Profiles of PATRICIA Tries

Abstract

A PATRICIA trie is a trie in which non-branching paths are compressed. The external profile B n,k , defined to be the number of leaves at level k of a PATRICIA trie on n nodes, is an important "summarizing" parameter, in terms of which several other parameters of interest can be formulated. Here we derive precise asymptotics for the expected value and variance of B n,k , as well as a central limit theorem with error bound on the characteristic function, for PATRICIA tries on n infinite binary strings generated by a memoryless source with bias p > 1/2 for k ∼ α log n with α ∈ (1/ log(1/q) + , 1/ log(1/p) − ) for any fixed > 0. In this range, E[B n,k ] = Θ(Var[B n,k ]), and both are of the form Θ(n β(α) / √ log n), where the Θ hides bounded, periodic functions in log n whose Fourier series we explicitly determine. The compression property leads to extra terms in the Poisson functional equations for the profile which are not seen in tries or digital search trees, resulting in Mellin transforms which are only implicitly given in terms of the moments of B m,j for various m and j. Thus, the proofs require information about the profile outside the main range of interest. Our derivations rely on analytic techniques, including Mellin transforms, analytic de-Poissonization, the saddle point method, and careful bounding of complex functions.

Asymmetric Rényi Problem

“…Here we recall some facts, worked out in detail in [15], which will form the starting point of the analysis in the present paper. In order to derive our main results, we need proper asymptotic information about E[B n,k ] and Var[B n,k ] at the boundaries of this region.…”

“…If, on the other hand, all objects are included or all are excluded from the first potential query (which happens with probability p n + q n ), then the partition element splitting constraint on the queries applies, the potential query is ignored as inconclusive, and the contribution is μ n,k . The tools that we use to solve this recurrence (for details see [15,17]) are similar to those of the analyses for digital trees [23] such as tries and digital search trees (though the analytical details differ significantly). We first derive a functional equation for the Poisson transform…”

“…where A k (s) is an infinite series arising from the contributions coming from the functionW k,G (z), and the fundamental strip ofG k (z) contains (−k − 1, ∞). It involves unknown μ m, j − μ m, j−1 for various m and j (see [15,16]), that is,…”

“…We now discuss our new results concerning the probabilistic behaviour of the external profile. We establish in [15,17] precise asymptotic expressions for the expected value and variance of B n,k in the central range, that is, with k ∼ α log n, where, for any fixed ε > 0, α ∈ (1/ log(1/q) + ε, 1/ log(1/p) − ε) (the left and right endpoints of this interval as ε → 0 are associated with F n and H n , respectively). Specifically, it was shown that both the mean and the variance are of the same (explicit) polynomial order of growth (with respect to n).…”

PATRICIA Bridges

“…We show in Proposition 3.7 that ( νR n ) n∈N is a Markov chain which we call a PATRICIA chain. Features of PATRICIA trees for random inputs were first studied in [15] and this topic has since been the subject of quite a large literature (see, for example, [17], [19], [29], [20], [18], [28], [10], [6], [14], [5], [3], [23], [2], [4], [27], [13]).…”