, Volume 12, Issue 1, pp 95–124 | Cite as

On the deque conjecture for the splay algorithm

  • Rajamani Sundar


Splay is a simple, efficient algorithm for searching binary search trees, devised by Sleator and Tarjan, that reorganizes the tree after each search by means of rotations. An open conjecture of Sleator and Tarjan states that Splay is, in essence, the fastest algorithm for processing any sequence of search operations on a binary search tree, using only rotations to reorganize the tree. Tarjan proved a basic special case of this conjecture, called theScanning Theorem, and conjectured a more general special case, called theDeque Conjecture. The Deque Conjecture states that Splay requires linear time to process sequences of deque operations on a binary tree. We prove the following results:
  1. 1.

    Almost tight lower and upper bounds on the maximum numbers of occurrences of various types of right rotations in a sequence of right rotations performed on a binary tree. In particular, the lower bound for right 2-turns refutes Sleator's Right Turn Conjecture.

  2. 2.

    A linear times inverse Ackerman upper bound for the Deque Conjecture. This bound is derived using the above upper bounds.

  3. 3.

    Two new proofs of the Scanning Theorem, one, a simple potential-based proof that solves Tarjan's problem of finding a potential-based proof for the theorem, the other, an inductive proof that generalizes the theorem.


AMS subject classification code (1991)

68 P 05 68 R 05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Cole: On the Dynamic Finger Conjecture for Splay Trees. In Proc. 22nd ACM STOC, 1990, 8–17.Google Scholar
  2. [2]
    R. Cole: On the Dynamic Finger Conjecture for Splay Trees. Part II: Finger searching. Courant Institute Technical Report No.472, 1989.Google Scholar
  3. [3]
    R. Cole, B. Mishra, J. Schmidt, andA. Siegel: On the Dynamic Finger Conjecture for Splay Trees. Part I: Splay-sorting logn-block sequences. Courant Institute Technical Report No.471, 1989.Google Scholar
  4. [4]
    K. Culik II, andD. Wood: A note on some tree similarity measures.Info. Proc. Lett. 15 (1982), 39–42.Google Scholar
  5. [5]
    S. Hart, andM. Sharir: Nonlinearity of Davenport-Shinzel sequences and of generalized path compression schemes.Combinatorica 6 (1986), 151–177.Google Scholar
  6. [6]
    J. M. Lucas: Arbitrary splitting in Splay Trees. Rutgers University Tech. Rept. No.TR-234, June 1988.Google Scholar
  7. [7]
    D. D. Sleator, andR. E. Tarjan: Self-adjusting binary search trees.J. ACM 32 (1985), 652–686.Google Scholar
  8. [8]
    D. D. Sleator, R. E. Tarjan, andW. P. Thurston: Rotation distance, Triangulations, and Hyperbolic Geometry.J. Amer. Math. Soc. 1 (1988), 647–681.Google Scholar
  9. [9]
    R. E. Tarjan: Sequential access in Splay Trees takes linear time.Combinatorica 5 (1985), 367–378.Google Scholar
  10. [10]
    R. E. Tarjan: Amortized computational complexity.SIAM J. Appl. Discrete Meth. 6 (1985), 306–318.Google Scholar
  11. [11]
    R. Wilber: Lower bounds for accessing binary search trees with rotations.SIAM J. on Computing 18 (1989), 56–67.Google Scholar

Copyright information

© Akadémiai Kiadó 1992

Authors and Affiliations

  • Rajamani Sundar
    • 1
  1. 1.Computer Science DepartmentPrinceton UniversityPrincetonU S A

Personalised recommendations