## 4 Power Control For Multiple Sink Cluster WSNs (MSCWSNs-PC)
### Distributed Power Allocation  
Assume deployment of \(M\) sink nodes in an \(X\times Y\) region of space, denoted as \(S_{i}\), and the location be set as (\(x_{i}\),\(y_{i}\)), \(i\in M\). Let \(Z(x\),\(y)\) be the rectangle area, where \(x\) is the length and \(y\) is the width. CS(\(x_{i}\),\(y_{i}\),\(r_{i}\)) is the circle area, where the point (\(x_{i}\), \(y_{i}\)) acts as the centre and \(r_{i}\) is the radio radius. To ensure the network's coverage rate up to 100%, the value \(r_{i}\), \(i\in M\), must satisfy the following condition:  
\[Z(X,Y)\subset CS(x_{1},y_{1},r_{1}^{\prime})\cup CS(x_{2},y_{2},r_{2}^{ \prime})\cup...\cup CS(x_{M},y_{M},r_{M}^{\prime}) \tag{6}\]  
In [26] the network is represented as the graph \(G=(N\),\(E)\), where \(N\) is a set of \(m\)+\(n\) nodes (containing \(m\) sinks and \(n\) sensor nodes), and \(S\subset N\) is the set of sink nodes (with \(|S|=m\)). The set of weighted edges is denoted as \(E\), and we note \(di\);\(j\) as the distance along the shortest path between \(i\) and \(j\). With each sink \(k\) we  
Figure 3: Frame format of _HELLO_ packet in MSCWSNs-PC. associate the _Voronoi cluster_\(V_{k}\) containing the nodes whose closest sink is \(k\). More formally, a _Voronoi cluster_\(V_{k}\) is defined as:  
\[V_{k}=\{i:\min_{j\in S}d_{i,j}=d_{i,k}\} \tag{7}\]  
In the _Voronoi_ algorithm, each node must be aware of the location of all sink and itself. In MSCWSNs, firstly each sink pre-detects its coarse position in the work area and the corresponding set of coordinates is (_X\({}_{\textit{Topside}}\)_, _Y\({}_{\textit{Topside}}\)_), (_X\({}_{\textit{Underside}}\)_, _Y\({}_{\textit{Underside}}\)_), (_X\({}_{\textit{Leftside}}\)_, _Y\({}_{\textit{Leftside}}\)_), (_X\({}_{\textit{Rightside}}\)_, _Y\({}_{\textit{Rightside}}\)_). The sink number of each group is _nunsink\({}_{\textit{Topside}}\)_nunsink\({}_{\textit{Underside}}\)_, _nunsink\({}_{\textit{Leftside}}\)_ and _nunsink\({}_{\textit{Rightside}}\)_. Secondly, we make four sequences in a given order, respectively, according to the value of _X\({}_{\textit{Topside}}\)_, _X\({}_{\textit{Underside}}\)_, _Y\({}_{\textit{Leftside}}\)_, and _Y\({}_{\textit{Rightside}}\)_. Then the sink with the lowest order in each group uses the distance between the corresponding vertices of the rectangle and itself as its radius, then each these sink has three cross points with the boundary of the rectangle, (e.g., node A in Figure 4). The other sink at least have two cross points with the corresponding nearest border by employing an enough transmission power, (e.g., node B in Figure 4). Set a current sink node _w_(_x\({}_{\textit{w}}\)_,_y\({}_{\textit{w}}\)_) wants to pre-detect its coarse position in work area. ```
0:for all\(x_{w}\), \(w\in M\)
0:if\(x_{w}\)>7/2 then
0:\(w\in\textit{Topside}\)
0:\(nunsink_{\textit{Topside}}=\textit{nunsink}_{\textit{Topside}}+1\)
0:\(X_{\textit{Topside}}(\textit{nunsink}_{\textit{Topside}})=x_{w}\)
0:\(Y_{\textit{Topside}}(\textit{nunsink}_{\textit{Topside}})=y_{w}\)
0:else
0:\(w\in\textit{Underside}\)
0:\(nunsink_{\textit{Underside}}=\textit{nunsink}_{\textit{Underside}}+1\)
0:\(X_{\textit{Underside}}(\textit{nunsink}_{\textit{Underside}})=x_{w}\)
0:\(Y_{\textit{Underside}}(\textit{nunsink}_{\textit{Underside}})=y_{w}\)
0:endif
0://
0:if\(x_{w}\)<7/2 then
```  
**Algorithm 1** sink \(w\) pre-decide its coarse position  
Figure 4: Sink A and B cross with rectangular borders. In our opinion, if the intersection of all circle cut off from the edges contained total rectangle borders, the coverage rate of the rectangle work area would be guaranteed. In this time, the \(r_{i}\), \(i\in M\), must satisfy as the following conditions:  
\[