# Bamboo Garden Trimming Problem: Priority Schedulings +
## 4 Experimental Results
### Fixed \(H\) Experimentation
This supports the conjecture on the worst case approximation factor that has been proved so far being an over estimation of the true approximation upper bound. Another rather surprising evidence is that also \(\mathtt{RFast}_{1}\) is always below \(2H\) in terms of \(M(E)\). This suggests that, for this strategy as well there could be a way to prove the worst case 2-approximation. However, \(\mathtt{RFast}_{1}\) seems to follow an asymptotic trend toward \(2H\) as \(n\) increases while a slowly decreasing trend for RMax when \(n\) increases can be observed. Concerning \(\mathtt{RFast}_{2}\), from [1] it is known that \(M(E)\) is guaranteed to be below \(4H\) and this is confirmed by the performed tests. Nonetheless, it can be observed that \(M(E)\) is always above \(2H\), which is expected since no actions are performed by \(\mathtt{RFast}_{2}\) when there are no bamboos having height above \(2H\). However, the strategy exhibits a rather uniform behaviour, with \(M(E)\) stabilising toward threshold \(3H\) without ever overpassing it. This suggests that perhaps also the bound of \(4H\) guaranteed for \(\mathtt{RFast}_{2}\) is an overestimation of the true bound. Finally, regarding RMin, for low/high values of \(n\) (see Figures 3a and 4a), it provides smaller values of \(M(E)\) with respect to \(\mathtt{RFast}_{2}\), whereas for higher values of \(H\) its performance gets worse in the opposite sense. However, there no evidence of RMin  
Figure 4: Experiments conducted on all possible ordered instances obtained by setting \(H=35\) and hence considering \(n\) varying in \(\{1,2,\ldots,35\}\). Panels **(a,c)** refer to maximum \(M(E)\), whereas panels **(b,d)** refer to maximum \(\lambda_{E}\). Panels **(c,d)** show strategies that, experimentally, exhibit 2-approximation. exhibiting a constant approximation, with values of \(M(E)\) varying quite a lot with \(n\). As a final remark, in Figure 5a, it can be observed that, for all strategies, values of maximum \(M(E)\) tend to have small variance among all instances having a same \(H\), with curves assuming rather similar (flat) trends, and values always being very close to the average. The only exception is algorithm RMin, whose values of \(M(E)\) are quite different across instances having a same \(H\). Moreover, RMax achieves values of \(M(E)\) that are far better than all other strategies, including RFast\({}_{1}\), being below \(2H\) and, in some cases, below \(\frac{3}{2}H\). This is even more evident in Figures 3c and 4c where the focus is restricted on RMax and RFast\({}_{1}\), i.e., on experiments where \(M(E)\) is observed to be always below \(2H\). Finally, a global view supporting the conclusions with respect to the performance of the strategies in terms of \(M(E)\) is given in the scatterplot chart of Figure 6a where all obtained values of \(M(E)\) for all strategies and for all values of \(H\) are reported and compared. On the Length of the Cycle.In Figures 3b and 4b, the maximum \(\lambda_{E}\) exhibited by all strategies is reported. Such values can be considered as proxy of the complexity of the periodic part of each execution, as higher values of \(\lambda_{E}\) correspond to larger spaces of configurations that are explored by the strategies. This translates to higher variance in terms of height and volume, which can be seen as an undesired behaviour. Moreover, there might be also a relationship between such length and the quality of the provided factor of approximation, and it would be worth studying such relationship to define new bounds on this factor. In more details, the data show a very big gap between the results obtained by RMax, RFast\({}_{1}\), and those measured for RFast\({}_{2}\), RMin, with the latter two tending to exhibit larger values with respect to the former two. In particular, as shown in Figure 4b, when \(n=18\), RMin cyclic part takes around \(10^{7}\) steps, whereas the worst case for RFast\({}_{2}\) is obtained for \(n=19\), with around \(10^{5}\) steps.