Binary search trees based on Weyl and Lehmer sequences

  • Luc Devroye
Part of the Lecture Notes in Statistics book series (LNS, volume 127)


This paper is based upon the presentation at the meeting in Salzburg. As a courtesy to those who attended the meeting, I will try to faithfully reproduce—with minor omissions and additions—what I said at that meeting. There are two basic background references for mathematical details, Devroye (1987) and Devroye and Goudjil (1996).


Golden Ratio Continue Fraction Expansion Binary Search Tree External Node Partial Quotient 
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.


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  1. Aho, A.V., Hoperoft, J.E., and Ullman, J.D., 1983. Data Structures and Algorithms. Reading, Mass.: Addison-Wesley.Google Scholar
  2. Beck, J., 1991. Randomness of nv (mod 1) and a Ramsey property of the hyperbola. Colloquia Mathematica Societatis János Bolyai 60. Budapest.Google Scholar
  3. Béjian, R., 1982. Minoration de la discrépance d’une suite quelconque sur T. Acta Arithmetica 41. 185–202.zbMATHGoogle Scholar
  4. Bennett, C.H., 1979. On random and hard-to-describe numbers. IBM Watson Research Center Report RC 7483 (no 32272). Yorktown Heights, N.Y.Google Scholar
  5. Bohl, P., 1909. Über ein in der Theorie der säkularen Störungen vorkommendes Problem. Jl. Reine und Angewandte Mathematik 135. 189–283.zbMATHGoogle Scholar
  6. Boyd, D.W. and Steele, J.M., 1978. Monotone subsequences in the sequence of fractional parts of multiples of an irrational. Jl. Reine und Angewandte Mathematik 204. 49–59.Google Scholar
  7. Chatterji, S.D., 1966. Masse, die von regelmässigen Kettenbrüchen induziert sind. Mathematische Annalen 164. 113–117.CrossRefzbMATHMathSciNetGoogle Scholar
  8. Chow, Y.S., and Teicher, H., 1978. Probability Theory. New York, N.Y.: Springer-Verlag.CrossRefzbMATHGoogle Scholar
  9. del Junco, A. and Steele, J.M., 1979. Hammersley’s law for the van der Corput sequence: an instance of probability theory for pseudorandom numbers. Annals of Probability 7. 267–275.CrossRefzbMATHMathSciNetGoogle Scholar
  10. Devroye, L., 1986. A note on the height of binary search trees. Journal of the ACM 33. 489–498.CrossRefzbMATHMathSciNetGoogle Scholar
  11. Devroye, L., 1987. Branching processes in the analysis of the heights of trees Acta Informatica 24. 277–298.CrossRefzbMATHMathSciNetGoogle Scholar
  12. Devroye, L., 1988. Applications of the theory of records in the study of random trees. Acta Informatica 26. 123–130.CrossRefzbMATHMathSciNetGoogle Scholar
  13. Devroye, L., and Goudjil, A., 1996. A study of random Weyl trees. Technical Report, School of Computer Science, McGill University, Montreal.Google Scholar
  14. Diaconis, P., 1977. The distribution of leading digits and uniform distribution mod 1. Annals of Probability 5. 72–81.CrossRefzbMATHMathSciNetGoogle Scholar
  15. Ellis, M.H., and Steele, J.M., 1981. Fast sorting of Weyl sequences using comparisons. SIAM Journal on Computing 10. 88–95.CrossRefzbMATHMathSciNetGoogle Scholar
  16. Franklin, J N, 1963. Deterministic simulation of random sequences. Mathematics of Computation 17. 28–59.CrossRefzbMATHMathSciNetGoogle Scholar
  17. Freiberger, W., and Grenander, U., 1971. A Short Course in Computational Probability and Statistics. New York: Springer-Verlag.CrossRefzbMATHGoogle Scholar
  18. Fushimi, M., and Tezuka, S., 1983. The k-distribution of generalized feedback shift register pseudorandom numbers. Communications of the ACM 26. 516–523.CrossRefzbMATHGoogle Scholar
  19. Fushimi, M., 1988. Designing a uniform random number generator whose subsequences are k-distributed. SIAM Journal on Computing 17. 89–99.CrossRefMathSciNetGoogle Scholar
  20. Galambos, J., 1972. The distribution of the largest coefficient in continued fraction expansions. Quarterly Journal of Mathematics Oxford Series 23.147–151.CrossRefzbMATHMathSciNetGoogle Scholar
  21. Graham, R.L., Knuth, D.E., and Patashnik, 0., 1989. Concrete Mathematics-A Foundation for Computer Science. Reading, MA: Addison-Wesley.Google Scholar
  22. Hlawka, E., 1984. The Theory of Uniform Distribution. Berkhamsted, U.K.: A B Academic.Google Scholar
  23. Kesten, H., 1960. Uniform distribution mod 1. Annals of Mathematics 71. 445–471.CrossRefzbMATHMathSciNetGoogle Scholar
  24. Khintchine, A., 1924. Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Mathematische Annalen 92. 115–125.CrossRefzbMATHMathSciNetGoogle Scholar
  25. Khintchine, A., 1935. Metrische Kettenbruchprobleme. Compositio Mathematica 1. 361–382.zbMATHMathSciNetGoogle Scholar
  26. Khintchine, A., 1936. Metrische Kettenbruchprobleme. Compositio Mathematica 2. 276–285.MathSciNetGoogle Scholar
  27. Khintchine, A., 1963. Continued Fractions. Groningen: Noordhoff.zbMATHGoogle Scholar
  28. Knuth, D.E., 1973. The Art of Computer Programming, Vol. 3: Sorting and Searching. Reading, MA: Addison-Wesley.Google Scholar
  29. Knuth, D.E., 1981. The Art of Computer Programming, Vol. 2, 2nd Ed.. Reading, Mass.: Addison-Wesley.Google Scholar
  30. Kuipers, L., and Niederreiter, H., 1974. Uniform Distribution of Sequences. New York: John Wiley.zbMATHGoogle Scholar
  31. Kusmin, R.O., 1928. On a problem of Gauss. Reports of the Academy of Sciences A. 375–380.Google Scholar
  32. Lang, S., 1966. Introduction to Diophantine Approximations. Reading, MA: Addison-Wesley.Google Scholar
  33. L’Ecuyer, P. and Blouin, F., 1988. Linear congruential generators of order k > 1. Proceedings of the 1988 Winter Simulation Conference, ed. M. Abrams P. Haigh and J. Comfort. 432–439. ACM.CrossRefGoogle Scholar
  34. L’Ecuyer, P., 1989. A tutorial on uniform variate generation. Proceedings of the 1989 Winter Simulation Conference, ed. E.A. MacNair, K.J. Musselman and P. Heidelberger. 40–49. ACM.CrossRefGoogle Scholar
  35. L’Ecuyer, P., 1990. Random numbers for simulation. Communications of the ACM 33. 85–97.CrossRefGoogle Scholar
  36. LeVeque, W.J., 1977. Fundamentals of Number Theory. Reading, MA: Addison-Wesley.Google Scholar
  37. Lévy, P., 1929. Sur les lois de probabilité dont dépendent les quotients complets et incomplets d’une fraction continue. Bulletin de la Société Mathématique de France 57. 178–193.zbMATHGoogle Scholar
  38. Lévy, P., 1937. Théorie de l’addition des variables aléatoires. Paris.Google Scholar
  39. Lynch, W.C., 1965. More combinatorial problems on certain trees. Computer Journal 7. 299–302.CrossRefMathSciNetGoogle Scholar
  40. Mahmoud, H.M., 1992. Evolution of Random Search Trees. New York: John Wiley.zbMATHGoogle Scholar
  41. Martin-Löf, P., 1966. The definition of random sequences. Information and Control 9. 602–619.CrossRefzbMATHMathSciNetGoogle Scholar
  42. Niederreiter, H., 1977. Pseudo-random numbers and optimal coefficients. Advances in Mathematics 26. 99–181.CrossRefzbMATHMathSciNetGoogle Scholar
  43. Niederreiter, H., 1978. Quasi-Monte Carlo methods and pseudo-random numbers. Bulletin of the American Mathematical Society 84. 957–1042.CrossRefzbMATHMathSciNetGoogle Scholar
  44. Niederreiter, H., 1991. Recent trends in random number and random vector generation. Annals of Operations Research 31. 323–346.CrossRefzbMATHMathSciNetGoogle Scholar
  45. Niederreiter, H., 1992. Random Number Generation and Quasi-Monte Carlo Methods 63. SIAM CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia: SIAM.CrossRefGoogle Scholar
  46. Peskun, P.H., 1980. Theoretical tests for choosing the parameters of the general mixed linear congruential pseudorandom number generator. Journal of Statistical Computation and Simulation 11. 281–305.CrossRefzbMATHMathSciNetGoogle Scholar
  47. Philipp, W., 1969. The central limit problem for mixing sequences of random variables. Zeitschrift für Wahrscheinlichkeitstheorie and verwandte Gebiete 12. 155–171.CrossRefzbMATHMathSciNetGoogle Scholar
  48. Philipp, W., 1970. Some metrical theorems in number theory II. Duke Mathematical Journal 38. 477–485.Google Scholar
  49. Robson, J.M., 1979. The height of binary search trees. The Australian Computer Journal 11. 151–153.MathSciNetGoogle Scholar
  50. Robson, J.M., 1982. The asymptotic behaviour of the height of binary search trees. Australian Computer Science Communications 88–88.Google Scholar
  51. Schmidt, W.M., 1972. Irregularities of distribution. Acta Arithmetica 21. 45–50. Sedgewick, R., 1977. The analysis of quicksort programs. Acta Informatica 4. 327–355.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Luc Devroye
    • 1
  1. 1.School of Computer ScienceMcGill University MontrealMontrealCanada

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