Node Profiles of Symmetric Digital Search Trees: Concentration Properties

TL;DR

This paper provides a detailed asymptotic analysis of the profiles of symmetric digital search trees, revealing their concentration properties and distributional behaviors for height and saturation level.

Contribution

It introduces advanced analytic techniques to analyze the variance and concentration of profiles, height, and saturation level in symmetric digital search trees.

Findings

Profiles concentrate around their means as size grows

Height and saturation level distributions are tightly concentrated at two points

Analytic methods successfully handle complex variance analysis

Abstract

We give a detailed asymptotic analysis of the profiles of random symmetric digital search trees, which are in close connection with the performance of the search complexity of random queries in such trees. While the expected profiles have been analyzed for several decades, the analysis of the variance turns out to be very difficult and challenging, and requires the combination of several different analytic techniques, including Mellin and Laplace transforms, analytic de-Poissonization, and Laplace convolutions. Our results imply concentration of the profiles in the range where the mean tends to infinity. Moreover, we also obtain a two-point concentration for the distributions of the height and the saturation level.