# An Energy Efficient Sink Location Service for Continuous Objects in Wireless Sensor Networks
## 3 Proposed Scheme  
### Overview of the Scheme  
A mobile sink should notify its current location to sources to collect data from the interested data. Generally, the location update message is directly delivered by unicasting. Namely, the location update cost increases proportional to the number of sources. So, in continuous object tracking, the location update consumes an amount of energy because a continuous object is detected by a number of sources. In energy constrained WSNs, reducing the cost of sink location update should be considered. In this context, the regional locality of sources bring up the possibility of integrating location updates. In order to reduce the location update cost, we divide the location update with the following two steps: the first step is from the mobile sink to a location agent and the second step is from the agent to sources. Figure 7 shows individual location updates and integrated location updates by a location agent. Intuitively, it seems that the integrated location update is efficient.For clarification, we modeled the individual location update and integrated location update on two-dimensional plane as shown in Figure 8. The center of the continuous object is at the point \((0,0)\) and its radius is 1. The black dots on the perimeter of the continuous object are sources. The black star indicates a sink located at point \((5,0)\) and the hatched square indicates a location agent. In WSNs, the communication cost between two nodes tends to be proportional to the distance between them. Therefore, we assume that the sum of dashed lines is the communication cost of individual location updates, whereas the sum of doted lines indicates the cost of integrated location update using the agent. While the location of the agent varies along the x-axis, the communication cost of the individual location update is not changed. We defined the cost value \(R\) for presenting the efficiency of the integrated location updated compared to the individual location update as  
\[R=\frac{\sum_{i=1}^{n}\sqrt{(x-x_{i})^{2}+y_{i}^{2}}+(5-x)}{\sum_{i=1}^{n} \sqrt{(5-x_{i})^{2}+y_{i}^{2}}} \tag{1}\]  
where \(x\) is the x-coordinate of the location agent, \((x_{i},y_{i})\) is the location of the boundary points, and \(n\) is the number of boundary points (i.e., in Figure 8, \(n=8\)). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Category** & **Construction Cost for Structures** & **Concentration of Messages** \\ \hline Source-based structure & Proportional to the number of sources & Distributed to each structure \\ \hline Independent structure & Constant & Concentrated on the structure \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Evaluation about sink mobility support schemes for continuous object tracking. Figure 9 shows the change of \(R\) according to the x-coordinate of the agent where the gray rectangular indicates the area within the continuous object. In the graph, the lower value of \(R\) means that the location agent is at the more efficient integration point. Here, there are two important observations. First, relatively row communication cost can be guaranteed (i.e., under 0.4 times of the individual location update) when the location agent is within the continuous object. The cost of the integrated location update tends to decrease as the agent is closer to the center of the continuous object whereas the value increases sharply as the agent far from the continuous object. Second, the location of the agent should be fixed and unique. In Figure 9, the x-coordinate of the agent resulting the lowest value of \(R\) is approximately 0.1 (i.e., the point indicated by the dotted line). It means the most efficient location of the agent is not deterministic but vary according to the shape of the continuous object and the location of the sink. However, calculating the most efficient location is extremely difficult because it has to consider both the location of a mobile sink and the locations of all sources in real time.