Bader Albader's EJ broadcast network repair achieves 100% recovery with minimal edges

Bader Albader's new paper tackles a foundational problem in dense Eisenstein-Jacobi (EJ) broadcast networks: how to quickly repair one-to-all broadcasting when nodes or links fail. The core innovation is the selected triple $(r, \theta, \Kcomp_{r,\theta})$—a chosen root, a coordinate-reduction orientation, and the healthy subgraph it induces. This triple becomes the unit of analysis, leading to a necessary and sufficient condition: hybrid repair works if and only if the healthy graph remains connected. When it does, a spanning tree of the component graph maps to exactly $c-1$ crossing edges, the theoretical minimum.

The method provides deterministic guarantees that real-world engineers can rely on: one or two faulty nodes are always placed on the distance-$t$ boundary by re-rooting; a single failed link is either avoided or fixed by exactly one crossing edge; and the repaired depth stays within $D_{r,\theta} \le 2t+1$ when using shallowest-layer entry selection. Albader backs these claims with a massive 260,000-trial validation campaign across five network sizes up to $N=120,601$ nodes, achieving 100% recovery and substantial repair-edge savings compared to fixed-source repair. Global-BFS, near-miss, and cap-sensitivity audits further clarify the tradeoffs between reachability, forwarding-state changes, and ranked root selection.