Mathematics > Probability
[Submitted on 21 Jan 2015 (v1), last revised 25 Feb 2016 (this version, v3)]
Title:Dependence and phase changes in random $m$-ary search trees
View PDFAbstract: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.
Submission history
From: Michael Fuchs [view email][v1] Wed, 21 Jan 2015 11:32:49 UTC (177 KB)
[v2] Sat, 30 Jan 2016 22:09:24 UTC (177 KB)
[v3] Thu, 25 Feb 2016 13:05:32 UTC (178 KB)
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