Optimal root recovery for uniform attachment trees and $d$-regular growing trees

We consider root-finding algorithms for random rooted trees grown by uniform attachment.

Given an unlabeled copy of the tree and a target accuracy $\varepsilon > 0$, such an algorithm outputs a set of nodes that contains the root with probability at least $1 - \varepsilon$.

We focus on the algorithm introduced by Bubeck, Devroye and Lugosi (2017) and proved to be optimal by Crane and Xu (2021).

We prove that, for the optimal algorithm, an output set of size $\exp(O(\log^{1/2}(1/\varepsilon)))$ suffices; this bound is sharp and answers a question of Bubeck, Devroye and Lugosi (2017).

We prove similar bounds for random regular trees that grow by uniform attachment, strengthening a result of Khim and Loh (2017).