Skip to main content
SpringerLink
Log in

Random sequential bisection and its associated binary tree

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Summary

Random sequential bisection is a process to divide a given interval into two, four, eight, ... parts at random. Each division point is uniformly distributed on the interval and conditionally independent of the others. To study the asymptotic behavior of the lengths of subintervals in random seqential bisection, the associated binary tree is introduced.

The number of internal or external nodes of the tree is asymptotically normal. The levels of the lowest and the highest external nodes are bounded with probability one or with probability increasing to one as the number of nodes increases to infinity.

The associated binary tree is closely related to random binary tree which arises in computer algorithms, such as binary search tree and quicksort, and one-dimensional packing or the parking problem.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bánkövi, G. (1962). On gaps generated by a random space filling procedure,Publ. Math. Inst. Hungar. Acad. Sci.,7, 395–407.

    MathSciNet  MATH  Google Scholar 

  2. Dvoretzky, A. and Robbins, H. (1964). On the ‘parking problem’,Publ. Math. Inst. Hungar. Acad. Sci.,9, 209–225.

    MathSciNet  MATH  Google Scholar 

  3. Itoh, Y. (1980). On the minimum of gaps generated by one-dimensional random packing,J. Appl. Prob.,17, 134–144.

    Article  MathSciNet  Google Scholar 

  4. Knuth, D. E. (1973).The Art of Computer Programming, Vol. I,Fundamental Algorithms (2nd edition), Vol. IIISorting and Searching, Addison-Wesley.

  5. Mahmoud, H. and Pittel, B. (1984). On the most probable shape of a binary search tree grown from a random permutation,SIAM J. Algebraic Discrete Methods,5, 69–81.

    Article  MathSciNet  Google Scholar 

  6. Pittel, B. (1984). On growing binary trees,J. Math. Analysis and Appl.,103, 461–480.

    Article  MathSciNet  Google Scholar 

  7. Rényi, A. (1958). On a one-dimensional problem concerning space-filling,Publ. Math. Inst. Hungar. Acad. Sci.,3, 109–127.

    MathSciNet  MATH  Google Scholar 

  8. Robson, J. M. (1979). The height of binary search trees,The Australian Computer J.,11(4), 151–153.

    MathSciNet  Google Scholar 

  9. Lootgieter, J. C. (1977).Ann. Inst. Henri Poincaré,13, 385–410.

    MathSciNet  Google Scholar 

  10. van Zwet, W. R. (1978).Ann. Prob.,6, 133–137.

    Article  Google Scholar 

  11. Slud, E. (1978).Zeit. Wahrscheinlichkeitsth.,41, 341–352.

    Article  MathSciNet  Google Scholar 

  12. Pyke, R. (1980).Ann. Prob.,8, 157–163. New results in random sequential bisection were obtained in

    Article  MathSciNet  Google Scholar 

  13. Devroye, L. (1986).J. Ass. Comput. Math.,33, 489–498.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Keio University The Institute of Statistical Mathematics, Keio

    Masaaki Sibuya & Yoshiaki Itoh

Authors
  1. Masaaki Sibuya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoshiaki Itoh
    View author publications

    You can also search for this author in PubMed Google Scholar

About this article

Cite this article

Sibuya, M., Itoh, Y. Random sequential bisection and its associated binary tree. Ann Inst Stat Math 39, 69–84 (1987). https://doi.org/10.1007/BF02491450

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02491450

Key words and phrases

Search

Navigation