# Bamboo Garden Trimming Problem: Priority Schedulings +
## 4 Experimental Results
### Fixed \(n\) Experimentation  
In this section, the results of further experimentation are reported where parameter \(n\) is fixed and all possible instances with \(H\geq n\) are considered, when \(H\in\{5,10,15,20,25,30,35\}\). For example, when \(n=15\), then \(H\) can assume values \(15\), \(20\) and \(25\). Instances are again chosen among all the integer partitions of specific values of \(H\). Figure 2 provides an overview of the total number of inputs considered in this experiment, as a function of \(n\) and \(H\), respectively. The results of this experiment are summarised in Figures 7 and 8 where the results of application of all considered algorithms on instances with \(n=10\) and \(n=15\), respectively, are shown. In particular, the maximum values of \(M(E)\) and \(\lambda_{E}\) obtained by all strategies as a function of \(H\) are reported. Reference lines \(2H\) and \(4H\) are again used to highlight the quality in terms of \(M(E)\) obtained by the different algorithms. Note that graphs for other possible values of \(n\) (e.g., \(n=20\)) are omitted since they lead to similar considerations with respect to those that follow. Figure 6: Distribution of values of: \(M(E)\) **(a)**; \(\lambda_{E}\) **(b)**; and number of days before stabilisation **(c)** as a function of \(H\), for all considered strategies and instances. On the Maximum HeightAs already observed above, in this experiment, it was also possible to notice that RMax and RFast\({}_{1}\) exhibit values of \(M(E)\) that are always below \(2H\) (see Figures 7 and 8). Moreover, again, RMax seems to be the best strategy, achieving values of maximum \(M(E)\) that are below \(2H\) and better than all other scheduling strategies, including RFast\({}_{1}\). This is again more evident if the attention is restricted to RMax and RFast\({}_{1}\) only (see Figure 7). Furthermore, the experiments confirm also the theoretical bound of RFast\({}_{2}\), with bamboos never exceeding \(4H\). Moreover, all strategies exhibit a trend of \(M(E)\) with respect to \(H\) that looks linear, with the only exception of RMin, where the curve looks slightly super-linear. On the Length of the CycleIn Figures 7b and 8b, the maximum \(\lambda_{E}\) obtained by all strategies in this setting is reported. The results confirm the trend observed in the case of fixed \(H\). However, here it can be seen how RMin starts growing significantly faster for values of \(H\) that are greater than 20. Shortest cycles are obtained by RMax and ReduceFast\({}_{1}\). Still, the length of the cycle seems dependent on the way ties among bamboos are resolved (see Figure 7d). For the sake of completeness, in Figures 9-12, the entire executions of the scheduling strategies are plotted on four sample instances whose size (i.e., \(n\) or \(H\)) is larger than all others considered in this paper, namely having:  
* \(n=30\) bamboos with \(H\in\{70,100\}\); and
* \(n=10\) with \(H\in\{40,200\}\). Figure 7: How maximum \(M(E)\) (**a,c**) and maximum \(\lambda_{E}\) (**b,d**) change as a function of \(H\) when instances are induced by all partitions of \(H\in\{10,15,20,25,30,35\}\) having cardinality \(n=10\). Panels (**c,d**) show strategies that, experimentally, exhibit 2-approximation. Figure 8: How maximum \(M(E)\) (**a**) and maximum \(\lambda_{E}\) (**b**) change as a function of \(H\) when instances are induced by all partitions of \(H\in\{15,20,25,30,35\}\) having cardinality \(n=15\). Panels (**c,d**) show strategies that, experimentally, exhibit 2-approximation. Figure 9: Cont. shows the beginning of the periodic phase while the horizontal lines show, respectively, \((\sum\limits_{t=1}^{\lambda_{E}}l^{t})/\lambda_{E}\) (i.e., \(H\)), \(2H\), \(4H\), and \(M(E)\) over the entire execution. Figure 10: Snapshot of the evolution of maximum heights when strategies: RMin (a); RMax (b); RFast\({}_{2}\) (c); and RFast\({}_{1}\) (d) are applied on an instance having \(n=30\) and \(H=100\) (in details, structured as follows \([70;2;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1;1]\)).