One-way communication complexity and non-adaptive decision trees

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 $f \circ IP$ equals $\Omega(n(b-1))$. 2) If $f$ is a partial Boolean function, the deterministic one-way communication complexity of $f \circ IP$ is at least $\Omega(b \cdot D_{dt}^{\rightarrow}(f))$, where $D_{dt}^{\rightarrow}(f)$ denotes the non-adaptive decision tree complexity of $f$. For our quantum lower bound, we show a lower bound on the VC-dimension of $f \circ IP$, 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 $f \circ XOR_2$ 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 $f \circ AND_2$. 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 $f \circ AND_2$. In view of the first bullet, 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 $f \circ AND_2$. The proof of the first bullet above uses the well-studied Odd-Max-Bit function.