Communication Complexities of XOR functions

Abstract: We call $$F:{0, 1}^n\times {0, 1}^n\to{0, 1}$$ a symmetric XOR function if for a function $S:{0, 1, ..., n}\to{0, 1}$, $F(x, y)=S(|x\oplus y|)$, for any $x, y\in{0, 1}<sup>n$, where $|x\oplus y|$ is the Hamming weight of the bit-wise XOR of $x$ and $y$. We show that for any such function, (a) the deterministic communication complexity is always $\Theta(n)$ except for four simple functions that have a constant complexity, and (b) up to a polylog factor, the error-bounded randomized and quantum communication complexities are $\Theta(r_0+r_1)$, where $r_0$ and $r_1$ are the minimum integers such that $r_0, r_1\leq n/2$ and $S(k)=S(k+2)$ for all $k\in[r_0, n-r_1)$.

• The paper establishes that the deterministic communication complexity for nontrivial symmetric XOR functions is Θ(n), with only four functions achieving constant complexity.
• It shows that the randomized and quantum communication complexities are, up to polylog factors, Θ(r0 + r1), where r0 and r1 mark symmetry transition thresholds.
• By leveraging Fourier analysis and a matrix rank argument, the work links nonzero Fourier coefficients to communication cost, offering insights for protocol optimization.

Communication Complexities of Symmetric XOR Functions

The paper "Communication Complexities of Symmetric XOR Functions" by Yaoyun Shi and Zhiqiang Zhang addresses the complexities involved in the communication required to evaluate symmetric XOR functions. These are functions of the form $F: \{0, 1\}^n \times \{0, 1\}^n \to \{0, 1\}$ where $F(x, y) = S(|x \oplus y|)$, with $|x \oplus y|$ being the Hamming weight of the bitwise XOR operation on $x$ and $y$.

Key Results and Contributions

1. Deterministic Communication Complexity: The authors determine that for any nontrivial symmetric XOR function, the deterministic communication complexity $D(F)$ is $\Theta(n)$. This result is significant because it refines the understanding of deterministic complexities for a wide sub-class of XOR functions. They show that only four specific functions have constant complexity, thus establishing almost tight bounds for all others.
2. Randomized and Quantum Communication Complexities: For these same functions, the paper establishes that the randomized ($R(F)$) and quantum ($Q(F)$) communication complexities are, up to polylogarithmic factors, $\Theta(r_0 + r_1)$. Here, $r_0$ and $r_1$ are minimal integers such that $S(k) = S(k+2)$ for all $k \in [r_0, n - r_1)$. The bounds significantly enhance the understanding of how randomized and quantum communication complexities behave for symmetric XOR functions.
3. Technical Contributions:
   • Fourier Analysis: The paper leverages Fourier analysis of Boolean functions to prove lower bounds on deterministic complexity, particularly noting the number of non-zero Fourier coefficients that play a crucial role.
   • Connections to Rank: It demonstrates that for XOR functions, $rank(M_F) = \|\tilde{f}\|_0$, where $\|\tilde{f}\|_0$ is the number of non-zero Fourier coefficients of $f$, thus relating communication complexity with matrix ranks effectively.

Theoretical and Practical Implications

The theoretical implications are profound given the tight characterization of communication complexities for symmetric XOR functions. This contributes towards a greater understanding of fundamental limits in communication models. Practically, the results offer a basis for optimizing communication protocols in distributed computing where XOR operations are prevalent, such as in error-detecting and correcting codes, and secure multi-party computation.

Such concrete complexity bounds help in designing efficient communication protocols. For instance, determining the communication overhead in networked systems that use XOR-based operations could directly benefit from the deterministic and randomized complexity results presented in the paper.

Future Directions

The research opens several avenues for future investigations:

1. Asymmetric XOR Functions: Extending these results to asymmetric XOR functions where $F(x, y) = f(x \oplus y)$ without the symmetry constraint.
2. Unbounded-Error Communication Complexity: Investigating the unbounded-error communication complexity for XOR functions remains an open problem. A conjecture is that this complexity might be closely related to the number of transitions in $S$ when advancing by 2.
3. One-Way Communication Complexity: The gap between one-way and two-way communication complexities, particularly for the one-way scenarios realized in practical applications, warrants further scrutiny.

Conclusion

Overall, the paper makes significant strides in understanding the communication complexities associated with symmetric XOR functions. By establishing tight bounds and leveraging sophisticated analytical methods, Shi and Zhang provide a comprehensive view that has both theoretical and practical implications, positioning their work as a vital reference for researchers and practitioners working with communication protocols.