arXiv subjects

Jean-François Marckert

Publications and source records attributed to Jean-François Marckert.

40 records · Page 3Linked to original sources

arXiv2006

A Quasi-Optimal Leader Election Algorithm in Radio Networks with Log-Logarithmic Awake Time Slots

Radio networks (RN) are distributed systems (\textit{ad hoc networks}) consisting in $n \ge 2$ radio stations. Assuming the number $n$ unknown, two distinct models of RN without collision detection (\textit{no-CD}) are addressed: the model with \textit{weak no-CD} RN and the one with \textit{strong no-CD} RN. We design and analyze two distributed leader election protocols, each one running in each of the above two (no-CD RN) models, respectively. Both randomized protocols are shown to elect a leader within $\BO(\log{(n)})$ expected time, with no station being awake for more than $\BO(\log{\log{(n)}})$ time slots (such algorithms are said to be \textit{energy-efficient}). Therefore, a new class of efficient algorithms is set up that matchthe $Ω(\log{(n)})$ time lower-bound established by Kushilevitz and Mansour.

cs.DC↗

arXiv2006

Globally centered discrete snakes

We consider branching random walks built on Galton-Watson trees with offspring distribution having a bounded support, conditioned to have $n$ nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of "globally centered discrete snake'' that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when $n$ goes to $+\infty$, "globally centered discrete snakes'' converge to the Brownian snake. The proof relies on a precise study of the "lineage'' of the nodes in a Galton-Watson tree conditioned by the size, and their links with a multinomial process. Some consequences concerning Galton-Watson trees conditioned by the size are also derived.

math.PR↗

arXiv2006

On the reduction of a random basis

For $g < n$, let $b\_1,...,b\_{n-g}$ be $n - g$ independent vectors in $\mathbb{R}^n$ with a common distribution invariant by rotation. Considering these vectors as a basis for the Euclidean lattice they generate, the aim of this paper is to provide asymptotic results when $n\to +\infty$ concerning the property that such a random basis is reduced in the sense of {\sc Lenstra, Lenstra & Lovász}. The proof passes by the study of the process $(r\_{g+1}^{(n)},r\_{g+2}^{(n)},...,r\_{n-1}^{(n)})$ where $r\_j^{(n)}$ is the ratio of lengths of two consecutive vectors $b^*\_{n-j+1}$ and $b^*\_{n-j}$ built from $(b\_1,...,b\_{n-g})$ by the Gram--Schmidt orthogonalization procedure, which we believe to be interesting in its own. We show that, as $n\to+\infty$, the process $(r\_j^{(n)}-1)\_j$ tends in distribution in some sense to an explicit process $({\mathcal R}\_j -1)\_j$; some properties of this latter are provided.

math.PR↗

arXiv2006

Asymptotics of Bernoulli random walks, bridges, excursions and meanders with a given number of peaks

A Bernoulli random walk is a random trajectory starting from 0 and having i.i.d. increments, each of them being $+1$ or -1, equally likely. The other families cited in the title are Bernoulli random walks under various conditionings. A peak in a trajectory is a local maximum. In this paper, we condition the families of trajectories to have a given number of peaks. We show that, asymptotically, the main effect of setting the number of peaks is to change the order of magnitude of the trajectories. The counting process of the peaks, that encodes the repartition of the peaks in the trajectories, is also studied. It is shown that suitably normalized, it converges to a Brownian bridge which is independent of the limiting trajectory. Applications in terms of plane trees and parallelogram polyominoes are also provided.

math.PR↗