Abstract
We consider a problem as follows: Given unit weights arriving in an online manner with the total cardinality unknown, upon each arrival we decide where to place it on the unit circle in \(\mathbb {R}^{2}\). The objective is to set the center of mass of the placed weights as close to the origin as possible. We apply competitive analysis defining the competitive difference as a performance measure. We first present an optimal strategy for placing unit weights which achieves a competitive difference of \(\frac{1}{5}\). We next consider a variant in which the destination of each weight must be chosen from a set of positions that equally divide the unit circle. We give a simple strategy whose competitive difference is 0.35. Moreover, in the offline setting, several conditions for the center of mass to lie at the origin are derived.
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Acknowledgments
We would like to thank the participants of the 15th Enumeration Algorithm Seminar held in Ikaho, Japan, for their helpful comments. This work was supported by KAKENHI (23700014 and 23500014).
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Fujiwara, H., Seki, T., Fujito, T. (2014). Online Weight Balancing on the Unit Circle. In: Akiyama, J., Ito, H., Sakai, T. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2013. Lecture Notes in Computer Science(), vol 8845. Springer, Cham. https://doi.org/10.1007/978-3-319-13287-7_6
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DOI: https://doi.org/10.1007/978-3-319-13287-7_6
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