Minimum lethal sets in grids and tori under 3-neighbour bootstrap percolation

Let $\mathcal{r} ≥ 1$ be any non negative integer and let $G = (V, E)$ be any undirected graph in which a subset $D ⊆ V$ of vertices are initially infected. We consider the following process in which, at every step, each non-infected vertex with at least $\mathcal{r}$ infected neighbours becomes and an infected vertex never becomes non-infected. The problem consists in determining the minimum size $s_r (G)$ of an initially infected vertices set $D$ that eventually infects the whole graph $G$. Note that $s_1 (G)$ = 1 for any connected graph $G$. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that $s_1(G) = 1$ for any connected graph $G$. The case when $G$ is the $n × n$ grid $G_{n×n}$ and $\mathcal{r} = 2$ is well known and appears in many puzzles books, in particular due to the elegant proof that shows that $s_2(G_{n×n})$ = $n$ for all $n$ ∈ $\mathbb{N}$. We study the cases of square grids $G_{n×n}$ and tori $T_{n×n}$ when $\mathcal{r}$ ∈ {3, 4}. We show that $s_3(G_{n×n})$ = $\lceil\frac{n^2+2n+4}{3}\rceil$ for every $n$ even and that $\lceil\frac{n^2 +2n}{3}\rceil$ ≤ $s_3(G_ {n×n})$ ≤ $\lceil\frac{n^2 +2n}{3}\rceil$ + 1 for any $n$ odd. When $n$ is odd, we show that both bounds are reached, namely $s_3(G_{n×n})$ = $\lceil\frac{n^2 +2n}{3}\rceil$ if $n$ ≡ 5 (mod 6) or $n$ = 2$^p$ − 1 for any $p$ ∈ $\mathbb{N}^*$, and $s_3(G_{n×n})$ = $\lceil\frac{n^2 +2n}{3}\rceil$ + 1 if $n$ ∈ {9, 13}. Finally, for all $n$ ∈ $\mathbb{N}$, we give the exact expression of $s_4(G_{n×n})$ and of $s_r(T_{n×n})$ when $\mathcal{r}$ ∈ {3, 4}

Data gathering and personalized broadcasting in radio grids with interference

International audienceIn the gathering problem, a particular node in a graph, the base station , aims at receiving messages from some nodes in the graph. At each step, a node can send one message to one of its neighbors (such an action is called a call ). However, a node cannot send and receive a message during the same step. Moreover, the communication is subject to interference constraints, more precisely, two calls interfere in a step, if one sender is at distance at most dI from the other receiver. Given a graph with a base station and a set of nodes having some messages, the goal of the gathering problem is to compute a schedule of calls for the base station to receive all messages as fast as possible, i.e., minimizing the number of steps (called makespan). The gathering problem is equivalent to the personalized broadcasting problem where the base station has to send messages to some nodes in the graph, with same transmission constraints.In this paper, we focus on the gathering and personalized broadcasting problem in grids. Moreover, we consider the non-buffering model: when a node receives a message at some step, it must transmit it during the next step. In this setting, though the problem of determining the complexity of computing the optimal makespan in a grid is still open, we present linear (in the number of messages) algorithms that compute schedules for gathering with dI∈{0,1,2}. In particular, we present an algorithm that achieves the optimal makespan up to an additive constant 2 when dI=0. If no messages are “close” to the axes (the base station being the origin), our algorithms achieve the optimal makespan up to an additive constant 1 when dI=0, 4 when dI=2, and 3 when both dI=1 and the base station is in a corner. Note that, the approximation algorithms that we present also provide approximation up to a ratio 2 for the gathering with buffering. All our results are proved in terms of personalized broadcasting

Graphes et Télécommunications

International audienceDe nombreux problèmes d'optimisation se posent dans les réseaux de télécommunications. Pour les résoudre, on utilise souvent les outils de la théorie des graphes. Un graphe est constitué d'un ensemble de sommets qui modélisent les noeuds et d'un ensemble d'arêtes qui modélisent les liens du réseau. Les noeuds peuvent être les abonnés ou utilisateurs du réseau mais aussi des équipements comme des ordinateurs, serveurs, routeurs, antennes, des pages Web... Les liens peuvent être des liens physiques (câbles, fibres), des liens radio voire des liens virtuels comme les liens hypertextes.Le rôle des réseaux de (télé)communications est notamment d’échanger de l’information. Par exemple, un mail entre deux personnes, le chargement d’une vidéo à partir d’un serveur ou l’accès à une page web sont appelés des requêtes entre deux nœuds d’un réseau. Router une requête consiste à trouver dans le graphe représentant le réseau un chemin entre les deux nœuds qui communiquent. Par exemple, dans la Figure 3(a), une requête entre les nœuds s et t est routée par le plus court chemin de longueur 2 (via le sommet a). Trouver un chemin fait partie des problèmes dits « faciles ». Ils peuvent être résolus en temps polynomial en la taille du réseau (voir l’article « TRANSPORT, FLOTS ET COUPES »). Dans la réalité, plusieurs requêtes doivent être routées simultanément, en satisfaisant de nombreuses contraintes (capacité ou bande passante limitée, interférences, traitement des informations,...). Ceci rend les problèmes bien plus compliqués. Nous présentons des problèmes de conception du réseau et de gestion via divers exemples en nous focalisant sur leurs applications, leur description, et leur modélisation sous forme de problèmes de graphes

Minimum lethal sets in grids and tori under 3-neighbour bootstrap percolation

International audienceLet r ≥ 1 be any non negative integer and let G = (V, E) be any undirected graph in which a subset D ⊆ V of vertices are initially infected. We consider the process in which, at every step, each non-infected vertex with at least r infected neighbours becomes infected and an infected vertex never becomes non-infected. The problem consists in determining the minimum size s r (G) of an initially infected vertices set D that eventually infects the whole graph G. This problem is closely related to cellular automata, to percolation problems and to the Game of Life studied by John Conway. Note that s 1 (G) = 1 for any connected graph G. The case when G is the n × n grid, G n×n , and r = 2 is well known and appears in many puzzle books, in particular due to the elegant proof that shows that s 2 (G n×n) = n for all n ∈ N. We study the cases of square grids, G n×n , and tori, T n×n , when r ∈ {3, 4}. We show that s 3 (G n×n) = n 2 +2n+4 3 for every n even and that n 2 +2n 3 ≤ s 3 (G n×n) ≤ n 2 +2n 3 + 1 for any n odd. When n is odd, we show that both bounds are reached, namely s 3 (G n×n) = n 2 +2n 3 if n ≡ 5 (mod 6) or n = 2 p-1 for any p ∈ N * , and s 3 (G n×n) = n 2 +2n 3 + 1 if n ∈ {9, 13}. Finally, for all n ∈ N, we give the exact expression of s 3 (T n×n)

Data Gathering and Personalized Broadcasting in Radio Grids with Interferences ✩

In the gathering problem, a particular node in a graph, the base station, aims at receiving messages from some nodes in the graph. At each step, a node can send one message to one of its neighbor (such an action is called a call). However, a node cannot send and receive a message during the same step. Moreover, the communication is subjet to interference constraints, more precisely, two calls interfere in a step, if one sender is at distance at most dI from the other caller. Given a graph with a base station and a set of nodes having some messages, the goal of the gathering problem is to compute a schedule of calls for the base station to receive all messages as fast as possible, i.e., minimizing the number of steps (called makespan). The gathering problem is equivalent to the personalized broadcasting problem where the base station has to send messages to some nodes in the graph, with same transmission constraints. In this paper, we focus on the gathering and personalized broadcasting problem in grids. Moreover, we consider the non-buffering model: when a node receives a message at some step, it must transmit it during the next step. In this setting, though the problem of determining the complexity of computing the optimal makespan in a grid is still open, we present linear (in the number of messages) algorithms that compute schedules for gathering with dI ∈ {0, 1, 2}. In particular, we present an algorithm that achieves the optimal makespan up to an additive constant 2 when dI = 0. If no messages are “close ” to the axes (the base station being the origin), our algorithms achieve the optimal makespa

FIB-SEM based 3D tomography of micro-electronic components: Application to automotive high-definition LED lighting systems

International audienceIn new high definition lighting modules, assembling more than thousand elements (pixels) with a single mm-sized motherboard is achieved by reflow soldering. Lead-free solders may constitute a weak part of electronic components and LED assemblies. During reflow, the distance between pixels and motherboard controls the solder heights (10 μm). After assembly, all the pixels are in a single plane. To ensure constant solder heights, this plane has to be parallel to the motherboard. A small disorientation during reflow may lead to severe solder damage prior to any use of the opto-electronic component. Present work presents a novel non-destructive method for characterizing any chosen solder in the assembled component without unpacking the latter. A femtosecond motorized laser source is used for digging a hole in the LED above the selected solder. Then a Focused Ion Beam facility allows to mill around this specific solder. The same FIB facility allows making several hundred slices of controlled thickness and orientation in the solder material. During this slicing operation, high-resolution inspection is done by FEG-SEM (Field Emission Gun - Scanning Electron Microscope) imaging. 3D tomographic reconstruction of the solder is obtained by classical commercial software AVIZO. Finally, 3D damage in solder material is illustrated by example results