## 4 Problem and Algorithm Description
### Algorithm Description
#### 4.2.2 PSA Process Description
In time \(T\), if \(s_{j}\) receives \(M_{c}\), then \(s_{j}\) becomes a common node and broadcasts \(M_{nc}\). If \(s_{j}\) receives \(M_{nc}\), then \(s_{j}\) ignores the distance a node sent and compares its distance to the sink node with that of the nodes, except the node sending \(M_{nc}\). \(s_{j}\) also becomes the cluster head node when \(s_{j}\) waits for time \(T\). The common node then joins into the cluster, which is the closest to it, and the nodes in current layer update the broadcast radius \(R_{b}(s_{j})\), \[R_{b}(s_{i})=\min(width(level(s_{i}))+width(level(s_{i})-1),R_{c}^{\max})\] (18)
4. Each node \(s_{j}\) in the current layer broadcasts the information (\(level(s_{j})\), cluster head node or not). For the node receiving the information, the node whose layer number is greater than \(level(s_{j})\) proceeds to Step 2 and clusters with the residual nodes. The node whose layer number is less than \(level(s_{j})\), relays the information (self is cluster head node or not) to the cluster head node and ignores the information the common nodes send. The cluster head node in the current layer selects the closest cluster head node in the last layer as the next-hop node. If it does not receive the information of the cluster head node in the last layer, then it selects the closest common node in the last layer as the next-hop node. Figure 2: Solution coding diagram. After a certain time, if the node has the initial layer number, then it broadcasts its maximum communication radius and enters the closest cluster. If the node has the initial layer number and only has neighbor nodes with the initial layer number in its maximum communication radius, then this node moves a distance in the direction of the sink node one time and subsequently finds its next-hop node. Distance (signed as \(d_{sink}\) in following paper) here refers to the average distance between the node and its closest node in the direction of the sink node minus the maximum communication radius. After Stage (2) is run, the network distribution is generally shown in Figure 3. The number of the cluster in the region near the sink node is greater than that in the region far from the sink node. The flowchart of network layering and clustering is presented in Figure 4.  
(3) Optimizing the Position of the Cluster-in Nodes  
After Stage (2), each cluster node (i.e., \(s_{k}\)) knows its own cluster-in node set \(S_{c}(s_{k})\) and their corresponding original positions \(P_{o}(s_{k})\). When the cluster head node, \(s_{k}\), receives the command \"calculating deployment location\", it calculates the deployment position of its cluster-in nodes. First, \(s_{k}\) updates the set of nodes whose deployment positions have been calculated in the network \(S_{ND}\) and the set of nodes whose deployment positions have been calculated in its cluster \(S_{CD}\) (the element in \(S_{CD}\) is only the cluster head node when the cluster head node begins to optimize the positions of its cluster-in nodes). Subsequently, \(s_{k}\) calculates the deployment positions of its cluster-in nodes by using PSOA, and the deployment position of each cluster-in node is determined using PSOA once. The detailed procedure follows. (1) Selecting Next Node to be Calculated  
The cluster head node \(s_{k}\) calculates the CRR values of its cluster-in nodes whose deployment positions in the network have not yet been calculated and then selects the node (i.e., \(s_{i}\)) with the minimum CRR as the next node to be calculated. Figure 3: Network Distribution Diagram. 2. Constructing the Fitness Function  
The cluster head node \(s_{k}\) firstly formulizes the deployment energy consumption that nodes move from the initial position to the deployment position and subsequently formulizes the increase in the network coverage.