把竹子切成小块
Cutting Bamboo Down to Size
摘要：
本文研究的问题，编程一个机器人熊猫园丁，以保持竹花园从阻碍你的房子的看法湖。花园由已知日生长率的$n$竹杆组成，园丁每天最多可以砍一根竹子。作为一名计算机科学家，你发现这个问题已经在[Gingusieniec et al.，SOFSEM'17]作为竹园修剪（BGT）问题，其中目标是计算永久时间表（即，切割竹子的顺序）以使机器人园丁遵循以最小化完工时间，即，竹子所能达到的最高高度两个自然策略是Reduce-Max和Reduce-Fastest（x）。Reduce-Max修剪当天最高的竹子，而Reduce-Fastest（x）修剪比$x$高的竹子中生长最快的竹子。众所周知，Reduce-Max和Reduce-Fastest（x）分别实现了$O(\log n)$和$4$的最大完工时间，用于$x=2$的最佳选择。证明了Reduce-Max的$9$的第一常数上界，并将Reduce-Fastest（x）的第一常数上界改进为$x=1+\frac{1}{\sqrt{5}}$的$\frac{3+\sqrt{5}}{2} < 2.62$.另一个关键方面源于这样一个事实，即您的机器人园丁的处理能力和内存有限。然后，重要的是算法能够快速确定下一根要切割的竹子，同时最多需要线性空间。我们将这一方面形式化为设计一个Trimming Oracle数据结构的问题，并提供了三个有效的Trimming Oracle实现不同的永久调度，包括Reduce-Max和Reduce-Fastest（x）产生的调度。
英文摘要：
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 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 makespan, i.e., the maximum height ever reached by a bamboo. Two natural strategies are Reduce-Max and Reduce-Fastest(x). Reduce-Max trims the tallest bamboo of the day, while Reduce-Fastest(x) trims the fastest growing bamboo among the ones that are taller than $x$. It is known that Reduce-Max and Reduce-Fastest(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 Reduce-Max and improve the one for Reduce-Fastest(x) to $\frac{3+\sqrt{5}}{2} < 2.62$ for $x=1+\frac{1}{\sqrt{5}}$. 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 quickly determine the next bamboo to cut while requiring at most linear space. We formalize this aspect as the problem of designing a Trimming Oracle data structure, and we provide three efficient Trimming Oracles implementing different perpetual schedules, including those produced by Reduce-Max and Reduce-Fastest(x).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
