Notes on the Complexity of Switching Networks

  • Hung Quang Ngo
  • Ding-Zhu Du
Part of the Network Theory and Applications book series (NETA, volume 5)

Abstract

There are various complexity measures for switching networks and communication networks in general. These measures include, but not limited to, the number of switching components, the delay time of signal propagating through the network, the complexity of path selection algorithms, and the complexity of physically designing the network. This chapter surveys the study of the first measure, and partially the second measure. It is conceivable that the number of switching components, or the “size” of a network, affects directly the third and fourth measures.

Keywords

Dition Sorting Prefix Extractor Zucker 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    A. V. Aho, J. E. Hopcroft, And J. D.Ullman, The design and analysis of computer algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Second printing, Addison-Wesley Series in Computer Science and Information Processing.Google Scholar
  2. [2]
    M. Ajtai, Recursive construction for 3-regular expanders, Combinatorica, 14 (1994), pp. 379–416.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    M. Ajtai, J. Komlós, And E. Szemerédi, Sorting in clogn parallel steps, Combinatorica, 3 (1983), pp. 1–19.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    M. Ajtai, J. Komlós, And E. Szemerédi, Generating expanders from two permutations, in A tribute to Paul Erdós, Cambridge Univ. Press, Cambridge, 1990, pp. 1–12.Google Scholar
  5. [5]
    N. Alon, Eigenvalues, geometric expanders and sorting in rounds, in Graph theory with applications to algorithms and computer science (Kalamazoo, Mich., 1984), Wiley, New York, 1985, pp. 15–24.Google Scholar
  6. [6]
    N. Alon, Eigenvalues and expanders, Combinatorica, 6 (1986), pp. 83–96. Theory of computing (Singer Island, Fla., 1984 ).Google Scholar
  7. [7]
    N. Alon, Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory, Combinator- ica, 6 (1986), pp. 207–219.MATHCrossRefGoogle Scholar
  8. [8]
    N. Alon, Z. Galil, And V. D. Milman, Better expanders and superconcentrators, J. Algorithms, 8 (1987), pp. 337–347.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    N. Alon And V. D. Milman, a1, isoperimetric inequalities for graphs, and superconcentrators, J. Combin. Theory Ser. B, 38 (1985), pp. 73–88.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    N. Alon And P. Pudlak, Superconcentrators of depths2 and3; odd levels help (rarely), J. Comput. System Sci., 48 (1994), pp. 194–202.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    N. Alon And Y. Roichman, Random Cayley graphs and expanders, Random Structures Algorithms, 5 (1994), pp. 271–284.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    D. Angluin, A note on a construction of Margulis, Inform. Process. Lett., 8 (1979), pp. 1719.MathSciNetCrossRefGoogle Scholar
  13. [13]
    L. A. Bassalygo, Asymptotically optimal switching circuits, Problemy Peredachi Informatsii, 17 (1981), pp. 81–88.MathSciNetGoogle Scholar
  14. [14]
    L. A. Bassalygo And M. S. Pinsker, The complexity of an optimal non-blocking commu- tation scheme without reorganization, Problemy Peredaëi Informacii, 9 (1973), pp. 84 - 87.MathSciNetGoogle Scholar
  15. [15]
    B. Beizer, The analysis and synthesis of signal switching networks, in Proc. Sympos. Math. Theory of Automata (New York, 1962), Polytechnic Press of Polytechnic Inst. of Brooklyn, Brooklyn, N.Y., 1963, pp. 563–576.Google Scholar
  16. [16]
    V. E. Bene, Mathematical theory of connecting networks and telephone traffic, Academic Press, New York, 1965. Mathematics in Science and Engineering, Vol. 17.Google Scholar
  17. [17]
    F. Bien, Constructions of telephone networks by group representations, Notices Amer. Math. Soc., 36 (1989), pp. 5–22.MathSciNetMATHGoogle Scholar
  18. [18]
    N. Biggs, Algebraic graph theory, Cambridge University Press, Cambridge, second ed., 1993.Google Scholar
  19. [19]
    M. Blum, R. M. Karp, O. Vornberger, C. H. Papadimitriou, And M. Yannakakis, The complexity of testing whether a graph is a superconcentrator, Inform. Process. Lett., 13 (1981), pp. 164–167.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    L. M. Brègman, Certain properties of nonnegative matrices and their permanents, Dokl. Akad. Nauk SSSR, 211 (1973), pp. 27–30.MathSciNetGoogle Scholar
  21. [21]
    A. E. Brouwer, A. M. Cohen, And A. Neumaier, Distance-regular graphs, Springer-Verlag, Berlin, 1989.MATHGoogle Scholar
  22. [22]
    M. W. Buck, Expanders and diffusers, SIAM J. Algebraic Discrete Methods, 7 (1986), pp. 282–304.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    D. G. Cantor, On non-blocking switching networks, Networks, 1 (1971/72), pp. 367–377.Google Scholar
  24. [24]
    F. R. K. Chung, On concentrators, superconcentrators, generalizers, and nonblocking networks, Bell System Tech. J., 58 (1979), pp. 1765–1777.MathSciNetMATHGoogle Scholar
  25. [25]
    F. R. K. Chung, Spectral graph theory, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997.MATHGoogle Scholar
  26. [26]
    C. Clos, A study of non-blocking switching networks, Bell System Tech. J., 32 (1953), pp. 406–424.Google Scholar
  27. [27]
    D. M. Cvetkovi, M. Doob, And H. Sachs, Spectra of graphs, Johann Ambrosius Barth, Heidelberg, third ed., 1995. Theory and applications.Google Scholar
  28. [28]
    N. G. De Bruihn, P. Erdós, And J. Spencer, Solution, 350, Nieuw Archief voor Wiskunde, (1974), pp. 94–109.Google Scholar
  29. [29]
    D. Dolev, C. Dwork, N. Pippenger, And A. Wigderson, Superconcentrators, generalizers and generalized connectors with limited depth (preliminary version), in Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, Boston, Massachusetts, apr 1983, pp. 42–51.Google Scholar
  30. [30]
    M. Eichler, Quaternare quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion, Arch. Math., 5 (1954), pp. 355–366.MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    P. Feldman, J. Friedman, And N. Pippenger, Wide-sense nonblocking networks, SIAM J. Discrete Math., 1 (1988), pp. 158–173.MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    J. Friedman, A lower bound on strictly nonblocking networks, Combinatorica, 8 (1988), pp. 185–188.MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    J. Friedman, J. Kahn, And E. Szemerédi, On the second eigenvalue in random regular graphs, in Proceedings of the 21st ACM STOC, 1989, pp. 587–598.Google Scholar
  34. [34]
    O. Gabber And Z. Galil, Explicit constructions of linear size superconcentrators, in 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979), IEEE, New York, 1979, pp. 364–370.Google Scholar
  35. [35]
    O. Gabber And Z. Galil, Explicit constructions of linear-sized superconcentrators, J. Comput. System Sci., 22 (1981), pp. 407–420. Special issued dedicated to Michael Machtey.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    C. D. Godsil, Algebraic combinatorics, Chapman & Hall, New York, 1993.MATHGoogle Scholar
  37. [37]
    R. A. Horn And C. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985.MATHGoogle Scholar
  38. [38]
    X. D. Hu And F. K. Hwang, An improved upper bound for the subarray partial concentrators, Discrete Appl. Math., 37 /38 (1992), pp. 341–346.MathSciNetCrossRefGoogle Scholar
  39. [39]
    J. H. Hui, Switching and traffic theory for integrated broadband networks, Kluwer Academic Publishers, Boston/Dordrecht/London, 1990.MATHGoogle Scholar
  40. [40]
    F. K. Hwang, The mathematical theory of nonblocking switching networks, World Scientific Publishing Co. Inc., River Edge, NJ, 1998.MATHGoogle Scholar
  41. [41]
    F. K. Hwang And G. W. Richards, The capacity of the subarray partial concentrators, Discrete Appl. Math., 39 (1992), pp. 231–240.MathSciNetMATHCrossRefGoogle Scholar
  42. [42]
    F. K.Hwang And G. W. Richards, A two-stage network with dual partial concentrators, Networks, 23 (1993), pp. 53–58.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    J.-I. Igusa, Fibre systems of Jacobian varieties. III. Fibre systems of elliptic curves, Amer. J. Math., 81 (1959), pp. 453–476.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    S. Jimbo And A. Maruoka, Expanders obtained from affine transformations, Combinatorica, 7 (1987), pp. 343–355.MathSciNetMATHCrossRefGoogle Scholar
  45. [45]
    A. E. Joel, On permutation switching networks, Bell System Tech. J., 47 (1968), pp. 813–822.MATHGoogle Scholar
  46. [46]
    N.Kahale, Eigenvalues and expansion of regular graphs, J. Assoc. Comput. Mach., 42 (1995), pp. 1091–1106.MathSciNetMATHGoogle Scholar
  47. [47]
    D. G. Kirkpatrick, M. Klawe, And N. Pippenger, Some graph-colouring theorems with applications to generalized connection networks, SIAM J. Algebraic Discrete Methods, 6 (1985), pp. 576–582.MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    D. E. Knuth, The art of computer programming. Volume3, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1973, Sorting and searching, Addison-Wesley Series in Computer Science and Information Processing.Google Scholar
  49. [49]
    G.Lev And L. G. Valiant, Size bounds for superconcentrators, Theoret. Comput. Sci., 22 (1983), pp. 233–251.MathSciNetMATHCrossRefGoogle Scholar
  50. [50]
    A. Lubotzky, Cayley graphs: eigenvalues, expanders and random walks, in Surveys in cornbinatorics, 1995 (Stirling), Cambridge Univ. Press, Cambridge, 1995, pp. 155–189.Google Scholar
  51. [51]
    A. Lubotzky, R. Phillips, And P. Sarnak, Ramanujan graphs, Combinatorica, 8 (1988), pp. 261–277.MathSciNetMATHCrossRefGoogle Scholar
  52. [52]
    G. A. Margulis, Explicit constructions of expanders, Problemy Peredaì=i Informacii, 9 (1973), pp. 71–80.MathSciNetMATHGoogle Scholar
  53. [53]
    G. A. Margulis, Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problemy Peredachi Informatsii, 24 (1988), pp. 51–60.MathSciNetGoogle Scholar
  54. [54]
    G. M. Masson And B. W. Jordan, Jr., Generalized multi-stage connection networks, Networks, 2 (1972), pp. 191–209.MathSciNetMATHCrossRefGoogle Scholar
  55. [55]
    R. Meshulam, A geometric construction of a superconcentrator of depth2, Theoret. Comput. Sci., 32 (1984), pp. 215–219.MathSciNetMATHCrossRefGoogle Scholar
  56. [56]
    H. Minc, Upper bounds for permanents of (0, 1)-matrices, Bull. Amer. Math. Soc., 69 (1963), pp. 789–791.MathSciNetMATHCrossRefGoogle Scholar
  57. [57]
    M. Morgenstern, Existence and explicit constructions of q + 1 regular Ramanujan graphs for every prime power q, J. Combin. Theory Ser. B, 62 (1994), pp. 44–62.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    M. Morgenstern, Ramanujan diagrams, SIAM J. Discrete Math., 7 (1994), pp. 560–570.MathSciNetMATHCrossRefGoogle Scholar
  59. [59]
    M. Morgenstern, Natural bounded concentrators, Combinatorica, 15 (1995), pp. 111–122.MathSciNetMATHCrossRefGoogle Scholar
  60. [60]
    D. Nassimi And S. Sahni, Parallel permutation and sorting algorithms and a new generalized connection network, J. Assoc. Comput. Mach., 29 (1982), pp. 642–667.MathSciNetMATHGoogle Scholar
  61. [61]
    A. Nilli, On the second eigenvalue of a graph, Discrete Math., 91 (1991), pp. 207–210.MathSciNetMATHCrossRefGoogle Scholar
  62. [62]
    J. P. Ofman, A universal automaton, Trudy Moskov. Mat. OW., 14 (1965), pp. 186–199.MathSciNetMATHGoogle Scholar
  63. [63]
    W. J. Paul, R. E. Tarjan, And J. R. Celoni, Space bounds for a game on graphs, Math. Systems Theory, 10 (1976/77), pp. 239–251.MathSciNetCrossRefGoogle Scholar
  64. [64]
    W. J. Paul, R. E. Tarjan, And J. R. Celoni, Correction to: “Space bounds for a game on graphs ”, Math. Systems Theory, 11 (1977/78), p. 85.Google Scholar
  65. [65]
    M. S. Pinsker, On the complexity of a concentrator, in Proceedings of the 7th International Teletraffic Conference, Stockholm, June 1973, 1973, pp. 318/1–318/4.Google Scholar
  66. [66]
    N. Pippenger, The complexity of switching networks, PhD thesis, Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1973.Google Scholar
  67. [67]
    N. Pippenger, Superconcentrators, SIAM J. Comput., 6 (1977), pp. 298–304.MathSciNetMATHCrossRefGoogle Scholar
  68. [68]
    N. Pippenger, Generalized connectors, SIAM J. Comput., 7 (1978), pp. 510–514.MathSciNetMATHCrossRefGoogle Scholar
  69. [69]
    N. Pippenger, On rearrangeable and nonblocking switching networks, J. Comput. System Sci., 17 (1978), pp. 145–162.MathSciNetMATHCrossRefGoogle Scholar
  70. [70]
    N. Pippenger, A new lower bound for the number of switches in rearrangeable networks, SIAM J. Algebraic Discrete Methods, 1 (1980), pp. 164–167.MathSciNetMATHCrossRefGoogle Scholar
  71. [71]
    N. Pippenger, Superconcentrators of depth2, J. Comput. System Sci., 24 (1982), pp. 82–90.MathSciNetMATHCrossRefGoogle Scholar
  72. [72]
    N. Pippenger, Communication networks, in Handbook of theoretical computer science, Vol. A, Else- vier, Amsterdam, 1990, pp. 805–833.Google Scholar
  73. [73]
    N. Pippenger, Self-routing superconcentrators, J. Comput. System Sci., 52 (1996), pp. 53–60.MathSciNetMATHCrossRefGoogle Scholar
  74. [74]
    N. Pippenger And L. G. Valiant, Shifting graphs and their applications, J. Assoc. Comput. Mach., 23 (1976), pp. 423–432.MathSciNetMATHGoogle Scholar
  75. [75]
    N. Pippenger And A. C. C. Yao, Rearrangeable networks with limited depth, SIAM J. Algebraic Discrete Methods, 3 (1982), pp. 411–417.MathSciNetMATHCrossRefGoogle Scholar
  76. [76]
    P. Pudlak, Communication in bounded depth circuits, Combinatorica, 14 (1994), pp. 203–216.MathSciNetMATHCrossRefGoogle Scholar
  77. [77]
    J. Radhakrishnan And A. Ta-Shma, Bounds for dispersera, extractors, and depth-two superconcentrators, SIAM J. Discrete Math., 13 (2000), pp. 2–24 (electronic).MathSciNetMATHCrossRefGoogle Scholar
  78. [78]
    H. Robbins, A remark on Stirling’s formula, Amer. Math. Monthly, 62 (1955), pp. 26–29.MathSciNetMATHCrossRefGoogle Scholar
  79. [79]
    Y. Roichman, Expansion properties of Cayley graphs of the alternating groups, J. Combin. Theory Ser. A, 79 (1997), pp. 281–297.MathSciNetMATHCrossRefGoogle Scholar
  80. [80]
    C. E. Shannon, Memory requirements in a telephone exchange, Bell System Tech. J., 29 (1950), pp. 343–349.MathSciNetGoogle Scholar
  81. [81]
    M. Sipser And D. A. Spielman, Expander codes, IEEE Trans. Inform. Theory, 42 (1996), pp. 1710–1722. Codes and complexity.MathSciNetMATHCrossRefGoogle Scholar
  82. [82]
    D. A. Spielman, Constructing error-correcting codes from expander graphs, in Emerging applications of number theory (Minneapolis, MN, 1996), Springer, New York, 1999, pp. 591–600.Google Scholar
  83. [83]
    R. M. Tanner, Explicit concentrators from generalized N-gons, SIAM J. Algebraic Discrete Methods, 5 (1984), pp. 287–293.MathSciNetMATHCrossRefGoogle Scholar
  84. [84]
    L. G. Valiant, On non-linear lower bounds in computational complexity, in Seventh Annual ACM Symposium on Theory of Computing (Albuquerque, N. M., 1975), Assoc. Comput. Mach., New York, 1975, pp. 45–53.Google Scholar
  85. [85]
    L. G. Valiant, Graph-theoretic properties in computational complexity, J. Comput. System Sci., 13 (1976), pp. 278–285. Working papers presented at the ACM-SIGACT Symposium on the Theory of Computing (Albuquerque, N. M., 1975 ).MathSciNetMATHCrossRefGoogle Scholar
  86. [86]
    J. H. Van Lint, Problem 350, Nieuw Archief voor Wiskunde, (1973), p. 179.Google Scholar
  87. [87]
    A. Waksman, A permutation network, J. Assoc. Comput. Mach. 15 (1968), 159–163; corrigendum, ibid., 15 (1968), p. 340.MathSciNetMATHGoogle Scholar
  88. [88]
    A. Wigderson And D. Zuckerman, Expanders that beat the eigenvalue bound: explicit construction and applications, Combinatorica, 19 (1999), pp. 125–138.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Hung Quang Ngo
    • 1
  • Ding-Zhu Du
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of MinnesotaMinneapolisUSA

Personalised recommendations