开关二叉树远程控制系统 -- II. 完美二叉树的平衡

Olivier Golinelli
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引用次数: 0

摘要

我们研究了 Guidon(2018)引入的树着色模型,该模型最初基于铁路货场远程控制系统的类比,被视为二叉树上的开关。对于给定的二叉树,我们形式化了着色的约束条件,尤其是节点在不同颜色之间的分布。按照吉登的方法,我们对平衡着色感兴趣,即着色能使按颜色分布的树节点子集的最大尺寸最小化。通过这种方法,我们提出了高度不超过 7 的树的平衡着色方法,但他的方法似乎难以应用于高度更大的树。此外,我们还提出了另一种方法,它可以给出任意大树的解决方案。我们用高度为 8 的平衡着色来说明。在附录中,我们给出了着色数作为树高函数的精确公式和渐近行为。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Remote control system of a binary tree of switches -- II. balancing for a perfect binary tree
We study a tree coloring model introduced by Guidon (2018), initially based on an analogy with a remote control system of a rail yard, seen as switches on a binary tree. For a given binary tree, we formalize the constraints on the coloring, in particular the distribution of the nodes among colors. Following Guidon, we are interested in balanced colorings i.e. colorings which minimize the maximum size of the subsets of the tree nodes distributed by color. With his method, we present balanced colorings for trees of height up to 7. But his method seems difficult to apply for trees of greater height. Also we present another method which gives solutions for arbitrarily large trees. We illustrate it with a balanced coloring for height 8. In the appendix, we give the exact formulas and the asymptotic behavior of the number of colorings as a function of the height of the tree.
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