Submodularity and Optimality of Fusion Rules 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 the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman’s equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a stringsubmodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor of the optimal strategy in terms of reduction in the total error probability.

💡 Research Summary

The paper tackles the distributed detection problem in a perfectly balanced binary relay tree, where each leaf node is a binary sensor and every internal node fuses the two binary messages received from its children into a single binary message that is forwarded upward. The root node acts as the final decision maker (fusion center). A key modeling assumption is that all fusion nodes situated at the same depth employ an identical fusion rule; consequently, a sequence of fusion rules—one for each level of the tree—constitutes a “fusion strategy”.

The authors formulate the objective as maximizing the total reduction in error probability between the leaf sensors (with error probability P₀) and the root (with error probability P_L). The reduction Δ = P₀ − P_L depends on the chosen fusion strategy. To find the optimal strategy, they cast the problem as a deterministic dynamic program (DP). The DP state at level ℓ is the prefix of the rule string selected up to level ℓ‑1, the action is the choice of a rule f_ℓ from a finite set ℱ, and the transition updates the error probability according to a deterministic function g that captures how a specific rule combines the error probabilities of its two inputs. The Bellman recursion is

V_ℓ(s_{0:ℓ‑1}) = max_{f_ℓ∈ℱ}