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Bamboo Garden Trimming Problem: Priority Schedulings †

Department of Information Engineering, Computer Science and Mathematics, University of L’Aquila; 67100 L’Aquila, Italy
Department of Mathematics and Computer Science, University of Perugia, 06123 Perugia, Italy
Author to whom correspondence should be addressed.
This paper is an extended version of our paper published in the 45th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM), Nový Smokovec, Slovakia, 27–30 January 2019.
Algorithms 2019, 12(4), 74;
Received: 18 January 2019 / Revised: 4 April 2019 / Accepted: 11 April 2019 / Published: 13 April 2019
The paper deals with the Bamboo Garden Trimming (BGT) problem introduced in [Gąsieniec et al., SOFSEM’17]. The problem is difficult to solved due to its close relationship to Pinwheel scheduling. The garden with n bamboos is an analogue of a system of n machines that have to be attended (e.g., serviced) with different frequencies. During each day, bamboo b i grows an extra height h i , for i = 1 , , n and, on the conclusion of the day, at most one bamboo has its entire height cut.The goal is to design a perpetual schedule of cuts to keep the height of the tallest ever bamboo as low as possible. The contribution in this paper is twofold, and is both theoretical and experimental. In particular, the focus is on understanding what has been called priority schedulings, i.e., cutting strategies where priority is given to bamboos whose current height is above a threshold greater than or equal to H = i = 1 n h i . Value H represents the total daily growth of the system and it is known that one cannot keep bamboos in the garden below this threshold indefinitely. As the first result, it is proved that, for any distribution of integer growth rates h 1 , , h n and any priority scheduling, the system stabilises in a fixed cycle of cuts. Then, the focus is on the so-called ReduceMax strategy, a greedy priority scheduling that each day cuts the tallest bamboo, regardless of the growth rates distribution. ReduceMax is known to provide a O ( log n ) -approximation, with respect to the lower bound H. One of the main results achieved is that, if ReduceMax stabilises in a round-robin type cycle, then it guarantees 2-approximation. Furthermore, preliminary results are provided relating the structure of the input instance, in terms of growth rates, and the behavior of ReduceMax when applied to such inputs. Finally, a conjecture that ReduceMax is 2-approximating for the BGT problem is claimed, hence an extended experimental evaluation was conducted to support the conjecture and to compare ReduceMax with other relevant scheduling algorithms. The obtained results show that ReduceMax : (i) provides 2-approximation in all considered inputs; and (ii) always outperforms other considered strategies, even those for which better worst case approximation guarantees have been proven. View Full-Text
Keywords: bamboo garden trimming; periodic scheduling; approximation algorithms; perpetual testing; experimental evaluation bamboo garden trimming; periodic scheduling; approximation algorithms; perpetual testing; experimental evaluation
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MDPI and ACS Style

D’Emidio, M.; Di Stefano, G.; Navarra, A. Bamboo Garden Trimming Problem: Priority Schedulings. Algorithms 2019, 12, 74.

AMA Style

D’Emidio M, Di Stefano G, Navarra A. Bamboo Garden Trimming Problem: Priority Schedulings. Algorithms. 2019; 12(4):74.

Chicago/Turabian Style

D’Emidio, Mattia, Gabriele Di Stefano, and Alfredo Navarra. 2019. "Bamboo Garden Trimming Problem: Priority Schedulings" Algorithms 12, no. 4: 74.

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