在有噪声的通信中计算奇偶校验有多难？
How Hard is Computing Parity with Noisy Communications?
摘要：
我们展示了一个严格的下限$Ω(N \log\log N)$上的传输数量计算的奇偶校验$N$输入位的$N$随机放置的传感器，每个有一个输入位和通信与他人使用本地传输功率附近的连接阈值的噪声通信网络中的恒定错误。这个结果解决了Ying、Srikant和Dullerud（WiOpt 06）留下的下限问题，他们展示了如何使用$O(N \log\log N)$传输计算所有$N$位的总和。Dutta和Radhakrishnan（FOCS，2008年）已经证明，同样的下限适用于许多其他函数，包括多数函数。大多数作品的通信网络的下界考虑大多是全广播模型，而不使用的事实，即在真正的网络中的通信是本地的，由发射机的功率确定。事实上，在全广播网络中计算奇偶校验需要$θ(N)$传输。为了获得我们的下限，我们采用Goyal，Kindler和Saks（FOCS 05）开发的技术，他们通过将问题简化为噪声决策树模型来显示全广播模型的下限。然而，为了捕捉在真正的传感器网络中的传输范围有限，我们适应他们的定义的嘈杂的决策树，并允许每个节点的树访问只有有限的一部分输入。我们的下界是通过利用特殊性质的奇偶校验计算在这样的噪音决策树。
英文摘要：
We show a tight lower bound of $Ω(N \log\log N)$ on the number of transmissions required to compute the parity of $N$ input bits with constant error in a noisy communication network of $N$ randomly placed sensors, each having one input bit and communicating with others using local transmissions with power near the connectivity threshold. This result settles the lower bound question left open by Ying, Srikant and Dullerud (WiOpt 06), who showed how the sum of all the $N$ bits can be computed using $O(N \log\log N)$ transmissions. The same lower bound has been shown to hold for a host of other functions including majority by Dutta and Radhakrishnan (FOCS 2008). Most works on lower bounds for communication networks considered mostly the full broadcast model without using the fact that the communication in real networks is local, determined by the power of the transmitters. In fact, in full broadcast networks computing parity needs $θ(N)$ transmissions. To obtain our lower bound we employ techniques developed by Goyal, Kindler and Saks (FOCS 05), who showed lower bounds in the full broadcast model by reducing the problem to a model of noisy decision trees. However, in order to capture the limited range of transmissions in real sensor networks, we adapt their definition of noisy decision trees and allow each node of the tree access to only a limited part of the input. Our lower bound is obtained by exploiting special properties of parity computations in such noisy decision trees.
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