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

TL;DR

This paper establishes optimal algorithms for root-finding in uniform attachment trees and regular growing trees, providing sharp bounds on the size of candidate sets needed to identify the root with high probability.

Contribution

It proves the optimal size bounds for root-finding algorithms in uniform attachment and regular growing trees, answering open questions and strengthening previous results.

Findings

Optimal root-finding set size is exponential in the square root of log(1/ε).

Bounds are sharp and improve previous results.

Results apply to both uniform attachment and regular growing trees.

Abstract

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 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).