Dependence and phase changes in random $m$-ary search trees

TL;DR

This paper investigates the asymptotic behavior of space and path length in random m-ary search trees, revealing phase transitions and oscillations in covariance and correlation as m varies, with extensions to other tree classes.

Contribution

It uncovers phase changes and oscillatory behavior in the covariance and correlation of key tree parameters, extending analysis to general shape parameters and other random log-trees.

Findings

Covariance changes from linear to higher order at m=14

Correlation coefficient oscillates for m>26

Methods combine asymptotic transfer and contraction method

Abstract

We study the joint asymptotic behavior of the space requirement and the total path length (either summing over all root-key distances or over all root-node distances) in random $m$-ary search trees. The covariance turns out to exhibit a change of asymptotic behavior: it is essentially linear when $3\le m\le 13$ but becomes of higher order when $m\ge14$. Surprisingly, the corresponding asymptotic correlation coefficient tends to zero when $3\le m\le 26$ but is periodically oscillating for larger $m$. Such a less anticipated phenomenon is not exceptional and we extend the results in two directions: one for more general shape parameters, and the other for other classes of random log-trees such as fringe-balanced binary search trees and quadtrees. The methods of proof combine asymptotic transfer for the underlying recurrence relations with the contraction method.