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2012 Novel characteristics of split trees by use of renewal theory
Cecilia Holmgren
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Electron. J. Probab. 17: 1-27 (2012). DOI: 10.1214/EJP.v17-1723


We investigate characteristics of random split trees introduced by Devroye [SIAM J Comput 28, 409-432, 1998]; split trees include e.g., binary search trees, $m$-ary search trees, quadtrees, median of $(2k+1)$-trees, simplex trees, tries and digital search trees. More precisely: We use renewal theory in the studies of split trees, and use this theory to prove several results about split trees. A split tree of cardinality n is constructed by distributing n balls (which often represent data) to a subset of nodes of an infinite tree. One of our main results is a relation between the deterministic number of balls n and the random number of nodes N. In Devroye [SIAM J Comput 28, 409-432, 1998] there is a central limit law for the depth of the last inserted ball so that most nodes are close to depth $\ln n/\mu+O(\ln n)^{1/2})$, where $\mu$ is some constant depending on the type of split tree; we sharpen this result by finding an upper bound for the expected number of nodes with depths $\geq \mu^{-1}\ln n-(\ln n)^{1/2+\epsilon}$ or depths $\leq\mu^{-1}\ln n+(\ln n)^{1/2+\epsilon}$ for any choice of $\epsilon>0$. We also find the first asymptotic of the variances of the depths of the balls in the tree.


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Cecilia Holmgren. "Novel characteristics of split trees by use of renewal theory." Electron. J. Probab. 17 1 - 27, 2012.


Accepted: 16 January 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1244.05058
MathSciNet: MR2878784
Digital Object Identifier: 10.1214/EJP.v17-1723

Primary: 05C05
Secondary: 05C80 , 60C05 , 68P05 , 68P10 , 68R10 , 68W40

Keywords: Random trees , renewal theory , Split Trees


Vol.17 • 2012
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