We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.
The use of firewalls between business and process control networks is often suggested as an ideal solution for plant floor cyber security. But research shows that few firewalls are properly configured and that many control system security incidents bypass the firewall. If firewalls are to be effective, guidance on how to deploy them in industrial settings… (More)
We present new links between some remarkable martin-gales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process.
A classical random walk (S t , t ∈ N) is defined by S t := t n=0 X n , where (X n) are i.i.d. When the increments (X n) n∈N are a one-order Markov chain, a short memory is introduced in the dynamics of (S t). This so-called " persistent " random walk is nolonger Markovian and, under suitable conditions, the rescaled process converges towards the integrated… (More)
We consider boolean functions over n variables. Any such function can be represented (and computed) by a complete binary tree with and or or in the internal nodes and a literal in the external nodes, and many different trees can represent the same function, so that a fundamental question is related to the so-called complexity of a boolean function: L(f) :=… (More)
We consider a two colors Pólya urn with balance S. Assume it is a large urn i.e. the second eigenvalue m of the replacement matrix satisfies 1/2 < m/S ≤ 1. After n drawings, the composition vector has asymptotically a first deterministic term of order n and a second random term of order n m/S. The object of interest is the limit distribution of this random… (More)
Let m ≥ 3 be an integer. The so-called m-ary search tree is a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when m ≤ 26, the asymptotic behavior of the process is Gaussian, but for m ≥ 27 it is no longer… (More)
The purpose of this article is to show that the distribution of the longest fragment in the random multisection problem after k steps and the height of m-ary search trees (and some extensions) are not only closely related in a formal way but both can be asymptotically described with the same distribution function that has to be shifted in a proper way… (More)
We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process.