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 = {w₁, …, wₙ}, 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 ≤ i ≤ n) is at most wᵢ. 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 ⌈(d(W) + 1) / 0.75⌉ channels, where d(W) = ∑(i=1 to n)(1/wᵢ). When d(W) ≤ 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) ≤ 1 for which windows scheduling on one channel (i.e., pinwheel scheduling) is impossible.

Publication
In Proceedings of the 49th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM)