arXiv subjects

Nader Mbarek

Publications and source records attributed to Nader Mbarek.

5 recordsLinked to original sources

arXiv2026

Constructing two completely independent spanning trees in the dual-cube

In this paper, we prove the existence of two completely independent spanning trees in the $n$-dimensional dual-cube $F_n$, a variant of the hypercube, for every $n \geq 5$. To this end, we use the hypercube structure of the clusters of $F_n$ to extend the construction of CIST from the $(n-1)$-dimensional hypercube to the dual-cube. In addition, we propose a recursive algorithm that builds the two trees while improving their diameters. Finally, we propose a conjecture concerning the existence of $k$ completely independent spanning trees in the dual-cube.

math.CO↗

arXiv2025

Completely Independent Spanning Trees in Split Graphs: Structural Properties and Complexity

We study completely independent spanning trees (CIST), \textit{i.e.}, trees that are both edge-disjoint and internally vertex-disjoint, in split graphs. We establish a correspondence between the existence of CIST in a split graph and some types of hypergraph colorings (panchromatic and bipanchromatic colorings) of its associated hypergraph, allowing us to obtain lower and upper bounds on the number of CIST. Using these relations, we prove that the problem of the existence of two CIST in a split graph is NP-complete. Finally, we formulate a conjecture on the bipanchromatic number of a hypergraph related to the results obtained for the number of CIST.

math.CO↗

arXiv2020

Leader Election And Local Identifiers For 3D Programmable Matter

In this paper, we present two deterministic leader election algorithms for programmable matter on the face-centered cubic grid. The face-centered cubic grid is a 3-dimensional 12-regular infinite grid that represents an optimal way to pack spheres (i.e., spherical particles or modules in the context of the programmable matter) in the 3-dimensional space. While the first leader election algorithm requires a strong hypothesis about the initial configuration of the particles and no hypothesis on the system configurations that the particles are forming, the second one requires fewer hypothesis about the initial configuration of the particles but does not work for all possible particles' arrangement. We also describe a way to compute and assign-local identifiers to the particles in this grid with a memory space not dependent on the number of particles. A-local identifier is a variable assigned to each particle in such a way that particles at distance at most each have a different identifier.

cs.DM↗

arXiv2018

Distributed leader election and computation of local identifiers for programmable matter

The context of this paper is programmable matter, which consists of a set of computational elements, called particles, in an infinite graph. The considered infinite graphs are the square, triangular and king grids. Each particle occupies one vertex, can communicate with the adjacent particles, has the same clockwise direction and knows the local positions of neighborhood particles. Under these assumptions, we describe a new leader election algorithm affecting a variable to the particles, called the k-local identifier, in such a way that particles at close distance have each a different k-local identifier. For all the presented algorithms, the particles only need a O(1)-memory space.

cs.DC↗

arXiv2017

Priority Levels Based Multi-hop Broadcasting Method for Vehicular Ad hoc Networks

This paper deals with broadcasting problem in vehicular ad hoc networks (VANETs). This communication mode is commonly used for sending safety messages and traffic information. However, designing an efficient broadcasting protocol is hard to achieve since it has to take into account some parameters related to the network environment, for example, the network density, in order to avoid causing radio interferences. In this paper, we propose a novel Autonomic Dissemination Method (ADM) which delivers messages in accordance with given priority and density levels. The proposed approach is based on two steps: an offline optimization process and an adaptation to the network characteristics. The first step uses a genetic algorithm to find solutions that fit the network context. The second one relies on the Autonomic Computing paradigm. ADM allows each vehicle to dynamically adapt its broadcasting strategy not only with respect to the network density, but also in accordance to the priority level of the message to send. The experimental results show that ADM effectively uses the radio resources even when there are globally many messages to send simultaneously. Moreover, ADM allows to increase the message delivery ratio and to reduce the latency and radio interferences.

cs.NI↗