# An Energy-Efficient Two-Stage Cooperative Routing Scheme in Wireless Multi-Hop Networks
## 3 Proposed Two-Stage Cooperative Routing
### Two-Stage Link Cost  
On the basis of the above two-stage cooperative process, the link cost of the link \(i\to j\) under core helper \(u\) consists of two parts:  
\[\mathcal{C}_{i,u,j}=\mathcal{C}_{i,u}+\mathcal{C}_{\mathcal{T}\left(u\right),j} \tag{14}\]  
where \(\mathcal{C}_{i,u}\) and \(\mathcal{C}_{\mathcal{T}\left(u\right),j}\) correspond to the cost of _Stage 1_ and _Stage 2_, respectively. Similar to Equations (7) and (9), the link cost of the first broadcast stage is defined as  
\[\mathcal{C}_{i,u}=\left(\frac{E_{\text{ini}}}{E_{i}}\right)^{x}\cdot P_{T}, \tag{15}\]  
where \(x\) is a parameter that governs the effect of normalized residual energy. The packet transmission in the second stage is performed in a general cooperative beamforming manner. Set the cooperative beamforming vector as \(\mathbf{w}=\left\{w_{1},w_{2},\cdots,w_{n}\right\}\), therefore, in accordance with Equation (15), the link cost of the second stage is defined as  
\[\mathcal{C}_{\mathcal{T}\left(u\right),j}=\min\left\{\sum\limits_{t\in \mathcal{T}\left(u\right)}\left(\frac{E_{\text{ini}}}{E_{t}}\right)^{x}\left| w_{t}\right|^{2}\left|\gamma_{j}\geq\gamma_{\min}\right.\right\}, \tag{16}\]  
which aims at minimizing the total weighted transmission power of \(\mathcal{T}\left(u\right)\) subject to the condition that node \(j\) can successfully decode the data. As considered in the Appendix A, by solving this optimization problem, the cooperative beamforming vector \(\mathbf{w}\) can be designed as  
\[\hat{w}_{i}=\frac{\left(E_{i}\right)^{x}h_{i,j}^{*}}{\sum\limits_{t\in \mathcal{T}\left(u\right)}\left(E_{t}\right)^{x}\left|h_{t,j}\right|^{2}} \sqrt{\gamma_{\min}\sigma^{2}},\quad i\in\mathcal{T}\left(u\right), \tag{17}\]  
Figure 2: The two-stage cooperative transmission model. where \(\left(\cdot\right)^{*}\) stands for the complex conjugate operation. In addition, the minimum value of the total weighted transmission power is given by  
\[\mathcal{C}_{\mathcal{T}\left(u\right),j}=\frac{\gamma_{\min} \sigma^{2}}{\sum\limits_{t\in\mathcal{T}\left(u\right)}(\frac{E_{t}}{E_{\min} })^{x}|h_{t,j}|^{2}}. \tag{18}\]  
The link cost \(\mathcal{C}_{i,u,j}\) defined in Equation (14) highly depends on the selection of core helper. By selecting the best core helper, the cooperative helpers of the link \(i\to j\) can be optimized and the number of cooperative nodes can also be determined. The optimized two-stage cost of link \(i\to j\) under proposed TSC transmission model is then defined as the minimum of \(\mathcal{C}_{i,u,j}\) over all possible core helpers  
\[\mathcal{C}_{i,j}^{\text{TSC}}=\min_{u\in\mathcal{U}_{i,j}}\big{\{} \mathcal{C}_{i,u,j}\big{\}}. \tag{19}\]  
Furthermore, the selection of core helper node is shown in Algorithm 1. ```
1:Obtain the set of candidate core helpers \(\mathcal{U}_{i,j}\) for link \(\mathcal{P}_{i,j}\) according to Equation (11). 2:Traverse \(u\in\mathcal{U}_{i,j}\), and calculate the related two-stage link cost \(\mathcal{C}_{i,u,j}\) by Equations (14-16). 3:Determine the core helper with the optimized two-stage link cost \(\mathcal{C}_{i,j}^{\text{TSC}}\) by Equation (19). 4:The link \(\mathcal{P}_{i,j}\) owns a core helper \(u_{i,j}^{*}\underset{u\in\mathcal{U}_{i,j}}{\overset{\ast}{\underset{u\in \mathcal{U}_{i,j}}{\min}}}\big{\{}\mathcal{C}_{i,u,j}\big{\}}\). ```  
**Algorithm 1** Selection of core helper node  
Algorithm 1 selects the candidate core helper with the optimized two-stage link cost as the core helper \(u^{*}\) for the link \(\mathcal{P}_{i,j}\), and the corresponding helper set \(\mathcal{T}\left(u^{*}\right)\) can be further determined. In case \(u^{*}=i\), node \(i\) will transmit data directly to \(j\) with the transmit power given by Equation (17). In this situation, the link cost is completely determined by the second stage and the last term in Equation (14).