We investigate Δn, the distance between randomly selected pairs of nodes among n keys in a random trie, which is a kind of digital tree. Analytical techniques, such as the Mellin transform and an excursion between poissonization and depoissonization, capture small fluctuations in the mean and variance of these random distances. The mean increases logarithmically in the number of keys, but curiously enough the variance remains O(1), as n→∞. It is demonstrated that the centered random variable Δn*=Δn−⌊2log2n⌋ does not have a limit distribution, but rather oscillates between two distributions.
Primary Subjects: 05C05, 60C05
Secondary Subjects: 60F05, 68P05, 68P10, 68P20
References
Coffman, E. and Eve, J. (1970). File structures using hashing functions. Comm. ACM 13 427--432, 436.
De La Briandais, R. (1959). File searching using variable length keys. In Proceedings of the Western Joint Computer Conference 295--298. AFIPS, San Francisco, CA.
Fill, J., Mahmoud, H. and Szpankowski, W. (1996). On the distribution for the duration of a randomized leader election algorithm. Ann. Appl. Probab. 6 1260--1283.
Fredkin, E. (1960). Trie memory. Communications of the ACM 3 490--499.
Gutman, I. and Polansky, O. (1986). Mathematical Concepts in Organic Chemistry. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR861119
Jacquet, P. and Régnier, M. (1986). Trie Partitioning Process: Limiting Distributions. Lecture Notes in Comput. Sci. 214 196--210. Springer, New York.
Mathematical Reviews (MathSciNet):
MR848795
Jacquet, P. and Szpankowski, W. (1998). Analytical depoissonization and its applications. Theoret. Comput. Sci. 201 1--62.
Kirschenhofer, P. and Prodinger, H. (1986). Some further results on digital search trees. Lecture Notes in Comput. Sci. 226 177--185. Springer, New York.
Mathematical Reviews (MathSciNet):
MR864680
Knuth, D. (1998). The Art of Computer Programming 3: Sorting and Searching, 2nd ed. Addison--Wesley, Reading, MA.
Mathematical Reviews (MathSciNet):
MR378456
Kuipers, L. and Niederreiter, H. (1974). Uniform Distribution of Sequences. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR419394
Mahmoud, H. (1992). Evolution of Random Search Trees. Wiley, New York.
Mahmoud, H. (2000). Sorting: A Distribution Theory. Wiley, New York.
Mahmoud, H. and Neininger, R. (2003). Distribution of distances in random binary search trees. Ann. Appl. Probab. 13 253--276.
Neininger, R. (2002). The Wiener index of random trees. Combin. Probab. Comput. 11 587--597.
Panholzer, A. and Prodinger, H. (2004). Spanning tree size in random binary search trees. Ann. Appl. Probab. 14 718--733.
Prodinger, H. (2004). Oscillations in the analysis of algorithms and their cancellations. Journal of the Iranian Statistical Society 3 251--270.
Pittel, B. (1986). Paths in a random digital tree: Limiting distribution. Adv. in Appl. Probab. 18 139--155.
Mathematical Reviews (MathSciNet):
MR827333
Szpankowski, W. (2001). Average Case Analysis of Algorithms on Sequences. Wiley, New York.
Trinajstić, N. (1992). Chemical Graph Theory. CRC Press, Boca Raton, FL.
