One-way communication complexity and non-adaptive decision trees

TL;DR

This paper explores the relationships between one-way communication complexity and decision tree complexity for composed functions, revealing limitations of existing lower bound techniques and establishing new bounds for specific function classes.

Contribution

It demonstrates that lower bounds on non-adaptive decision tree complexity do not necessarily imply similar bounds on one-way communication complexity for AND-composed functions, and provides new upper bounds based on matrix rank.

Findings

Quantum one-way communication complexity of $f \,\circ\, IP$ is $\,\Omega(n(b-1))$ for total functions.

Deterministic one-way communication complexity of $f \,\circ\, IP$ is at least $\,\Omega(b \cdot D_{dt}^{\rightarrow}(f))$ for partial functions.

Rank-based upper bound shows the deterministic one-way communication complexity of $f \,\circ\, AND_2$ is at most proportional to the rank of its communication matrix.

Abstract

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. - 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 $\Omega(n(b-1))$. - 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 $f$. Montanaro and Osborne [arXiv'09] observed 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 $AND_2$. - There exists a function for which even the quantum non-adaptive AND decision tree complexity of $f$ is exponentially large in the deterministic one-way communication complexity of $f \circ AND_2$. - 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 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 $f \circ AND_2$. In our final result we show that for all $f$, the deterministic one-way communication complexity of $F = f \circ AND_2$ is at most $(rank(M_{F}))(1 - \Omega(1))$, where $M_{F}$ denotes the communication matrix of $F$. This shows that the rank upper bound on one-way communication complexity (which can be tight in general) is not tight for AND-composed functions.