computational complexity

, Volume 1, Issue 4, pp 311–329 | Cite as

Randomized vs. deterministic decision tree complexity for read-once Boolean functions

  • Rafi Heiman
  • Avi Wigderson


We consider the deterministic and the randomized decision tree complexities for Boolean functions, denotedDC(f) andRC(f), respectively. A major open problem is how smallRC(f) can be with respect toDC(f). It is well known thatRC(f)DC(f)0.5 for every Boolean functionf (called “0.5-exponent”). On the other hand, some Boolean functionf is known to haveRC(f) = Θ(DC(f))0.753...) (or “0.753...-exponent”). It is not known whether there is a Boolean function with exponent smaller than 0.753... Likewise, no lower bound for arbitrary Boolean functions with exponent greater than 0.5 is known.

Our result is a 0.51 lower bound on the exponent for everyread-once function. Read-once means that each input variable appears exactly once in the Boolean formula representing the function. To obtain this result we generalize an existing lower bound technique and combine it with restriction arguments. This result provides a lower bound ofn0.51 on the number of positions that have to be evaluated by any randomized α-β pruning algorithm computing the value of any two-person zero-sum game tree withn final positions.

Key words

Boolean decision trees Randomized complexity Readonce formulae 

Subject classifications



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  1. [1]
    D. Angluin, L. Hellerstein, and M. Karpinski. Learning read once formulae with queries. Technical Report UCB CSD 89-528, U.C. Berkeley, 1989. To appear in J. Assoc. Comput. Mach.Google Scholar
  2. [2]
    R. B. Boppana. Amplification of probabilistic Boolean formulae, volume 5 ofAdvances in Computer Research, editor S. Micali, pages 27–45, JAI press, 1989.Google Scholar
  3. [3]
    P. Hajnal. On the power of randomness in the decision tree model. InProc. 5th Structure in Complexity Theory Conf., pages 66–77, 1990.Google Scholar
  4. [4]
    R. Heiman.Randomized Decision Tree Complexity for Read-Once Boolean Functions. PhD thesis, submitted to The Feinberg Graduate School, The Weizmann Institute of Science, Rehovot 76100, Israel, 1991.Google Scholar
  5. [5]
    R. Heiman, I. Newman, and A. Wigderson. On read-once threshold formulae and their randomized decision tree complexity. InProc. 5th Structure in Complexity Theory Conf., pages 78–87, 1990. To appear in Theoretical Computer Science.Google Scholar
  6. [6]
    V. King. Lower bounds on the complexity of graph properties. InProc. 20th ACM Symp. on Theory of Computing, pages 468–476, 1988.Google Scholar
  7. [7]
    J. Pearl. The solution for the branching factor of the alpha beta pruning algorithm and its optimality.Comm. Assoc. Comput. Math., 25:559–564, 1982.Google Scholar
  8. [8]
    M. Saks and A. Wigderson. Probabilistic Boolean decision trees and the complexity of evaluating game trees. InProc. 27th IEEE Symp. on Foundations of Computer Science, pages 29–38, 1986.Google Scholar
  9. [9]
    M. Snir. Lower bounds for probabilistic linear decision trees.Theoretical Computer Science, 38:69–82, 1985.Google Scholar
  10. [10]
    M. Tarsi. Optimal search on some game trees.J. Assoc. Comput. Mach., 3:69–82, 1983.Google Scholar
  11. [11]
    L. G. Valiant. Short monotone formulae for the majority function.J. Algorithms, 5:363–366, 1984.Google Scholar
  12. [12]
    A. C. Yao. Probabilistic computations: towards a unified measure of complexity. InProc. 18th IEEE Symp. on Foundations of Computer Science, pages 222–227, 1977.Google Scholar
  13. [13]
    A. C. Yao. Lower bounds to randomized algorithms for graph properties. InProc. 28th IEEE Symp. on Foundations of Computer Science, pages 393–400, 1987.Google Scholar

Copyright information

© Birkhäuser Verlag 1991

Authors and Affiliations

  • Rafi Heiman
    • 1
  • Avi Wigderson
    • 2
  1. 1.Bell Communication ResearchMorristown
  2. 2.Department of Computer SciencePrinceton UniversityPrinceton

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