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Compacted binary trees admit a stretched exponential
Journal of Combinatorial Theory Series A ( IF 1.1 ) Pub Date : 2020-08-25 , DOI: 10.1016/j.jcta.2020.105306
Andrew Elvey Price , Wenjie Fang , Michael Wallner

A compacted binary tree is a directed acyclic graph encoding a binary tree in which common subtrees are factored and shared, such that they are represented only once. We show that the number of compacted binary trees of size n grows asymptotically likeΘ(n!4ne3a1n1/3n3/4), where a12.338 is the largest root of the Airy function. Our method involves a new two parameter recurrence which yields an algorithm of quadratic arithmetic complexity for computing the number of compacted trees up to a given size. We use empirical methods to estimate the values of all terms defined by the recurrence, then we prove by induction that these estimates are sufficiently accurate for large n to determine the asymptotic form. Our results also lead to new bounds on the number of minimal finite automata recognizing a finite language on a binary alphabet. As a consequence, these also exhibit a stretched exponential.



中文翻译:

压缩的二叉树允许拉伸指数

压缩的二叉树是编码二叉树的有向无环图,在其中对公共子树进行分解和共享,以使它们仅被表示一次。我们显示大小为n的压缩二叉树的数量渐近增长,如Θñ4ñË3一种1个ñ1个/3ñ3/4 哪里 一种1个-2.338是Airy函数的最大根源。我们的方法涉及一个新的两参数递归,该递归产生了二次算术复杂度的算法,用于计算达到给定大小的压缩树的数量。我们使用经验方法来估计递归定义的所有项的值,然后通过归纳证明这些估计值对于大n足以确定渐近形式是足够准确的。我们的结果也为识别二进制字母上的有限语言的最小有限自动机的数量带来了新的界限。结果,它们也表现出拉伸的指数。

更新日期:2020-08-25
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