Abstract
Energy efficient algorithms are becoming critically important, as huge data centers and server farms have increasing impact on monetary and environmental costs. Motivated by such issues, we study online load balancing from an energy perspective. Our framework extends recent work by Khuller, Li, and Saha (SODA 2010) to the online model. We are given m machines, each with some energy activation cost c i and d dimensions (i.e., components). There are n jobs which arrive online and must be assigned to machines. Each job induces a load on its assigned machine along each dimension. We must select machines to activate so that the total activation cost of the machines falls within a budget B and the largest load over all machines and dimensions (i.e., the makespan) by assigning jobs to active machines is at most Λ.
We first study the model in which machines are unrelated and can have arbitrary activation cost. In this problem, which we call Machine Activation, we extend previous work to handle jobs which arrive online. We consider a variant where the target makespan Λ and budget B are given. The first main result is an online algorithm which is O(log(md) log(nm))-competitive on the load Λ and O(dlog2 (nm))-competitive on the energy budget B. We also address cases where one parameter is given and we are asked to minimize the other, or where we want to minimize a convex combination of the two. Running our previous algorithm in phases gives results for these variants. We prove lower bounds indicating that the effect on the competitive ratio due to multiple phases is necessary.
Our second main result is in the same setting except all machines are identical and have no activation cost. We call this problem Vector Load Balancing, our objective is to minimize the largest load induced over all machines and dimensions (makespan) and the sum of the largest induced load on each machine (energy). We give an online algorithm that is O(logd)-competitive on makespan, which improves even on the best prior offline result, and O(logd)-competitive on the sum of induced loads if the target makespan is given; without this knowledge we show that it is impossible to get a competitive ratio independent of m.
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
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: The online set cover problem. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing (2003)
Aspnes, J., Azar, Y., Fiat, A., Plotkin, S., Waarts, O.: On-line load balancing with applications to machine scheduling and virtual circuit routing. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing (1993)
Awerbuch, B., Azar, Y., Grove, E., Kao, M., Krishnan, P., Vitter, J.: Load balancing in the l p norm. In: Proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science (1995)
Azar, Y., Bhaskar, U., Fleischer, L., Panigrahi, D.: Online mixed packing and covering. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (2013)
Azar, Y., Cohen, I.R., Kamara, S., Shepherd, B.: Tight bounds for online vector bin packing. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing (2013)
Bansal, N., Caprara, A., Sviridenko, M.: Improved approximation algorithms for multidimensional bin packing problems. In: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (2006)
Bartal, Y., Fiat, A., Karloff, H., Vohra, R.: New algorithms for an ancient scheduling problem. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing (1992)
Buchbinder, N., Naor, J.: Fair online load balancing. In: Proceedings of the 18th Annual ACM Symposium on Parallelism in Algorithms and Architectures (2006)
Chekuri, C., Khanna, S.: On multi-dimensional packing problems. In: Proceedings of the 10th Annual ACM-SIAM Symposium on Discrete Algorithms (1999)
DeWitt, D., Gray, J.: Parallel database systems: The future of high performance database systems. Communications of the ACM (1992)
Garofalakis, M., Ioannidis, Y.: Scheduling issues in multimedia query optimization. ACM Computing Surveys (1995)
Garofalakis, M., Ioannidis, Y.: Multi-dimensional resource scheduling for parallel queries. In: Proceedings of 1996 ACM SIGMOD International Conference on Management of Data (1996)
Garofalakis, M., Ioannidis, Y.: Parallel query scheduling and optimization with time- and space-shared resources. In: Proceedings of the 23rd International Conference on Very Large Data Bases (1997)
Hochbaum, D., Shmoys, D.: Using dual approximation algorithms for scheduling problems: theoretical and practical results. Journal of the ACM 34(1), 144–162 (1987)
Irani, S., Pruhs, K.: Algorithmic problems in power management. SIGACT News 36(2), 63–76 (2005)
Karp, R., Luby, M., Marchetti-Spaccamela, A.: A probabilistic analysis of multidimensional bin packing problems. In: Proceedings of the 16th Annual ACM Symposium on Theory of Computing (1984)
Khuller, S., Li, J., Saha, B.: Energy efficient scheduling via partial shutdown. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (2010)
Kou, L., Markowsky, G.: Multidimensional bin packing algorithms. IBM Journal of Research and Development (1977)
Lavi, R., Swamy, C.: Truthful mechanism design for multi-dimensional scheduling via cycle monotonicity. In: Proceedings of the 8th Annual ACM Conference on Electronic Commerce (2007)
Li, J., Khuller, S.: Generalized machine activation problems. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (2011)
Lo, S.-T., Chen, R.-M., Huang, Y.-M.: Multi-constraint system scheduling using dynamic and delay ant colony system. In: Okuno, H.G., Ali, M. (eds.) IEA/AIE 2007. LNCS (LNAI), vol. 4570, pp. 655–664. Springer, Heidelberg (2007)
Moore, J., Chase, J., Ranganathan, P., Sharma, R.: Making scheduling “cool”: temperature-aware workload placement in data centers. In: Proceedings of the USENIX Annual Technical Conference (2005)
Pruhs, K., Sgall, J., Torng, E.: Online scheduling. In: Handbook of Scheduling: Algorithms, Models, and Performance Analysis (2004)
Rofouei, M., Stathopoulos, T., Ryffel, S., Kaiser, W., Sarrafzadeh, M.: Energy aware high performance computing with graphical processing units. In: Proceedings of USENIX 1st Workshop on Power Aware Computing Systems (2008)
Ryffel, S., Stathopoulos, T., McIntire, D., Kaiser, W., Thiele, L.: Accurate energy attribution and accounting for multi-core systems. Technical Report 67, Center for Embedded Network Sensing (2009)
Shmoys, D., Tardos, E.: An approximation algorithm for the generalized assignment problem. Math. Program. 62(3), 461–474 (1993)
Stathopoulos, T., McIntire, D., Kaiser, W.: The energy endoscope: Real-time detailed energy accounting for wireless sensor nodes. In: Proceedings of the 7th Annual International Conference on Information Processing in Sensor Networks (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Meyerson, A., Roytman, A., Tagiku, B. (2013). Online Multidimensional Load Balancing. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_21
Download citation
DOI: https://doi.org/10.1007/978-3-642-40328-6_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40327-9
Online ISBN: 978-3-642-40328-6
eBook Packages: Computer ScienceComputer Science (R0)