  For any $n$-bit boolean function $f$, we show that the randomized
communication complexity of the composed function $f\circ g^n$, where $g$ is an
index gadget, is characterized by the randomized decision tree complexity of
$f$. In particular, this means that many query complexity separations involving
randomized models (e.g., classical vs. quantum) automatically imply analogous
separations in communication complexity.
