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

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  5. Azar, Y., Naor, J., Rom, R.: The competitiveness of on-line assignments. J. Algorithms 18(2), 221–237 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  10. Bender, M.A., Muthukrishnan, S., Rajaraman, R.: Improved algorithms for stretch scheduling. In: 13th Symposium on discrete algorithms (SODA), pp. 762–771 (2002)

  11. Chekuri, C., Moseley, B.: Online scheduling to minimize the maximum delay factor. In: 20th Symposium on discrete algorithms (SODA), pp. 1116–1125 (2009)

  12. Cynthia, A.P., Stein, C., Torng, E., Wein, J.: Optimal time-critical scheduling via resource augmentation. Algorithmica 32(2), 163–200 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

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Correspondence to Tobias Friedrich.

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A preliminary conference version [4] without most of the proofs appeared in the 40th International Colloquium on Automata, Languages and Programming (ICALP 2013). This work was done while all five authors were visiting the Department of Computer Science and Engineering of IIT Delhi, India. Naveen Garg is supported by the Indo-German Max Planck Center for Computer Science (IMPECS).

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Anand, S., Bringmann, K., Friedrich, T. et al. Minimizing Maximum (Weighted) Flow-Time on Related and Unrelated Machines. Algorithmica 77, 515–536 (2017). https://doi.org/10.1007/s00453-015-0082-y

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