Abstract
We consider the problem of finding a schedule for n identical malleable tasks on p identical processors with minimal completion time. This problem arises while using the branch & bound or the divide & conquer strategy to solve a problem on a parallel system. If nothing is known about the sub-problems, then they are assumed to be identical. We assume that the execution time decreases with the number of processors while the computational work increases. We give an algorithm with running time exponential in p which computes an optimal schedule. In order to approximate an optimal schedule, we use the concept of phase-by-phase schedules. Here schedules consist of phases in which every job uses the same number of processors. We prove that one can approximate an optimal schedule up to a factor \(\frac{5}{4}\) using constant time, and we show that this is optimal. Furthermore, we give an ε-approximation algorithm if the speed-up is optimal up to a constant factor.
Partly supported by the DFG-Sonderforschungsbereich 376 Massive Parallelität: Algorithmen, Entwurfsmethoden, Anwendungen, and by the IST Programme of the EU under contract number IST-1999-14186 (ALCOM-FT).
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References
Baker, B., Coffman, E., Rivest, R.: Orthogonal packings in two dimensions. SIAM Journal on Computing 9(4), 846–855 (1980)
Baker, B.S., Brown, D.J., Katseff, H.P.: A 5/4 algorithm for two-dimensional packing. Journal of Algorithms 2, 348–368 (1981)
Belkhale, K.P., Banerjee, P.: Partitionable independent task scheduling problem. In: Proc. of the 1990 International Conference on Parallel Processing (ICPP 1990), vol. 1, pp. 72–75 (1990)
Blazewicz, J., Machowiak, M., Mounié, G., Trystram, D.: Approximation algorithms for scheduling independent malleable tasks. In: Sakellariou, R., Keane, J.A., Gurd, J.R., Freeman, L. (eds.) Euro-Par 2001. LNCS, vol. 2150, pp. 191–197. Springer, Heidelberg (2001)
Brucker, P.: Scheduling Algorithms. Springer, Heidelberg (1995)
Fernandez de la Vega, W., Zissimopoulos, V.: An approximation scheme for strip packing of rectangles with bounded dimensions. Discrete Applied Mathematics 82, 98–101 (1998)
Decker, T.: Ein universelles Lastverteilungssystem und seine Anwendung bei der Isolierung reeller Nullstellen. Dissertation (2000)
Decker, T., Krandick, W.: Parallel real root isolation using the descartes method. In: Banerjee, P., Prasanna, V.K., Sinha, B.P. (eds.) HiPC 1999. LNCS, vol. 1745, pp. 261–268. Springer, Heidelberg (1999)
Decker, T., Lücking, T., Monien, B.: A 5/4-approximation algorithm for scheduling identical malleable tasks. Technical report, tr-rsfb-02-071, University of Paderborn (2002)
Du, J., Leung, Y.-T.: Complexity of scheduling parallel task systems. SIAM Journal on Discrete Mathematics 2(4), 473–487 (1989)
Dyckhoff, H.: A typology of cutting and packing problems. European Journal of Operational Research 44, 145–159 (1990)
Garey, M., Graham, R.: Bounds for multiprocessor scheduling with resource constraints. SIAM Journal on Computing 4(2), 187–200 (1975)
Jansen, K.: Scheduling malleable parallel tasks: An asymptotic fully polynomialtime approximation scheme. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 562–573. Springer, Heidelberg (2002)
Jansen, K., Porkolab, L.: Linear time approximation schemes for scheduling malleable parallel tasks. Algorithmica 32(3), 507–520 (2002)
Kenyon, C., Remila, E.: Approximate strip packing. In: Proc. of the 37th Annual Symposium on Foundations of Computer Science (FOCS 1996), pp. 142–154 (1996)
Khanna, S., Muthukrishnan, S., Paterson, M.: On approximating rectangular tiling and packing. In: Proc. of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1998), pp. 384–393 (1998)
Krishnamurti, R., Ma, E.: The processor partitioning problem in special-purpose partitionable systems. In: Proc. of the 1988 International Conference on Parallel Processing (ICPP 1988), vol. 1, pp. 434–443 (1988)
Mounié, G., Rapine, C., Trystram, D.: Efficient approximation algorithms for scheduling malleable tasks. In: Proc. of the 11th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA 1999), pp. 23–32 (1999)
Mounié, G., Rapine, C., Trystram, D.: A \(\frac{3}{2}\) -approximation algorithm for independent scheduling malleable tasks (2001) (submitted for publication)
Steinberg, A.: A strip-packing algorithm with absolute performance bound 2. SIAM Journal on Computing 26(2), 401–409 (1997)
Turek, J., Wolf, J.L., Yu, P.S.: Approximate algorithms for scheduling parallelizable tasks. In: Proc. of the 4th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA 1992), pp. 323–332 (1992)
Veltman, B., Lageweg, B.J., Lenstra, J.K.: Multiprocessor scheduling with communication delays. Parallel Computing 16, 173–182 (1990)
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Decker, T., Lücking, T., Monien, B. (2004). A \(\frac{5}{4}\)-Approximation Algorithm for Scheduling Identical Malleable Tasks. In: Solis-Oba, R., Jansen, K. (eds) Approximation and Online Algorithms. WAOA 2003. Lecture Notes in Computer Science, vol 2909. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24592-6_8
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DOI: https://doi.org/10.1007/978-3-540-24592-6_8
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