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

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Abstract

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.

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Funding

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.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, China

    Weiguang Peng

  2. Department of Mathematics, Wuhan University of Technology, Wuhan, China

    NingNing Peng

  3. Mathematical Institute, Tohoku University, Tohoku, Japan

    Kazuyuki Tanaka

Authors
  1. Weiguang Peng
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  2. NingNing Peng
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  3. Kazuyuki Tanaka
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There was an equal amount of contributions from all three authors. All authors read and approved the manuscript.

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Correspondence to NingNing Peng.

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Peng, W., Peng, N. & Tanaka, K. The Eigen-Distribution for Multi-Branching Weighted Trees on Independent Distributions. Methodol Comput Appl Probab 24, 277–287 (2022). https://doi.org/10.1007/s11009-021-09849-7

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  • DOI: https://doi.org/10.1007/s11009-021-09849-7

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