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

Fault-tolerant broadcasting in dense Gaussian networks can be recovered by re-rooting the broadcast at a new source that is at maximum graph distance from the faulty nodes. This paper extends the published re-rooting framework by replacing its boundary-search source-selection step with a quotientlattice- aware algebraic construction. The first contribution is a constant-time counting method for valid new sources. The counting problem is formulated as an intersection of two diameter-k boundary sets in the Gaussian quotient. A compact piecewise expression is retained for the local, unshifted boundary intersection,and the exact quotient-network count is obtained by a fixed union of side-pair intervals over the nine relevant quotient-lattice copies. This gives a closed-form constant-size counting procedure without scanning either the network or the boundary. The second contribution is a shifted direct selector for two arbitrary faulty nodes. Given faulty nodes A and B, the problem is translated to C = modGk (B−A), and the selector finds a point P satisfying d(P, 0) = d(P, C) = k. For each of nine quotient-lattice shifts, the selector checks sixteen signed linear systems. Nonparallel systems are solved using Cramer’s rule, while parallel systems are handled by interval-endpoint selection. Thus, at most 9×16 = 144 shifted sign cases are evaluated, giving O(1) source selection under the standard word-RAM model. Computational validation reports zero count mismatches over 26,623 tested nodes, 500,000 valid outputs over 500,000 sampled fault pairs, and 40,000 successful re-rooted broadcast trials. Runtime results show that the shifted selector has a small fixed overhead for small networks but remains nearly stable as k increases, achieving a 5.92× speedup over boundary search at k = 200. These results strengthen the re-rooting approach by making new-source selection algebraic, bounded, and independent of the network size.