Abstract
Let f : {0, 1}n → {0, 1}. Let μ be a product probability measure on {0, 1}n. For ϵ ≥ 0, we define Dϵ(f), the ϵ-approximate decision tree complexity of f, to be the minimum depth of a decision tree T with μ(T(x) ≠ f(x)) ≤ ϵ. For j = 0 or 1 and for δ ≥ 0, we define Cj,δ(f), the δ-approximate j-certificate complexity of f, to be the minimum certificate complexity of a set A ⊆ Ω with μ(AΔf−1(j)) ≤ ϵ. Note that if μ(x) > 0 for all x then D0(f) = D(f) and Cj,0(f) = Cj(f) are the ordinary decision tree and j-certificate complexities of f, respectively. We extend the well-known result, D(f) ≤ C1(f)C0(f) [Blum and Impagliazzo 1987; Hartmanis and Hemachandra 1991; Tardos 1989], proving that for all ϵ > 0 there exists a δ > 0 and a constant K = K(ϵ, δ) > 0 such that for all n, μ, f, Dϵ(f) ≤ K C1,δ(f)C0,δ (f). We also give a partial answer to a related question on query complexity raised by Tardos [1989]. We prove generalizations of these results to general product probability spaces.
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Index Terms
Approximate Query Complexity
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