Bader Albader's O(1) algorithm enables instant fault recovery in dense networks

Fault-tolerant broadcasting in dense Eisenstein–Jacobi networks traditionally required scanning boundary nodes or testing candidates when two faults disrupted the original broadcast tree. Bader Albader's new paper solves this with a closed-form, constant-time algorithm that counts and selects a valid new source without any runtime search. The approach transforms the two-fault problem into a boundary-intersection problem: treating the network as a quotient structure over a hexagonal lattice, the distance-$t$ boundary is partitioned into six directed sides. By solving intersection equations modulo the defining lattice across all $6\times 6$ side pairs, the algorithm evaluates at most 252 algebraic systems. Each system is either a simple $2\times 2$ linear system yielding at most one candidate or a parallel system producing an integer interval of candidates. The count function enumerates all valid new sources for faults at the origin and a second point, while the selector picks one for arbitrary fault pairs.

The entire method runs in constant time ($O(1)$) under the fixed-word arithmetic model, a critical improvement over prior heuristic or iterative approaches. Computational validation over 500,000 sampled fault pairs and 40,000 re-rooting trials confirmed that the selector always returns a valid new source and that recovered broadcast messages reach all non-faulty nodes. The paper also outlines how the closed-form solution enables predictable, low-latency recovery in dense network topologies used in high-performance computing clusters and sensor networks. By eliminating network scanning, the algorithm reduces recovery overhead to a handful of algebraic operations, making it practical for real-time fault tolerance. The preprint is available on arXiv and Zenodo.