Advertisement

On the Advantage of Network Coding

  • Rudolf AhlswedeEmail author
Chapter
Part of the Foundations in Signal Processing, Communications and Networking book series (SIGNAL, volume 15)

Abstract

Network coding for achieving the maximum information flow in the multicast networks has been proposed by us (Ahlswede, Cai, Li, and Yeung). Therefore now these networks are called ACLY-networks. We demonstrated that the conventional network switching, without resort to network coding, is in general not able to achieve the optimum information flow that has been promised by network coding. A basic problem arising here is that, for a given multicast network, what is the switching gap of the network defined as the ratio of the maximum information flow in the multicast network with network coding to that only with network switching.

References

  1. 1.
    R.W. Yeung, A First Course in Information Theory, Information Technology: Transmission, Processing, and Storage (Kluwer Academic/Plenum Publishers, New York, 2002)CrossRefGoogle Scholar
  2. 2.
    S.Y. Li, R.W. Yeung, N. Cai, Linear network coding. IEEE Trans. Inf. Theory 49, 371–381 (2003)MathSciNetCrossRefGoogle Scholar
  3. 3.
    R. Kötter, M. Médard, An algebraic approach to network coding. Trans. Netw. 11(5), 782–795 (2003)CrossRefGoogle Scholar
  4. 4.
    P. Sanders, S. Egner, L. Tolhuizen, Polynomial time algorithms for network information flow, in Proceedings of the 15th Annual ACM Symposium on Parallel Algorithms and Architectures, San Diego, CA, USA (2003), pp. 286–294Google Scholar
  5. 5.
    S. Jaggi, P. Sanders, P.A. Chou, M. Effros, S. Egner, K. Jain, L. Tolhuizen, Polynomial time algorithms for multicast network code construction. IEEE Trans. Inf. Theory 51(6), 1973–1982 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Y. Zhu, B. Li, J. Guo, Multicast with network coding in application-layer overlay networks. IEEE J. Sel. Areas Commun. 22(1) (2004)CrossRefGoogle Scholar
  7. 7.
    L. Song, R.W. Yeung, N. Cai, Zero-error coding for acyclic network, submitted to IEEE Trans. ITGoogle Scholar
  8. 8.
    R.J. McEliece, Information multicasts, Lee Center Meeting, California Institute of Technology, Pasadena, CA, 22 Nov 2004, http://www.ee.caltech.edu/EE/Faculty/rjm/papers/Lunch.pdf
  9. 9.
    K. Jain, M. Mahdian, M. Salavatipour, Packing Steiner trees, in Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD (Jan. 2003), pp. 266–274Google Scholar
  10. 10.
    Y. Wu, P.A. Chou, K. Jain, A comparison of network coding and tree packing, in Proceedings of the IEEE International Symposium on Information Theory, Chicago, IL (Jun./Jul. 2004), p. 145Google Scholar
  11. 11.
    A. Agarwal, M. Charikar, On the advantage of network coding for improving network throughput, in Proceedings of the 2004 IEEE Information Theory Workshop, San Antonio, TX (Oct. 2004), pp. 247–249Google Scholar
  12. 12.
    R. Ahlswede, N. Cai, S.Y.R. Li, R.W. Yeung, Network information flow, Preprint 98–033, SFB 343 Diskrete Strukturen in der Mathematik, Universität Bielefeld. IEEE Trans. Inf. Theory 46(4), 1204–1216 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D.F. Robinson, L.R. Foulds, Digraphs: Theory and Techniques (Gordon and Breach, New York, 1980)zbMATHGoogle Scholar
  14. 14.
    W. Rudin, Principles of Mathematical Analysis, 3rd edn. (McGraw-Hill, New York, 1976)zbMATHGoogle Scholar
  15. 15.
    J.C.C. McKinsey, Introduction to the Theory of Games (McGraw-Hill, New York, 1952). Also: Dover, Mineaole, 2003Google Scholar
  16. 16.
    G. Owen, Game Theory, 3rd edn. (Academic, San Diego, 1995)zbMATHGoogle Scholar
  17. 17.
    J. Szép, F. Forgó, Introduction to the Theory of Games (Reidel, Dordrecht, 1985)CrossRefGoogle Scholar
  18. 18.
    V.F. Dem’yanov, V.N. Malozemov, Introduction to Minimax (Wiley, New York, 1974)zbMATHGoogle Scholar
  19. 19.
    C.K. Ngai, R.W. Yeung, Network coding gain of combination networks, in Proceedings of the IEEE Information Theory Workshop, San Antonio, TX (Oct. 2004), pp. 283–287Google Scholar
  20. 20.
    J. Cannons, R. Dougherty, C. Freiling, K. Zeger, Network routing capacity. IEEE Trans. Inf. Theory 52(3), 777–788 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    G.W. Brown, Iterative solution of games by fictitious play, in Activity Analysis of Production and Allocation ed. by T.C. Koopmans (Wiley, New York, 1951), pp. 374–376Google Scholar
  22. 22.
    J. Robinson, An iterative method of solving a game. Ann. Math. 54(2), 296–301 (1951)MathSciNetCrossRefGoogle Scholar
  23. 23.
    M.J. Todd, The many facets of linear programming. Math. Program. Ser. B 91, 417–436 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    D.P. Bertsekas, A. Nedić, A.E. Ozdaglar, Convex Analysis and Optimization (Athena Scientific, Belmont, 2003)zbMATHGoogle Scholar
  25. 25.
    S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)CrossRefGoogle Scholar
  26. 26.
    A. Kreinin, M. Sidelnikova, Regularization algorithms for transition matrices. Algo. Res. Q. 4(1/2), 23–40 (2001)Google Scholar
  27. 27.
    L. Merkoulovitch, The projection on the standard simplex, Algorithmics Inc., Working PaperGoogle Scholar
  28. 28.
    H.J.H. Tuenter, The minimum \(L\)-distance projection onto the canonical simplex: a simple algorithm. Algo. Res. Q. 4, 53–55 (2001)Google Scholar
  29. 29.
    B.C. Eaves, On the basic theorem of complementarity. Math. Program. 1, 68–75 (1971)MathSciNetCrossRefGoogle Scholar
  30. 30.
    P.T. Harker, J.S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications. Math. Program. Ser. B 48, 161–220 (1990)MathSciNetCrossRefGoogle Scholar
  31. 31.
    B. He, H. Yang, A neural-network model for monotone linear asymmetric variational inequalities. IEEE Trans. Neural Netw. 11(1), 3–16 (2000)CrossRefGoogle Scholar
  32. 32.
    J.M. Peng, M. Fukushima, A hybrid Newton method for solving the variational inequality problem via the \(D\)-gap function. Math. Program. Ser. A 86, 367–386 (1999)MathSciNetCrossRefGoogle Scholar
  33. 33.
    E.M. Gafni, D.P. Bertsekas, Two-metric projection methods for constrained optimization. SIAM J. Control Optim. 22, 936–964 (1984)MathSciNetCrossRefGoogle Scholar
  34. 34.
    V.V. Vazirani, Approximation Algorithms (Springer, Berlin, 2003)CrossRefGoogle Scholar
  35. 35.
    R.A. Brualdi, Introductory Combinatorics, 4th edn. (Pearson/Prentice-Hall, Upper Saddle River, 2004)zbMATHGoogle Scholar
  36. 36.
    G. Chartrand, L. Lesniak, Graphs & Digraphs, 2nd edn. (Wadsworth, Belmont, 1986)zbMATHGoogle Scholar
  37. 37.
    L.R. Ford, D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, 1962)zbMATHGoogle Scholar
  38. 38.
    P. Elias, A. Feinstein, C.E. Shannon, A note on the maximum flow through a network. IEEE Trans. Inf. Theory 11 (1956)Google Scholar
  39. 39.
    D.S. Johnson, Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)MathSciNetCrossRefGoogle Scholar
  40. 40.
    L. Lovász, On the ratio of optimal integral and fractional covers. Discret. Math. 13, 383–390 (1975)MathSciNetCrossRefGoogle Scholar
  41. 41.
    V.V. Chvatal, A greedy heuristic for the set-covering problem. Math. Oper. Res. 4(3), 233–235 (1979)MathSciNetCrossRefGoogle Scholar
  42. 42.
    R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice-Hall, Englewood Cliffs, 1993)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.BielefeldGermany

Personalised recommendations