One-Way Communication Complexity of Partial XOR Functions

TL;DR

This work investigates the one-way communication complexity of partial XOR functions $F(x,y)=f(x\oplus y)$, establishing precise conditions under which $D_{cc}^{\rightarrow}(F)$ equals $NADT^{\oplus}(f)$ and where substantial gaps occur. It introduces a linear-algebraic framework that substitutes Fourier methods (used for total functions) with coset and subspace analyses on the Boolean cube, connecting one-way protocols to partition and graph structures. A key result shows that if $f$ has few undefined inputs, specifically at most $O\left(\frac{2^{n-t}}{\sqrt{n-t}}\right)$ with $t=D_{cc}^{\rightarrow}(F)$, then $D_{cc}^{\rightarrow}(F)=NADT^{\oplus}(f)$; however, for a wide range of parameters there exist partial functions with $D_{cc}^{\rightarrow}(F)<NADT^{\oplus}(f)$, including exponential gaps. These separations extend to two-sided complexity and refute a straightforward extrapolation of total-function results to partial functions, highlighting the limitations of Fourier-based techniques and underscoring the value of the new linear-algebraic approach and covering-code reductions.

Abstract

Boolean function $F(x,y)$ for $x,y \in \{0,1\}^n$ is an XOR function if $F(x,y)=f(x\oplus y)$ for some function $f$ on $n$ input bits, where $\oplus$ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing Fourier analytic technique. For total XOR functions it is known that deterministic communication complexity of $F$ is closely related to parity decision tree complexity of $f$. Montanaro and Osbourne (2009) observed that one-sided communication complexity $D_{cc}^{\rightarrow}(F)$ of $F$ is exactly equal to nonadaptive parity decision tree complexity $NADT^{\oplus}(f)$ of $f$. Hatami et al. (2018) showed that unrestricted communication complexity of $F$ is polynomially related to parity decision tree complexity of $f$. We initiate the studies of a similar connection for partial functions. We show that in case of one-sided communication complexity whether these measures are equal, depends on the number of undefined inputs of $f$. On the one hand, if $D_{cc}^{\rightarrow}(F)=t$ and $f$ is undefined on at most $O(\frac{2^{n-t}}{\sqrt{n-t}})$, then $NADT^{\oplus}(f)=t$. On the other hand, for a wide range of values of $D_{cc}^{\rightarrow}(F)$ and $NADT^{\oplus}(f)$ (from constant to $n-2$) we provide partial functions for which $D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)$. In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-sided communication complexity as well, in particular showing that the result of Hatami et al. (2018) cannot be generalized to partial functions. Previous results for total functions heavily rely on Boolean Fourier analysis and the technique does not translate to partial functions. For the proofs of our results we build a linear algebraic framework instead. Separation results are proved through the reduction to covering codes.

Figures (1)

• Figure 1: Example of total h-induced graph

Theorems & Definitions (52)

• Definition 1
• Definition 2: Hales order harper
• Theorem 3: Harper's theorem harper
• Lemma 4
• Definition 5
• Theorem 6: CohenHLL97
• Theorem 7: CohenHLL97
• Definition 8
• Lemma 9: CohenHLL97
• Lemma 10