Convergence of the QuickVal Residual

TL;DR

This paper analyzes the asymptotic behavior of the residual cost in the QuickVal algorithm, a probabilistic variant of QuickSelect, showing it converges to a Gaussian mixture under mild conditions.

Contribution

It introduces the QuickVal residual, derives its asymptotic distribution, and extends previous results on QuickSort residuals to a broader class of cost functions and quantiles.

Findings

The scaled residual  converges to a limit random variable.

The  residual scaled by  converges in law to a Gaussian mixture.

The paper provides exact and asymptotic  residual norms for the special case =0.

Abstract

QuickSelect (aka Find), introduced by Hoare (1961), is a randomized algorithm for selecting a specified order statistic from an input sequence of $n$ objects, or rather their identifying labels usually known as keys. The keys can be numeric or symbol strings, or indeed any labels drawn from a given linearly ordered set. We discuss various ways in which the cost of comparing two keys can be measured, and we can measure the efficiency of the algorithm by the total cost of such comparisons. We define and discuss a closely related algorithm known as QuickVal and a natural probabilistic model for the input to this algorithm; QuickVal searches (almost surely unsuccessfully) for a specified population quantile $\alpha \in [0, 1]$ in an input sample of size $n$. Call the total cost of comparisons for this algorithm $S_n$. We discuss a natural way to define the random variables $S_1, S_2, \ldots$ on a common probability space. For a general class of cost functions, Fill and Nakama (2013) proved under mild assumptions that the scaled cost $S_n / n$ of QuickVal converges in $L^p$ and almost surely to a limit random variable $S$. For a general cost function, we consider what we term the QuickVal residual: \[\rho_n := \frac{S_n}n - S.\] The residual is of natural interest, especially in light of the previous analogous work on the sorting algorithm QuickSort. In the case $\alpha = 0$ of QuickMin with unit cost per key-comparison, we are able to calculate -- \`a la Bindjeme and Fill (2012) for QuickSort -- the exact (and asymptotic) $L^2$-norm of the residual. We take the result as motivation for the scaling factor $\sqrt{n}$ for the QuickVal residual for general population quantiles and for general cost. We then prove in general (under mild conditions on the cost function) that $\sqrt{n}\,\rho_n$ converges in law to a scale-mixture of centered Gaussians, and we also prove convergence of moments.