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Leaf Stripping on Uniform Attachment Trees

Louigi Addario-Berry, Anna Brandenberger, Simon Briend, Nicolas Broutin, Gábor Lugosi

Abstract

In this note we analyze the performance of a simple root-finding algorithm in uniform attachment trees. The leaf-stripping algorithm recursively removes all leaves of the tree for a carefully chosen number of rounds. We show that, with probability $1 - ε$, the set of remaining vertices contains the root and has a size only depending on $ε$ but not on the size of the tree.

Leaf Stripping on Uniform Attachment Trees

Abstract

In this note we analyze the performance of a simple root-finding algorithm in uniform attachment trees. The leaf-stripping algorithm recursively removes all leaves of the tree for a carefully chosen number of rounds. We show that, with probability , the set of remaining vertices contains the root and has a size only depending on but not on the size of the tree.
Paper Structure (9 sections, 6 theorems, 35 equations, 4 figures)

This paper contains 9 sections, 6 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries
  4. Embedding in the Ulam--Harris tree
  5. Tail bounds on the height of random recursive trees
  6. Proof of Theorem 1
  7. Detection of the root
  8. Size of the confidence set
  9. Proof of Theorem 2

Key Result

Theorem 1

There exist $c, \gamma > 0$ and $c' > 0$ such that for all $\varepsilon \in (0,1)$, setting $k = \lceil c \log(2/\varepsilon)\rceil$, for all $n$ sufficiently large, $R_{k}(T_n)$ satisfies

Figures (4)

  • Figure 1: An instance of an increasing tree on $\{1, \dots, 14\}$ and its embedding via $\varphi$ into the Ulam--Harris tree.
  • Figure 2: Image of the tree on $\{1,\dots,14\}$ from Figure \ref{['fig:phi-embedding']} under the tree flipping involution $\ell^{-1} \circ f_2 \circ \ell$, i.e., flipping up to zone $z = 2$.
  • Figure 3: The Ulam--Harris tree (left) and its corresponding child-sibling binary tree (right) given by the bijection $\ell$ in \ref{['eq:child-sibling']}, both with zones 1 through 4 illustrated in different colours.
  • Figure 4: Illustration of $f_3$: the second and third coordinates are flipped for every node. Notice that for each node in zone 3, its subtree remains identical under the map.

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 2 more