The Distributional Tail of Worst-Case Quickselect

TL;DR

This paper analyzes the tail behavior of the limit distribution of the worst-case number of comparisons in Quickselect, providing explicit bounds and a method for estimating its mean.

Contribution

It introduces explicit tail bounds for the distribution of the Quickselect limit variable and develops a scheme for computing upper bounds on its mean.

Findings

The tail probability satisfies $- ext{log} P(S>t) = t ext{log} t + O(t ext{log} ext{log} t)$ as $t o fty$.

An explicit Chernoff majorant for the tail is derived.

A distribution-function scheme for upper bounding the mean of $S$ is proposed.

Abstract

We study the almost surely finite random variable $S$ defined by the distributional fixed-point equation \[ S \stackrel{d}{=} 1 + \max\{US', (1-U)S''\}, \qquad U \sim \mathrm{Unif}(0,1), \] where $S'$ and $S''$ are independent copies of $S$, independent of $U$. This random variable arises as the almost sure limit of the normalized worst-case number of key comparisons used by classical Quickselect with uniformly chosen pivots in the model of Devroye. Our first contribution concerns the right tail of $S$. We prove explicit one-sided bounds for the rate function $-\log \mathbb{P}(S>t)$ and, in particular, identify its first-order asymptotic growth: \[ -\log \mathbb{P}(S>t) = t \log t + O(t \log \log t), \qquad t \to \infty. \] The argument combines a binary-search-tree embedding and a one-level second-moment method with a moment-generating-function comparison inspired by ideas of Alsmeyer and Dyszewski for the nonhomogeneous smoothing transform. As a byproduct, we obtain an explicit pointwise Chernoff majorant for the tail. Our second contribution is a distribution-function scheme for deriving explicit upper bounds on $\mathbb{E}[S]$. Starting from the fixed-point equation at the level of the distribution function, we construct an order-preserving lower iteration and a conservative mesh discretization suited to computer-assisted upper bounds on the mean. We illustrate the latter numerically in floating-point arithmetic, but do not pursue a certified numerical proof here.