再访竹子修剪：简单算法也能做得很好
Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too
摘要：
竹子修剪问题考虑生长速率$h_1, h_2, \ldots, h_n$满足$\sum_i h_i = 1$的$n$竹子。在给定的时间单位内，每根竹子都以$h_i$为单位生长，然后竹子修剪算法将其中一根竹子修剪回零高度。目标是最小化最高竹子的高度，也称为积压。竹子修剪问题与许多调度问题密切相关，可以看作是广泛研究的固定费率杯游戏的变体，但具有恒定因素的资源增加。过去的工作已经给出了复杂的风车算法，实现了最佳积压2竹修剪问题。它仍然是一个悬而未决的问题，但是，是否存在一个简单的算法具有相同的保证-最近的工作已经投入了相当大的理论和实验努力来回答这个问题。特别是，有两种算法作为自然候选者出现：Reduce-Max算法（总是切割最高的竹子）和Reduce-Fastest $(x)$算法（从高度至少为$x$的竹子中切割生长最快的竹子）。这两种算法都是为了实现backlog 2而设计的。本文改进了Reduce-Fastest和Reduce-Max的界。在其他结果中，我们证明了Reduce-Fastest $(x)$的确切最优积压是所有$x \ge 2$的$x + 1$（在$x = 2$的情况下证明了D 'Aldio，Di Stefano和Navarra的猜想），并且我们证明了Reduce-Fastest $(1)$没有实现积压2（反驳了D' Aldio，Di Stefano和Navarra的猜想）。最后，我们展示了一种不同的算法，我们称之为截止日期驱动策略，它既非常简单，又能实现最优积压2。这解决了问题，是否存在一个简单的最坏情况下的优化算法的竹子修剪问题。
英文摘要：
The bamboo trimming problem considers $n$ bamboo with growth rates $h_1, h_2, \ldots, h_n$ satisfying $\sum_i h_i = 1$. During a given unit of time, each bamboo grows by $h_i$, and then the bamboo-trimming algorithm gets to trim one of the bamboo back down to height zero. The goal is to minimize the height of the tallest bamboo, also known as the backlog. The bamboo trimming problem is closely related to many scheduling problems, and can be viewed as a variation of the widely-studied fixed-rate cup game, but with constant-factor resource augmentation. Past work has given sophisticated pinwheel algorithms that achieve the optimal backlog of 2 in the bamboo trimming problem. It remained an open question, however, whether there exists a simple algorithm with the same guarantee -- recent work has devoted considerable theoretical and experimental effort to answering this question. Two algorithms, in particular, have appeared as natural candidates: the Reduce-Max algorithm (which always cuts the tallest bamboo) and the Reduce-Fastest$(x)$ algorithm (which cuts the fastest-growing bamboo out of those that have height at least $x$). Both algorithms are conjectured to achieve backlog 2. This paper improves the bounds for both Reduce-Fastest and Reduce-Max. Among other results, we show that the exact optimal backlog for Reduce-Fastest$(x)$ is $x + 1$ for all $x \ge 2$ (proving a conjecture of D'Emidio, Di Stefano, and Navarra in the case of $x = 2$), and we show that Reduce-Fastest$(1)$ does not achieve backlog 2 (disproving a conjecture of D'Emidio, Di Stefano, and Navarra). Finally, we show that there is a different algorithm, which we call the Deadline-Driven Strategy, that is both very simple and achieves the optimal backlog of 2. This resolves the question as to whether there exists a simple worst-case optimal algorithm for the bamboo trimming problem.
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。
