一种解决多方P2P通信复杂性的新方法
A New Approach to Multi-Party Peer-to-Peer Communication Complexity
摘要：
我们介绍了新的模型和新的信息论措施的研究通信复杂性在自然对等，多方，号码在手设置。我们证明了我们的新模型和措施的一些性质，然后，为了验证它们的有效性，我们用它们来证明两个下界。更详细的是$Ω(kn)$关于$k$ -播放器、$n$ -位Disjointness函数的多方对等随机通信复杂性的严格下限。另一个是关于$k$ -播放器、$n$ -位逐位奇偶校验函数的多方对等随机通信复杂性的$Ω(kn)$的紧下界。当${n=Ω(k)}$。不相交性的下限比可以从Braverman等人的结果推断出的下限有所改进。(FOCS 2013），其在协调器模型中被证明并且可以在对等模型中产生$Ω(kn/\log k)$的下限。据我们所知，我们的下限是第一个严格的（非平凡的）下限通信复杂性的自然{ \em点对点}多方设置。除了上述结果的通信复杂性，我们还证明，使用相同的工具，一个$Ω(n)$下界的随机位数所需的（信息理论）私人计算的$k$播放器，$n$位不相交函数。
英文摘要：
We introduce new models and new information theoretic measures for the study of communication complexity in the natural peer-to-peer, multi-party, number-in-hand setting. We prove a number of properties of our new models and measures, and then, in order to exemplify their effectiveness, we use them to prove two lower bounds. The more elaborate one is a tight lower bound of $Ω(kn)$ on the multi-party peer-to-peer randomized communication complexity of the $k$-player, $n$-bit Disjointness function. The other one is a tight lower bound of $Ω(kn)$ on the multi-party peer-to-peer randomized communication complexity of the $k$-player, $n$-bit bitwise parity function. Both lower bounds hold when ${n=Ω(k)}$. The lower bound for Disjointness improves over the lower bound that can be inferred from the result of Braverman et al.~(FOCS 2013), which was proved in the coordinator model and can yield a lower bound of $Ω(kn/\log k)$ in the peer-to-peer model. To the best of our knowledge, our lower bounds are the first tight (non-trivial)lower bounds on communication complexity in the natural {\em peer-to-peer} multi-party setting. In addition to the above results for communication complexity, we also prove, using the same tools, an $Ω(n)$ lower bound on the number of random bits necessary for the (information theoretic) private computation of the $k$-player, $n$-bit Disjointness function .
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