Cutting bamboo down to size

This paper studies the problem of programming a robotic panda gardener to keep a bamboo garden from obstructing the view of the lake by your house. The garden consists of $n$ bamboo stalks with known daily growth rates and the gardener can cut at most one bamboo per day. As a computer scientist, you found out that this problem has already been formalized in [Gąsieniec et al., SOFSEM'17] as the emph{Bamboo Garden Trimming (BGT) problem}, where the goal is that of computing a perpetual schedule (i.e., the sequence of bamboos to cut) for the robotic gardener to follow in order to minimize the emph{makespan}, i.e., the maximum height ever reached by a bamboo. Two natural strategies are educemax and educefastest{x}. educemax trims the tallest bamboo of the day, while educefastest{x} trims the fastest growing bamboo among the ones that are taller than $x$. It is known that educemax and educefastest{x} achieve a makespan of $O(log n)$ and $4$ for the best choice of $x=2$, respectively. We prove the first constant upper bound of $9$ for educemax and improve the one for educefastest{x} to $rac{3+sqrt{5}}{2} < 2.62$ for $x=1+rac{1}{sqrt{5}}$. Another critical aspect stems from the fact that your robotic gardener has a limited amount of processing power and memory. It is then important for the algorithm to be able to emph{quickly} determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a emph{Trimming Oracle} data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by educemax and educefastest{$x$}.