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How to Design Graphs with Low Forwarding Index and Limited Number of Edges

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Combinatorial Algorithms (IWOCA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9538))

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Abstract

The (edge) forwarding index of a graph is the minimum, over all possible routings of all the demands, of the maximum load of an edge. This metric is of a great interest since it captures the notion of global congestion in a precise way: the lesser the forwarding-index, the lesser the congestion. In this paper, we study the following design question: Given a number e of edges and a number n of vertices, what is the least congested graph that we can construct? and what forwarding-index can we achieve? Our problem has some distant similarities with the well-known \((\varDelta ,D)\) problem, and we sometimes build upon results obtained on it. The goal of this paper is to study how to build graphs with low forwarding indices and to understand how the number of edges impacts the forwarding index. We answer here these questions for different families of graphs: general graphs, graphs with bounded degree, sparse graphs with a small number of edges by providing constructions, most of them asymptotically optimal. For instance, we provide an asymptotically optimal construction for \((n,n+k)\) cubic graphs - its forwarding index is \(\sim \frac{n^2}{3k} \log _2(k)\). Our results allow to understand how the forwarding-index drops when edges are added to a graph and also to determine what is the best (i.e. least congested) structure with e edges. Doing so, we partially answer the practical problem that initially motivated our work: If an operator wants to power only e links of its network, in order to reduce the energy consumption (or wiring cost) of its networks, what should be those links and what performance can be expected?

This work has been partially supported by ANR project Stint under reference ANR-13-BS02-0007, ANR program Investments for the Future under reference ANR-11-LABX-0031-01, ANR VISE, CNRS-FUNCAP project GAIATO, the associated Inria team AlDyNet, the project ECOS-Sud Chile.

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References

  1. Araujo, J., Giroire, F., Liu, Y., Modrzejewski, R., Moulierac, J.: Energy efficient content distribution. In: IEEE International Conference on Communications (ICC 2013), pp. 4233–4238. IEEE (2013)

    Google Scholar 

  2. Baliga, J., Tucker, R., Ayre, R., Hinton, K.W., Sorin, W.: Energy consumption in IP networks. In: 34th European Conference on Optical Communication, ECOC 2008, p. 1 (2008)

    Google Scholar 

  3. Bouabdallah, A., Sotteau, D.: On the edge forwarding index problem for small graphs. Networks 23(4), 249–255 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Restrepo, J. C. C., Gruber, C. G., Machuca, C. M.: Energy profile aware routing. In: Communications Workshops of IEEE International Conference on Communications (ICC), pp. 1–5 (2009)

    Google Scholar 

  5. Chiaraviglio, L., Mellia, M., Neri, F.: Energy-aware umts core network design. In: The 11th International Symposium on Wireless Personal Multimedia Communications (2008)

    Google Scholar 

  6. Chung, F.R.K., Coffman Jr., E.G., Reiman, M.I., Simon, B.: The forwarding index of communication networks. IEEE Trans. Inf. Theory 33(2), 224–232 (1987)

    Google Scholar 

  7. Giroire, F., Mazauric, D., Moulierac, J., Onfroy, B.: Minimizing routing energy consumption: from theoretical to practical results. In: IEEE/ACM International Conference on Green Computing and Communications (GreenCom 2010), Hangzhou, China, p. 8 (2010)

    Google Scholar 

  8. Giroire, F., Perennes, S., Tahiri, I.: Grid spanners with low forwarding index for energy efficient networks. In: International Network Optimization Conference (INOC), Warsaw, Poland, May 2015

    Google Scholar 

  9. Giroire, F., Pérennes, S., Tahiri, I.: Graphs with optimal forwarding indices: what is the best throughput you can get with a given number of edges? Research Report RR-8752, INRIA Sophia Antipolis, INRIA, June 2015. https://hal.inria.fr/hal-01172725

  10. Heydemann, M.C., Meyer, J.C., Sotteau, D.: On forwarding indices of networks. Discrete Appl. Math. 23(2), 103–123 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Xu, J.-M., Xu, M.: The forwarding indices of graphs - a survey. xarchiv (2012)

    Google Scholar 

  12. Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM 46(6), 787–832 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15, 577–591 (1994)

    MathSciNet  Google Scholar 

  14. Manoussakis, Y., Tuza, Z.: The forwarding index of directed networks. Discrete Appl. Math. 68(3), 279–291 (1996). http://www.sciencedirect.com/science/article/pii/0166218X9500072Y

    Article  MathSciNet  MATH  Google Scholar 

  15. Meringer, M.: Small cubic graphs, flinders Univ projet. http://www.flinders.edu.au/science_engineering/csem/research/programs/flinders-hamiltonian-cycle-project/graph-database.cfm

  16. Miller, M., Širán, J.: Moore graphs and beyond: a survey of the degree/diameter problem. Electron. J. Comb. 61, 1–63 (2005)

    Google Scholar 

  17. Mohar, B.: Some applications of laplace eigenvalues of graphs. In: Hahn, G., Sabidussi, G. (eds.) Graph Symmetry. NATO ASI Series, vol. 497, pp. 225–275. Springer, Amsterdam (1997)

    Chapter  Google Scholar 

  18. Orlowski, S., Wessäly, R., Pióro, M., Tomaszewski, A.: Sndlib 1.0—survivable network design library. Networks 55(3), 276–286 (2010)

    Google Scholar 

  19. Shahrokhi, F., Matula, D.W.: The maximum concurrent flow problem. J. ACM 37(2), 318–334 (1990). http://doi.acm.org/10.1145/77600.77620

    Article  MathSciNet  MATH  Google Scholar 

  20. Sinclair, A.: Improved bounds for mixing rates of markov chains and multicommodity flow. Comb. Probab. Comput. 1, 351–370 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Solé, P.: Expanding and forwarding. Discrete Appl. Math. 58(1), 67–78 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tassiulas, L., Ephremides, A.: Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks. IEEE Trans. Autom. Control 37(12), 1936–1948 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. de la Vega, W.F., Manoussakis, Y.: The forwarding index of communication networks with given connectivity. Discrete Appl. Math. 37–38, 147–155 (1992)

    Article  Google Scholar 

  24. de la Vega, F., Gordones, L.M.: The forwarding indices of random graphs. Random Struct. Algorithms 3(1), 107–116 (1992)

    Article  MATH  Google Scholar 

  25. Xu, J.M., Xu, M.: The forwarding indices of graphs - a survey. CoRR abs/1204.2604 (2012)

    Google Scholar 

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Correspondence to Frédéric Giroire .

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Giroire, F., Pérennes, S., Tahiri, I. (2016). How to Design Graphs with Low Forwarding Index and Limited Number of Edges. In: Lipták, Z., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2015. Lecture Notes in Computer Science(), vol 9538. Springer, Cham. https://doi.org/10.1007/978-3-319-29516-9_19

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  • DOI: https://doi.org/10.1007/978-3-319-29516-9_19

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