Does freezing impede the growth of random recursive trees?

TL;DR

This work analyzes the height of uniform attachment with freezing, a dynamics that freezes vertices to impede attachment. Using time-reversal based constructions and coupling techniques, the authors show that the height $\mathsf{H}(\mathcal{T}(\mathbf{x}))$ is, for any admissible choice sequence, at least of order $e\log n$, comparable to the height of the classic random recursive tree $\mathcal{R}_n$, so freezing does not shrink height asymptotically. They prove that local modifications to the update sequence can both increase heights (removing a freezing or an attachment step) and that the first consecutive such steps, when removed, stochastically decrease height; these results are supported by alternating constructions (Algorithm 1 and Algorithm 2) and a coupling (Algorithm 3) between different sequences. A two-pronged analysis, based on high-value and low-value regimes for the active-vertex count, yields high-probability lower bounds of the form $\mathsf{H}(\mathcal{T}(\mathbf{x})) \ge e\log n - 5\log\log n$ for all $\mathbf{x}\in\mathbb{X}_n$. The work culminates in open questions on minimal height gaps and domination thresholds between $\mathcal{T}(\mathbf{x})$ and $\mathcal{R}_n$, highlighting future directions in the interplay between freezing and growth in random trees.

Abstract

Uniform attachment with freezing is an extension of the classical model of random recursive trees, in which trees are recursively built by attaching new vertices to old ones. In the model of uniform attachment with freezing, vertices are allowed to freeze, in the sense that new vertices cannot be attached to already frozen ones. We study the impact of removing attachment and/or freezing steps on the height of the trees. We show in particular that removing an attachment step can increase the expected height, and that freezing cannot substantially decrease the height of random recursive trees. Our methods are based on coupling arguments.

Figures (4)

• Figure 1: An illustration of the coupling of $\mathcal{T}(\mathrm{\mathbf{x}})$ and $\mathcal{T}(\hat{\mathrm{\mathbf{x}}})$ in the proof of Proposition \ref{['prop:increaseheight']}(i). The empty vertex is frozen.
• Figure 2: An illustration of the coupling of $\mathcal{T}(\mathrm{\mathbf{x}})$ and $\mathcal{T}(\hat{\mathrm{\mathbf{x}}})$ in the proof of Proposition \ref{['prop:increaseheight']}(ii). The empty vertices are frozen.
• Figure 3: An illustration of the coupling between $\mathcal{R}_{n}$, $\mathcal{T}(\mathrm{\mathbf{x}})$ and $\mathcal{T}(\hat{\mathrm{\mathbf{x}}})$ in the proof of Proposition \ref{['prop:increaseheight']}(iii). The root is blue, and the empty vertices are frozen. The probability of each configuration is indicated.
• Figure 4: Example of the coupling for $k=4$, $\mathrm{\mathbf{x}}=(+1,+1,+1,+1,-1,\ldots)$ and $\hat{\mathrm{\mathbf{x}}}= (+1,+1,+1,\ldots)$. The vertex in red is the vertex ${}$ added in the $0$-th position in the initialization step.

Theorems & Definitions (10)

• Proposition 1.1
• Theorem 1.2
• Proposition 1.3
• Lemma 2.2
• proof : Proof of Proposition \ref{['prop:increaseheight']}(i)
• proof : Proof of Lemma \ref{['lem:estimates']}
• proof : Proof of Proposition \ref{['prop:increaseheight']}(ii).
• Proposition 2.3: Bennett's inequality
• proof : Proof of Theorem \ref{['prop:decreaseheight']}
• proof : Proof of Theorem \ref{['thm:main']}.