Skip to main content
SpringerLink
Log in

The Ubiquitous Digital Tree

  • Conference paper
Book cover STACS 2006 (STACS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3884))

Included in the following conference series:

Abstract

The digital tree also known as trie made its first appearance as a general-purpose data structure in the late 1950’s. Its principle is a recursive partitioning based on successive bits or digits of data items. Under various guises, it has then surfaced in the management of very large data bases, in the design of efficient communication protocols, in quantitative data mining, in the leader election problem of distributed computing, in data compression, as well as in some corners of computational geometry. The algorithms are invariably very simple, easy to implement, and in a number of cases surprisingly efficient. The corresponding quantitative analyses pose challenging mathematical problems and have triggered a flurry of research works. Generating functions and symbolic methods, singularity analysis, the saddle-point method, transfer operators of dynamical systems theory, and the Mellin transform have all been found to have a bearing on the probabilistic behaviour of trie algorithms. We offer here a perspective on the rich algorithmic, analytic, and probabilistic aspects of tries, culminating with a connection between a sorting problem and the Riemann hypothesis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. David, J., Aldous, D.J.: Ultimate instability of exponential back-off protocol for acknowledgement-based transmission control of random access communication channels. IEEE Transactions on Information Theory 33(2), 219–223 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. Journal of Computer and System Sciences 58(1), 137–147 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avnaim, F., Boissonnat, J.-D., Devillers, O., Preparata, F.P., Yvinec, M.: Evaluating signs of determinants using single-precision arithmetic. Algorithmica 17(2), 111–132 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beeler, M., Gosper, R.W., Schroeppel, R.: HAKMEM, Memorandum 239, M.I.T, Artificial Intelligence Laboratory (February 1972), Available on the WorldWide Web at: http://www.inwap.com/-pdp10/-hbaker/-hakmem/-hakmem.html

  5. Bentley, J., Sedgewick, R.: Fast algorithms for sorting and searching strings. In: Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM Press, Philadelphia (1997)

    Google Scholar 

  6. Bentley, J.L.: Multidimensional binary search trees used for associative searching. Communications of the ACM 18(9), 509–517 (1975)

    Article  MATH  Google Scholar 

  7. Clément, J., Flajolet, P., Vallée, B.: The analysis of hybrid trie structures. In: Proceedings of the Ninth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 531–539. SIAM Press, Philadelphia (1998)

    Google Scholar 

  8. Clément, J., Flajolet, P., Vallée, B.: Dynamical sources in information theory: A general analysis of trie structures. Algorithmica 2(1/2), 307–369 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. Crochemore, M., Mignosi, F., Restivo, A., Salemi, S.: Text compression using antidictionaries. In: Wiedermann, J., Van Emde Boas, P., Nielsen, M. (eds.) ICALP 1999. LNCS, vol. 1644, Springer, Heidelberg (1999)

    Google Scholar 

  10. de Bruijn, N.G.: Asymptotic methods in analysis. A reprint of the third North Holland edition, Amsterdam (1970); (1st edn., 1958)

    Google Scholar 

  11. Devroye, L.: A probabilistic analysis of the height of tries and of the complexity of triesort. Acta Informatica 21, 229–237 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  12. Devroye, L.: An analysis of random LC tries. Random Structures & Algorithms 19, 359–375 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Devroye, L., Szpankowski, W.: Probabilistic behaviour of level compressed tries. Random Structures & Algorithms 27(2), 185–200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Durand, M., Flajolet, P.: LogLog counting of large cardinalities. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 605–617. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  15. Estan, C., Varghese, G.: New directions in traffic measurement and accounting: Focusing on the elephants, ignoring the mice. ACM Transactions on Computer Systems 21(3), 270–313 (2003)

    Article  Google Scholar 

  16. Fagin, R., Nievergelt, J., Pippenger, N., Strong, R.: Extendible hashing: A fast access method for dynamic files. A.C.M. Transactions on Database Systems 4, 315–344 (1979)

    Article  Google Scholar 

  17. Fayolle, G., Flajolet, P., Hofri, M.: On a functional equation arising in the analysis of a protocol for a multiaccess broadcast channel. Advances in Applied Probability 18, 441–472 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fayolle, G., Flajolet, P., Hofri, M., Jacquet, P.: Analysis of a stack algorithm for random access communication. IEEE Transactions on Information Theory IT-31(2), 244–254 (1985) (Special Issue on Random Access Communication, Massey, J. (ed.))

    Article  MathSciNet  MATH  Google Scholar 

  19. Fayolle, J.: An average-case analysis of basic parameters of the suffix tree. In: Drmota, M., et al. (eds.) Mathematics and Computer Science III: Algorithms, Trees, Combinatorics and Probabilities. Trends in Mathematics, pp. 217–227. Birkhäuser Verlag, Basel (2004)

    Chapter  Google Scholar 

  20. Fill, J.A., Mahmoud, H.M., Szpankowski, W.: On the distribution for the duration of a randomized leader election algorithm. The Annals of Applied Probability 6(4), 1260–1283 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Finkel, R.A., Bentley, J.L.: Quad trees, a data structure for retrieval on composite keys. Acta Informatica 4, 1–9 (1974)

    Article  MATH  Google Scholar 

  22. Flajolet, P.: On the performance evaluation of extendible hashing and trie searching. Acta Informatica 20, 345–369 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Flajolet, P.: Counting by coin tossings. In: Maher, M.J. (ed.) ASIAN 2004. LNCS, vol. 3321, pp. 1–12. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  24. Flajolet, P., Gourdon, X., Dumas, P.: Mellin transforms and asymptotics: Harmonic sums. Theoretical Computer Science 144(1-2), 3–58 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  25. Flajolet, P., Martin, G.N.: Probabilistic counting algorithms for data base applications. Journal of Computer and System Sciences 31(2), 182–209 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  26. Flajolet, P., Puech, C.: Partial match retrieval of multidimensional data. Journal of the ACM 33(2), 371–407 (1986)

    Article  MathSciNet  Google Scholar 

  27. Flajolet, P., Régnier, M., Sotteau, D.: Algebraic methods for trie statistics. In: Ausiello, G., Lucertini, M. (eds.) Analysis and Design of Algorithms for Combinatorial Problems. Annals of Discrete Mathematics, vol. 25, pp. 145–188 (1985)

    Google Scholar 

  28. Flajolet, P., Richmond, B.: Generalized digital trees and their difference–differential equations. Random Structures & Algorithms 3(3), 305–320 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Flajolet, P., Sedgewick, R.: Digital search trees revisited. SIAM Journal on Computing 15(3), 748–767 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Flajolet, P., Sedgewick, R.: Mellin transforms and asymptotics: finite differences and Rice’s integrals. Theoretical Computer Science 144(1–2), 101–124 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Flajolet, P., Sedgewick, R.: Analytic combinatorics. Chapters I–IX of a book to be published, 688p.+x, available electronically from P. Flajolet’s home page (October 2005)

    Google Scholar 

  32. Flajolet, P., Vallée, B.: Continued fractions, comparison algorithms, and fine structure constants. In: Théra, M. (ed.) Constructive, Experimental, and Nonlinear Analysis (Providence). Canadian Mathematical Society Conference Proceedings, vol. 27, pp. 53–82. American Mathematical Society (2000)

    Google Scholar 

  33. Goulden, I.P., Jackson, D.M.: Combinatorial enumeration. John Wiley, New York (1983)

    MATH  Google Scholar 

  34. Guibas, L.J., Odlyzko, A.M.: String overlaps, pattern matching, and nontransitive games. Journal of Combinatorial Theory. Series A 30(2), 183–208 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hofri, M.: Analysis of algorithms: Computational methods and mathematical tools. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  36. Hwang, H.-K.: On convergence rates in the central limit theorems for combinatorial structures. European Journal of Combinatorics 19(3), 329–343 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  37. Jacquet, P., Régnier, M.: Trie partitioning process: Limiting distributions. In: Franchi-Zannettacci, P. (ed.) CAAP 1986. LNCS, vol. 214, pp. 196–210. Springer, Heidelberg (1986)

    Google Scholar 

  38. Jacquet, P., Szpankowski, W.: Autocorrelation on words and its applications: analysis of suffix trees by string-ruler approach. Journal of Combinatorial Theory. Series A 66(2), 237–269 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. Jacquet, P., Szpankowski, W.: Asymptotic behavior of the Lempel-Ziv parsing scheme and digital search trees. Theoretical Computer Science 144(1–2), 161–197 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. Jacquet, P., Szpankowski, W.: Analytical de-Poissonization and its applications. Theoretical Computer Science 201(1-2), 1–62 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kirschenhofer, P., Prodinger, H.: On some applications of formulæ of Ramanujan in the analysis of algorithms. Mathematika 38, 14–33 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  42. Kirschenhofer, P., Prodinger, H., Szpankowski, W.: On the variance of the external path length in a symmetric digital trie. Discrete Applied Mathematics 25, 129–143 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  43. Knuth, D.E.: The art of computer programming, 2nd edn. Sorting and Searching, vol. 3. Addison-Wesley, Reading (1998)

    MATH  Google Scholar 

  44. Larson, P.A.: Dynamic hashing. BIT 18, 184–201 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  45. Mahmoud, H.M.: Evolution of random search trees. John Wiley, New York (1992)

    MATH  Google Scholar 

  46. Massey, J.L (ed.).: Special issue on random-access communications. IEEE Transactions on Information Theory IT-31(2) (March 1985)

    Google Scholar 

  47. Mathys, P., Flajolet, P.: Q–ary collision resolution algorithms in random access systems with free or blocked channel access. IEEE Transactions on Information Theory IT-31(2), 217–243 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  48. Nilsson, S., Karlsson, G.: IP–address lookup using LC tries. IEEE Journal on Selected Areas in Communications 17(6), 1083–1092 (1999)

    Article  Google Scholar 

  49. Prodinger, H.: How to select a loser. Discrete Mathematics 120, 149–159 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  50. Régnier, M.: On the average height of trees in in digital search and dynamic hashing. Information Processing Letters 13, 64–66 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  51. Régnier, M.: Analysis of grid file algorithms. BIT 25, 335–357 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  52. Samet, H.: Applications of spatial data structures. Addison–Wesley, Reading (1990)

    Google Scholar 

  53. Samet, H.: The design and analysis of spatial data structures. Addison–Wesley, Reading (1990)

    Google Scholar 

  54. Sedgewick, R.: Algorithms in C, Parts 1–4, 3rd edn. Addison–Wesley, Reading (1998)

    MATH  Google Scholar 

  55. Sedgewick, R., Flajolet, P.: An introduction to the analysis of algorithms. Publishing Company. Addison-Wesley Publishing Company, Reading (1996)

    MATH  Google Scholar 

  56. Stanley, R.P.: Stanley, Enumerative combinatorics, vol, vol. II. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  57. Szpankowski, W.: Average-case analysis of algorithms on sequences. John Wiley, New York (2001)

    Book  MATH  Google Scholar 

  58. Pardo, L.T.: Set representation and set intersection, Tech. report, Stanford University (1978)

    Google Scholar 

  59. Vallée, B.: Dynamical sources in information theory: Fundamental intervals and word prefixes. Algorithmica 29(1/2), 262–306 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  60. Vallée, B.: Euclidean dynamics. Discrete and Continuous Dynamical Systems, 69 (submitted, October 2005)

    Google Scholar 

  61. Whittaker, E.T., Watson, G.N.: A course of modern analysis, 4th edn. Cambridge University Press, Cambridge (1927) (reprinted, 1973)

    MATH  Google Scholar 

  62. Wilf, H.S.: Generatingfunctionology. Academic Press, London (1990)

    MATH  Google Scholar 

  63. Yao, A.C.C.: A note on the analysis of extendible hashing. Information Processing Letters 11, 84–86 (1980)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Algorithms Project, INRIA Rocquencourt, F-78153, Le Chesnay, France

    Philippe Flajolet

Authors
  1. Philippe Flajolet
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LIF, CNRS & Univ. de Provence,, Marseille, France

    Bruno Durand

  2. RWTH Aachen University,, Lehrstuhl Informatik 7,, 52056,, Aachen,, Germany

    Wolfgang Thomas

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Flajolet, P. (2006). The Ubiquitous Digital Tree. In: Durand, B., Thomas, W. (eds) STACS 2006. STACS 2006. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11672142_1

Download citation

  • DOI: https://doi.org/10.1007/11672142_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-32301-3

  • Online ISBN: 978-3-540-32288-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation