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arXiv:0808.1762

Communication Complexities of XOR functions

Yaoyun Shi, Zhiqiang Zhang

Published Aug 13, 2008• 11 citations
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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 $x, y\in\{0, 1\}^n$, 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 $Θ(n)$ 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 $Θ(r_0+r_1)$, 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 $k\in[r_0, n-r_1)$.

DOI
10.48550/arxiv.0808.1762

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We call F:{0,1}n×{0,1}n→{0,1}F:\{0, 1\}^n\times \{0, 1\}^n\to\{0, 1\}F:{0,1}n×{0,1}n→{0,1} a symmetric XOR function if for a function S:{0,1,...,n}→{0,1}S:\{0, 1, ..., n\}\to\{0, 1\}S:{0,1,...,n}→{0,1}, F(x,y)=S(∣x⊕y∣)F(x, y)=S(|x\oplus y|)F(x,y)=S(∣x⊕y∣), for any x,y∈{0,1}nx, y\in\{0, 1\}^nx,y∈{0,1}n, where ∣x⊕y∣|x\oplus y|∣x⊕y∣ is the Hamming weight of the bit-wise XOR of xxx and yyy. We show that for any such function, (a) the deterministic communication complexity is always Θ(n)Θ(n)Θ(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 Θ(r0+r1)Θ(r_0+r_1)Θ(r0​+r1​), where r0r_0r0​ and r1r_1r1​ are the minimum integers such that r0,r1≤n/2r_0, r_1\leq n/2r0​,r1​≤n/2 and S(k)=S(k+2)S(k)=S(k+2)S(k)=S(k+2) for all k∈[r0,n−r1)k\in[r_0, n-r_1)k∈[r0​,n−r1​).
We call F:{0,1}n×{0,1}n→{0,1}F:\{0, 1\}^n\times \{0, 1\}^n\to\{0, 1\}F:{0,1}n×{0,1}n→{0,1} a symmetric XOR function if for a function S:{0,1,...,n}→{0,1}S:\{0, 1, ..., n\}\to\{0, 1\}S:{0,1,...,n}→{0,1}, F(x,y)=S(∣x⊕y∣)F(x, y)=S(|x\oplus y|)F(x,y)=S(∣x⊕y∣), for any x,y∈{0,1}nx, y\in\{0, 1\}^nx,y∈{0,1}n, where ∣x⊕y∣|x\oplus y|∣x⊕y∣ is the Hamming weight of the bit-wise XOR of xxx and yyy. We show that for any such function, (a) the deterministic communication complexity is always Θ(n)\Theta(n)Θ(n) except for four simple functions that have a constant complexity, and (b) up to a polylog factor, both the error-bounded randomized complexity and quantum communication with entanglement complexity are Θ(r0+r1)\Theta(r_0+r_1)Θ(r0​+r1​), where r0r_0r0​ and r1r_1r1​ are the minimum integers such that r0,r1≤n/2r_0, r_1\leq n/2r0​,r1​≤n/2 and S(k)=S(k+2)S(k)=S(k+2)S(k)=S(k+2) for all k∈[r0,n−r1)k\in[r_0, n-r_1)k∈[r0​,n−r1​).
Boolean function F(x,y)F(x,y)F(x,y) for x,y∈{0,1}nx,y \in \{0,1\}^nx,y∈{0,1}n is an XOR function if F(x,y)=f(x⊕y)F(x,y)=f(x\oplus y)F(x,y)=f(x⊕y) for some function fff on nnn 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 FFF is closely related to parity decision tree complexity of fff. Montanaro and Osbourne (2009) observed that one-sided communication complexity Dcc→(F)D_{cc}^{\rightarrow}(F)Dcc→​(F) of FFF is exactly equal to nonadaptive parity decision tree complexity NADT⊕(f)NADT^{\oplus}(f)NADT⊕(f) of fff. Hatami et al. (2018) showed that unrestricted communication complexity of FFF is polynomially related to parity decision tree complexity of fff. 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 fff. On the one hand, if Dcc→(F)=tD_{cc}^{\rightarrow}(F)=tDcc→​(F)=t and fff is undefined on at most O(2n−tn−t)O(\frac{2^{n-t}}{\sqrt{n-t}})O(n−t​2n−t​), then NADT⊕(f)=tNADT^{\oplus}(f)=tNADT⊕(f)=t. On the other hand, for a wide range of values of Dcc→(F)D_{cc}^{\rightarrow}(F)Dcc→​(F) and NADT⊕(f)NADT^{\oplus}(f)NADT⊕(f) (from constant to n−2n-2n−2) we provide partial functions for which Dcc→(F)<NADT⊕(f)D_{cc}^{\rightarrow}(F) < NADT^{\oplus}(f)Dcc→​(F)<NADT⊕(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.
Boolean function F(x,y) for x,y ∈ {0,1}ⁿ is an XOR function if F(x,y) = f(x⊕ y) for some function f on n input bits, where ⊕ is a bit-wise XOR. XOR functions are relevant in communication complexity, partially for allowing the 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-way communication complexity D_{cc}^{→}(F) of F is exactly equal to non-adaptive parity decision tree complexity NADT^{⊕}(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 study of a similar connection for partial functions. We show that in the case of one-way communication complexity whether these measures are equal, depends on the number of undefined inputs of f. More precisely, if D_{cc}^{→}(F) = t and f is undefined on at most O((2^{n-t})/(√{n-t})) inputs, then NADT^{⊕}(f) = t. We also provide stronger bounds in extreme cases of small and large complexity. We show that the restriction on the number of undefined inputs in these results is unavoidable. That is, for a wide range of values of D_{cc}^{→}(F) and NADT^{⊕}(f) (from constant to n-2) we provide partial functions (with more than Ω((2^{n-t})/(√{n-t})) undefined inputs, where t = D_{cc}^{→}) for which D_{cc}^{→}(F) < NADT^{⊕}(f). In particular, we provide a function with an exponential gap between the two measures. Our separation results translate to the case of two-way 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 the Boolean Fourier analysis and thus, 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.
We study the two-party communication complexity of functions with large outputs, and show that the communication complexity can greatly vary depending on what output model is considered. We study a variety of output models, ranging from the open model, in which an external observer can compute the outcome, to the XOR model, in which the outcome of the protocol should be the bitwise XOR of the players' local outputs. This model is inspired by XOR games, which are widely studied two-player quantum games. We focus on the question of error-reduction in these new output models. For functions of output size k, applying standard error reduction techniques in the XOR model would introduce an additional cost linear in k. We show that no dependency on k is necessary. Similarly, standard randomness removal techniques, incur a multiplicative cost of 2k2^k2k in the XOR model. We show how to reduce this factor to O(k). In addition, we prove analogous error reduction and randomness removal results in the other models, separate all models from each other, and show that some natural problems, including Set Intersection and Find the First Difference, separate the models when the Hamming weights of their inputs is bounded. Finally, we show how to use the rank lower bound technique for our weak output models.