Abstract
The worst-case behavior of the “critical path” (CP) algorithm for multiprocessor scheduling with an out-tree task dependency structure is studied. The out-tree is not known in advance and the tasks are released on-line over time (each task is released at the completion time of its direct predecessor task in the out-tree). For each task, the processing time and the remainder (the length of the longest chain of the future tasks headed by this task) become known at its release time. The tight worst-case ratio and absolute error are derived for this strongly clairvoyant on-line model. For out-trees with a specific simple structure, essentially better worst-case ratio and absolute error are derived. Our bounds are given in terms of t max, the length of the longest chain in the out-tree, and it is shown that the worst-case ratio asymptotically approaches 2 for large t max when the number of processors \(m=\widetilde {\tau}(\widetilde{\tau}+1)/2-2\), where \(\widetilde{\tau}\) is an integer close to \(\sqrt {t_{\max}}\). A non-clairvoyant on-line version (without knowledge of task processing time and remainder at the release time of the task) is also considered and is shown that the worst-case behavior of width-first search is better or the same as that of the depth-first search.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. van den Akker, J. Hoogeveen, and N. Vakhania, “Restarts can help in on-line minimization of the maximum delivery time on a single machine,” Journal of Scheduling, vol. 3, pp. 333–341, 2000.
S. Albers, “Better bounds for online scheduling,” Proc. of the 29th Ann. ACM Symp. on Theory of Computing, ACM, 1997, pp. 130–139.
N.F. Chen and C.L. Liu, “On a class of scheduling algorithms for multiprocessors computing systems,” Parallel Processing, Lecture Notes in Computer Science, T.Y. Feng (Ed.), Springer: Berlin, vol. 24, pp. 1–16, 1975.
B. Chen, C.N. Potts, and G.J. Woeginger, “A review of machine scheduling: Complexity, algorithms and approx-imability,” Handbook of Combinatorial Optimization, vol. 3, D.Z. Du and P. Pardalos (Eds.). Kluwer Academic Publishers: Dordrecht, 1998, pp. 21–169.
E.G. Coffman, Jr and R.L. Graham, “Optimal scheduling for two-processor systems,” Acta Informatica, vol. 1, pp. 200–213, 1972.
D.K. Freisen, “Tighter bounds for the multifit processor scheduling algorithm,” SIAM J. on Comput., vol. 13, pp. 170–181, 1984.
M. Fujii, T. Kasami, and K. Nimomiya, “Optimal sequencing of two equivalent processors,” SIAM J. Appl. Math., vol. 14, pp.784–789, 1969.
H.N. Gabow and R.E. Tarjan, “A linear-time algorithm for a special case of disjoint set union,” J. Comput. System Sci., vol. 30, pp. 209–221, 1985.
G. Galambos and G.J. Woeginger, “An on-line scheduling heuristic with better worst case ratio than Graham's list scheduling,” SIAM J. Comput., vol. 22, pp. 349–355, 1993.
R.L. Graham, “Bounds for certain multiprocessing anomalies,” Bell Syst. Tech. J., vol. 45, pp. 1563–1581, 1966.
R.L. Graham, “Bounds on multiprocessing timing anomalies,” SIAM J. Applied Math., vol. 17, pp. 416–429, 1969.
D.S. Hochbaum and D.B. Shmoys, “A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach,” SIAM J. Comput., vol. 17, no. 3, pp. 539–551, 1988.
N.C. Hsu, “Elementary proof of Hu's theorem on isotone mappings,” Proc. Amer. Math. Soc., vol. 17, pp. 11–114, 1966.
T.C. Hu, “Parallel sequencing and assembly line problems,” Oper. Res., vol. 9, pp. 841–848, 1961.
D.R. Karger, S.J. Phillips, and E. Torng, “A better algorithm for ancient scheduling problem,” J. of Algorithms, vol. 20, pp. 400–430, 1996.
M.T. Kaufman, “An almost-optimal algorithm for the assembly line scheduling problem,” IEEE Trarns. on Computers, C-23, pp. 1170–1174, 1974.
M. Kunde, “Beste Schranken biem LP-scheduling,” Institut fur Informatic und Prakt. Math., Bericht 7603, Universitat Kiel, 1976.
M. Kunde, “Nonpreemptive LP-scheduling on homogeneous multiprocessor system,” SIAM J. Comput., vol. 10, pp. 151–173, 1981.
S. Lam and R. Sethi, “Worst case analysis of two scheduling algorithms,” SIAM J. Comput., vol. 6, pp. 518–536, 1977.
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys, “Sequencing and scheduling: Algorithms and complexity,” Handbooks in Operations Research and Management Science, vol. 4, S.C. Graves, A.H.G. Rinnooy Kan, and P. Zipkin (Eds.), North Holland: Amsterdam, 1993.
J.K. Lenstra and A.H.G. Rinnooy Kan, “Complexity of scheduling under precedence constraints,” Oper. Res., vol. 26, pp. 22–35, 1978.
R. Sethi, “Algorithms for minimal-length schedules,” Computer and Job/Shop Scheduling Theory, E.G. Coffman Jr. (Ed.), Wiley: New York, 1976, pp. 51–59.
J. Sgall, “On-line scheduling-A servey,” A. Fiat and G.J. Woeginger (Eds.), Online algorithms: The state of the art, Springer, 1998.
J.D. Ullman, “NP-complete scheduling problems,” J. Computer System Sci., vol. 10, pp. 384–393, 1975.
M. Yue, “On the exact upper bound for the multifit processor scheduling algorithm,” Ann. Oper. Res., vol. 24, pp. 233–259, 1990.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Vakhania, N. Tight Performance Bounds of CP-Scheduling on Out-Trees. Journal of Combinatorial Optimization 5, 445–464 (2001). https://doi.org/10.1023/A:1011676725533
Issue Date:
DOI: https://doi.org/10.1023/A:1011676725533