One-way communication complexity and non-adaptive decision trees
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We study the relationship between various one-way communication complexity measures of a composed function with the analogous decision tree complexity of the outer function. We consider two gadgets: the AND function on 2 inputs, and the Inner Product on a constant number of inputs. Let IP denote Inner Product on 2b bits. 1) If f is a total Boolean function that depends on all of its inputs, the bounded-error one-way quantum communication complexity of f ∘ IP equals Ω(n(b-1)). 2) If f is a partial Boolean function, the deterministic one-way communication complexity of f ∘ IP is at least Ω(b · D_dt^→(f)), where D_dt^→(f) denotes the non-adaptive decision tree complexity of f. For our quantum lower bound, we show a lower bound on the VC-dimension of f ∘ IP, and then appeal to a result of Klauck [STOC'00]. Our deterministic lower bound relies on a combinatorial result due to Frankl and Tokushige [Comb.'99]. It is known due to a result of Montanaro and Osborne [arXiv'09] that the deterministic one-way communication complexity of f ∘ XOR_2 equals the non-adaptive parity decision tree complexity of f. In contrast, we show the following with the gadget AND_2. 1) There exists a function for which even the randomized non-adaptive AND decision tree complexity of f is exponentially large in the deterministic one-way communication complexity of f ∘ AND_2. 2) For symmetric functions f, the non-adaptive AND decision tree complexity of f is at most quadratic in the (even two-way) communication complexity of f ∘ AND_2. In view of the first point, a lower bound on non-adaptive AND decision tree complexity of f does not lift to a lower bound on one-way communication complexity of f ∘ AND_2. The proof of the first point above uses the well-studied Odd-Max-Bit function.
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