# On the Stack-Size of General Tries

Jérémie Bourdon; Markus Nebel; Brigitte Vallée

RAIRO - Theoretical Informatics and Applications (2010)

- Volume: 35, Issue: 2, page 163-185
- ISSN: 0988-3754

## Access Full Article

top## Abstract

top## How to cite

topBourdon, Jérémie, Nebel, Markus, and Vallée, Brigitte. "On the Stack-Size of General Tries." RAIRO - Theoretical Informatics and Applications 35.2 (2010): 163-185. <http://eudml.org/doc/222100>.

@article{Bourdon2010,

abstract = {
Digital trees or tries are a general purpose flexible data structure
that implements dictionaries built on words. The present paper is focussed
on the average-case analysis of
an important parameter of this
tree-structure, i.e., the stack-size.
The stack-size of a tree is the memory needed
by a storage-optimal preorder traversal. The analysis is carried out
under a general model in which words are produced by
a source (in the information-theoretic sense)
that emits symbols.
Under some natural assumptions that encompass all commonly
used data models (and more), we obtain a precise average-case
and probabilistic analysis of
stack-size.
Furthermore, we study
the dependency between the stack-size and the ordering on symbols in
the alphabet:
we establish that, when the source emits independent symbols, the
optimal ordering arises
when the most probable symbol is the last one in this order.
},

author = {Bourdon, Jérémie, Nebel, Markus, Vallée, Brigitte},

journal = {RAIRO - Theoretical Informatics and Applications},

keywords = {Average-case analysis of data-structures; information theory; trie; Mellin analysis.; digital trees; stack-size},

language = {eng},

month = {3},

number = {2},

pages = {163-185},

publisher = {EDP Sciences},

title = {On the Stack-Size of General Tries},

url = {http://eudml.org/doc/222100},

volume = {35},

year = {2010},

}

TY - JOUR

AU - Bourdon, Jérémie

AU - Nebel, Markus

AU - Vallée, Brigitte

TI - On the Stack-Size of General Tries

JO - RAIRO - Theoretical Informatics and Applications

DA - 2010/3//

PB - EDP Sciences

VL - 35

IS - 2

SP - 163

EP - 185

AB -
Digital trees or tries are a general purpose flexible data structure
that implements dictionaries built on words. The present paper is focussed
on the average-case analysis of
an important parameter of this
tree-structure, i.e., the stack-size.
The stack-size of a tree is the memory needed
by a storage-optimal preorder traversal. The analysis is carried out
under a general model in which words are produced by
a source (in the information-theoretic sense)
that emits symbols.
Under some natural assumptions that encompass all commonly
used data models (and more), we obtain a precise average-case
and probabilistic analysis of
stack-size.
Furthermore, we study
the dependency between the stack-size and the ordering on symbols in
the alphabet:
we establish that, when the source emits independent symbols, the
optimal ordering arises
when the most probable symbol is the last one in this order.

LA - eng

KW - Average-case analysis of data-structures; information theory; trie; Mellin analysis.; digital trees; stack-size

UR - http://eudml.org/doc/222100

ER -

## References

top- J. Clément, P. Flajolet and B. Vallée, Dynamical Sources in Information Theory: A General Analysis of Trie Structures. Algorithmica29 (2001) 307-369. Zbl1035.68039
- H. Daudé, P. Flajolet and B. Vallée, An average-case analysis of the Gaussian algorithm for lattice reduction. Combina. Probab. Comput.6 (1997) 397-433. Zbl0921.11072
- N.G. De Bruijn, D.E. Knuth and S.O. Rice, The average height of planted plane trees, Graph Theory and Computing. Academic Press (1972) 15-22. Zbl0247.05106
- L. Devroye and P. Kruszewski, On the Horton-Strahler number for Random Tries. RAIRO: Theoret. Informatics Appl.30 (1996) 443-456. Zbl0867.68087
- P. Flajolet, On the performance of evaluation of extendible hashing and trie searching. Acta Informatica20 (1983) 345-369. Zbl0515.68048
- P. Flajolet, X. Gourdon and P. Dumas, Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci.144 (1995) 3-58. Zbl0869.68057
- P. Flajolet and C. Puech, Partial match retrieval of multidimensional data. J. ACM33 (1986) 371-407.
- E.H. Fredkin, Trie Memory. Comm. ACM3 (1990) 490-500.
- G.H. Gonnet and R. Baeza-Yates, Handbook of Algorithms and Data Structures: in Pascal and C. Addison-Wesley (1991). Zbl0719.68001
- A. Grothendieck, Produit tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 16 (1955).
- A. Grothendieck, La Théorie de Fredholm. Bull. Soc. Math. France 84, 319-384.
- P. Jacquet and W. Szpankowski, Analytical Depoissonization and its Applications. Theoret. Comput. Sci. 201 in ``Fundamental Study'' (1998) 1-62. Zbl0902.68087
- P. Kirschenhofer and H. Prodinger, On the Recursion Depth of Special Tree Traversal Algorithms. Inform. and Comput.74 (1987) 15-32. Zbl0625.68048
- R. Kemp, The average height of Zbl06657087
- D.E. Knuth, The Art of Computer Programming, Vol. 3: Sorting and Searching. Addison-Wesley (1973). Zbl0302.68010
- M; Krasnoselskii, Positive solutions of operator equations. P. Noordhoff, Groningen (1964).
- H.M. Mahmoud, Evolution of Random Search Trees. Wiley-Interscience Series (1992). Zbl0762.68033
- M.E. Nebel, The Stack-Size of Tries, a Combinatorial Study. Theoret. Comput. Sci. (to appear). Zbl0988.68137
- M.E. Nebel, The Stack-Size of Uniform Random Tries Revisited (submitted). Zbl0995.68071
- M.E. Nebel, On the Horton-Strahler Number for Combinatorial Tries. RAIRO: Theoret. Informatics Appl.34 (2000) 279-296. Zbl0966.05019
- M. Régnier, Trie hashing analysis, in Proc. 4th Int.Conf. Data Eng.. Los Angeles, CA (1988) 377-387.
- M. Régnier, On the average height of trees in in digital search and dynamic hashing. Inform. Process. Lett.13 (1982) 64-66. Zbl0472.68058
- R.L. Rivest, Partial match retrieval algorithms. SIAM J. Comput.5 (1976) 19-50. Zbl0331.68064
- R. Sedgewick, Algorithms. Addison-Wesley (1988).
- W. Szpankowski, On the height of digital trees and related problem. Algorithmica6 (1991) 256-277. Zbl0711.68035
- W. Szpankowski, Some results on Zbl0637.68072
- L. Trabb Pardo, Set representation and set intersection, Technical Report. Stanford University (1998).
- B. Vallée, Dynamical Sources in Information Theory: Fundamental Intervals and Word Prefixes. Algorithmica29 (2001) 162-306. Zbl1009.94003
- X.G. Viennot, Trees Everywhere, in Proc. CAAP'90. Springer, Lecture Notes in Comput. Sci. 431 (1990) 18-41.
- A. Yao, A note on the analysis of extendible hashing. Inform. Process. Lett.11 (1980) 84-86. Zbl0447.68077

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.