## 4 Proposed Protocol
### Obtaining the Optimal Route  
The feasible routing set \(\mathcal{R}_{i}\) including many valid routes provides a solution space in which we obtain the optimal route from the transmitter \(v_{i}\) to the BS \(v_{0}\) according to the node energy property. Therefore, the procedure of obtaining the optimal route can be formulated as a combinatorial optimization problem. As we know, the bucket effect refers to the law that the shortest piece of wood constrains the amount of water in a bucket; the same principle also applies to obtain the optimal route. Hence, the sensor node with the minimal residual energy along a valid route limits the transmission capacity of the route. In order to achieve energy usage evenly, the maximum-minimum criterion is used to obtain the especial sensor node defined by \(v_{(p^{*},j^{*})}^{(i)}\), which has the minimum residual energy for the sensor nodes along the optimal route \(r_{p^{*}}^{(i)}\), and whose routing transmission capacity is maximized in the feasible routing set \(\mathcal{R}_{i}\). At the transmission round \(t\), the residual energy of the sensor nodes in the feasible routing set \(\mathcal{R}_{i}\) can build an energy matrix named \(\Psi_{i}(t)\) that is shown as  
\[\Psi_{i}(t)= \begin{bmatrix}\varepsilon_{(1,k(i))}^{(i)}(t)&\cdots&\varepsilon_ {(1,j)}^{(i)}(t)&\cdots&\varepsilon_{(1,1)}^{(i)}(t)\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \varepsilon_{(p,k(i))}^{(i)}(t)&\cdots&\varepsilon_{(p,j)}^{(i)}(t)&\cdots& \varepsilon_{(p,1)}^{(i)}(t)\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \varepsilon_{(N_{i},k(i))}^{(i)}(t)&\cdots&\varepsilon_{(N_{i},j)}^{(i)}(t)& \cdots&\varepsilon_{(N_{i},1)}^{(i)}(t)\end{bmatrix}  \tag{10}\]  
where \(\varepsilon_{(p,j)}^{(i)}(t)\) represents the residual energy of the sensor node \(v_{(p,j)}^{(i)}\) at the transmission round \(t\). In order to obtain the especial sensor node \(v_{(p^{*},j^{*})}^{(i)}\), we calculate the transmission capacity for each route based on the node residual energy, and then obtain the route with the maximal transmission capacity from the feasible routing set \(\mathcal{R}_{i}\). The procedures of solving for the especial sensor node \(v_{(p^{*},j^{*})}^{(i)}\) and the optimal route \(r_{p^{*}}^{(i)}\) for the transmitter \(v_{i}\) are modeled as follows:  
\[ \begin{array}{l}(p^{*},j^{*})=\text{ arg }\max_{p}\left\{\min_{j}\ \varepsilon_{(p,j)}^{(i)} \left(t\right)\right\}=\text{ arg }\max_{p}\ \varepsilon_{(p,j^{*})}^{(i)}\left(t\right);\\ \\ p\in\left\{1,2,\cdots,N_{i}\right\},\quad j\in\left\{1,2,\cdots,k\left(i\right) \right\}\end{array}  \tag{11}\]The value of the subscript \(p^{*}\) can locate the optimal route \(r_{p^{*}}^{(i)}\), while values of the subscripts \(p^{*}\) and \(j^{*}\) can jointly locate the especial sensor node \(v_{(p^{*},j^{*})}^{(i)}\). Thus the optimal route for the transmitter \(v_{i}\) is given by  
\[r_{p^{*}}^{(i)}=\left[v_{(p^{*},k(i))}^{(i)}\rightarrow\cdots\to v _{(p^{*},j^{*})}^{(i)}\rightarrow\cdots\to v_{(p^{*},2)}^{(i)}\to v _{(p^{*},1)}^{(i)}\right] \tag{12}\]  
Finally, at the transmission round \(t\), the transmitter \(v_{i}\) sends information gathered to the BS \(v_{0}\) in line with \(r_{p^{*}}^{(i)}\), and the sensor nodes along the optimal route \(r_{p^{*}}^{(i)}\) will decrease in energy based on Equations (4) and (5). Then the remaining energy of the sensor nodes along the optimal route \(r_{p^{*}}^{(i)}\) is given by  
\[f\left(\Psi\left(t\right),r_{p^{*}}^{(i)}\right)=\left[ \begin{array}{c} \varepsilon_{(p^{*},k(i))}^{(i)}(t)-0-E_{T}\left(l,d\left(v_{(p^{*},k(i))}^{(i) },v_{(p^{*},k(i)-1)}^{(i)}\right)\right)\\ \vdots\\ \varepsilon_{(p^{*},j^{*})}^{(i)}(t)-E_{R}\left(l\right)-E_{T}\left(l,d\left( v_{(p^{*},j^{*})}^{(i)},v_{(p^{*},j^{*}-1)}^{(i)}\right)\right)\\ \vdots\\ \varepsilon_{(p^{*},2)}^{(i)}\left(t\right)-E_{R}\left(l\right)-E_{T}\left(l,d \left(v_{(p^{*},2)}^{(i)},v_{(p^{*},1)}^{(i)}\right)\right)\\ \varepsilon_{(p^{*},1)}^{(i)}(t)-E_{R}\left(l\right)-0\end{array}