## 6 Simulation Results
### HECTOR Results
The optimal hop length (based on energy consumption) is plotted as a reference. Results are similar for other choices of the number of landmarks and the topology. We can notice that VCost and HECTOR follow edges whose lengths are close to the optimal one [17] in every case. The mean hop length is greater than the optimal one with HECTOR because the choice of the next hop is conditioned by the progress made over \(T\) labels, which leads to greater hop length because of tree construction. Indeed, because of the labeling process, close nodes are mainly at the same level in the tree and thus the progress they provide is null. Therefore, since nodes try to minimize the cost over progress ratio, they generally try to maximize the progress, thus reaching farther nodes. **Path Length. ** Figure 10 draws the path length in number of hops when VCost succeeds. We can notice that VCost is the algorithm that provides the shortest paths. This is because it is the only algorithm to select the next hop in the forwarding direction at each hop. LTP is the one achieving the longest paths since it follows the path in the tree, sometimes with shortcuts between branches but which is rarely the shortest path. The tree construction affects the mean hop length of LTP. The impact of having an energy efficient tree or a tree with optimized range is left to future works. As already mentioned, HECTOR tries to stick to VCost when it is possible. HECTOR' acts as VCost but since it is not energy-aware, HECTOR' takes long edges and thus gets shorter paths than HECTOR. Note that in the worst cases when HECTOR can never provide a progress over \(V\) coordinates and that there is only one candidate that provides a progress over \(T\) labels, it also follows the tree. In this latter case, the path length provided by HECTOR is longer than the one achieved by LTP since they both follow more or less the same route, but LTP takes longer edges and HECTOR tries to fit the optimal range. Figure 9: Mean hop length. When VCost fails, HECTOR and HECTOR' make decision based on \(T\) labels only, and thus HECTOR and HECTOR' stick to LTP. Thereby, HECTOR provides longer paths than LTP and HECTOR', which are not energy aware and take long links while HECTOR favors edges less energy-costly. This behavior is highlighted in Figure 11, which shows the path length in number of hops when VCost fails. Another interesting feature to point out is that globally, HECTOR provides longer paths than HECTOR' and VCost while it spends less energy. This also shows that HECTOR distributes the energy spending over the nodes on the paths. Figure 11: Path length in number of hops when VCost does not succeed. For VCost: number of hops before failing (\(3\) landmarks). Figure 10: Path length in number of hops when VCost succeeds (\(3\) landmarks). **Path shapes. ** Figure 12 shows example of the paths followed by each protocol in a network with a crescent hole. Five landmarks are randomly chosen and the tree root is the red/black node in the middle of the network. Source and destination are also randomly chosen. These schemes clearly show the behavior of each algorithm. We can see in Figure 12 that VCost and HECTOR follow exactly the same path. This means that every hop provides a progress on both \(V\) and \(T\) coordinates. It is nevertheless worth noting that this would not appear in the general case. Indeed, for HECTOR to follow exactly the same path that VCost, a progress has to be made at each step on both sets of coordinates. Even if a progress is made on \(V\) coordinates by a node \(u\), to be chosen, this node \(u\) has to also provide a progress on \(T\) labels, which is strongly related to the tree root, the source and the destination position. In the contrary, HECTOR' does not try to minimize the COP and thus the first hop is different from that in VCost and directs it toward the hole. From it, in order to provide a progress regarding \(T\) labels and avoiding the dead end, HECTOR' has to follow the path in the tree, as in LTP.