Electronic Journal of Probability

Novel characteristics of split trees by use of renewal theory

Cecilia Holmgren

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Random Trees Split Trees Renewal Theory

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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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  • Asmussen, Søren. Applied probability and queues. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1987. x+318 pp. ISBN: 0-471-91173-9
  • C.J. Bell, phAn Investigation into the Principles of the Classification and Analysis of Data on an Automatic Digital Computer. phPhD Thesis 1965.
  • Clément, J.; Flajolet, P.; Vallée, B. Dynamical sources in information theory: a general analysis of trie structures. Average-case analysis of algorithms (Princeton, NJ, 1998). Algorithmica 29 (2001), no. 1-2, 307–369.
  • E. G. Coffman, and J. Eve, File structures using hashing functions. phCommunications of the ACM 13 (1970), 427–436.
  • Devroye, Luc. Universal limit laws for depths in random trees. SIAM J. Comput. 28 (1999), no. 2, 409–432.
  • Devroye, Luc. Applications of Stein's method in the analysis of random binary search trees. Stein's method and applications, 247–297, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, Singapore Univ. Press, Singapore, 2005.
  • Feller, William. Fluctuation theory of recurrent events. Trans. Amer. Math. Soc. 67, (1949). 98–119.
  • Feller, William. An introduction to probability theory and its applications. Vol. I. Third edition John Wiley & Sons, Inc., New York-London-Sydney 1968 xviii+509 pp.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • R.A. Finkel and J.L. Bentley, Quad trees, a data structure for retrieval on composite keys. phActa Inform 4 (1974), 1–9.
  • E. Fredkin, Trie memory. phCommunications of the ACM 3 (1960), no. 9, 490–499.
  • Gut, Allan. Stopped random walks. Limit theorems and applications. Applied Probability. A Series of the Applied Probability Trust, 5. Springer-Verlag, New York, 1988. x+199 pp. ISBN: 0-387-96590-4
  • Gut, Allan. Probability: a graduate course. Springer Texts in Statistics. Springer, New York, 2005. xxiv+603 pp. ISBN: 0-387-22833-0
  • Hoare, C. A. R. Quicksort. Comput. J. 5 1962 10–15.
  • Holmgren, Cecilia. A weakly 1-stable distribution for the number of random records and cuttings in split trees. Adv. in Appl. Probab. 43 (2011), no. 1, 151–177.
  • Mahmoud, Hosam M.; Pittel, Boris. Analysis of the space of search trees under the random insertion algorithm. J. Algorithms 10 (1989), no. 1, 52–75.
  • Pyke, R. Spacings. (With discussion.) J. Roy. Statist. Soc. Ser. B 27 1965 395–449.
  • Mohamed, Hanène; Robert, Philippe. A probabilistic analysis of some tree algorithms. Ann. Appl. Probab. 15 (2005), no. 4, 2445–2471.