Asymptotic variance of random symmetric digital search trees

TL;DR

This paper introduces a novel analytical approach to determine the asymptotic variance of shape parameters in random symmetric digital search trees, simplifying complex calculations and clarifying previously uncertain results.

Contribution

It proposes a new normalization method using Poisson generating functions combined with Laplace and Mellin transforms, enabling easier analysis of variances in digital search trees.

Findings

Derived an asymptotic $n( ext{log } n)^2$-variance for total path-length.

Provided a simplified, adaptable methodology for variance analysis in binomially distributed problems.

Clarified previously uncertain variance behaviors in digital search trees.

Abstract

Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising $n(\log n)^2$-variance for certain notions of total path-length is also clarified.