Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T22:06:23.968Z Has data issue: false hasContentIssue false
  • English
  • Français

DEGREE PROFILE OF m-ARY SEARCH TREES: A VEHICLE FOR DATA STRUCTURE COMPRESSION

Published online by Cambridge University Press:  14 December 2015

Ravi Kalpathy  and
Hosam Mahmoud
Ravi Kalpathy
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts Amherst, MA 01003, USA E-mail: rkalpathy@math.umass.edu
Hosam Mahmoud
Affiliation:
Department of Statistics, The George Washington University, Washington, DC 20052, USA E-mail: hosam@gwu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We revisit the random m-ary search tree and study a finer profile of its node outdegrees with the purpose of exploring possibilities of data structure compression. The analysis is done via Pólya urns. The analysis shows that the number of nodes of each individual node outdegree has a phase transition: Up to m = 26, the number of nodes of outdegree k, for k = 0, 1, …, m, is asymptotically normal; that behavior changes at m = 27. Based on the analysis, we propose a compact m-ary tree that offers significant space saving.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 1 , January 2016 , pp. 113 - 123
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Athreya, K. & Karlin, S. (1968). Embedding of urn schemes into continuous time Markov branching process and related limit theorems. The Annals of Mathematical Statistics 39: 1801–1817.CrossRefGoogle Scholar
2.Chauvin, B. & Pouyanne, N. (2004). m–ary Search trees when m ≥ 27: A strong asymptotics for the space requirements. Random Structures and Algorithms 24: 133–154.CrossRefGoogle Scholar
3.Cheon, G. & Shapiro, L. (2008). Protected points in ordered trees. Applied Mathematics Letters 21: 516–520.CrossRefGoogle Scholar
4.Cormen, T., Leiserson, C., Rivest, R., & Stein, C. (2001). Introduction to Algorithms. 2nd edn.New York: McGraw-Hill.Google Scholar
5.Devroye, L. (1991). Limit laws for local counters in random binary search trees. Random Structures & Algorithms 2: 303–315.CrossRefGoogle Scholar
6.Devroye, L. & Janson, S. (2014). Protected nodes and fringe subtrees in some random trees. Electronic Communications in Probability 19: Article 6, 1–10.CrossRefGoogle Scholar
7.Drmota, M., Gittenberger, B., & Panholzer, A. (2012). The degree distribution of thickened trees. In DMTCS Proceedings, Fifth Colloquium on Mathematics and Computer Science, AI: 149–162.Google Scholar
8.Du, R. & Prodinger, H. (2012). Notes on protected nodes in digital search trees. Applied Mathematics Letters 25: 1025–1028.CrossRefGoogle Scholar
9.Fill, J. & Kapur, N. (2004). The space requirement of m–ary search trees: Distributional asymptotics for m ≥ 27. In Proceedings of the Seventh Iranian Statistical Conference. Tehran, Iran.Google Scholar
10.Frobenius, G. (1912). Über Matrizen aus nicht negativen elementen. Sitzungsber, Königl. Preuss. Akad. Wiss. 456–477.Google Scholar
11.Holmgren, C. & Janson, S. (2015). Asymptotic distribution of two-protected nodes in ternary search trees. Electronic Journal of Probability 20: 1–20.CrossRefGoogle Scholar
12.Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya Urns. Stochastic Processes and Applications 110: 177–245.CrossRefGoogle Scholar
13.Janson, S. (2005). Asymptotic degree distribution in random recursive trees. Random Structures & Algorithms 26: 69–83.CrossRefGoogle Scholar
14.Knuth, D. (1998). The Art of Computer Programming, Vol. 3: Sorting and Searching. 3rd edn.Reading, MA: Addison–Wesley.Google Scholar
15.Kuba, M. & Panholzer, A. (2007). On the degree distribution of the nodes in increasing trees. Journal of Combinatorial Theory, Series A, 114: 597–618.CrossRefGoogle Scholar
16.Mahmoud, H. (1992). Evolution of Random Search Trees. New York: Wiley.Google Scholar
17.Mahmoud, H. (2000). Sorting: A Distribution Theory. New York: Wiley.CrossRefGoogle Scholar
18.Mahmoud, H. (2008). Pólya Urn Models. Orlando, FL: Chapman–Hall.CrossRefGoogle Scholar
19.Mahmoud, H. & Pittel, B. (1989). Analysis of the space of search trees under the random insertion algorithm. Journal of Algorithms 10: 52–75.CrossRefGoogle Scholar
20.Mahmoud, H. & Ward, M. (2012). Asymptotic distribution of two-protected nodes in random binary search trees. Applied Mathematics Letters 25: 2218–2222.CrossRefGoogle Scholar
21.Mahmoud, H. & Ward, M. (2015). Asymptotic properties of protected nodes in random recursive trees. Journal of Applied Probability 52: 290–297.CrossRefGoogle Scholar
22.Mansour, T. (2011). Protected points in k-ary trees. Applied Mathematics Letters 24: 478–480.CrossRefGoogle Scholar
23.Perron, O. (1907). Zur theorie der matrices. Mathematische Annalen 64: 248–263.CrossRefGoogle Scholar
24.Smythe, R. (1996). Central limit theorems for urn models. Stochastic Processes and Their Applications 65: 115–137.CrossRefGoogle Scholar