计算布尔阈值函数的最优传输顺序
Optimal ordering of transmissions for computing Boolean threhold functions
摘要：
我们解决了一个顺序决策问题，出现在分布式数据的对称布尔函数的计算。我们考虑一个并置网络，其中每个节点的传输可以被每个其他节点听到。每个节点都有一个布尔测量，我们希望计算这些测量的给定布尔函数。我们假设测量是独立的，伯努利分布。因此，最优计算的问题变成了最优排序节点的传输的问题，以便最小化总的期望比特数。我们解决了布尔阈值函数类的排序问题。最优排序是动态的，即，它可能潜在地取决于先前发送的比特的值。此外，它只取决于边际概率的排序，而不是它们的确切值。这为最优策略提供了一个优雅的结构。对于每个节点都有一个测量块的情况下，这个问题是非常困难的，我们推测的最佳策略。
英文摘要：
We address a sequential decision problem that arises in the computation of symmetric Boolean functions of distributed data. We consider a collocated network, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and we wish to compute a given Boolean function of these measurements. We suppose that the measurements are independent and Bernoulli distributed. Thus, the problem of optimal computation becomes the problem of optimally ordering node's transmissions so as to minimize the total expected number of bits. We solve the ordering problem for the class of Boolean threshold functions. The optimal ordering is dynamic, i.e., it could potentially depend on the values of previously transmitted bits. Further, it depends only on the ordering of the marginal probabilites, but not on their exact values. This provides an elegant structure for the optimal strategy. For the case where each node has a block of measurements, the problem is significantly harder, and we conjecture the optimal strategy.
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