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
Article

Abstract

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

68C25 

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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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