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Algorithmica

, Volume 80, Issue 1, pp 209–233 | Cite as

On the Complexity of Compressing Two Dimensional Routing Tables with Order

  • Frédéric Giroire
  • Frédéric Havet
  • Joanna Moulierac
Article

Abstract

Motivated by routing in telecommunication network using Software Defined Network (SDN) technologies, we consider the following problem of finding short routing lists using aggregation rules. We are given a set of communications \(\mathcal {X}\), which are distinct pairs \((s,t)\subseteq S\times T\), (typically S is the set of sources and T the set of destinations), and a port function \(\pi :\mathcal {X} \rightarrow P\) where P is the set of ports. A routing list \(\mathcal {R}\) is an ordered list of triples which are of the form (stp), \((*,t,p)\), \((s,*,p)\) or \((*,*,p)\) with \(s\in S\), \(t\in T\) and \(p\in P\). It routes the communication (st) to the port \(r(s,t) =p\) which appears on the first triple in the list \(\mathcal {R}\) that is of the form (stp), \((*,t,p)\), \((s,*,p)\) or \((*,*,p)\). If \(r(s,t)=\pi (s,t)\), then we say that (st) is properly routed by \(\mathcal {R}\) and if all communications of \(\mathcal {X}\) are properly routed, we say that \(\mathcal {R}\) emulates \((\mathcal {X}, \pi )\). The aim is to find a shortest routing list emulating \((\mathcal {X}, \pi )\). In this paper, we carry out a study of the complexity of the two dual decision problems associated to it. Given a set of communication \(\mathcal {X}\), a port function \(\pi \) and an integer k, the first one called Routing List (resp. the second one, called List Reduction) consists in deciding whether there is a routing list emulating \((\mathcal {X}, \pi )\) of size at most k (resp. \(|\mathcal {X}| -k\)). We prove that both problems are NP-complete. We then give a 3-approximation for List Reduction, which can be generalized to higher dimensions. We also give a 4-approximation for Routing List in the fundamental case when there are only two ports (i.e. \(|P|=2\)), \(\mathcal {X}=S\times T\) and \(|S|=|T|\).

Keywords

Routing Routing tables Order Priority Software Defined Networks Complexity Approximation algorithm Compact tables 

References

  1. 1.
    Arora, S., Frieze, A., Kaplan, H.: A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Math. Program. 92(1), 1–36 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Buddhikot, M.M., Suri, S., Waldvogel, M.: Space Decomposition Techniques for Fast Layer-4 Switching. Springer, Boston (2000)CrossRefGoogle Scholar
  3. 3.
    Cohen, R., Lewin-Eytan, L., Naor, J., Raz, D.: On the effect of forwarding table size on SDN network utilization. In: IEEE INFOCOM, pp. 1734–1742 (2014)Google Scholar
  4. 4.
    Eppstein, D., Muthukrishnan, S.: Internet packet filter management and rectangle geometry. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 827–835 (2001)Google Scholar
  5. 5.
    Gallai, T.: Maximum-minimum sätze über graphen. Acta Math. Hung. 9(3), 395–434 (1958)CrossRefzbMATHGoogle Scholar
  6. 6.
    Giroire, F., Havet, F., Moulierac, J.: Compressing Two-dimensional Routing Tables with Order. Research Report RR-8658, INRIA Sophia Antipolis (2014)Google Scholar
  7. 7.
    Giroire, F., Havet, F., Moulierac, J.: Compressing two-dimensional routing tables with order. In: INOC (International Network Optimization Conference). Varsovie (2015). https://hal.inria.fr/hal-01162724
  8. 8.
    Giroire, F., Moulierac, J., Khoa Phan, T.: Optimizing Rule Placement in Software-Defined Networks for Energy-aware Routing. In: IEEE GLOBECOM. IEEE, Austin (2014)Google Scholar
  9. 9.
    Guo, J., Hüffner, F., Moser, H.: Feedback arc set in bipartite tournaments is np-complete. Inf. Process. Lett. 102(2), 62–65 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hari, A., Suri, S., Parulkar, G.: Detecting and resolving packet filter conflicts. In: INFOCOM 2000, pp. 1203–1212. IEEE (2000)Google Scholar
  11. 11.
    Hoffman, A.J., Kruskal, J.B.: Integral boundary points of convex polyhedra. In: 50 Years of Integer Programming 1958–2008, pp. 49–76. Springer (2010)Google Scholar
  12. 12.
    Kang, N., Liu, Z., Rexford, J., Walker, D.: Optimizing the “one big switch abstraction” in software-defined networks. In: Proceedings of CoNEXT, pp. 13–24. ACM, New York (2013)Google Scholar
  13. 13.
    Kanizo, Y., Hay, D., Keslassy, I.: Palette: Distributing tables in software-defined networks. In: Proceedings of IEEE INFOCOM, 2013, pp. 545–549 (2013)Google Scholar
  14. 14.
    Kann, V.: On the approximability of np-complete optimization problems. Ph.D. thesis, Royal Institute of Technology Stockholm (1992)Google Scholar
  15. 15.
    Karp, R.M.: Reducibility Among Combinatorial Problems. Springer, New York (1972)CrossRefzbMATHGoogle Scholar
  16. 16.
    Kogan, K., Nikolenko, S.I., Rottenstreich, O., Culhane, W., Eugster, P.: Exploiting order independence for scalable and expressive packet classification. IEEE/ACM Trans. Netw. 24(2), 1251–1264 (2016)CrossRefGoogle Scholar
  17. 17.
    Lakshman, T., Stiliadis, D.: High-speed policy-based packet forwarding using efficient multi-dimensional range matching. ACM SIGCOMM Compu. Commun. Rev. 28(4), 203–214 (1998)CrossRefGoogle Scholar
  18. 18.
    McKeown, N., Anderson, T., Balakrishnan, H., Parulkar, G., Peterson, L., Rexford, J., Shenker, S., Turner, J.: Openflow: Enabling innovation in campus networks. SIGCOMM Comput. Commun. Rev. 38(2), 69–74 (2008)CrossRefGoogle Scholar
  19. 19.
    Narayanan, R., Kotha, S., Lin, G., Khan, A., Rizvi, S., Javed, W., Khan, H., Khayam, S.: Macroflows and microflows: Enabling rapid network innovation through a split sdn data plane. In: 2012 European Workshop on Software Defined Networking (EWSDN), pp. 79–84 (2012)Google Scholar
  20. 20.
    Rifai, M., Huin, N., Caillouet, C., Giroire, F., Lopez-Pacheco, D., Moulierac, J., Urvoy-Keller, G.: Too many sdn rules? Compress them with minnie. In: 2015 IEEE Global Communications Conference (GLOBECOM), pp. 1–7. IEEE (2015)Google Scholar
  21. 21.
    Rottenstreich, O., Keslassy, I., Hassidim, A., Kaplan, H., Porat, E.: Optimal in/out tcam encodings of ranges. IEEE/ACM Trans. Netw. 24(1), 555–568 (2016)CrossRefGoogle Scholar
  22. 22.
    Rottenstreich, O., et al.: Lossy compression of packet classifiers. In: Proceedings of the Eleventh ACM/IEEE Symposium on Architectures for networking and communications systems, pp. 39–50. IEEE Computer Society (2015)Google Scholar
  23. 23.
    Stephens, B., Cox, A., Felter, W., Dixon, C., Carter, J.: Past: Scalable ethernet for data centers. In: Proceedings of CoNEXT, pp. 49–60. ACM, New York (2012)Google Scholar
  24. 24.
    Suri, S., Sandholm, T., Warkhede, P.: Compressing two-dimensional routing tables. Algorithmica 35(4), 287–300 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Taylor, D.E.: Survey and taxonomy of packet classification techniques. ACM Comput. Surv. (CSUR) 37(3), 238–275 (2005)CrossRefGoogle Scholar
  26. 26.
    Van Zuylen, A.: Linear programming based approximation algorithms for feedback set problems in bipartite tournaments. In: Chen, J., Cooper, S.B. (eds.) Theory and Applications of Models of Computation: 6th Annual Conference, TAMC 2009, Changsha, China, May 18-22, 2009. Proceedings, vol 5532, pp 370–379. Springer Berlin Heidelberg (2009)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Université Côte d’Azur, CNRS, Inria, I3SSophia Antipolis CedexFrance

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