# Bamboo Garden Trimming Problem: Priority Schedulings +
## 4 Experimental Results
### Fixed \(H\) Experimentation  
In the first set of experiments, parameter \(H\) was fixed and all possible input instances of \(n\) bamboos were considered whose growth rates sum up to \(H\). Such instances were generated by considering, for a given \(H\in\mathbb{N}\), all integer partitions of \(H\). Hence, clearly, the resulting \(n\) was in \(\{1,2,\ldots,H\}\), where for \(n=1\) there was only one bamboo having growth rate equal to \(H\) while for \(n=H\) all bamboos had unitary growth rate (e.g., for \(H=3\), we had instances \([3]\), \([2,1]\) and \([1,1,1]\)). Note that, for \(H\), values in the set \(\{5,10,15,20,25,30,35\}\) were selected. This choice was dictated by the fact that the number of integer partitions, and therefore instances to consider, grows very quickly as \(H\) increases and hence too large values of \(H\) induce a computationally prohibitive number of simulations. In particular, it is known that the number of integer partitions \(p(k)\) of a natural number \(k\) grows asymptotically as \(p(k)\approx\frac{1}{4k\sqrt{3}}e^{\pi\sqrt{\frac{2k}{3}}}\) as \(k\) approaches infinity [30]. For instance, \(p(100)=190\),569,292 while \(p(1000)\approx 2.41\cdot 10^{31}\). Figure 2b provides an overview of the total number of inputs considered in this experiment. However, the trend of the obtained results does not seem to be affected for \(H\) growing The results of the application of all the considered scheduling algorithms, on instances induced by the integer partitions \(H=30\) (\(H=35\), respectively) are shown in Figure 3 (Figure 4, respectively). Panels on the left side show how the maximum \(M(E)\), obtained for all configurations and strategies, varies as a function of \(n\) (see Figures 3a,c and 4a,c, respectively). Reference lines \(2H\) and \(4H\) are plotted to emphasise performance of the strategies in terms of \(M(E)\). Panels on the right instead, instead, show how the achieved maximum value of \(\lambda_{E}\) changes as a function of \(n\), given that all strategies are guaranteed to stabilise into cycles. Note that the maximum measured value is reported as it is possible to have many instances having the same \(n\). Notice also that the results for \(H\in\{5,10,15,20,25\}\) lead to similar considerations with respect to those provided below, hence they have been omitted. Figure 3: Experiments conducted on all possible ordered instances obtained by setting \(H=30\) and hence considering \(n\) varying in \(\{1,2,\ldots,30\}\). 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. Figure 2: (**a**) Distribution of instances with respect to \(n\) for all partitions of integer \(H\in\{5,10,15,20,25,30,35\}\); and (**b**) distribution of instances with respect to \(H\) for all partitions of integer \(H\in\{5,10,15,20,25,30,35\}\). An alternative view of the results of the experiments for \(H=35\) is given in Figure 5 where it is shown how the obtained values of \(M(E)\) and \(\lambda_{E}\) are distributed over the considered instances. In detail, each value on the \(x\)-axis in these diagrams simply represents one instance, to which the corresponding values of \(M(E)\) and \(\lambda_{E}\) has been associated on the \(y\)-axis. Instances are sorted on the \(x\)-axis in non-decreasing order according to their values on the \(y\)-axis to highlight the amount of inputs providing a same value of \(M(E)\) and \(\lambda_{E}\). This information is condensed in Figure 4 where \(n\) is on the \(x\)-axis and the maximum values are plot only for each set of instances sharing the same value of \(n\). _On the Maximum Height._ The main and the most interesting outcome of this experiment is that, notwithstanding the \(O(\log n)\) approximation factor [1], RMax exhibits properties of a 2-approximation, which can be observed as \(M(E)<2H\).