Abstract
This paper is devoted to the ongoing study on the equilibrium points of AND-OR trees. Liu and Tanaka (2007, 2007a) characterized the eigen-distributions that achieve the distributional complexity, and among others, they proved the uniqueness of eigen-distribution for a uniform binary tree. Later, Suzuki and Nakamura (2012) showed that the uniqueness fails if only directional algorithms are allowed. Peng et al. (2016) extended the studies on eigen-distributions to balanced multi-branching trees of height 2. But, it remains open whether the uniqueness still holds or not for general multi-branching trees. To this end, we introduce the weighted trees, namely, trees with weighted cost depending on the value of a leaf. Using such models, we prove that for balanced multi-branching trees, the uniqueness of eigen-distribution holds w.r.t. all deterministic algorithms, but fails w.r.t. only directional algorithms.
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Acknowledgement
We would like to express our sincere appreciations for Prof. ChenGuang Liu (Northwestern Polytechnical University) for his original insight and helpful suggestions on this topic. We are also grateful to Prof. Yue Yang (National University of Singapore) for his useful discussions and valuable comments. This work was supported in part by the JSPS KAKENHI Grant Numbers 26540001 and 15H03634, and by National Natural Science Foundation of China Grant Number 11701438.
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Okisaka, S., Peng, W., Li, W., Tanaka, K. (2017). The Eigen-Distribution of Weighted Game Trees. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10627. Springer, Cham. https://doi.org/10.1007/978-3-319-71150-8_25
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DOI: https://doi.org/10.1007/978-3-319-71150-8_25
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