异或函数的奇偶决策树复杂度与四方通信复杂性多项式等价
Parity Decision Tree Complexity and 4-Party Communication Complexity of XOR-functions Are Polynomially Equivalent
摘要：
本文研究了布尔函数$f$的奇偶性决策树复杂度（记为$\mathrm{D}_{\oplus}(f)$）与异或函数$F(x_1,\ldots, x_k)= f(x_1\oplus\cdots\oplus x_k)$的$k$（记为$\mathrm{CC}^{(k)}(F)$）的多方通信复杂性之间的关系.已知$\mathrm{CC}^{(k)}(F)\leq k\cdot\mathrm{D}_{\oplus}(f)$，因为参与者可以模拟计算$f$的奇偶性决策树。在本说明中，我们表明，\[\mathrm{D}_{\oplus}(f)\leq O\big(\mathrm{CC}^{(4)}(F)^5\big).\]我们的主要工具是最近的结果添加剂组合由于桑德斯。由于$\mathrm{CC}^{(k)}(F)$不随$k$的增长而减少，因此只要$k\geq 4$，$f$的奇偶性决策树复杂度和对应的$k$ -自变量XOR函数的通信复杂性就多项式等价。备注：在这篇论文的第一个版本完成后，我们发现哈塔米和洛维特几年前就已经发现了同样的结果，没有写下来。
英文摘要：
In this note, we study the relation between the parity decision tree complexity of a boolean function $f$, denoted by $\mathrm{D}_{\oplus}(f)$, and the $k$-party number-in-hand multiparty communication complexity of the XOR functions $F(x_1,\ldots, x_k)= f(x_1\oplus\cdots\oplus x_k)$, denoted by $\mathrm{CC}^{(k)}(F)$. It is known that $\mathrm{CC}^{(k)}(F)\leq k\cdot\mathrm{D}_{\oplus}(f)$ because the players can simulate the parity decision tree that computes $f$. In this note, we show that \[\mathrm{D}_{\oplus}(f)\leq O\big(\mathrm{CC}^{(4)}(F)^5\big).\] Our main tool is a recent result from additive combinatorics due to Sanders. As $\mathrm{CC}^{(k)}(F)$ is non-decreasing as $k$ grows, the parity decision tree complexity of $f$ and the communication complexity of the corresponding $k$-argument XOR functions are polynomially equivalent whenever $k\geq 4$. Remark: After the first version of this paper was finished, we discovered that Hatami and Lovett had already discovered the same result a few years ago, without writing it up.
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