Electronic Journal of Probability

Novel characteristics of split trees by use of renewal theory

Cecilia Holmgren

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Abstract

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.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 5, 27 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062327

Digital Object Identifier
doi:10.1214/EJP.v17-1723

Mathematical Reviews number (MathSciNet)
MR2878784

Zentralblatt MATH identifier
1244.05058

Subjects
Primary: 05C05: Trees
Secondary: 05C80: Random graphs [See also 60B20] 68W40: Analysis of algorithms [See also 68Q25] 68P10: Searching and sorting 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35] 60C05: Combinatorial probability 68P05: Data structures

Keywords
Random Trees Split Trees Renewal Theory

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Holmgren, Cecilia. Novel characteristics of split trees by use of renewal theory. Electron. J. Probab. 17 (2012), paper no. 5, 27 pp. doi:10.1214/EJP.v17-1723. https://projecteuclid.org/euclid.ejp/1465062327


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