# Improved upper bounds for online malleable job scheduling

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## Abstract

In this paper, we study online algorithms that schedule *malleable* jobs, i.e., jobs that can be parallelized on any subset of the available \(m\) identical machines. We study a model that accounts for the tradeoff between multiprocessor speedup and overhead time, namely, if job \(j\) has processing requirement \(p_j\) and is assigned to run on \(k_j\) machines, then \(j\)’s execution time becomes \(p_j/k_j + (k_j -1)c\), where \(c\) is a constant parameter to the problem. For \(m=2\), we present an online algorithm OCS that has a strong competitive ratio of 3/2, matching a previously established lower bound. We also present an online algorithm ASYM2 that is asymptotically \(((4-\epsilon )/(3-\epsilon ))\)-competitive when \(m=2\), where \(0 < \epsilon \le 2\) is a parameter to the algorithm, improving upon an existing asymptotically (3/2)-competitive algorithm. Finally, we present an online algorithm OTO that is strongly \(2\)-competitive when \(m = 3\), improving upon the previous best upper bound of \(9/4\).

## Keywords

Online algorithms Scheduling Parallel jobs Identical machines Overhead time## References

- Baker, B. S., Coffman, E. G, Jr, & Rivest, R. L. (1980). Orthogonal packings in two dimensions.
*SIAM Journal on Computing*,*9*(4), 846–855.CrossRefGoogle Scholar - Bartal, Y., Fiat, A., Karloff, H. J., & Vohra, R. (1995). New algorithms for an ancient scheduling problem.
*Journal of Computer and System Sciences*,*51*(3), 359–366.CrossRefGoogle Scholar - Carroll, T. E., & Grosu, D. (2010). Incentive compatible online scheduling of malleable parallel jobs with individual deadlines. In
*Proceedings of ICPP*(pp. 516–524). IEEE Computer Society.Google Scholar - Chen, B., van Vliet, A., & Woeginger, G. J. (1994). New lower and upper bounds for on-line scheduling.
*Operations Research Letters*,*16*(4), 221–230.CrossRefGoogle Scholar - Coffman, E. G, Jr, Garey, M. R., Johnson, D. S., & Tarjan, R. E. (1980). Performance bounds for level-oriented two-dimensional packing algorithms.
*SIAM Journal on Computing*,*9*(4), 808–826.CrossRefGoogle Scholar - Du, J., & Leung, J. (1989). Complexity of scheduling parallel task systems.
*SIAM Journal on Discrete Mathematics*,*2*(4), 473–487.CrossRefGoogle Scholar - Dutton, R. A., & Mao, W. (2007). Online scheduling of malleable parallel jobs. In
*Proceedings of ICPDCS*(pp. 1–6). IASTED.Google Scholar - Dutton, R. A., Mao, W., Chen, J., & Watson, W. A. III (2008). Parallel job scheduling with overhead: A benchmark study. In
*Proceedings of NAS*(pp. 326–333). IEEE Computer Society.Google Scholar - Faigle, U., Kern, W., & Turán, G. (1989). On the performance of on-line algorithms for partition problems.
*Acta Cybernetica*,*9*(2), 107–119.Google Scholar - Garey, M. R., & Graham, R. L. (1975). Bounds for multiprocessor scheduling with resource constraints.
*SIAM Journal on Computing*,*4*(2), 187–200.CrossRefGoogle Scholar - Graham, R. L. (1969). Bounds on multiprocessing timing anomalies.
*SIAM Journal on Applied Mathematics*,*17*(2), 416–429.CrossRefGoogle Scholar - Guo, S., & Kang, L. (2010). Online scheduling of malleable parallel jobs with setup times on two identical machines.
*European Journal of Operational Research*,*206*(3), 555–561.CrossRefGoogle Scholar - Harren, R., & Kern, W. (2011). Improved lower bound for online strip packing—(extended abstract). In
*Proceedings of WAOA*, LNCS (Vol. 7164, pp. 211–218). Springer.Google Scholar - Havill, J. T., & Mao, W. (2008). Competitive online scheduling of perfectly malleable jobs with setup times.
*European Journal of Operational Research*,*187*(3), 1126–1142.CrossRefGoogle Scholar - Havill, J. T. (2010). Online malleable job scheduling for m\(\le \)3.
*Information Processing Letters*,*111*(1), 31–35.CrossRefGoogle Scholar - Hurink, J. L., & Paulus, J. J. (2011). Improved online algorithms for parallel job scheduling and strip packing.
*Theoretical Computer Science*,*412*(7), 583–593.CrossRefGoogle Scholar - Johannes, B. (2006). Scheduling parallel jobs to minimize the makespan.
*Journal of Scheduling*,*9*(5), 433–452.CrossRefGoogle Scholar - Karloff, H. J., Suri, S., & Vassilvitskii, S. (2010). A model of computation for mapreduce. In
*Proceedings of SODA*(pp. 938–948). SIAM.Google Scholar - Moseley, B., Dasgupta, A., Kumar, R., & Sarlós, T. (2011). On scheduling in map-reduce and flow-shops. In
*Proceedings of SPAA*(pp. 289–298). ACM.Google Scholar - Nagarajan, V., Wolf, J. L., Balmin, A., & Hildrum, K. (2013). Flowflex: Malleable scheduling for flows of mapreduce jobs. In
*Proceedings of Middleware*, LNCS (Vol. 8275, pp. 103–122). Springer.Google Scholar - Naroska, E., & Schwiegelshohn, U. (2002). On an on-line scheduling problem for parallel jobs.
*Information Processing Letters*,*81*(6), 297–304.CrossRefGoogle Scholar - Schwiegelshohn, U., Ludwig, W., Wolf, J. L., Turek, J., & Yu, P. S. (1998). Smart smart bounds for weighted response time scheduling.
*SIAM Journal on Computing*,*28*(1), 237–253.Google Scholar - Sgall, J. (1994). On-line scheduling of parallel jobs. In
*Proceedings of MFCS*, LNCS (vol. 841, pp. 159–176). Springer.Google Scholar - Sleator, D. D. (1980). A 2.5 times optimal algorithm for packing in two dimensions.
*Information Processing Letters*,*10*(1), 37–40.CrossRefGoogle Scholar - Turek, J., Wolf, J. L., & Yu, P. S. (1992). Approximate algorithms scheduling parallelizable tasks. In
*Proceedings of SPAA*(pp. 323–332). ACM.Google Scholar - Ye, D., & Zhang, G. (2007). On-line scheduling of parallel jobs in a list.
*Journal of Scheduling*,*10*(6), 407–413.CrossRefGoogle Scholar - Ye, D., Han, X., & Zhang, G. (2009). A note on online strip packing.
*Journal of Combinatorial Optimization*,*17*(4), 417–423.CrossRefGoogle Scholar