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

Louigi Addario-Berry, Catherine Fontaine, Robin Khanfir, Louis-Roy Langevin, Simone Têtu

TL;DR

It is proved 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).

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).

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

TL;DR

It is proved that, for the optimal algorithm, an output set of size suffices; this bound is sharp and answers a question of Bubeck, Devroye and Lugosi (2017).

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 , such an algorithm outputs a set of nodes that contains the root with probability at least . We prove that, for the optimal algorithm, an output set of size 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).

Paper Structure

This paper contains 19 sections, 22 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A sketch of the proof
  3. Overview of the paper
  4. Setup and general results
  5. The Ulam--Harris formalism
  6. The centrality measure and the competitive ratio
  7. Preflows and flows on $\mathbb{U}$
  8. Some distributional definitions and identities
  9. Regular trees
  10. Presentation and asymptotics of the model
  11. Number of competitors in small subtrees
  12. Competitive ratio
  13. The dependence on $d$ in Theorem \ref{['thm:dary']}
  14. Uniform attachment trees
  15. Presentation and asymptotics of the model
  16. ...and 4 more sections

Key Result

Theorem 1.1

There exist $C^*,c^*>0$ such that the following holds. For $\varepsilon > 0$, let $K= K(\varepsilon)=C^*\exp(c^*\sqrt{\log1/\varepsilon}))$. Then for all $n \in \mathbb{N}_1$, for $T_n \sim \mathrm{UA}(n)$, it holds that $\mathbb{P}(\mathrm{\o}\not\in \mathcal{A}_K(T_n)) \le \varepsilon$.

Figures (1)

  • Figure 1: Illustration of the map $\chi$ and of the variables $Z_u$. The colour of each edge indicates the ratios of the values taken by the flow between the upper-end and the lower-end of the edge. Two vertices $a$ and $b$ are marked in the tree at the top. In the tree at the bottom, three nodes have $a$ as their $\chi$-image (marked with squares), meaning that $Z_a = 3$. Four nodes have $b$ as their $\chi$-image (marked with triangles), meaning that $Z_b = 4$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • proof : Proof of Theorem \ref{['thm:dary']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 34 more