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arXiv · 1501.05135

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

Hua-Huai Chern·Michael Fuchs·Hsien-Kuei Hwang·Ralph Neininger
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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.

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Hua-Huai Chern, Michael Fuchs, Hsien-Kuei Hwang, Ralph Neininger. 2016-02-25. Dependence and phase changes in random $m$-ary search trees. https://arxiv.org/abs/1501.05135

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