Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too

TL;DR

This paper investigates simple algorithms for the bamboo trimming problem, proving bounds for existing algorithms and introducing a new simple strategy that guarantees an optimal backlog of 2, resolving longstanding open questions.

Contribution

It establishes the exact backlog bounds for Reduce-Fastest algorithms and introduces the Deadline-Driven Strategy, a simple optimal algorithm for the problem.

Findings

Reduce-Fastest(x) has an optimal backlog of x+1 for x ≥ 2.

Reduce-Fastest(1) does not achieve backlog 2.

The Deadline-Driven Strategy achieves the optimal backlog of 2.

Abstract

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.