Sequential access in splay trees takes linear time


Sleator and Tarjan have invented a form of self-adjusting binary search tree called thesplay tree. On any sufficiently long access sequence, splay trees are as efficient, to within a constant factor, as both dynamically balanced and static optimum search trees. Sleator and Tarjan have made a much stronger conjecture; namely, that on any sufficiently long access sequence and to within a constant factor, splay trees are as efficient asany form of dynamically updated search tree. Thisdynamic optimality conjecture implies as a special case that accessing the items in a splay tree in sequential order takes linear time, i.e.O(1) time per access. In this paper we prove this special case of the conjecture, generalizing an unpublished result of Wegman. Oursequential access theorem not only supports belief in the dynamic optimality conjecture but provides additional insight into the workings of splay trees. As a corollary of our result, we show that splay trees can be used to simulate output-restricted deques (double-ended queues) in linear time. We pose several open problems related to our result.

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  1. [1]

    K. Culik II andD. Wood, A note on some tree similarity measures,Info. Proc. Lett. 15 (1982), 39–42.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    G. Gordon,System Simulation, Prentice-Hall, Englewood Cliffs, NJ. 1969.

    Google Scholar 

  3. [3]

    D. E. Knuth,The Art of Computer Programming Volume I: Fundamental Algorithms, Second Edition, Addison-Wesley, Reading, MA, 1973.

    Google Scholar 

  4. [4]

    T. H. Naylor, J. L. Balinty, D. S. Burdick, andK. Chu,Computer Simulation Techniques, Wiley, New York, NY, 1966.

    Google Scholar 

  5. [5]

    D. D. Sleator andR. E. Tarjan, Self-adjusting binary trees,Proc. Fifteenth Annual ACM Symp. on Theory of Computing (1983), 235–245.

  6. [6]

    D. D. Sleator andR. E. Tarjan, Self-adjusting binary search trees,J. Assoc. Comput. Mach.,32 (1885), 652–686.

    MathSciNet  Google Scholar 

  7. [7]

    R. E. Tarjan,Data Structures and Network Algorithms, CMBS 44, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983.

    Google Scholar 

  8. [8]

    R. E. Tarjan, Amortized computational complexity,SIAM J. Alg. Disc. Meth., to 6 (1985), 545–568.

    Google Scholar 

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Tarjan, R.E. Sequential access in splay trees takes linear time. Combinatorica 5, 367–378 (1985).

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AMS subject classification (1980)

  • 68 E 05
  • 68 C 25
  • 05 C 05
  • 68 E 10