Abstract
Online scheduling of parallelizable jobs has received a significant amount of attention recently. Scalable algorithms are known—that is, algorithms that are (1+\(\varepsilon \))-speed O(1)-competitive for any fixed \(\varepsilon >0\). Previous research has focused on the case where each job’s parallelizability can be expressed as a concave speedup curve. However, there are cases where a job’s speedup curve can be convex. Considering convex speedup curves has received attention in the offline setting, but, to date, there are no positive results in the online model. In this work, we consider scheduling jobs with convex or concave speedup curves for the first time in the online setting. We give a new algorithm that is (1+\(\varepsilon \))-speed O(1)-competitive. There are strong lower bounds on the competitive ratio if the algorithm is not given resource augmentation over the adversary, and thus this is essentially the best positive result one can show for this setting.
This research was supported in part by NSF grants CNS 1408695, CCF 1439084, IIS 1247726, IIS 1251137, and CCF 1217708.
Samuel McCauley was also supported in part by Sandia National Laboratories.
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Notes
- 1.
Specifically, [17] gives an \(\varOmega (\log n/m)\) lower bound.
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Acknowledgements
We would like to thank Michael Bender for helpful discussions, and Bertrand Simon for informing us of reference [2]
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Ebrahimi, R., McCauley, S., Moseley, B. (2015). Scheduling Parallel Jobs Online with Convex and Concave Parallelizability. In: Sanità, L., Skutella, M. (eds) Approximation and Online Algorithms. WAOA 2015. Lecture Notes in Computer Science(), vol 9499. Springer, Cham. https://doi.org/10.1007/978-3-319-28684-6_16
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