Asymptotic variance of random symmetric digital search trees

Abstract

Dedicated to the 60th birthday of Philippe Flajolet International audience Asymptotics of the variances of many cost measures in random digital search trees are often notoriously messy and involved to obtain. A new approach is proposed to facilitate such an analysis for several shape parameters on random symmetric digital search trees. Our approach starts from a more careful normalization at the level of Poisson generating functions, which then provides an asymptotically equivalent approximation to the variance in question. Several new ingredients are also introduced such as a combined use of the Laplace and Mellin transforms and a simple, mechanical technique for justifying the analytic de-Poissonization procedures involved. The methodology we develop can be easily adapted to many other problems with an underlying binomial distribution. In particular, the less expected and somewhat surprising n (logn)(2)-variance for certain notions of total path-length is also clarified.

An analytic approach to the asymptotic variance of trie statistics and related structures

“…The key, crucial step of our approach to the asymptotic variance of trie statistics is to introduce, as in [54], the corrected Poissonized variance of the form…”

“…Such a consideration results in simpler Fourier series expansions for the periodic functions that appear in the asymptotic approximations due to the no-cancelation character, especially in the symmetric case. We will enhance this approach by introducing the class of JS-admissible functions as in our previous paper [54], a notion formulated from Jacquet and Szpankowski's works on analytic de-Poissonization (see [61]) and mostly inspired from Hayman's classical work [50] on saddle-point method (see also [37,§VIII.5]), via which many asymptotic approximations can be derived by checking only simple criteria of admissibility. The combined use leads to a very effective, systematic approach that can be easily adapted for diverse contexts where a similar type of analytic problems is encountered; see Sections 5 and 6 for some examples.…”

“…We begin with recalling the definition and a few fundamental properties from [54] of JSadmissibility (a framework combining ideas from [50,61]).…”

“…Technical content of this paper. This paper is a sequel to [54] and we will develop an analytic approach that is especially useful for characterizing the asymptotics of the mean and the variance of additive statistics of random tries under the Bernoulli model; such statistics can often be computed recursively by…”

“…It turns out that the equation ( 6) covers almost all cases we collected (a few hundred of publications) in the analysis of BSPs the majority of which correspond to the case b = λ = 0. The cases when b = 0 and λ > 0 are thoroughly treated in [22,80,21,56], and the cases when b 1 are discussed in detail in [54] (see also the references cited there). We focus on b = λ = 0 in this paper.…”