Advertisement

Makespan Minimization for Parallel Jobs with Energy Constraint

  • Alexander Kononov
  • Yulia KovalenkoEmail author
Conference paper
  • 141 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12095)

Abstract

We are given a set of parallel jobs that have to be executed on a set of speed-scalable processors varying their speeds dynamically. Running a job at a slower speed is more energy efficient, however it takes longer time and affects the performance. Every job is characterized by the processing volume and the number of the required processors. Our objective is to minimize the maximum completion time so that the energy consumption is not greater than a given energy budget. For various particular cases we propose polynomial-time approximation algorithms, consisting of two stages. At the first stage, we give an auxiliary convex program. By solving this problem in polynomial time, we find processing times of jobs and a lower bound on the makespan. Then, at the second stage, we transform our problem to the classical problem without speed scaling and construct a feasible schedule.

Keywords

Parallel job Speed scaling Scheduling Approximation algorithm 

References

  1. 1.
    Albers, S., Müller, F., Schmelzer, S.: Speed scaling on parallel processors. Algorithmica 68(2), 404–425 (2014)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bampis, E., Letsios, D., Lucarelli, G.: A note on multiprocessor speed scaling with precedence constraints. In: 26th ACM symposium on Parallelism in algorithms and architectures, SPAA 2014, pp. 138–142. ACM (2014)Google Scholar
  3. 3.
    Bunde, D.: Power-aware scheduling for makespan and flow. J. Sched. 12, 489–500 (2009).  https://doi.org/10.1007/s10951-009-0123-yMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Du, J., Leung, J.T.: Complexity of scheduling parallel task systems. SIAM J. Discrete Math. 2(4), 472–478 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gerards, M.E.T., Hurink, J.L., Hölzenspies, P.K.F.: A survey of offline algorithms for energy minimization under deadline constraints. J. Sched. 19(1), 3–19 (2016).  https://doi.org/10.1007/s10951-015-0463-8MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Graham, R.L.: Bounds for certain multiprocessor anomalies. SIAM J. Appl. Math. 17(2), 416–429 (1966)CrossRefGoogle Scholar
  7. 7.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimizations, 2nd edn. Springer, Heidelberg (1993).  https://doi.org/10.1007/978-3-642-78240-4CrossRefzbMATHGoogle Scholar
  8. 8.
    Kononov, A., Kovalenko, Y.: On speed scaling scheduling of parallel jobs with preemption. In: Kochetov, Y., Khachay, M., Beresnev, V., Nurminski, E., Pardalos, P. (eds.) DOOR 2016. LNCS, vol. 9869, pp. 309–321. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44914-2_25CrossRefGoogle Scholar
  9. 9.
    Kuhn, H., Tucker, A.: Nonlinear programming. In: The Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
  10. 10.
    Li, K.: Analysis of the list scheduling algorithm for precedence constrained parallel tasks. J. Comb. Opt. 3, 73–88 (1999).  https://doi.org/10.1023/A:1009817206440MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Naroska, E., Schwiegelshohn, U.: On an on-line scheduling problem for parallel jobs. Inform. Process. Lett. 81(6), 297–304 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Nesterov, E.: Methods of Convex Optimization. Moscow (2010). (in Russian)Google Scholar
  13. 13.
    Pruhs, K., van Stee, R.: Speed scaling of tasks with precedence constraints. Theory Comput. Syst. 43, 67–80 (2007).  https://doi.org/10.1007/s00224-007-9070-1MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shabtay, D., Kaspi, M.: Parallel machine scheduling with a convex resource consumption function. Eur. J. Oper. Res. 173, 92–107 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Sobolev Institute of Mathematics SB RASNovosibirskRussia
  2. 2.Sobolev Institute of Mathematics SB RAS, Omsk DepartmentOmskRussia
  3. 3.Dostoevsky Omsk State UniversityOmskRussia

Personalised recommendations