Asymptotic Normality for the Size of Graph Tries built from M-ary Tree Labelings

TL;DR

This paper proves a central limit theorem for the size of graph tries built from random labelings of M-ary trees, confirming a conjecture and providing asymptotic normality results.

Contribution

It verifies Jacquet's conjecture on the asymptotic normality of graph try size using the method of moments.

Findings

Confirmed asymptotic normality of graph try size

Derived asymptotic expansion for mean and variance

Validated conjecture through rigorous proof

Abstract

Graph tries are a new and interesting data structure proposed by Jacquet in 2014. They generalize the classical trie data structure which has found many applications in computer science and is one of the most popular data structure on words. For his generalization, Jacquet considered the size (or space requirement) and derived an asymptotic expansion for the mean and the variance when graph tries are built from $n$ independently chosen random labelings of a rooted $M$-ary tree. Moreover, he conjectured a central limit theorem for the (suitably normalized) size as the number of labelings tends to infinity. In this paper, we verify this conjecture with the method of moments.