Algorithmica

, Volume 77, Issue 2, pp 515–536 | Cite as

Minimizing Maximum (Weighted) Flow-Time on Related and Unrelated Machines

  • S. Anand
  • Karl Bringmann
  • Tobias Friedrich
  • Naveen Garg
  • Amit Kumar
Article
  • 249 Downloads

Abstract

In this paper we initiate the study of job scheduling on related and unrelated machines so as to minimize the maximum flow time or the maximum weighted flow time (when each job has an associated weight). Previous work for these metrics considered only the setting of parallel machines, while previous work for scheduling on unrelated machines only considered \(L_p, p<\infty \) norms. Our main results are: (1) we give an \(\mathcal {O}({\varepsilon }^{-3})\)-competitive algorithm to minimize maximum weighted flow time on related machines where we assume that the machines of the online algorithm can process \(1+{\varepsilon }\) units of a job in 1 time-unit (\({\varepsilon }\) speed augmentation). (2) For the objective of minimizing maximum flow time on unrelated machines we give a simple \(2/{\varepsilon }\)-competitive algorithm when we augment the speed by \({\varepsilon }\). For m machines we show a lower bound of \({\varOmega }(m)\) on the competitive ratio if speed augmentation is not permitted. Our algorithm does not assign jobs to machines as soon as they arrive. To justify this “drawback” we show a lower bound of \({\varOmega }(\log m)\) on the competitive ratio of immediate dispatch algorithms. In both these lower bound constructions we use jobs whose processing times are in \(\left\{ 1,\infty \right\} \), and hence they apply to the more restrictive subset parallel setting. (3) For the objective of minimizing maximum weighted flow time on unrelated machines we establish a lower bound of \({\varOmega }(\log m)\)-on the competitive ratio of any online algorithm which is permitted to use \(s=\mathcal {O}(1)\) speed machines. In our lower bound construction, job j has a processing time of \(p_j\) on a subset of machines and infinity on others and has a weight \(1/p_j\). Hence this lower bound applies to the subset parallel setting for the special case of minimizing maximum stretch.

Keywords

Scheduling Minimizing flow-time Online algorithms  Competitive analysis 

References

  1. 1.
    Ambühl, C., Mastrolilli, M.: On-line scheduling to minimize max flow time: an optimal preemptive algorithm. Oper. Res. Lett. 33(6), 597–602 (2005)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anand, S., Garg, N., Megow, N.: Meeting deadlines: how much speed suffices? In: 38th International colloquium on automata, languages and programming (ICALP), pp. 232–243 (2011)Google Scholar
  3. 3.
    Anand, S., Garg, N., Kumar, A.: Resource augmentation for weighted flow-time explained by dual fitting. In: 23rd Symposium on discrete algorithms (SODA), pp. 1228–1241 (2012)Google Scholar
  4. 4.
    Anand, S., Bringmann, K., Friedrich, T., Garg, N., Kumar, A.: Minimizing maximum (weighted) flow-time on related and unrelated machines. In: 40th International colloquium on automata, languages and programming (ICALP), pp. 13–24 (2013)Google Scholar
  5. 5.
    Azar, Y., Naor, J., Rom, R.: The competitiveness of on-line assignments. J. Algorithms 18(2), 221–237 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Azar, Y., Kalyanasundaram, B., Plotkin, S.A., Pruhs, K., Waarts, O.: On-line load balancing of temporary tasks. J. Algorithms 22(1), 93–110 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bansal, N., Pruhs, K.: Server scheduling in the \(\ell _{p}\) norm: a rising tide lifts all boats. In: 35th Symposium on theory of computing (STOC), pp. 242–250 (2003)Google Scholar
  8. 8.
    Bansal, N., Pruhs, K.: Server scheduling in the weighted \(\ell _{p}\) norm. In: 6th Latin American theoretical informatics conference (LATIN), pp. 434–443 (2004)Google Scholar
  9. 9.
    Bender, M.A., Chakrabarti, S., Muthukrishnan, S.: Flow and stretch metrics for scheduling continuous job streams. In: 9th Symposium on discrete algorithms (SODA), pp. 270–279 (1998)Google Scholar
  10. 10.
    Bender, M.A., Muthukrishnan, S., Rajaraman, R.: Improved algorithms for stretch scheduling. In: 13th Symposium on discrete algorithms (SODA), pp. 762–771 (2002)Google Scholar
  11. 11.
    Chekuri, C., Moseley, B.: Online scheduling to minimize the maximum delay factor. In: 20th Symposium on discrete algorithms (SODA), pp. 1116–1125 (2009)Google Scholar
  12. 12.
    Cynthia, A.P., Stein, C., Torng, E., Wein, J.: Optimal time-critical scheduling via resource augmentation. Algorithmica 32(2), 163–200 (2002)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Golovin, D., Gupta, A., Kumar, A., Tangwongsan, K.: All-norms and all-\(\ell _{p}\)-norms approximation algorithms. In: 28th Conference foundations of software technology and theoretical computer science (FSTTCS), pp. 199–210 (2008)Google Scholar
  14. 14.
    Im, S., Moseley, B.: An online scalable algorithm for minimizing \(\ell _{k}\)-norms of weighted flow time on unrelated machines. In: 22nd Symposium on discrete algorithms (SODA), pp. 95–108 (2011)Google Scholar
  15. 15.
    Lam, T.W., To, K.-K.: Trade-offs between speed and processor in hard-deadline scheduling. In: 10th Symposium discrete algorithms (SODA), pp. 623–632 (1999)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • S. Anand
    • 1
  • Karl Bringmann
    • 2
  • Tobias Friedrich
    • 3
  • Naveen Garg
    • 4
  • Amit Kumar
    • 4
  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.ETH ZürichZurichSwitzerland
  3. 3.Hasso Plattner InstitutePotsdamGermany
  4. 4.Computer Science and EngineeringIndian Institute of Technology DelhiHauz Khas, New DelhiIndia

Personalised recommendations