Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T23:31:10.041Z Has data issue: false hasContentIssue false
  • English
  • Français

An Analysis of the Height of Tries with Random Weights on the Edges

Published online by Cambridge University Press:  01 March 2008

N. BROUTIN  and
L. DEVROYE
N. BROUTIN
Affiliation:
School of Computer Science, McGill University, Montreal H3A2K6Canada (e-mail: nicolas.broutin@m4x.org, luc@cs.mcgill.ca)
L. DEVROYE
Affiliation:
School of Computer Science, McGill University, Montreal H3A2K6Canada (e-mail: nicolas.broutin@m4x.org, luc@cs.mcgill.ca)
Get access Rights & Permissions [Opens in a new window]

Abstract

We analyse the weighted height of random tries built from independent strings of i.i.d. symbols on the finite alphabet {1, . . .d}. The edges receive random weights whose distribution depends upon the number of strings that visit that edge. Such a model covers the hybrid tries of de la Briandais and the TST of Bentley and Sedgewick, where the search time for a string can be decomposed as a sum of processing times for each symbol in the string. Our weighted trie model also permits one to study maximal path imbalance. In all cases, the weighted height is shown to be asymptotic to c log n in probability, where c is determined by the behaviour of the core of the trie (the part where all nodes have a full set of children) and the fringe of the trie (the part of the trie where nodes have only one child and form spaghetti-like trees). It can be found by maximizing a function that is related to the Cramér exponent of the distribution of the edge weights.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 2 , March 2008 , pp. 161 - 202
Copyright
Copyright © Cambridge University Press 2007

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]Alon, N., Spencer, J. and Erdős, P. (2000) The Probabilistic Method, 2nd edn, Wiley, New York.CrossRefGoogle Scholar
[2]Archibald, M. and Clément, J. (2006) Average depth in binary search tree with repeated keys. In Fourth Colloquium on Mathematics and Computer Science, pp. 309–320.Google Scholar
[3]Athreya, K. B. and Ney, P. E. (1972) Branching Processes, Springer, Berlin.Google Scholar
[4]Bentley, J. L. and Sedgewick, R. (1997) Fast algorithm for sorting and searching strings. In Eighth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 360–369.Google Scholar
[5]Broutin, N. (2007) {Shedding new light on random trees}. PhD thesis, McGill University, Montreal.Google Scholar
[6]Broutin, N. and Devroye, L. (2006) Large deviations for the weighted height of an extended class of trees. Algorithmica 46 271–297.Google Scholar
[7]Broutin, N. and Devroye, L. (2007) The core of a trie. In International Conference on Analysis of Algorithms, to appear.Google Scholar
[8]Broutin, N. and Devroye, L. (2007) The height of list tries and {TST}. In International Conference on Analysis of Algorithms, to appear.Google Scholar
[9]Broutin, N., Devroye, L. and McLeish, E. (2007) Weighted height of random trees. Manuscript.Google Scholar
[10]Brown, G. G. and Shubert, B. O. (1984) On random binary trees. Math. Oper. Research 9 43–65.Google Scholar
[11]Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493–507.CrossRefGoogle Scholar
[12]Christophi, C. A. and Mahmoud, H. M. (2007) One-sided variations on tries: Path imbalance, climbing, and key sampling. In Proc. International Conference on Analysis of Algorithms (AofA), pp. 301–310.CrossRefGoogle Scholar
[13]Clampett, H. A. (1964) Randomized binary searching with tree structures. Comm. Assoc. Comput. Mach. 7 163–165.Google Scholar
[14]Clément, J., Flajolet, P. and Vallée, B. (1998) The analysis of hybrid trie structures. In 9th Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM Press, Philadelphia, pp. 531–539.Google Scholar
[15]Clément, J., Flajolet, P. and Vallée, B. (2001) Dynamical source in information theory: A general analysis of trie structures. Algorithmica 29 307–369.Google Scholar
[16]Coffman, E. G. and Eve, J. (1970) File structures using hashing functions. Comm. Assoc. Comput. Mach. 13 427–436.Google Scholar
[17]Cramér, H. (1938) Sur un nouveau théorème-limite de la théorie des probabilités. In Colloque Consacré à la Théorie des Probabilités, Vol. 736, Hermann, Paris, pp. 5–23.Google Scholar
[18]de la Briandais, R. (1959) File searching using variable length keys. In Proc. Western Joint Computer Conference, Montvale, NJ, USA, AFIPS Press.Google Scholar
[19]Dembo, A. and Zeitouni, O. (1998) Large Deviation Techniques and Applications, 2nd edn, Springer.Google Scholar
[20]den Hollander, F. (2000) Large Deviations. AMS, Providence, RI.Google Scholar
[21]Deuschel, J.-D. and Stroock, D. W. (1989) Large Deviations, AMS, Providence, RI.Google Scholar
[22]Devroye, L. (1984) A probabilistic analysis of the height of tries and of the complexity of triesort. Acta Informatica 21 229–237.Google Scholar
[23]Devroye, L. (2002) Laws of large numbers and tail inequalities for random tries and patricia trees. J. Comput. Appl. Math. 142 27–37.CrossRefGoogle Scholar
[24]Devroye, L. (2005) Universal asymptotics for random tries and patricia trees. Algorithmica 42 11–29.Google Scholar
[25]Devroye, L., Szpankowski, W. and Rais, B. (1992) A note on the height of suffix trees. SIAM J. Comput. 21 48–53.Google Scholar
[26]Flajolet, P. (1983) On the performance evaluation of extendible hashing and trie searching. Acta Informatica 20 345–369.CrossRefGoogle Scholar
[27]Flajolet, P. (2006) The ubiquitous digital tree. In STACS 2006, Annual Symposium on Theoretical Aspects of Computer Science (B. Durand and W. Thomas, eds), Vol. 3884 of Lecture Notes in Computer Science, pp. 1–22.Google Scholar
[28]Flajolet, P. and Steyaert, J. M. (1982) A branching process arising in dynamic hashing, trie searching and polynomial factorization. In Automata, Languages and Programming: Proc. 9th ICALP Conference (M. Nielsen and E. M. Schmidt, eds), Vol. 140 of Lecture Notes in Computer Science, Springer, pp. 239–251.CrossRefGoogle Scholar
[29]Fredkin, E. (1960) Trie memory. Comm. Assoc. Comput. Mach. 3 490–499.Google Scholar
[30]Janson, S. (2006) Left and right pathlengths in random binary trees. Algorithmica 46 419–429.Google Scholar
[31]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley, New York.Google Scholar
[32]Kuba, M. and Panholzer, A. (2007) The left-right-imbalance of binary search trees. Theoret. Comput. Sci. 370 265–278.CrossRefGoogle Scholar
[33]Lynch, W. C. (1965) More combinatorial properties of certain trees. Comput. J. 7 299–302.CrossRefGoogle Scholar
[34]Mahmoud, H. (1992) Evolution of Random Search Trees, Wiley, New York.Google Scholar
[35]Mahmoud, H. M. (2007) Imbalance in random digital trees. Submitted.Google Scholar
[36]Park, G., Hwang, H. K., Nicodème, P. and Szpankowski, W. (2006) Profile of tries. Manuscript.Google Scholar
[37]Pittel, B. (1985) Asymptotic growth of a class of random trees. Ann. Probab. 13 414–427.Google Scholar
[38]Régnier, M. (1981) On the average height of trees in digital search and dynamic hashing. Inform. Process. Lett. 13 64–66.Google Scholar
[39]Rockafellar, R. (1970) Convex Analysis. Princeton University Press, Princeton, NJ.Google Scholar
[40]Sedgewick, R. and Flajolet, P. (1996) An Introduction to the Analysis of Algorithm, Addison-Wesley, Reading, MA.Google Scholar
[41]Szpankowski, W. (2001) Average Case Analysis of Algorithms on Sequences, Wiley, New York.Google Scholar
[42]Szpankowski, W. (1991) On the height of digital trees and related problems. Algorithmica 6 256–277.CrossRefGoogle Scholar