Analysis of dynamic algorithms in D.E.Knuth's model

  • J. Françon
  • B. Randrianarimanana
  • R. Schott
Part of the Lecture Notes in Computer Science book series (LNCS, volume 299)


This paper analyses the average behaviour of algorithms that operate on dynamically varying data structures subject to insertions I, deletions D, positive (resp. negative) queries Q+ (resp.Q) under the following assumptions: if the size of the data structure is k (k ε N), then the number of possibilities for the operations D and Q+ is a linear function of k, whereas the number of possibilities for the i-th insertion or negative query is equal to i. This statistical model was introduced by J.Françon (6), (7) and D.E.Knuth(11) and differes from the model used in previous analyses (2), (3), (4), (5), (6), (7). Integrated costs for these dynamic structures are defined as averages of costs taken over the set of all their possible histories (i.e. evolutions considered up to order isomorphism) of length n. We show that the costs can be calculated for the data structures serving as implementations of linear lists, priority queues and dictionaries. The problem of finding the limiting distributions is also considered and the linear list case is treated in detail. The method uses continued fractions and orthogonal polynomials but in a paper in preparation, we show that the same results can be recovered with the help of a probabilistic model.


Markovian Model Orthogonal Polynomial Average Cost Priority Queue Dynamic Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L.Cheno Profils limites d'histoires sur les dictionnaires et les files de priorité. Application aux files binomiales. Thèse de 3è cycle, Université d'Orsay, 1981.Google Scholar
  2. [2]
    L.Cheno, P.Flajolet, J.Françon, C.Puech, J.Vuillemin Finite files, limiting profiles and variance analysis. Proceedings 18th Allerton Conf. on Com. Control and Computing (Illinois 1980).Google Scholar
  3. [3]
    P. Flajolet, J. Françon, J. Vuillemin Sequence of operations analysis for dynamic data structures. J. of algorithms 1, 111–141, 1980.Google Scholar
  4. [4]
    P. Flajolet, C. Puech, J. Vuillemin The analysis of simple lists structures. Inf. Sc. 38, 121–146, 1986.Google Scholar
  5. [5]
    P.Flajolet Analyse d'algorithmes de manipulation d'arbres et de fichiers. B.U.R.O. cahier 34–35, 1981.Google Scholar
  6. [6]
    J.Françon Combinatoire des structures de données. Thèse de doc. d'Etat. Université de Strasbourg, 1979Google Scholar
  7. [7]
    J. Françon Histoires de fichiers. RAIRO Inf. Th. 12, 49–62, 1978.Google Scholar
  8. [8]
    J.Françon, B.Randrianarimanana, R.Schott Analysis of dynamic data structures in D.E.Knuth's model. Rapport C.R.I.N. 1986.(submitted)Google Scholar
  9. [9]
    A. Jonassen, D.E. Knuth A trivial algorithm whose analysis isn't. J. of Comp. and System Sc. 16, 301–332, 1978.Google Scholar
  10. [10]
    G.D.Knott Deletion in binary storage trees. Report Stan-CS, 75–491, may 1975.Google Scholar
  11. [11]
    D.E.Knuth Deletions that preserve randomness. Trans. Software Eng, 351–359, 1977.Google Scholar
  12. [12]
    D.E.Knuth The art of computer programming: Sorting and Searching, vol.3, second printing, 1975.Google Scholar
  13. [13]
    G. Louchard Random walks, Gaussian processes and list structures. Th. Comp. Sc. 53, 99–124, 1987.Google Scholar
  14. [14]
    B.Randrianarimanana Analyse des structures de données dynamiques dans le modèle de D.E.Knuth. Thèse de 3ème cycle, Université Nancy 1, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • J. Françon
    • 1
  • B. Randrianarimanana
    • 2
  • R. Schott
    • 2
  1. 1.Département InformatiqueUniversité Louis PasteurStrasbourgFrance
  2. 2.C.R.I.N. Université Nancy 1Vandoeuvre-lès-NancyFrance

Personalised recommendations