# Bamboo Garden Trimming Problem (Perpetual Maintenance of Machines with Different Attendance Urgency Factors)

• Leszek Gąsieniec
• Ralf Klasing
• Christos Levcopoulos
• Andrzej Lingas
• Jie Min
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10139)

## Abstract

A garden G is populated by $$n\ge 1$$ bamboos $$b_1, b_2, ..., b_n$$ with the respective daily growth rates $$h_1 \ge h_2 \ge \dots \ge h_n$$. It is assumed that the initial heights of bamboos are zero. The robotic gardener or simply a robot maintaining the bamboo garden is attending bamboos and trimming them to height zero according to some schedule. The Bamboo Garden Trimming Problem, or simply BGT, is to design a perpetual schedule of cuts to maintain the elevation of bamboo garden as low as possible. The bamboo garden is a metaphor for a collection of machines which have to be serviced with different frequencies, by a robot which can service only one machine during a visit. The objective is to design a perpetual schedule of servicing the machines which minimizes the maximum (weighted) waiting time for servicing.

We consider two variants of BGT. In discrete BGT the robot is allowed to trim only one bamboo at the end of each day. In continuous BGT the bamboos can be cut at any time, however, the robot needs time to move from one bamboo to the next one and this time is defined by a weighted network of connections.

For discrete BGT, we show a simple 4-approximation algorithm and, by exploiting relationship between BGT and the classical Pinwheel scheduling problem, we obtain also a 2-approximation and even a closer approximation for more balanced growth rates. For continuous BGT, we propose approximation algorithms which achieve approximation ratios $$O(\log (h_1/h_n))$$ and $$O(\log n)$$.

## Keywords

Virtual Machine Approximation Ratio Steiner Tree Online Schedule Daily Growth Rate
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Alshamrani, S., Kowalski, D.R., Gąsieniec, L.: How reduce max algorithm behaves with symptoms appearance on virtual machines in clouds. In: Proceedings of IEEE International Conference CIT/IUCC/DASC/PICOM, pp. 1703–1710 (2015)Google Scholar
2. 2.
Baruah, S.K., Cohen, N.K., Plaxton, C.G., Varvel, D.A.: Proportionate progress: a notion of fairness in resource allocation. Algorithmica 15(6), 600–625 (1996)
3. 3.
Baruah, S.K., Lin, S.-S.: Pfair scheduling of generalized pinwheel task systems. IEEE Trans. Comput. 47(7), 812–816 (1998)
4. 4.
Bender, M.A., Fekete, S.P., Kröller, A., Mitchell, J.S.B., Liberatore, V., Polishchuk, V., Suomela, J.: The minimum backlog problem. Theor. Comput. Sci. 605, 51–61 (2015)
5. 5.
Bodlaender, M.H.L., Hurkens, C.A.J., Kusters, V.J.J., Staals, F., Woeginger, G.J., Zantema, H.: Cinderella versus the Wicked Stepmother. In: Baeten, J.C.M., Ball, T., Boer, F.S. (eds.) TCS 2012. LNCS, vol. 7604, pp. 57–71. Springer, Heidelberg (2012). doi:
6. 6.
Chan, M.Y., Chin, F.Y.L.: General schedulers for the pinwheel problem based on double-integer reduction. IEEE Trans. Comput. 41(6), 755–768 (1992)
7. 7.
Chan, M.Y., Chin, F.: Schedulers for larger classes of pinwheel instances. Algorithmica 9(5), 425–462, May 1993Google Scholar
8. 8.
Chrobak, M., Csirik, J., Imreh, C., Noga, J., Sgall, J., Woeginger, G.J.: The buffer minimization problem for multiprocessor scheduling with conflicts. In: Orejas, F., Spirakis, P.G., Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 862–874. Springer, Berlin (2001). doi:
9. 9.
Collins, A., Czyzowicz, J., Gąsieniec, L., Kosowski, A., Kranakis, E., Krizanc, D., Martin, R., Ponce, O.M.: Optimal patrolling of fragmented boundaries. In: Proceedings of the Twenty-fifth Annual ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2013, New York, USA, pp. 241–250 (2013)Google Scholar
10. 10.
Czyzowicz, J., Gąsieniec, L., Kosowski, A., Kranakis, E.: Boundary patrolling by mobile agents with distinct maximal speeds. In: Demetrescu, C., Halldórsson, M.M. (eds.) ESA 2011. LNCS, vol. 6942, pp. 701–712. Springer, Heidelberg (2011). doi:
11. 11.
Czyzowicz, J., Gasieniec, L., Kosowski, A., Kranakis, E., Krizanc, D., Taleb, N.: When Patrolmen Become Corrupted: Monitoring a Graph Using Faulty Mobile Robots. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 343–354. Springer, Heidelberg (2015). doi:
12. 12.
Fishburn, P.C., Lagarias, J.C.: Pinwheel scheduling: achievable densities. Algorithmica 34(1), 14–38, September 2002Google Scholar
13. 13.
Holte, R., Mok, A., Rosier, L., Tulchinsky, I., Varvel, D.: The pinwheel: a real-time scheduling problem. In: II: Software Track, Proceedings of the Twenty-Second Annual Hawaii International Conference on System Sciences, vol. 2, pp. 693–702, January 1989Google Scholar
14. 14.
Holte, R., Rosier, L., Tulchinsky, I., Varvel, D.: Pinwheel scheduling with two distinct numbers. Theor. Comput. Sci. 100(1), 105–135 (1992)
15. 15.
Kawamura, A., Kobayashi, Y.: Fence patrolling by mobile agents with distinct speeds. Distrib. Comput. 28(2), 147–154 (2015)
16. 16.
Lin, S.-S., Lin, K.-J.: A pinwheel scheduler for three distinct numbers with a tight schedulability bound. Algorithmica 19(4), 411–426, December 1997Google Scholar
17. 17.
Ntafos, S.: On gallery watchmen in grids. Inf. Process. Lett. 23(2), 99–102 (1986)
18. 18.
Romer, T.H., Rosier, L.E.: An algorithm reminiscent of euclidean-gcd for computing a function related to pinwheel scheduling. Algorithmica 17(1), 1–10 (1997)
19. 19.
Serafini, P., Ukovich, W.: A mathematical model for periodic scheduling problems. SIAM J. Discrete Math. 2(4), 550–581 (1989)
20. 20.
Urrutia, J.: Art gallery and illumination problems. In: Handbook of Computational Geometry, vol. 1, no. 1, pp. 973–1027 (2000)Google Scholar

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## Authors and Affiliations

• Leszek Gąsieniec
• 1
• Ralf Klasing
• 2
Email author
• Christos Levcopoulos
• 3
• Andrzej Lingas
• 3
• Jie Min
• 1