## 3 The Proposed Scheme
### Tree Construction  
Each node will have its own position coordinate, and the coordinate of the cell. Each node also has the position coordinates of other nodes. In the process of constructing the grid structure, each cell selects the sensor node with the highest remaining energy as the cell head, which has an information table to store the relevant information of the cell. After the grid construction is completed, the tree construction begins. The BS is used as the root node and the edge with the least energy consumption is selected to join the tree, until all sensors are added. In the process of building the tree structure, it is important to check whether the cell head already exists in the tree. If the cell head already exists in the tree, select the edges with the least energy consumption in the remaining cell heads to join the tree. The energy model in this work uses the First Order Radio Model [23]. In the transmitting unit of the transmitting node, the transmitted electronic unit is used for processing and the signal is amplified by the amplifier. The receiving unit at the receiving node will send the data to the receiving unit of the node. \(E_{elc}\) is the energy consumption of the sensor used in the transmitter or receiver circuit. \(E_{amp}\) is the energy consumption of the amplifier used to send the data. The transmitted data will have \(d^{2}\) energy loss after a distance, \(d\). Therefore, the wireless transmission energy consumption of the transmitting node to transmit \(k\)-bit packets over distance \(d\) is as follows:  
\[E_{TX}\left(k,d\right)=E_{dec}\times k+E_{amp}\times k\times d^{2} \tag{1}\]  
The energy consumption of wireless transmission received by the receiving node is as follows:  
\[E_{RX}(k)=E_{elec}\times k \tag{2}\]  
The energy consumption of the cell head is determined by the following two equations:  
\[C_{ij}(k)=2\times E_{elec}\times k+E_{amp}\times k\times{d_{ij}}^{2} \tag{3}\]  
\[C_{iB}(k)=E_{dec}\times k+E_{amp}\times k\times{d_{iB}}^{2} \tag{4}\]  
where \(Cij\) represents the consumed energy for sending a \(k\)-bit packet from the \(i\)-th cell head to the \(j\)-th cell head on the current tree structure. \(d_{ij}\) represents the distance between the \(i\)-th cell head and the \(j\)-th cell head on the current tree structure. \(C_{iB}\) represents the energy consumed by sending a \(k\)-bit packet from the \(i\)-th cell head to the BS. \(d_{iB}\) represents the distance between the \(i\)-th cell head and the BS. The length of the transmission distance will affect the energy consumption of the sensors. Therefore, a grid structure is built in the sensing area, so that the sensors in the cell only need to send data to the cell head, thus reducing the overall energy consumption. Assuming that cell head \(A\) is the head with the least energy consumption for transmitting data to the BS, the BS broadcasts the tree construction request packet to cell head \(A\). When cell head \(A\) receives the tree construction request packet (sent by the BS on the tree structure), cell head \(A\) will judge whether it has already been added to the tree. If it does not exist in the tree, cell head \(A\) will send the request packet to a node on the tree structure to add cell head \(A\) to the tree. For the rest of the cell heads not added to the tree structure, the cell heads with the least energy consumption edge are repeatedly selected to be added to the tree until the structure is completed. The tree structure of GB-PEDAP is shown in Figure 1.  
Figure 1: Tree structure of GB-PEDAP. The proposed GB-PEDAP mainly uses the tree construction algorithm to continuously select cell heads until all cell heads have been added to the tree. The GB-PEDAP tree construction algorithm is shown in Algorithm 1.