## 4 Proposed Routing Protocol
### Relay Node Selection and Routing Update  
In the proposed scheme, each node dynamically selects the next-hop relay based on its current neighbor and routing tables upon receiving a data packet, as illustrated in Figure 3. This on-demand routing avoids pre-defined paths and adapts to changes in topology and energy status. By combining local and global state updates, the proposed protocol ensures robust data delivery in dynamic underwater environments. A probability-based relay-selection mechanism is adopted to enhance transmission reliability and energy efficiency. It considers multiple factors, including residual energy, neighbor density, average residual energy of neighbors, and Link Energy Stability Factor. This approach balances the network lifetime, delay, and energy consumption, selecting the next hop based on the forwarding probability derived from these metrics. For instance, as shown in Figure 4, node N1 may choose N3 over N2 based on energy metrics alone; however, if N3 has no neighbors at a shallower depth, a routing void occurs. Inspired by the consideration of multi-hop link characteristics [28] and the energy-aware void-avoidance strategy proposed in [29], we introduce a Link Energy Stability Factor (LESF) that integrates both the energy status of neighbor nodes and the historical link quality, aiming to enhance routing stability and energy balancing. It is defined as:  
\[f_{l,j}=\gamma\frac{\sum\limits_{j\in N_{i}}E_{j}}{|N_{i}|\cdot E_{max}}+(1- \gamma)f_{l,j} \tag{10}\]  
where \(\gamma\) adjusts the balance between current energy and historical stability, and \(N_{i}\) is the neighbor set of node \(i\), \(E_{j}\) is the residual energy of neighbor node, \(f_{l,j}\) is the stability factor of neighbor. This mechanism enables routing decisions to dynamically sense the health of the surrounding nodes, avoid excessive load concentration, and significantly reduce communication voids caused by premature node failures. The overall relay-selection utility is  
\[f_{l}=\alpha E_{ab}+\beta E_{re}+(1-\alpha-\beta)f_{l,j} \tag{11}\]  
Figure 4: Avoiding local optimal. Figure 3: Relay node selection process. The first term, \(\alpha E_{ab}\), represents the energy efficiency of transmission from node A to node B, as defined in Equation (9), with \(\alpha\) controlling the weight of communication quality. The second term, \(\beta E_{re}\), represents the residual energy of node B, with \(\beta\) determining the importance of energy awareness. The third term reflects the energy status of the neighbors by normalizing their residual energy relative to the network's maximum energy, thereby assessing link stability. Parameters \(\alpha\), \(\beta\) control trade-offs among link quality, energy, and stability. The parameters are dynamically adjusted based on each node's local information, including the energy status and neighborhood topology, to adapt to changing network conditions. Upon receiving a data or control packet, the receiving node first measures the received signal strength and extracts the embedded information. If the sender exists in the neighbor table, the node updates its information; otherwise, it evaluates based on depth whether the sender qualifies as a next-hop candidate. If the criteria are met, the sender is added to the neighbor table; otherwise, it is discarded. In the proposed routing protocol, all packet types, except for void notification packets, can be used to update the routing information. In the proposed routing protocol, all packets (except void notification packets) can trigger routing table updates. As shown in Figure 5, during data transmission, node N1 sends a data packet to its neighbor node N2. Upon receiving the packet, N2 uses the forwarding probability factors in its neighbor table to select the optimal next-hop node from the candidate relay nodes N3, N4, and N5--choosing N4. N2 then compares the transmission power required to send the packet to N1 and N4 and forwards the packet to N4 using the more energy-efficient option.