Arbres digitaux

  • Brigitte Chauvin
  • Julien Clément
  • Danièle Gardy
Part of the Mathématiques et Applications book series (MATHAPPLIC, volume 83)


Dans ce chapitre, nous nous concentrerons uniquement sur la structure de trie. Les autres types d’arbres digitaux ne seront pas abordés sauf en exercice.


  1. 19.
    J.L. Bentley, R. Sedgewick, Fast algorithms for sorting and searching strings, in Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’97 (Society for Industrial and Applied Mathematics, Philadelphia, 1997), pp. 360–369zbMATHGoogle Scholar
  2. 20.
    B. Bercu, D. Chafaï. Modélisation stochastique et simulation : cours et applications. Sciences sup. (Dunod, Paris, 2007). Série << Mathématiques appliquées pour le Master / SMAI >>.Google Scholar
  3. 33.
    E. Cesaratto, B. Vallée. Gaussian distribution of trie depth for strongly tame sources. Comb. Probab. Comput. 24(1), 54–103 (2015)MathSciNetCrossRefGoogle Scholar
  4. 45.
    J. Clément, P. Flajolet, B. Vallée, The analysis of hybrid trie structures, in Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, 25–27 January 1998, pp. 531–539Google Scholar
  5. 46.
    J. Clément, P. Flajolet, B. Vallée. Dynamical sources in information theory: a general analysis of trie structures. Algorithmica 29(1), 307–369 (2001)MathSciNetCrossRefGoogle Scholar
  6. 47.
    J. Clément, T. Nguyen Thi, B. Vallée. A general framework for the realistic analysis of sorting and searching algorithms. Application to some popular algorithms, in STACS (2013), pp. 598–609Google Scholar
  7. 48.
    J. Clément, T.H.N. Thi, B. Vallée. Towards a realistic analysis of some popular sorting algorithms. Comb. Probab. Comput. 24(1), 104–144 (2015). Special issue dedicated to the memory of Philippe FlajoletMathSciNetCrossRefGoogle Scholar
  8. 55.
    L. Devroye, A probabilistic analysis of the height of tries and of the complexity of triesort. Acta Inform. 21, 229–237 (1984)MathSciNetCrossRefGoogle Scholar
  9. 56.
    L. Devroye, Branching processes in the analysis of the height of trees. Acta Inform. 24, 277–298 (1987)MathSciNetCrossRefGoogle Scholar
  10. 58.
    L. Devroye, A study of trie-like structures under the density model. Ann. Appl. Probab. 2(2), 402–434 (1992)MathSciNetCrossRefGoogle Scholar
  11. 59.
    L. Devroye, Branching processes and their applications in the analysis of tree structures and tree algorithms, in Probabilistic Methods for Algorithmic Discrete Mathematics, ed. by M. Habib, C. McDiarmid, J. Ramirez-Alfonsin, B. Reed (Springer, Berlin, 1998)Google Scholar
  12. 79.
    P. Flajolet, On the performance evaluation of extendible hashing and trie searching. Acta Inform. 20, 345–369 (1983)MathSciNetCrossRefGoogle Scholar
  13. 83.
    P. Flajolet, The ubiquitous digital tree, in Proceedings of 23rd Annual Symposium on Theoretical Aspects of Computer Science, STACS 2006, Marseille, 23–25 February 2006. Lecture Notes in Computer Science, vol. 3884 (Springer, Berlin, 2006), pp. 1–22Google Scholar
  14. 91.
    P. Flajolet, B. Richmond, Generalized digital trees and their difference—differential equations. Random Struct. Algoritm. 3(3), 305–320 (1992)MathSciNetCrossRefGoogle Scholar
  15. 92.
    P. Flajolet, R. Sedgewick, Digital search trees revisited. SIAM J. Comput. 15(3), 748–767 (1986)MathSciNetCrossRefGoogle Scholar
  16. 93.
    P. Flajolet, R. Sedgewick, Mellin transforms and asymptotics: finite differences and Rice’s integrals. Theor. Comput. Sci. 144(1&2), 101–124 (1995)MathSciNetCrossRefGoogle Scholar
  17. 94.
    P. Flajolet, R. Sedgewick, Analytic Combinatorics (Cambridge University Press, Cambridge, 2009)CrossRefGoogle Scholar
  18. 96.
    P. Flajolet, J.-M. Steyaert, A branching process arising in dynamic hashing, trie searching and polynomial factorization, in Automata, Languages and Programming, ed. by M. Nielsen, E.M. Schmidt. Lecture Notes in Computer Science, vol. 140 (Springer, Berlin, 1982), pp. 239–251. Proceedings of 9th ICALP Colloquium, Aarhus, July 1982Google Scholar
  19. 97.
    P. Flajolet, M. Régnier, D. Sotteau, Algebraic methods for trie statistics. Ann. Discret. Math. 25, 145–188 (1985). In Analysis and Design of Algorithms for Combinatorial Problems, ed. by G. Ausiello, M. Lucertini (Invited Lecture)Google Scholar
  20. 100.
    P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics: harmonic sums. Theor. Comput. Sci. 144(1–2), 3–58 (1995)MathSciNetCrossRefGoogle Scholar
  21. 105.
    P. Flajolet, M. Roux, B. Vallée. Digital trees and memoryless sources: from arithmetics to analysis, in Proceedings of AofA’10, DMTCS, Proc AM (2010), pp. 231–258Google Scholar
  22. 113.
    M. Fuchs, H.-K. Hwang, V. Zacharovas, An analytic approach to the asymptotic variance of trie statistics and related structures. Theor. Comput. Sci. 527, 1–36 (2014)MathSciNetCrossRefGoogle Scholar
  23. 135.
    K. Hun, B. Vallée. Typical depth of a digital search tree built on a general source, in 2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics, ANALCO 2014, Portland, 6 January 2014, pp. 1–15Google Scholar
  24. 136.
    H.-K. Hwang, Théorémes limites pour les structures combinatoires et les fonctions arithmétiques. PhD thesis, École polytechnique, Palaiseau, 1994Google Scholar
  25. 137.
    H.-K. Hwang, On convergence rates in the central limit theorems for combinatorial structures. Eur. J. Comb. 19(3), 329–343 (1998)MathSciNetCrossRefGoogle Scholar
  26. 141.
    P. Jacquet, M. Régnier, Trie partitioning process: limiting distributions, in CAAP’86, ed. by P. Franchi-Zanetacchi. Proceedings of the 11th Colloquium on Trees in Algebra and Programming, Nice, LNCS, vol. 214 (Springer, Berlin, 1986), pp. 196–210Google Scholar
  27. 142.
    P. Jacquet, M. Régnier, New results on the size of tries. IEEE Trans. Inf. Theory 35(1), 203–205 (1989)MathSciNetCrossRefGoogle Scholar
  28. 143.
    P. Jacquet, W. Szpankowski, Analysis of digital tries with markovian dependency. IEEE Trans. Inf. Theory 37(5), 1470–1475 (1991)CrossRefGoogle Scholar
  29. 144.
    P. Jacquet, W. Szpankowski, Autocorrelation on words and its applications: analysis of suffix trees by string-ruler approach. J. Comb. Theory (A) 66(2), 237–269 (1994)MathSciNetCrossRefGoogle Scholar
  30. 145.
    P. Jacquet, W. Szpankowski, Asymptotic behavior of the Lempel-Ziv parsing scheme and digital search trees. Theor. Comput. Sci. 144(1–2), 161–197 (1995)MathSciNetCrossRefGoogle Scholar
  31. 150.
    S. Janson, W. Szpankowski, Analysis of an asymmetric leader election algorithm. Electron. J. Comb. 9 (1997)Google Scholar
  32. 166.
    G. Louchard, Trie size in a dynamic list structure. Random Struct. Algoritm. 5(5), 665–702 (1994)MathSciNetCrossRefGoogle Scholar
  33. 172.
    H. Mahmoud, Evolution of Random Search Trees (Wiley, New York, 1992)zbMATHGoogle Scholar
  34. 196.
    N.E. Nörlund. Leçons sur les équations linéaires aux différences finies, in Collection de monographies sur la théorie des fonctions (Gauthier-Villars, Paris, 1929)Google Scholar
  35. 197.
    N.E. Nörlund. Vorlesungen über Differenzenrechnung (Chelsea Publishing Company, New York, 1954)zbMATHGoogle Scholar
  36. 204.
    G. Park, H.-K. Hwang, P. Nicodème, W. Szpankowski, Profiles of tries. SIAM J. Comput. 38(5), 1821–1880 (2009)MathSciNetCrossRefGoogle Scholar
  37. 208.
    B. Pittel, Paths in a random digital tree: limiting distributions. Adv. Appl. Probab. 18, 139–155 (1986)MathSciNetCrossRefGoogle Scholar
  38. 222.
    R.F. Rice, Some practical universal noiseless coding techniques. Technical Report JPL-79–22, JPL, Pasadena (1979).
  39. 226.
    M. Roux, B. Vallée, Information theory: sources, Dirichlet series, and realistic analyses of data structures, in Proceedings 8th International Conference Words 2011. EPTCS, vol. 63 (2011), pp. 199–214Google Scholar
  40. 227.
    D. Salomon, Data Compression: The Complete Reference (Springer, Berlin, 2007). With contributions by G. Motta and D. BryantGoogle Scholar
  41. 232.
    R. Sedgewick, P. Flajolet, Introduction to the Analysis of Algorithms (Addison-Wesley, Reading, 1996)zbMATHGoogle Scholar
  42. 233.
    R. Sedgewick, K. Wayne, Algorithms, 4th edn. (Addison-Wesley, Reading, 2011)Google Scholar
  43. 239.
    W. Szpankowski, Average Case Analysis of Algorithms on Sequences (Wiley, New York, 2001)CrossRefGoogle Scholar
  44. 245.
    B. Vallée, Dynamical sources in information theory: fundamental intervals and word prefixes. Algorithmica 29(1), 262–306 (2001)MathSciNetCrossRefGoogle Scholar
  45. 246.
    B. Vallée, J. Clément, J.A. Fill, P. Flajolet, The number of symbol comparisons in quicksort and quickselect, in ICALP 2009, ed. by S. Albers et al. Part I, LNCS, vol. 5555 (Springer, Berlin, 2009), pp. 750–763CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Brigitte Chauvin
    • 1
  • Julien Clément
    • 2
  • Danièle Gardy
    • 3
  1. 1.Laboratoire de MathématiquesUniversité Versailles, Saint-Quentin-en-YvelinesVersailles CedexFrance
  2. 2.GREYC, CNRS UMR 6072Normandie UniversitéCaen CedexFrance
  3. 3.Laboratoire DAVIDUniversité Versailles, Saint-Quentin-en-YvelinesVersailles CedexFrance

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