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 (s, t, p), \((*,t,p)\), \((s,*,p)\) or \((*,*,p)\) with \(s\in S\), \(t\in T\) and \(p\in P\). It routes the communication (s, t) to the port \(r(s,t) =p\) which appears on the first triple in the list \(\mathcal {R}\) that is of the form (s, t, p), \((*,t,p)\), \((s,*,p)\) or \((*,*,p)\). If \(r(s,t)=\pi (s,t)\), then we say that (s, t) 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 \(|S|=|T|\).
Similar content being viewed by others
References
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)
Buddhikot, M.M., Suri, S., Waldvogel, M.: Space Decomposition Techniques for Fast Layer-4 Switching. Springer, Boston (2000)
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)
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)
Gallai, T.: Maximum-minimum sätze über graphen. Acta Math. Hung. 9(3), 395–434 (1958)
Giroire, F., Havet, F., Moulierac, J.: Compressing Two-dimensional Routing Tables with Order. Research Report RR-8658, INRIA Sophia Antipolis (2014)
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
Giroire, F., Moulierac, J., Khoa Phan, T.: Optimizing Rule Placement in Software-Defined Networks for Energy-aware Routing. In: IEEE GLOBECOM. IEEE, Austin (2014)
Guo, J., Hüffner, F., Moser, H.: Feedback arc set in bipartite tournaments is np-complete. Inf. Process. Lett. 102(2), 62–65 (2007)
Hari, A., Suri, S., Parulkar, G.: Detecting and resolving packet filter conflicts. In: INFOCOM 2000, pp. 1203–1212. IEEE (2000)
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)
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)
Kanizo, Y., Hay, D., Keslassy, I.: Palette: Distributing tables in software-defined networks. In: Proceedings of IEEE INFOCOM, 2013, pp. 545–549 (2013)
Kann, V.: On the approximability of np-complete optimization problems. Ph.D. thesis, Royal Institute of Technology Stockholm (1992)
Karp, R.M.: Reducibility Among Combinatorial Problems. Springer, New York (1972)
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)
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)
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)
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)
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)
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)
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)
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)
Suri, S., Sandholm, T., Warkhede, P.: Compressing two-dimensional routing tables. Algorithmica 35(4), 287–300 (2003)
Taylor, D.E.: Survey and taxonomy of packet classification techniques. ACM Comput. Surv. (CSUR) 37(3), 238–275 (2005)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
A preliminary short version of this work has appeared in [7]. This work has been partially supported by ANR program ANR-11-LABX-0031-01 and by ANR Grant ANR-13-BS02-0007 STINT.
Rights and permissions
About this article
Cite this article
Giroire, F., Havet, F. & Moulierac, J. On the Complexity of Compressing Two Dimensional Routing Tables with Order. Algorithmica 80, 209–233 (2018). https://doi.org/10.1007/s00453-016-0243-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-016-0243-7