Near-optimal distributed detection in balanced binary relay trees

We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes an overall decision. Every other node in the tree is a relay node that fuses binary messages from its two child nodes into a new binary message and sends it to the parent node at the next level. We assume that the relay nodes at the same level use identical fusion rule. The goal is to find a string of fusion rules used at all the levels in the tree that maximizes the reduction in the total error probability between the leaf nodes and the fusion center. We formulate this problem as a deterministic dynamic program and express the optimal strategy in terms of Bellman's equation. Moreover, we use the notion of string-submodularity to show that the reduction in the total error probability is a string-submodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, performs at least within a factor (1 - 1/e) of the optimal strategy in terms of reduction in the total error probability, even if the nodes and links in the trees are subject to random failures.