TY - JOUR

T1 - The Eigen-Distribution for Multi-Branching Weighted Trees on Independent Distributions

AU - Peng, Weiguang

AU - Peng, Ning Ning

AU - Tanaka, Kazuyuki

N1 - Funding Information:
This work was supported by National Natural Science Foundation of China Grant Number 11701438 and by Fundamental Research Funds for the Central Universities SWU118128, and Fundamental Research Funds for the Central Universities(WUT:2019IB011). Also supported by the JSPS KAKENHI Grant Numbers 26540001.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

PY - 2022/3

Y1 - 2022/3

N2 - Okisaka et al. (2017) investigated the eigen-distribution for multi-branching trees weighted with (a,b) on correlated distributions, which is a weak version of Saks and Wigderson’s (1986) weighted trees. In the present work, we concentrate on the studies of eigen-distribution for multi-branching weighted trees on independent distributions. In particular, we generalize our previous results in Peng et al. (Inform Process Lett 125:41–45, 2017) to weighted trees where the cost of querying each leaf is associated with the leaf and its Boolean value. For a multi-branching weighted tree, we define a directional algorithm and show it is optimal among all the depth-first algorithms with respect to the given independent distribution. For some balanced multi-branching trees weighted with (a,b) on the assumption 0 < r < 1 (r is the probability that the root has value 0), we further prove that if an independent distribution d achieves the distributional complexity, then d turns out to be an independent and identical distribution.

AB - Okisaka et al. (2017) investigated the eigen-distribution for multi-branching trees weighted with (a,b) on correlated distributions, which is a weak version of Saks and Wigderson’s (1986) weighted trees. In the present work, we concentrate on the studies of eigen-distribution for multi-branching weighted trees on independent distributions. In particular, we generalize our previous results in Peng et al. (Inform Process Lett 125:41–45, 2017) to weighted trees where the cost of querying each leaf is associated with the leaf and its Boolean value. For a multi-branching weighted tree, we define a directional algorithm and show it is optimal among all the depth-first algorithms with respect to the given independent distribution. For some balanced multi-branching trees weighted with (a,b) on the assumption 0 < r < 1 (r is the probability that the root has value 0), we further prove that if an independent distribution d achieves the distributional complexity, then d turns out to be an independent and identical distribution.

KW - Alpha-beta pruning algorithm

KW - Computational complexity

KW - Game trees with weights

KW - Independent distribution

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U2 - 10.1007/s11009-021-09849-7

DO - 10.1007/s11009-021-09849-7

M3 - Article

AN - SCOPUS:85100667220

VL - 24

SP - 277

EP - 287

JO - Methodology and Computing in Applied Probability

JF - Methodology and Computing in Applied Probability

SN - 1387-5841

IS - 1

ER -