Electronic Journal of Probability

Asymptotic distribution of two-protected nodes in ternary search trees

Cecilia Holmgren and Svante Janson

Full-text: Open access

Abstract

We study protected nodes in $m-$ary search trees, by putting them in context of generalized Pólya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to $m-$ary search trees with larger $m$ as well, although the size of the matrices used in the calculations grow rapidly with $m$; we conjecture that the method yields an asymptotically normal distribution for all $m \leq 26$.

The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler urn (that is similar to the one that has earlier been used to study the total number of nodes in $m-$ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all $m \leq 26$.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 9, 20 pp.

Dates
Accepted: 5 February 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067115

Digital Object Identifier
doi:10.1214/EJP.v20-3577

Mathematical Reviews number (MathSciNet)
MR3311222

Zentralblatt MATH identifier
1327.60032

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 05C05: Trees 60F05: Central limit and other weak theorems 68P05: Data structures

Keywords
Random Trees Polya urns Normal limit laws M-ary search trees

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

Citation

Holmgren, Cecilia; Janson, Svante. Asymptotic distribution of two-protected nodes in ternary search trees. Electron. J. Probab. 20 (2015), paper no. 9, 20 pp. doi:10.1214/EJP.v20-3577. https://projecteuclid.org/euclid.ejp/1465067115


Export citation

References

  • Bóna, Miklós. $k$-protected vertices in binary search trees. Adv. in Appl. Math. 53 (2014), 1–11.
  • Chauvin, Brigitte; Pouyanne, Nicolas. $m$-ary search trees when $m\ge27$: a strong asymptotics for the space requirements. Random Structures Algorithms 24 (2004), no. 2, 133–154.
  • Cheon, Gi-Sang; Shapiro, Louis W. Protected points in ordered trees. Appl. Math. Lett. 21 (2008), no. 5, 516–520.
  • Chern, Hua-Huai; Hwang, Hsien-Kuei. Phase changes in random $m$-ary search trees and generalized quicksort. Analysis of algorithms (Krynica Morska, 2000). Random Structures Algorithms 19 (2001), no. 3-4, 316–358.
  • Devroye, Luc. Limit laws for local counters in random binary search trees. Random Structures Algorithms 2 (1991), no. 3, 303–315.
  • Devroye, Luc; Janson, Svante. Protected nodes and fringe subtrees in some random trees. Electron. Commun. Probab. 19 (2014), no. 6, 10 pp.
  • Drmota, Michael. Random trees. An interplay between combinatorics and probability. SpringerWienNewYork, Vienna, 2009. xviii+458 pp. ISBN: 978-3-211-75355-2
  • Du, Rosena R. X.; Prodinger, Helmut. Notes on protected nodes in digital search trees. Appl. Math. Lett. 25 (2012), no. 6, 1025–1028.
  • Fill, James Allen; Kapur, Nevin. Transfer theorems and asymptotic distributional results for $m$-ary search trees. Random Structures Algorithms 26 (2005), no. 4, 359–391.
  • Heimbürger, Axel. Asymptotic distribution of two-protected nodes in m-ary search trees. Master thesis, Stockholm University and KTH (2014).
  • Holmgren, Cecilia; Janson, Svante. Limit laws for functions of fringe trees for binary search trees and random recursive trees. Electron. J. Probab. 20 (2015), no. 4, 1–51.
  • Janson, Svante. Functional limit theorems for multitype branching processes and generalized Pólya urns. Stochastic Process. Appl. 110 (2004), no. 2, 177–245.
  • Lew, William; Mahmoud, Hosam M. The joint distribution of elastic buckets in multiway search trees. SIAM J. Comput. 23 (1994), no. 5, 1050–1074.
  • Mahmoud, Hosam M. Evolution of random search trees. Wiley-Interscience Series in Discrete Mathematics and Optimization. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1992. xii+324 pp. ISBN: 0-471-53228-2
  • Mahmoud, Hosam M. The size of random bucket trees via urn models. Acta Inform. 38 (2002), no. 11-12, 813–838.
  • Mahmoud, Hosam M. Pólya urn models. Texts in Statistical Science Series. CRC Press, Boca Raton, FL, 2009. xii+290 pp. ISBN: 978-1-4200-5983-0
  • 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.
  • Mahmoud, Hosam M.; Ward, Mark Daniel. Asymptotic distribution of two-protected nodes in random binary search trees. Appl. Math. Lett. 25 (2012), no. 12, 2218–2222.
  • Mansour, Toufik. Protected points in $k$-ary trees. Appl. Math. Lett. 24 (2011), no. 4, 478–480.