Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory

Abstract

Expanding graphs are relevant to theoretical computer science in several ways. Here we show that the points versus hyperplanes incidence graphs of finite geometries form highly (nonlinear) expanding graphs with essentially the smallest possible number of edges. The expansion properties of the graphs are proved using the eigenvalues of their adjacency matrices.

These graphs enable us to improve previous results on a parallel sorting problem that arises in structural modeling, by describing an explicit algorithm to sortn elements ink time units using\(O(n^{\alpha _k } )\) parallel processors, where, e.g., α2=7/4, α3=8/5, α4=26/17 and α5=22/15.

Our approach also yields several applications to Ramsey Theory and other extremal problems in combinatorics.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    H. Abelson, A note on time space tradeoffs for computing continuous functions,Infor. Proc. Letters 8 (1979), 215–217.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    M. Ajtai, J. Komlós andE. Szemerédi, Sorting inc logn parallel steps,Combinatorica 3 (1983), 1–9.

    MATH  MathSciNet  Google Scholar 

  3. [3]

    N. Alon, Expanders, sorting in rounds and superconcentrators of limited depth,Proc. 17th Annual ACM Symp. on Theory of Computing, Providence, RI (1985), 98–102.

  4. [4]

    N. Alon, Eigenvalues and expanders,Combinatorica,6 (1986), 83–96.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    N. Alon, Z. Galil andV. D. Milman, Better expanders and superconcentrators,J. of Algorithms, to appear.

  6. [6]

    N. Alon andV. D. Milman, λ1, isoperimetric inequalities for graphs and superconcentrators,J. Combinatorial Theory Ser. B,38 (1985), 73–88.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    N. Alon andV. D. Milman, Eigenvalues, expanders and superconcentrators,Proc. 25 th Annual Symp. on Foundations of Comp. Sci., Florida (1984), 320–322.

  8. [8]

    B. Bollobás,Extremal Graph Theory, Academic Press, London and New York (1978).

    Google Scholar 

  9. [9]

    N. G. de Bruijn andP. Erdős, On a combinatorial problem,Indagationes Math. 20 (1948), 421–423.

    Google Scholar 

  10. [10]

    B. Bollobás andP. Hell, Sorting and Graphs, in:Graphs and Order, (I. Rival, ed.) D. Reidel (1985), 169–184.

  11. [11]

    B. Bollobás andM. Rosenfeld, Sorting in one round,Israel J. Math. 38 (1981), 154–160.

    MATH  MathSciNet  Google Scholar 

  12. [12]

    B. Bollobás andA. Thomason, Parallel sorting,Discrete Appl. Math. 6 (1983), 1–11.

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    F. R. K. Chung, On concentrators, superconcentrators, generalizers, and nonblocking networks,Bell Sys. Tech. J. 58 (1978), 1765–1777.

    Google Scholar 

  14. [14]

    P. Erdős, Problems and results in Graph Theory,in: Proc. Inter. 4th Conf. on the theory and applications of graphs (G. Chartrand et al. eds.), Kalamazoo, Michigan (1980), pp. 331–341.

  15. [15]

    P. Erdős, Extremal problems in Number Theory,Combinatorics and Geometry, Proc. Inter. Conf. in Warsaw, 1983, to appear.

  16. [16]

    P. Erdős andA. Hajnal, On complete topological subgraphs of certain graphs,Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 7 (1964), 143–149.

    Google Scholar 

  17. [17]

    P. Erdős andA. Rényi, On a problem in the theory of graphs,Publ. Math. Inst. Hungar. Acad. Sci. 7 (1962), 215–235 (in Hungarian).

    Google Scholar 

  18. [18]

    P. Frankl, V. Rődl andR. M. Wilson, The number of submatrices of given type in a Hadamard martrix,J. of Combinatorial Theory B, to appear.

  19. [19]

    P. Frankl andR. M. Wilson, Intersection theorems with geometric consequences,Combinatorica 1 (1981), 357–368.

    MATH  MathSciNet  Google Scholar 

  20. [20]

    O. Gabber andZ. Galil, Explicit construction of linear sized superconcentrators,J. Comp. and Sys. Sci. 22 (1981), 407–420.

    MATH  Article  MathSciNet  Google Scholar 

  21. [21]

    S. Golomb,Shift Register Sequences, Holden Day, Inc., San Francisco, 1967.

    Google Scholar 

  22. [22]

    R. K. Guy andS. Znam, A problem of Zarankiewicz, in:Recent Progress in Combinatorics (W. T. Tutte, ed.) Academic Press, 1969, 237–243.

  23. [23]

    M. Hall, Jr.,Combinatorial Theory, Wiley and Sons, New York and London, 1967.

    Google Scholar 

  24. [24]

    R. Häggkvist andP. Hell, Graphs and parallel comparison algorithms.Congr. Num. 29 (1980), 497–509.

    Google Scholar 

  25. [25]

    R. Häggkvist andP. Hell, Parallel sorting with constant time for comparisons,SIAM J. Comp. 10, (1981), 465–472.

    MATH  Article  Google Scholar 

  26. [26]

    R. Häggkvist andP. Hell, Sorting and merging in rounds,SIAM J. Alg. and Disc. Meth. 3 (1982), 465–473.

    MATH  Google Scholar 

  27. [27]

    J. Ja’Ja, Time space tradeoffs for some algebraic problems,Proc. 12 th Ann. ACM Symp. on Theory of Computing, 1980, 339–350.

  28. [28]

    M. Klawe, Non-existence of one-dimensional expanding graphs,Proc. 22 nd Ann. Symp. Found. Comp. Sci. Nashville (1981). 109–113.

  29. [29]

    T. Lengauer andR. E. Tarjan, Asymptotically tight bounds on time space trade-offs in a pebble game,J. ACM 29 (1982), 1087–1130.

    MATH  Article  MathSciNet  Google Scholar 

  30. [30]

    G. A. Margulis, Explicit constructions of concentrators,Prob. Per. Infor. 9 (1973), 71–80, (English translation inProblems of Infor. Trans. (1975), 325–332).

    MathSciNet  MATH  Google Scholar 

  31. [31]

    R. Meshuluam, A geometric construction of a superconcentrator of depth 2,preprint.

  32. [32]

    M. Pinsker, On the complexity of a concentrator,7th International Teletraffic Conference, Stockholm, June 1973, 318/1–318/4.

  33. [33]

    N. Pippenger, Superconcentrators,SIAM J. Computing 6 (1977), 298–304.

    MATH  Article  MathSciNet  Google Scholar 

  34. [34]

    N. Pippenger, Advances in pebbling, Internat.Colloq. on Autom. Lang. and Prog. 9 (1982), 407–417.

    Article  Google Scholar 

  35. [35]

    N. Pippenger, Explicit construction of highly expanding graphs,preprint.

  36. [36]

    W. J. Paul, R. E. Tarjan andJ. R. Celoni, Space bounds for a game on graphs,Math. Sys. Theory 20 (1977), 239–251.

    Google Scholar 

  37. [37]

    S. Scheele,Final report to office of environmental education. Dept. of Health, Education and Welfare, Social Engineering Technology, Los Angeles, CA 1977.

    Google Scholar 

  38. [38]

    J. Spencer, Asymtotic lower bounds for Ramsey functions,Discrete Math. 20 (1977), 69–76.

    Article  MathSciNet  Google Scholar 

  39. [39]

    R. M. Tanner, Explicit construction of concentrators from generalizedN-gons,SIAM J. Alg. Discr. Meth.,5 (1984), 287–293.

    MATH  MathSciNet  Article  Google Scholar 

  40. [40]

    M. Tompa, Time space tradeoffs for computing functions, using connectivity properties of their circuits,J. Comp. and Sys. Sci. 20 (1980), 118–132.

    MATH  Article  MathSciNet  Google Scholar 

  41. [41]

    L. G. Valiant, Paralelism in comparison networks.SIAM J. Comp. 4 (1975), 348–355.

    MATH  Article  MathSciNet  Google Scholar 

  42. [42]

    L. G. Valiant, Graph theoretic properties in computational complexity,J. Com. and Sys. Sci. 13 (1976), 278–285.

    MATH  MathSciNet  Google Scholar 

  43. [43]

    N. Alon, Y. Azar, andU. Vizhkin, Tight complexity bounds for parallel comparison sorting,Proc. 27th FOCS, to appear.

  44. [44]

    A. Lubotzky, R. Phillips, andP. Sarnak, Ramanujan graphs,to appear.

  45. [45]

    N. Pippenger, Sorting and selecting in rounds,preprint.

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Alon, N. Eigenvalues, geometric expanders, sorting in rounds, and ramsey theory. Combinatorica 6, 207–219 (1986). https://doi.org/10.1007/BF02579382

Download citation

AMS subject classification (1980)

  • 68 E 10
  • 68 E 05
  • 05 B 25
  • 05 C 55