collection: Topical Collection on Recent Trends in Reactive Systems
Guest Editor: Sebastian Lahaye
This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissement d’Avenir.
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Gaujal, B., Girault, A. & Plassart, S. A pseudo-linear time algorithm for the optimal discrete speed minimizing energy consumption. Discrete Event Dyn Syst 31, 163–184 (2021). https://doi.org/10.1007/s10626-020-00327-9
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DOI: https://doi.org/10.1007/s10626-020-00327-9