Bamboo Trimming Revisited: Simple Algorithms Can Do Well Too
John Kuszmaul

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 bamboo with growth rates satisfying . During a given unit of time, each bamboo grows by , 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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · VLSI and FPGA Design Techniques
