Random walks on trees

  • R. Schott
Probability On Trees
Part of the Lecture Notes in Computer Science book series (LNCS, volume 214)

Abstract

Random walks or Brownian motions appear as a useful tool in algorithm analysis. Recently P. Flajolet ([2]) obtained a complete and detailed analysis of the two stacks problem with the help of properties of simple random walks on lattices. G. Louchard ([7], [8]) proved that the Brownian motion permits to give easily asymptotic results on the complexity of manipulation algorithms for sorted tables, dictonaries and priority queues. In [4], J. Françon and the author proved that random walks on some homogeneous trees can be analysed with simple combinatorial technics : generating functions, continued and multicontinued fractions, orthogonal polynomials, theorem of Darboux etc.. In this paper we show that random walks on more general trees can be related to random walks on N and that on general Cayley graphs (i.e. graphs corresponding to finitely generated groups with relations between the generators) the asymptotic behavior of the random walks can be obtained using proporties of the Brownian motions on Riemannian manifolds and a simple criteria can be given in terms of γ(n) the number of different words was length is less than or equal to n .

Keywords

Random walk tree generating function generator asymptotic behavior dynamic data structure method of Darboux 

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References

  1. [1]
    Flajolet P. (1981): Analyse d'algorithmes de manipulation d'arbres et de fichiers — Cahier du B.U.R.O. 34–35.Google Scholar
  2. [2]
    Flajolet P. (1984): The evolution of two stacks in bounded space and random walks in a triangle. Publication I.N.R.I.A. Roquencourt.Google Scholar
  3. [3]
    Françon J. (1985: Une approche quantitative de l'exclusion mutuelle. Proceedings of the conference S.T.A.C.S.Google Scholar
  4. [4]
    Françon J. and Schott R. (1984): Multicontinued fractions and combinatorial of words on finitely generated groups. N.A.T.O. advanced workshop "combinatorial algorithms on words", Moratea, Italy, May 18–22.Google Scholar
  5. [5]
    Guivarc'H Y., Keane M., Roynette B.: Marches aléatoires sur les groupes de Lie — Lecture Notes in mathematics no 624, Springer VerlagGoogle Scholar
  6. [6]
    Jonassen A. and Knuth D.E. (1978): A trivial algorithm whose analysis isn't. Journal of Comp. and System Sc. 16, 323–332.Google Scholar
  7. [7]
    Louchard G. (1983): The Brownian notion a neglected tool for the complexity analysis of sorted tables manipulations. R.A.I.R.O., vol. 17, 4, 365–385.Google Scholar
  8. [8]
    Louchard G. (1985): Random walks, Gaussian processes and list structures Technical report 152. Université libre de Bruxelles. Laboratoire d'informatique théorique.Google Scholar
  9. [9]
    Milnor J. (1968): A note on curvature and fundamental group J. Diff. Geometry 2, 1–7.Google Scholar
  10. [10]
    Vanopoulos N.Th. (1981): Potential theory and diffusion on Riemannian manifolds. Zygmund 80th Birsthday Volume.Google Scholar
  11. [11]
    Wall H.S. (1948): Analytic theory of continued fractions Von Nostrand, Toronto.Google Scholar
  12. [12]
    Wolf J.A. (1968): Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Diff. Geom. 2, 421–446.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • R. Schott
    • 1
  1. 1.C.R.I.N. U.E.R. Sciences MathématiquesVandoeuvre Les Nancy CedexFrance

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