Dense Gaussian networks provide degree-4 interconnection topologies with small diameter and regular structure, making them suitable for efficient one-to-all broadcasting. However, node failures can disrupt the broadcast process when faulty nodes occupy internal forwarding positions. This paper proposes a lightweight fault-tolerant broadcasting method based on dynamic source relocation, or re-rooting. Instead of constructing redundant spanning trees or backup routing structures, the proposed method selects a new source node so that the faulty nodes are located at graph distance \(k\), the network diameter, from the new source. Consequently, faulty nodes become leaf-level nodes in the broadcast process and are not required to forward the message. For the single-fault case, the new source is selected directly from the graph-distance-\(k\) boundary of the faulty node. For the two-fault case, we prove that for any pair of faulty nodes in \(G(k+(k+1)i)\), there exists a node whose graph distance from both faulty nodes is exactly \(k\). The source-selection procedure requires \(O(k)\) time. Since the original one-to-all broadcast completes in \(k\) parallel steps and the relocation distance is at most \(k\), the proposed method completes in at most \(2k\) steps in the worst case. We also show that the two-fault guarantee does not generally extend to arbitrary three-fault configurations by giving a counterexample in \(G(3+4i)\). Simulation results confirm complete delivery to all non-faulty nodes under the tested one- and two-node failure scenarios, while the baseline broadcast may fail when faulty nodes occur at internal forwarding positions.

翻译：稠密高斯网络提供度数为4的互连拓扑结构，具有小直径和规则特性，使其适用于高效的一对全广播。然而，当故障节点占据内部转发位置时，节点故障会中断广播过程。本文提出一种基于动态源重定位（即重定根）的轻量级容错广播方法。该方法无需构建冗余生成树或备份路由结构，而是选择一个新的源节点，使得故障节点位于与新源节点的图距离 \(k\)（网络直径）处。因此，故障节点在广播过程中成为叶节点，无需转发消息。针对单故障情况，新源节点直接从故障节点的图距离 \(k\) 边界中选择。针对双故障情况，我们证明在 \(G(k+(k+1)i)\) 中任意一对故障节点均存在一个节点，其与这两个故障节点的图距离恰好为 \(k\)。源选择过程需要 \(O(k)\) 时间。由于原始的一对全广播在 \(k\) 个并行步骤中完成，且重定位距离至多为 \(k\)，所提方法在最坏情况下最多需 \(2k\) 步骤完成。通过给出 \(G(3+4i)\) 中的反例，我们还证明双故障保证通常不扩展到任意三故障配置。仿真结果确认在测试的单节点和双节点故障场景下，所有非故障节点均能完全完成投递，而基线广播在故障节点位于内部转发位置时可能失败。