Abstract
We consider the on-line load balancing problem where there are m identical machines (servers). Jobs arrive at arbitrary times, where each job has a weight and a duration. A job has to be assigned upon its arrival to exactly one of the machines. The duration of each job becomes known only upon its termination (this is called temporary tasks of unknown durations). Once a job has been assigned to a machine it cannot be reassigned to another machine. The goal is to minimize the maximum over time of the sum (over all machines) of the squares of the loads, instead of the traditional maximum load.
Minimizing the sum of the squares is equivalent to minimizing the load vector with respect to the ℓ2 norm. We show that for the ℓ2 norm greedy algorithm performs within at most 1.50 of the optimum. We show (an asymptotic) lower bound of 1.33 on the competitive ratio of the greedy algorithm. We also show a lower bound of 1.20 on the competitive ratio of any deterministic algorithm.
We extend our techniques and analyze the competitive ratio of greedy with respect to the ℓ p norm. We show that the greedy algorithm performs within at most 2-Ω(1/p) of the optimum. We also show a lower bound of 2 − O(ln p /p) on the competitive ratio of any on-line algorithm.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Albers, S.: Better bounds for online scheduling. SIAM Journal on Computing 29, 459–473 (1999)
Alon, N., Azar, Y., Woeginger, G., Yadid, T.: Approximation schemes for scheduling on parallel machines. Journal of Scheduling 1(1), 55–66 (1998)
Avidor, A., Azar, Y., Sgall, J.: Ancient and new algorithms for load balancing in the \({\it l_p}\) norm. Algorithmica 29, 422–441 (2001); Also in Proc. 9th ACM-SIAMSODA, pp. 426–435 (1998)
Azar, Y., Epstein, L.: On-line load balancing of temporary tasks on identical machines. In: 5th Israeli Symp. on Theory of Computing and Systems, pp. 119–125 (1997)
Azar, Y., Regev, O., Sgall, J., Woeginger, G.: Off-line temporary tasks assignment. Theoretical Computer Science 287, 419–428 (2002)
Bartal, Y., Fiat, A., Karloff, H., Vohra, R.: New algorithms for an ancient scheduling problem. Journal of Computer and System Sciences 51(3), 359–366 (1995)
Bartal, Y., Karloff, H., Rabani, Y.: A better lower bound for on-line scheduling. Information Processing Letters 50, 113–116 (1994)
Chandra, A.K., Wong, C.K.: Worst-case analysis of a placement algorithm related to storage allocation. SIAM Journal on Computing 4(3), 249–263 (1975)
Faigle, U., Kern, W., Turan, G.: On the performance of online algorithms for partition problems. Acta Cybernetica 9, 107–119 (1989)
Fleischer, R., Wahl, M.: Online scheduling revisited. Journal of Scheduling 3(5), 343–353 (2000)
Gormley, T., Reingold, N., Torng, E., Westbrook, J.: Generating adversaries for request-answer games. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 564–565 (2000)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell System Technical Journal 45, 1563–1581 (1966)
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math 17, 416–429 (1969)
Hochbaum, D.S., Shmoys, D.B.: Using dual approximation algorithms for scheduling problems: Theoretical and practical results. J. Assoc. Comput. Mach. 34(1), 144–162 (1987)
Rudin III., J.F.: Improved bounds for the on-line scheduling problem. PhD thesis, The University of Texas at Dallas (May 2001)
Karger, D., Phillips, S., Torng, E.: A better algorithm for an ancient scheduling problem. Journal of Algorithms 20(2), 400–430 (1996)
Leung, J.Y.T., Wei, W.D.: Tighter bounds on a heuristic for a partition problem. Information Processing Letters 56, 51–57 (1995)
Sahni, S.: Algorithms for scheduling independent tasks. Journal of the Association for Computing Machinery 23, 116–127 (1976)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Azar, Y., Epstein, A., Epstein, L. (2004). Load Balancing of Temporary Tasks in the ℓ p Norm. 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_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-24592-6_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-21079-5
Online ISBN: 978-3-540-24592-6
eBook Packages: Springer Book Archive