## 4 Repeal Flooding Boundary
### Finding the Repeal-Flooding Boundary
#### 4.1.3 Finding the Repeal-Flooding Boundary in Practice  
The previously described operation is challenging to carry out in practice. In a real wireless ad hoc and sensor network, node density and the network's shape and size are generally unknown, let alone the target node's position within the network [32; 33]. Consequently, in practice, it is not straightforward for any node to estimate its repeal-flooding boundary accurately. Figure 6 shows the number of signaling packets generated by F&C versus the distance separating source and target nodes and two different arbitrary repeal-flooding boundary values (i.e., 11 and 7 hops). The simulations settings considered the IEEE 802.11g and randomly placed 500 nodes in a 3500 \(\times\) 3500 m\({}^{2}\) network. The transmission range is 250 m. In Figure 6a, all nodes located within 11 hops from the source node initiated chase flooding. In many cases, this operation resulted in more signaling packets than traditional flooding (i.e., points found above the 500 packet line). On the other hand, Figure 6b shows the results when nodes initiated chase flooding only if reached within seven hops from the source node. In this case, the vast majority of points are below the 500 packet line. These figures illustrate how the repeal-flooding boundary value estimated by each node impacts the performance of F&C. Nodes need some information regarding the network's size and their location within it to estimate their repeal-flooding boundary. Initially, we assumed that nodes have an infinite repeal-flooding boundary, meaning they will chase any incoming flooding while estimating a more realistic boundary. Now, for node \(i\) (represented by \(n_{i}\)) to be able to estimate its repeal-flooding boundary \(FB_{i}\), \(n_{i}\) needs a clue that helps it compute \(FB_{i}\) with accuracy while requiring minimal intervention from the other nodes. In F&C, this clue takes the form of an unbounded version of F&C protocol; this is, a target node that has not previously estimated its repeal-flooding boundary launches a chase flooding after receiving an RQ packet. This initial behavior of F&C is similar to an unbounded repeal-based protocol. Once a target node chases the initial flooding with RP packets, other nodes overhearing both RQ and RP packets can estimate their repeal-flooding boundary according to the procedure described below. By doing this, they can make the best decision as to whether or not to chase the original flooding in the future. The evaluation section shows that it takes as few as one chase flooding in the best of cases, and only a few chase floodings on average, for all nodes in the network to estimate their repeal-flooding boundary with reasonable accuracy. Figure 7 illustrates the way nodes estimate their repeal-flooding boundary. This figure illustrates two different resource searches: a long distance route  
Figure 5: Repeal-flooding boundary versus target node’s distance to the network’s center. resource (16 hops) from node \(A\) to node \(B\) (Figure on the left), and a short distance route resource (3 hops) from node \(D\) to node \(C\) (Figure on the right). In both resource searches, we assumed that each node in the network increases by one #RQ or #RP counter every time they forward those packets, respectively. The square brackets next to each node in Figure 7 show the pair [#RQ,#RP] as a result of using F&C for each resource search. Now \(n_{i}\) can estimate the radius of the network (\(R_{i}\)) and its distance to the center of the network (\(r_{i}\)) as follows:  
\[R_{i}\approx\frac{\#RQ+\#RP}{2} \tag{13}\]  
\[r_{i}\approx R_{i}-min[\#RQ,\#RP] \tag{14}\]  
In simple terms, each node guesses if it lies somewhere along the shortest route resource connecting the source and target nodes. The diameter of the network is thus #RQ + #RP. Then, it follows that the network radius is just half the diameter, and the node's distance to the center of the network is the estimated network radius minus the minimum of either #RQ or #RP.