Parity Decision Tree Complexity is Greater Than Granularity

We prove a new lower bound on the parity decision tree complexity D_⊕(f) of a Boolean function f. Namely, granularity of the Boolean function f is the smallest k such that all Fourier coefficients of f are integer multiples of 1/2^k. We show that D_⊕(f)≥ k+1. This lower bound is an improvement of the known lower bound through the sparsity of f. Using our lower bound we determine the exact parity decision tree complexity of several important Boolean functions including majority, recursive majority and MOD^3 function. For majority the complexity is n - B(n)+1, where B(n) is the number of ones in the binary representation of n. For recursive majority the complexity is n+1/2. For MOD^3 the complexity is n-1 for n divisible by 3 and is n otherwise. Finally, we provide an example of a function for which our lower bound is not tight.


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