## 4 Repeal Flooding Boundary
### Finding the Repeal-Flooding Boundary
#### 4.1.2 Lens-Shaped Scenario  
As the source node moves away from the network's center (i.e., between 0.3\(R_{net}\) and \(R_{net}\)), the previously described disk shape increases its radius up to a point where it intersects the network boundary, creating the lens shape shown in Figure 4b. The lens shape results from the overlapping of two disks. The first disk is the network boundary, having an \(R_{net}\) radius, while the chase disk's center is at the location of the target node, and its radius is the distance between the source and target nodes (see Figure 4b). The lens area's size depends on the distance separating the target node from the network's center, denoted as \(r\), as well as the angles between the origins and intersection points of the two disks, named \(\alpha\) and \(\gamma\) (see Figure 4b). The lens region area results from adding two segments named \(S_{1}\) and \(S_{2}\), which can be computed as follows:  
\[S_{1}=\frac{1}{2}R_{net}^{2}(\theta-\sin\theta)\]  
\[S_{2}=\frac{1}{2}R_{2F}^{2}(\gamma-\sin\gamma)\]  
Figure 4: Maximum area (solid line) where launching a chase flooding generates similar overhead compared with traditional flooding. (**a**) Disk shaped network; (**b**) Lens shaped network. with both \(\theta\) and \(\gamma\) expressed in radians, therefore the overlapping area (i.e., \(S_{1}+S_{2}\)) becomes:  
\[A_{1F}=A_{2F}=\frac{1}{2}R_{net}^{2}(\theta-sin\theta)+\frac{1}{2}R_{1F}^{2}( \gamma-sin\gamma) \tag{6}\]  
Applying the law of cosines to either triangle in Figure 4b, these angles can be computed as follows:  
\[\theta=2\sec(\frac{r^{2}+R_{net}^{2}-R_{2F}^{2}}{2 aR_{net}}) \tag{7}\]  
and  
\[\gamma=2\sec(\frac{r^{2}+R_{2F}^{2}-R_{net}^{2}}{2 aR_{net}}) \tag{8}\]  
Similar to the disk-shaped case, the maximum radius of the chase flooding (i.e., repeal-flooding boundary), in which the combined signaling of the first and chase flooding generates the same number of signaling packets as traditional flooding, becomes:  
\[A_{1F}+A_{2F}=A_{net} \tag{9}\]  
or equivalently  
\[2(S_{1}+S_{2})=A_{net} \tag{10}\]  
Solving for \(FB\) provides:  
\[FB=\frac{1}{\sqrt{2}}R_{net}+0.64(r-0.3) \tag{11}\]  
Finally, the repeal-flooding boundary (\(FB\)) becomes:  
\[FB= \begin{cases}\frac{1}{\sqrt{2}}R_{net},&0<r<0.3R_{net}\\ \frac{1}{\sqrt{2}}R_{net}+0.64(r-0.3),&0.3R_{net}<r<R_{net}\end{cases}  \tag{12}\]  
where, again, \(r\) is the target node's distance to the center of the network, and \(R_{net}\) is the radius of the network. The next section will explain how any node can estimate both distances in practice, so each node can compute its flooding boundary using Equation (12). Figure 5 shows how the flooding boundary (i.e., the point where the signaling overhead of F&C equals the signaling overhead of traditional flooding) changes as the source node moves away from the network's center. As this Figure shows, this boundary remains constant at \(\frac{1}{\sqrt{2}}R_{net}\) when the target node is within 0 and \(0.3R_{net}\) from the network's center, which corresponds to the case where the overlapping area is disk-shaped. Beyond \(0.3R_{net}\), the overlapping area is a lens, and the value of the repeal-flooding boundary increases linearly with the distance up to a value of \(1.17R_{net}\), where the target node reaches the edge of the network. So far, in the derivation of the optimized repeal-flooding boundary, distance was considered in the analysis rather than hop count since the latter depends on both the distance and node density, which is more challenging to deal with. However, hop count units will help to establish the optimized repeal-flooding boundary in practice in Section 4.1.3. The graph shown in Figure 5 is used as follows: Whenever a source node needs a route, it launches slow flooding to search for the target. Upon being reached, the target node uses its estimated repeal-flooding boundary, according to Figure 5, to decide whether or not it should launch a chase flooding.