Trie partitioning process: limiting distributions
CAAP '86 Proceedings of the 11th colloquium on trees in algebra and programming
How many random questions are necessary to identify n distinct objects?
Journal of Combinatorial Theory Series A
A generalized suffix tree and its (un)expected asymptotic behaviors
SIAM Journal on Computing
Mellin transforms and asymptotics: harmonic sums
Theoretical Computer Science - Special volume on mathematical analysis of algorithms (dedicated to D. E. Knuth)
Asymptotic behavior of the Lempel-Ziv parsing scheme and digital search trees
Theoretical Computer Science - Special volume on mathematical analysis of algorithms (dedicated to D. E. Knuth)
Patricia tries again revisited
Journal of the ACM (JACM)
Asymptotic enumeration methods
Handbook of combinatorics (vol. 2)
Algorithms on strings, trees, and sequences: computer science and computational biology
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Analytical depoissonization and its applications
Theoretical Computer Science
The art of computer programming, volume 3: (2nd ed.) sorting and searching
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Average Case Analysis of Algorithms on Sequences
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Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
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A universal predictor based on pattern matching
IEEE Transactions on Information Theory
On the number of full levels in tries
Random Structures & Algorithms
Load-balancing performance of consistent hashing: asymptotic analysis of random node join
IEEE/ACM Transactions on Networking (TON)
B-tries for disk-based string management
The VLDB Journal — The International Journal on Very Large Data Bases
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We consider digital trees, namely, PATRICIA tries built from n random strings generated by an unbiased memoryless source (i.e., all symbols are equally likely). We study limit laws of the height, which is defined as the longest path in such trees. We shall identify three regions of the height distributions. In the region where most of the probability mass is concentrated, the asymptotic distribution of the PATRICIA height concentrates on one or two points around the most probable value k1 = ⌊log2 n + √2log2n - 3/2⌋ + 1. For most n all the mass is concentrated at k1, however, there exist subsequences of n such that the mass is on the two points k1 - 1 and k1, or k1 and k1 + 1. This should be contrasted with the limiting distribution of height for standard tries that is of extreme value type (i.e., double exponential distribution). We derive our results by a combination of analytic methods such as generating functions, Mellin transform, the saddle point method and ideas of applied mathematics such as linearization, asymptotic matching and the WKB principle. We make certain assumptions about the forms of some asymptotic expansions and their asymptotic matching. We also present some numerical verification of our results.