有向图和无向图中的最优函数计算
Optimal Function Computation in Directed and Undirected Graphs
摘要：
我们认为在传感器网络中的信息聚合的问题，其中一个感兴趣的是在计算的传感器测量的函数。我们允许块处理和研究在有向图和无向图中的网络函数计算。我们研究功能的结构如何影响编码策略，以及交互信息交换的效果。我们首先考虑传感器节点上的有向图G =（V，E），其中目标是确定在收集器节点处实现函数计算的每个边缘上的最佳编码器。我们的目标是表征R^{ |E| }中的速率区域，即，存在具有给定速率的可行编码器的点的集合，该编码器对于渐进大的块长度实现零误差计算。我们确定有向树的解决方案，指定每个边缘的最佳编码器和解码器。对于一般的有向无环图，我们提供了一个外边界的速率区域，找到消歧的要求，为每个切割，并描述的例子，这个外边界是紧的。接下来，我们解决的情况下，节点连接在一个无向树网络，每个节点希望计算一个给定的对称布尔函数的传感器数据。无向边允许交互式计算，因此，我们研究的聚合和通信策略的相互作用的效果。我们专注于总和阈值函数，并确定最坏情况下的最小总位数交换在每个边缘。最佳策略涉及递归的网络内聚合，这让人想起消息传递。在一般图的情况下，我们提出了一个割集下界，和一个可实现的计划的基础上沿树聚集。对于完全图，我们证明了该方案的复杂性不超过最优方案的两倍。
英文摘要：
We consider the problem of information aggregation in sensor networks, where one is interested in computing a function of the sensor measurements. We allow for block processing and study in-network function computation in directed graphs and undirected graphs. We study how the structure of the function affects the encoding strategies, and the effect of interactive information exchange. We begin by considering a directed graph G = (V, E) on the sensor nodes, where the goal is to determine the optimal encoders on each edge which achieve function computation at the collector node. Our goal is to characterize the rate region in R^{|E|}, i.e., the set of points for which there exist feasible encoders with given rates which achieve zero-error computation for asymptotically large block length. We determine the solution for directed trees, specifying the optimal encoder and decoder for each edge. For general directed acyclic graphs, we provide an outer bound on the rate region by finding the disambiguation requirements for each cut, and describe examples where this outer bound is tight. Next, we address the scenario where nodes are connected in an undirected tree network, and every node wishes to compute a given symmetric Boolean function of the sensor data. Undirected edges permit interactive computation, and we therefore study the effect of interaction on the aggregation and communication strategies. We focus on sum-threshold functions, and determine the minimum worst-case total number of bits to be exchanged on each edge. The optimal strategy involves recursive in-network aggregation which is reminiscent of message passing. In the case of general graphs, we present a cutset lower bound, and an achievable scheme based on aggregation along trees. For complete graphs, we prove that the complexity of this scheme is no more than twice that of the optimal scheme.
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