Pruned Discrete Random Samples

Let X i ,i ∈ ℕ, be independent and identically distributed random variables with values in ℕ0. We transform (‘prune’) the sequence {X 1,…,X n },n∈ ℕ, of discrete random samples into a sequence {0,1,2,…,Y n }, n∈ ℕ, of contiguous random sets by replacing X n+1 with Y n +1 if X n+1 >Y n . We consider the asymptotic behaviour of Y n as n→∞. Applications include path growth in digital search trees and the number of tables in Pitman's Chinese restaurant process if the latter is conditioned on its limit value.

Pruned Discrete Random Samples

Let Xi,i ∈ ℕ, be independent and identically distributed random variables with values in ℕ0. We transform (‘prune’) the sequence {X1,…,Xn},n∈ ℕ, of discrete random samples into a sequence {0,1,2,…,Yn}, n∈ ℕ, of contiguous random sets by replacing Xn+1 with Yn +1 if Xn+1 >Yn. We consider the asymptotic behaviour of Yn as n→∞. Applications include path growth in digital search trees and the number of tables in Pitman's Chinese restaurant process if the latter is conditioned on its limit value.

Approximate Counting via the Poisson-Laplace-Mellin Method

International audience Approximate counting is an algorithm that provides a count of a huge number of objects within an error tolerance. The first detailed analysis of this algorithm was given by Flajolet. In this paper, we propose a new analysis via the Poisson-Laplace-Mellin approach, a method devised for analyzing shape parameters of digital search trees in a recent paper of Hwang et al. Our approach yields a different and more compact expression for the periodic function from the asymptotic expansion of the variance. We show directly that our expression coincides with the one obtained by Flajolet. Moreover, we apply our method to variations of approximate counting, too.

Digital search trees with m trees: Level polynomials and insertion costs

Analysis of Algorithms International audience We adapt a novel idea of Cichon's related to Approximate Counting to the present instance of Digital Search Trees, by using m (instead of one) such trees. We investigate the level polynomials, which have as coefficients the expected numbers of data on a given level, and the insertion costs. The level polynomials can be precisely described, thanks to formulae from q-analysis. The asymptotics of expectation and variance of the insertion cost are fairly standard these days and done with Rice's method.