## 2 Algorithm for Optimal Deployment
### Optimal Deployment Algorithm for the Sensor Nodes  
In general, the performance of the algorithm for optimal deployment of the sensor nodes is evaluated in terms of maximizing the communication coverage rate and maintaining connectivity among the sensor nodes with a given number of nodes in the targeted area. Here, a hybrid method combining the advantages of the virtual force algorithm (VFA) [19; 20] and the particle swarm optimization (PSO) [21; 22] is used as a search algorithm, which is referred to as the VFPSO [23]. The VFA is a self-organizing search algorithm to determine the optimal distances among sensor nodes, which attempts to maximize the communication coverage using a combination of attractive and repulsive forces. As an initial step, the sensors are randomly placed in the targeted area. The distances between one sensor node and its neighboring nodes are compared to a communication threshold. In this work, the communication threshold was defined to be the smaller of the communication radii at two sensor node positions for two-way communication. The communication radius at a certain node position can be extracted from the communication PS map. If the distance is smaller than the threshold, a repulsive force arises between the two sensor nodes. In contrast, if the distance is too far apart, an attractive force arises. This process is performed for all neighboring nodes based on one sensor node, and a weighted sum is calculated to obtain the total force exerted on the node. This is repeated for each node. As the iteration of the algorithm continues, all sensors nodes are positioned, maintaining optimal distances between the sensor nodes. The PSO is a stochastic search algorithm that mimics social behavior of animals moving in flocks in order to find optimal positions. It has been widely used because it is an efficient optimization algorithm for solving dynamic optimization problems having fast convergence and robustness [21; 22]. The PSO uses particles, which correspond to the individual sensor nodes in this study, and the set of all sensor nodes is referred to as the swarm in the PSO algorithm. After being initialized with random positions and velocities, the particles estimate their communication radii at their positions and memorize this information. At each generation, particles evaluate the best positions by comparison with the positions achieved at previous generations. The velocity of each particle is adjusted based on the experiences of the particle and its companions. The position in the next generation is then updated by the sum of the present position and the adjusted velocity [23]. The VFA has outstanding performance in adjusting the distance among the sensor nodes. However, there may be a limit because the strong attractive and repulsive forces among the sensor nodes may hinder the nodes from settling at optimal positions. In contrast, the PSO has the ability to find a global optimal solution. However, it does not have the ability to adjust the distances among the sensor nodes, which hinders the maximization of the communication coverage rate. Here, the VFPSO algorithm is applied to find the optimal positions of the sensor nodes in the UWASN, which is expressed by [23]. \[\overset{\rightarrow}{F_{i}}\ =\ \sum_{j\ =\ 1,\ j\ \
eq\ i}^{k}\ \overset{ \rightarrow}{F_{i,\ j}} \tag{1}\]  
\[\overset{\rightarrow}{F_{i,\ j}}\ =\ \left\{ \begin{array}{c}\left[w_{A}\left(d_{i,\ j} -\ d_{th}\right),\ \alpha_{i,\ j}\right]\ \text{if}\ d_{i,\ j}\ >\ d_{th}\\ 0\ \text{if}\ d_{i,\ j}\ =\ d_{th}\\ \left[w_{R}\left(\frac{1}{d_{i,\ j}}-\ \frac{1}{d_{th}}\right),\ \alpha_{i,\ j}+\ \pi\ \right]\ \text{if}\ d_{i,\ j}\ <\ d_{th}\end{array}