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Scheduling shared continuous resources on many-cores

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Abstract

We consider the problem of scheduling a number of jobs on m identical processors sharing a continuously divisible resource. Each job j comes with a resource requirement . The job can be processed at full speed if granted its full resource requirement. If receiving only an x-portion of \(r_j\), it is processed at an x-fraction of the full speed. Our goal is to find a resource assignment that minimizes the makespan (i.e., the latest completion time). Variants of such problems, relating the resource assignment of jobs to their processing speeds, have been studied under the term discrete–continuous scheduling. Known results are either very pessimistic or heuristic in nature. In this article, we suggest and analyze a slightly simplified model. It focuses on the assignment of shared continuous resources to the processors. The job assignment to processors and the ordering of the jobs have already been fixed. It is shown that, even for unit size jobs, finding an optimal solution is NP-hard if the number of processors is part of the input. Positive results for unit size jobs include a polynomial-time algorithm for any constant number of processors. Since the running time is infeasible for practical purposes, we also provide more efficient algorithm variants: an optimal algorithm for two processors and a -approximation algorithm for m processors.

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Notes

  1. This is also known as speed scaling (cf. Yao et al. 1995).

  2. This reduces our analysis to a smaller problem instance.

  3. One could also consider resource requirements \({>}1\). However, the most natural extension of our model can easily be shown to reduce to non-unit size jobs with resource requirements \({\le }1\) (rescale jobs with resource requirement \(r>1\) and workload p such that it has resource requirement \(1/r \times r=1\) and workload \(r\times p\)).

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Authors and Affiliations

  1. Computer Science Institute, Johannes Gutenberg-Universität Mainz, Mainz, Germany

    Ernst Althaus

  2. Zentrum für Datenverarbeitung, Johannes Gutenberg-Universität Mainz, Mainz, Germany

    André Brinkmann, Lars Nagel & Tim Süß

  3. School of Computing Science, Simon Fraser University, Burnaby, Canada

    Peter Kling

  4. Heinz Nixdorf Institute and Computer Science Department, Paderborn University, Paderborn, Germany

    Friedhelm Meyer auf der Heide & Sören Riechers

  5. Computer Science Institute, Charles University, Prague, Czech Republic

    Jiří Sgall

Authors
  1. Ernst Althaus
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  2. André Brinkmann
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  3. Peter Kling
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  4. Friedhelm Meyer auf der Heide
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  5. Lars Nagel
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  6. Sören Riechers
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  7. Jiří Sgall
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  8. Tim Süß
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Corresponding author

Correspondence to Lars Nagel.

Additional information

Supported by the Federal Ministry of Education and Research (BMBF) under Grant 01IH13004 (Project “FAST”), by the German Research Foundation (DFG) within the Collaborative Research Center “On-The-Fly Computing” (SFB 901), by the Center of Excellence—ITI (Project P202/12/G061 of GA ČR), and by the Pacific Institute of Mathematical Sciences (PIMS).

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Althaus, E., Brinkmann, A., Kling, P. et al. Scheduling shared continuous resources on many-cores. J Sched 21, 77–92 (2018). https://doi.org/10.1007/s10951-017-0518-0

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  • DOI: https://doi.org/10.1007/s10951-017-0518-0

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