Fractional Bamboo Trimming and Distributed Windows Scheduling

Abstract

This paper studies two related scheduling problems: fractional bamboo trimming and distributed windows scheduling. In the fractional bamboo trimming problem, we are given n bamboos with different growth rates and cut fractions. At the end of each day, we can cut a fraction of one bamboo. The goal is to design a perpetual schedule of cuts to minimize the height of the tallest bamboo ever. For this problem, we present a 2-approximation algorithm. In addition, we prove upper bounds on the approximation factors of well-known algorithms Reduce-Max and Reduce-Fastest(x) for this problem. In the closely related windows scheduling problem, given a multiset of positive integers $W = \{w_1, ..., w_n\}$, we want to schedule n pages on broadcasting channels such that the time interval between any two consecutive appearances of the i-th page ($1 \le i \le n$) is at most $w_i$. The goal of this problem is to minimize the number of channels. We provide an algorithm for the windows scheduling problem that uses at most $\left\lceil \frac{d(W) + 1}{0.75} \right\rceil$ channels, where $d(W) = \sum _{i=1}^{n}{\frac{1}{w_i}}$. When $d(W) \le 46$, our algorithm guarantees a smaller upper bound on the number of channels than the best-known algorithm in the literature. We also describe the first approximation algorithm for the windows scheduling problem in a distributed setting, where input data is partitioned among a set of m machines. Furthermore, we introduce patterns of some multisets with $d(W) \le 1$ for which windows scheduling on one channel (i.e., pinwheel scheduling) is impossible.

This work was partially funded by NSERC Discovery Grants.