ABSTRACT
We study the scheduling of flows on a switch with the goal of optimizing metrics related to the response time of the flows. The input is a sequence of flow requests on a switch, where the switch is represented by a bipartite graph with a capacity on each vertex (port), and a flow request is an edge with associated demand. In each round, a subset of edges can be scheduled under the constraint that the total demand of the scheduled edges incident on any vertex is at most the capacity of the vertex. This class of scheduling problems has applications in datacenter networks, and has been extensively studied. Previous work has essentially settled the complexity of metrics based on completion time. The objective of average or maximum response time, however, is more challenging. To the best of our knowledge, there are no prior approximation algorithms results for these metrics in the context of flow scheduling.
We present the first approximation algorithms for flow scheduling over a switch to optimize response time based metrics. For the average response time metric, whose NP-hardness follows directly from past work, we present an offline O(1 + O(log(n))/c) approximation algorithm for unit flows, assuming that the port capacities of the switch can be increased by a factor of 1 + c, for any given positive integer c. For the maximum response time metric, we first establish that it is NP-hard to achieve an approximation factor of better than 4/3 without augmenting capacity. We then present an offline algorithm that achieves optimal maximum response time, assuming the capacity of each port is increased by at most 2 dmax - 1, where dmax is the maximum demand of any flow. Both algorithms are based on linear programming relaxations. We also study the online version of flow scheduling using the lens of competitive analysis, and present preliminary results along with experiments that evaluate the performance of fast online heuristics.
- S. Ahmadi, S. Khuller, M. Purohit, and S. Yang. On scheduling coflows. In IPCO, 2017.Google ScholarCross Ref
- M. Alizadeh, S. Yang, M. Sharif, S. Katti, N. McKeown, B. Prabhakar, and S. Shenker. pfabric: Minimal near-optimal datacenter transport. SIGCOMM Comput. Commun. Rev., 43(4):435--446, August 2013.Google ScholarDigital Library
- C. Ambühl and M. Mastrolilli. On-line scheduling to minimize max flow time: an optimal preemptive algorithm. Operations Research Letters, 33(6):597 -- 602, 2005.Google ScholarDigital Library
- K. R. Baker. Introduction to Sequencing and Scheduling. Wiley, New York, 1974.Google Scholar
- Hitesh Ballani, Paolo Costa, Thomas Karagiannis, and Ant Rowstron. Towards predictable datacenter networks. SIGCOMM Comput. Commun. Rev., 41(4):242--253, August 2011.Google ScholarDigital Library
- N. Bansal. Algorithms for Flow Time Scheduling. PhD thesis, School of Computer Science, Carnegie Mellon University, December 2003.Google ScholarDigital Library
- N. Bansal and H. Chan. Weighted flow time does not admit o(1)-competitive algorithms. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1238--1244, 2009.Google ScholarDigital Library
- N. Bansal and J. Kulkarni. Minimizing flow-time on unrelated machines. In Proceedings of the Forty-seventh Annual ACM Symposium on Theory of Computing, STOC '15, pages 851--860, New York, NY, USA, 2015. ACM.Google ScholarDigital Library
- J. Batra, N. Garg, and A. Kumar. Constant factor approximation algorithm for weighted flow time on a single machine in pseudo-polynomial time. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 778--789, Oct 2018.Google ScholarCross Ref
- M. Bender, S. Chakrabarti, and S. Muthukrishnan. Flow and stretch metrics for scheduling continuous job streams. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 270--279, January 1998.Google ScholarDigital Library
- K. Benzekki, A. El Fergougui, and Abdelbaki Elbelrhiti E. Software-defined networking (sdn): a survey. Security and Communication Networks, 9(18):5803--5833, 2016.Google ScholarCross Ref
- E. W. Biersack, B. Schroeder, and G. Urvoy-Keller. Scheduling in practice. SIGMETRICS Performance Evaluation Review, 34(4):21--28, 2007.Google ScholarDigital Library
- D. Birkhoff. Tres observaciones sobre el algebra lineal. Universidad Nacional de Tucuman Revista, Serie A, 5:147--151, 1946.Google Scholar
- C. Chekuri, S. Khanna, and A. Zhu. Algorithms for minimizing weighted flow time. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 84--93, 2001.Google ScholarDigital Library
- M. Chowdhury, S. Khuller, M. Purohit, S. Yang, and J. You. Near optimal coflow scheduling in networks. In Christian Scheideler and Petra Berenbrink, editors, The 31st ACM on Symposium on Parallelism in Algorithms and Architectures, SPAA 2019, Phoenix, AZ, USA, June 22--24, 2019, pages 123--134. ACM, 2019.Google Scholar
- M. Chowdhury, Y. Zhong, and I. Stoica. Efficient coflow scheduling with varys. SIGCOMM, Comput. Commun. Rev., 44(4):443--454, August 2014.Google ScholarDigital Library
- Michael Dinitz and Ben Moseley. Scheduling for weighted flow and completion times in reconfigurable networks. In IEEE INFOCOM 2020 - IEEE Conference on Computer Communications, 2020. Forthcoming.Google ScholarDigital Library
- J. Du, J. Y.-T. Leung, and G. H. Young. Minimizing mean flow time with release time constraint. Theoretical Computer Science, 75:347--355, 1990.Google ScholarDigital Library
- N. Dukkipati and N. McKeown. Why flow-completion time is the right metric for congestion control. SIGCOMM Comput. Commun. Rev., 36(1):59--62, January 2006.Google ScholarDigital Library
- S. Even, A. Itai, and A. Shamir. On the complexity of timetable and multicommodity flow problems. SIAM J. Comput., 5:691--703, 12 1976.Google ScholarDigital Library
- U. Feige, Janardhan Kulkarni, and Shi Li. A polynomial time constant approximation for minimizing total weighted flow-time. In Timothy M. Chan, editor, Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, California, USA, January 6--9, 2019, pages 1585--1595. SIAM, 2019.Google ScholarDigital Library
- N. Garg and A. Kumar. Better algorithms for minimizing average flow-time on related machines. In Michele Bugliesi, Bart Preneel, Vladimiro Sassone, and Ingo Wegener, editors, Automata, Languages and Programming, pages 181--190, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg.Google Scholar
- N. Garg and A. Kumar. Minimizing average flow-time: Upper and lower bounds. In Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS '07, pages 603--613, Washington, DC, USA, 2007. IEEE Computer Society.Google ScholarDigital Library
- N. Garg, A. Kumar, and V. N. Muralidhara. Minimizing total flow-time: The unrelated case. In Seok-Hee Hong, Hiroshi Nagamochi, and Takuro Fukunaga, editors, Algorithms and Computation, pages 424--435, Berlin, Heidelberg, 2008. Springer Berlin Heidelberg.Google ScholarDigital Library
- P. Giaccone, B. Prabhakar, and D. Shah. Randomized scheduling algorithms for high-aggregate bandwidth switches. IEEE Journal on Selected Areas in Communications, 21(4):546--559, 2003.Google ScholarDigital Library
- K. Giaro, M. Kubale, M. Malafiejski, and K. Piwakowski. Chromatic scheduling of dedicated 2-processor uet tasks to minimize mean flow time. In 1999 7th IEEE International Conference on Emerging Technologies and Factory Automation. Proceedings ETFA '99, volume 1, pages 343--347 vol.1, Oct 1999.Google ScholarCross Ref
- L. Gong, P. Tune, L. Liu, S. Yang, and J. Xu. Queue-proportional sampling: A better approach to crossbar scheduling for input-queued switches. In Proceedings of the ACM on Measurement and Analysis of Computing Systems (SIGMETRICS), 2017.Google ScholarDigital Library
- P. Goransson and C. Black. Software Defined Networks: A Comprehensive Approach. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1st edition, 2014.Google Scholar
- A. Greenberg, J. R. Hamilton, N. Jain, S. Kandula, C. Kim, P. Lahiri, D. A. Maltz, P. Patel, and S. Sengupta. Vl2: A scalable and flexible data center network. In Proceedings of the ACM SIGCOMM 2009 Conference on Data Communication, SIGCOMM '09, pages 51--62, New York, NY, USA, 2009. ACM.Google ScholarDigital Library
- I. Grosof, Z. Scully, and M. Harchol-Balter. SRPT for multiserver systems. Perform. Eval., 127--128:154--175, 2018.Google ScholarCross Ref
- H. Jahanjou, E. Kantor, and R. Rajaraman. Asymptotically optimal approximation algorithms for coflow scheduling. In Proceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '17, pages 45--54, New York, NY, USA, 2017. ACM.Google ScholarDigital Library
- H. Jahanjou, R. Rajaraman, and D. Stalfa. Scheduling flows on a switch to optimize response times. arXiv: https://arxiv.org/abs/2005.09724, May 2020.Google Scholar
- S. Jia, X. Jin, G. Ghasemiesfeh, J. Ding, and J. Gao. Competitive analysis for online scheduling in software-defined optical wan. In Proc. of the IEEE INFOCOM Conference, pages 1--9, 05 2017.Google ScholarCross Ref
- B. Kalyanasundaram and K. Pruhs. Minimizing flow time nonclairvoyantly. In Proceedings of the 38th IEEE Symposium on Foundations of Computer Science, pages 345--352, 1997.Google ScholarCross Ref
- N. Kang, Z. Liu, J. Rexford, and D. Walker. Optimizing the "one big switch" abstraction in software-defined networks. In Proceedings of the Ninth ACM Conference on Emerging Networking Experiments and Technologies, CoNEXT '13, pages 13--24, New York, NY, USA, 2013. ACM.Google ScholarDigital Library
- R. M. Karp, F. T. Leighton, R. L. Rivest, C. D. Thompson, U. V. Vazirani, and V. V. Vazirani. Global wire routing in two-dimensional arrays, 1987.Google ScholarDigital Library
- H. Kellerer, T. Tautenhahn, and G. J. Woeginger. Approximability and nonapproximability results for minimizing total flow time on a single machine. In Proceedings of the 28th Annual ACM Symposium on Theory of Computing, pages 418--426, May 1996.Google ScholarDigital Library
- S. Khuller and M. Purohit. Improved approximation algorithms for scheduling co-flows. In SPAA, 2016. Brief Announcement.Google ScholarDigital Library
- D. Kreutz, F. M. V. Ramos, P. E. Veríssimo, C. E. Rothenberg, S. Azodolmolky, and S. Uhlig. Software-defined networking: A comprehensive survey. Proceedings of the IEEE, 103(1):14--76, Jan 2015.Google ScholarCross Ref
- M. Kubale and H. Krawczyk. An approximation algorithm for diagnostic test scheduling in multicomputer systems. IEEE Transactions on Computers, 34:869--872, 09 1985.Google Scholar
- J. Kulkarni. Personal communication.Google Scholar
- E. L. Lawler and J. Labetoulle. On preemptive scheduling of unrelated parallel processors by linear programming. J. ACM, 25(4):612--619, October 1978.Google ScholarDigital Library
- S. Leonardi and D. Raz. Approximating total flow time on parallel machines. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pages 110--119, May 1997.Google ScholarDigital Library
- L. Luo, K. Foerster, H. Yu, and S. Schmid. Splitcast: Optimizing multicast flows in reconfigurable datacenter networks. In IEEE INFOCOM 2020 - IEEE Conference on Computer Communications, 2020. Forthcoming.Google ScholarDigital Library
- M. Mastrolilli. Scheduling to Minimize Max Flow Time: Offline and Online Algorithms, pages 49--60. Springer Berlin Heidelberg, Berlin, Heidelberg, 2003.Google Scholar
- R. Niranjan Mysore, A. Pamboris, N. Farrington, N. Huang, P. Miri, S. Radhakrishnan, V. Subramanya, and A. Vahdat. Portland: A scalable fault-tolerant layer 2 data center network fabric. In Proceedings of the ACM SIGCOMM 2009 Conference on Data Communication, SIGCOMM '09, pages 39--50, New York, NY, USA, 2009. ACM.Google ScholarDigital Library
- Z. Qiu, C. Stein, and Y. Zhong. Minimizing the total weighted completion time of coflows in datacenter networks. In Proceedings of the 27th ACM Symposium on Parallelism in Algorithms and Architectures, SPAA '15, pages 294--303, New York, NY, USA, 2015. ACM.Google ScholarDigital Library
- Z. Qiu, C. Stein, and Y. Zhong. Minimizing the total weighted completion time of coflows in datacenter networks. In SPAA, pages 294--303, 2015.Google ScholarDigital Library
- Mehrnoosh Shafiee and Javad Ghaderi. An improved bound for minimizing the total weighted completion time of coflows in datacenters. IEEE/ACM Trans. Netw., 26(4):1674--1687, 2018.Google ScholarDigital Library
- D. Shah and J. Shin. Randomized scheduling algorithm for queueing networks. The Annals of Applied Probability, 22(1):128--171, 2012.Google ScholarCross Ref
- Y. Zhao, K. Chen, W. Bai, M. Yu, C. Tian, Y. Geng, Y. Zhang, D. Li, and S. Wang. Rapier: Integrating routing and scheduling for coflow-aware data center networks. In INFOCOM, pages 424--432, 2015.Google ScholarCross Ref
Index Terms
- Scheduling Flows on a Switch to Optimize Response Times
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