[Paper Review] Martingales and Profile of Binary Search Trees

Brigitte Chauvin, Thierry Klein+2 more|arXiv (Cornell University)|Oct 7, 2004

Stochastic processes and statistical mechanics28 references4 citations

TL;DR

This paper uses a continuous-time embedding of the binary search tree (BST) process into a Yule birth process to analyze the asymptotic profile of BSTs under the random permutation model. By leveraging martingale convergence and probability tilting techniques, it establishes the almost sure convergence of the profile martingale and derives precise asymptotic behavior for the number of nodes at level $k \simeq 2z\log n$, resolving critical cases $z = z_c^{\pm}$ where previous methods failed.

ABSTRACT

We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.

Motivation & Objective

• To study the asymptotic behavior of the profile of binary search trees (BSTs) under the random permutation model.
• To resolve the open problem of the behavior of the profile martingale at the critical values $z = z_c^{\pm}$, where previous $L^2$ methods failed.
• To establish a connection between the discrete BST process and the continuous-time Yule process via a coupling that preserves the law of the tree at stopping times.
• To derive the limiting distribution of the number of nodes at level $k \simeq 2z\log n$ across the full range $z \in (z_c^-, z_c^+)$.
• To investigate the depth of insertion and the behavior of a biased line of descent in BSTs using change-of-measure techniques.

Proposed method

• Embed the discrete BST process into a continuous-time Yule process via random time change, where the $n$th insertion time $\tau_n$ corresponds to the $n+1$st leaf in the Yule tree.
• Use the independence structure of the Yule process to analyze the profile martingale $\mathcal{M}_n(z)$ via the associated continuous-time martingale $M_t(z)$.
• Apply probability tilting (Gärtner-Ellis theorem) to study the behavior of the martingale under biased measures, particularly for $z = z_c^{\pm}$.
• Derive the joint generating function $\mathbb{E}[z^{d(t)} s^{N_t}]$ for the depth $d(t)$ and number of leaves $N_t$ in the Yule process, enabling limit theorems.
• Use the convergence of the martingale $\mathcal{M}_n(z)$ and its derivative to derive the splitting formula (41), which generalizes the Quicksort equation.
• Apply the change-of-measure technique under $\mathbb{P}^{(2z)}$ to show that $e^{-t}(N_t - 1 + 2z)$ is a martingale under the tilted measure, enabling convergence results for $N_t$.

Experimental results

Research questions

• RQ1 What is the almost sure asymptotic behavior of the number of nodes at level $k \simeq 2z\log n$ in a random BST?
• RQ2 How does the profile martingale $\mathcal{M}_n(z)$ behave at the critical values $z = z_c^{\pm}$, where previous $L^2$ methods fail?
• RQ3 Can the embedding of the BST process into the Yule process be used to derive new convergence results for the profile and depth of insertion?
• RQ4 What is the limiting distribution of the depth of insertion $d_n$ in the BST, and how does it behave almost surely?
• RQ5 How does biasing the evolution of the BST (favoring a specific line of descent) affect the global structure of the tree?

Key findings

• The profile martingale $\mathcal{M}_n(z)$ converges almost surely to a limit $\mathcal{M}_\infty(z)$ that is positive for $z \in (z_c^-, z_c^+)$ and zero otherwise.
• At the critical points $z = z_c^{\pm}$, the limit $\mathcal{M}_\infty(z^{\pm}) = 0$ almost surely, resolving a previously open question.
• The derivative of the martingale limit satisfies a splitting formula (41), which for $z = 1$ reduces to the Quicksort equation.
• The depth of insertion $d_n$ satisfies a law of large numbers: $\frac{d_n}{2\log n} \to 1$ in probability, and $\frac{d_n - 2\log n}{\sqrt{2\log n}} \to \mathcal{N}(0,1)$ in distribution.
• Almost surely, $\liminf_{n \to \infty} \frac{d_n}{2\log n} = z_c^-$ and $\limsup_{n \to \infty} \frac{d_n}{2\log n} = z_c^+$, showing that the central limit theorem does not hold almost surely.
• Under the tilted measure $\mathbb{P}^{(2z)}$, the process $e^{-t}(N_t - 1 + 2z)$ is a martingale, and for $2z > 1$, it converges almost surely; for $2z < 1$, $e^{-t}N_t$ is a positive supermartingale and converges a.s.