Closed-Form and Constant-Time New-Source Selection for Fault-Tolerant Broadcasting in Dense Eisenstein–Jacobi Networks

Fault-tolerant broadcasting in dense Eisenstein–Jacobi networks requires an efficient recovery mechanism when faulty nodes interrupt the original broadcast structure. A recently published re-rootingbased broadcasting method for dense Eisenstein–Jacobi networks proves that, for any two faulty nodes, recovery can be performed by selecting a new source that is at maximum graph distance from both faults. However, the recovery step still benefits from a direct method that selects such a source without scanning the network or testing all boundary candidates. This paper develops a self-contained closed-form and constant-time new-source counting and selection method for dense Eisenstein–Jacobi networks. The two-fault problem is translated to an equivalent boundary-intersection problem involving the origin and a difference node. The distance-t boundary, where t is the network diameter, is partitioned into six directed sides of the Eisenstein–Jacobi hexagon. Because the network is a quotient network, the intersection equations must be solved modulo the defining Eisenstein–Jacobi lattice. Therefore, the proposed algorithms evaluate seven possible quotient-lattice shifts together with the 6×6 side pairs, giving at most 7·6·6 = 252 algebraic systems. For faults 0 and A, the first algorithm counts all valid new sources exactly. For two arbitrary faults, the second algorithm selects one valid new source by solving translated side-pair systems, verifying the candidate, and shifting it back. Each system is either a nonparallel two-by-two linear system with at most one candidate, or a parallel system whose feasible candidates form an integer interval. Since the number of systems is fixed, both algorithms run in O(1) time under the fixed-word arithmetic model. Computational validation over 500,000 sampled fault pairs and 40,000 re-rooting trials confirms that the direct selector always returns a valid new source and that the recovered broadcast reaches all non-faulty nodes in the tested settings.