Lali Barrière

Robust position-based routing in wireless ad hoc networks with irregular transmission ranges†

Several papers considered the problem of routing in ad hoc wireless networks using the positions of the mobile hosts. Perimeter routing1, 2 gives an algorithm that guarantees delivery of messages in such networks without the use of flooding of control packets. However, this protocol is likely to fail if the transmission ranges of the mobile hosts vary because of natural or man-made obstacles. It may fail because either some connections are not considered, which effectively results in a disconnection of the network, or because some crossing connections are used, which could misdirect the message. In this paper, we describe a robust routing protocol, a variant of perimeter routing, which tolerates up to 40% of variation in the transmission ranges of the mobile hosts. More precisely, our protocol guarantees message delivery in a connected ad hoc wireless network without the use of message flooding whenever the ratio of the maximum transmission range to the minimum transmission range is at most √2. Copyright

Robust position-based routing in wireless Ad Hoc networks with unstable transmission ranges

Several papers showed how to perform routing in ad hoc wireless networks based on the positions of the mobile hosts. However, all these protocols are likely to fail if the transmission ranges of the mobile hosts vary due to natural or man-made obstacles or weather conditions. These protocols may fail because in routing either some connections are not considered which effectively results in disconnecting the network, or the use of some connections causes livelocks. In this paper, we describe a robust routing protocol that tolerates up to roughly 40% of variation in the transmission ranges of the mobile hosts. More precisely, our protocol guarantees message delivery in a connected adhoc network whenever the ratio of the maximum transmission range to the minimum transmission range is at most √2.

Searching Is Not Jumping

This paper is concerned with the graph searching game: we are given a graph containing a fugitive (or lost) entity or item; the goal is to clear the edges of the graph, using searchers; an edge is clear if it cannot contain the searched entity, contaminated otherwise. The search numbers(G) of a graph G is the smallest number of searchers required to “clear” G. A search strategy is monotone (m) if no recontamination ever occurs. It is connected (c) if the set of clear edges always forms a connected subgraph. It is internal (i) if the removal of searchers is not allowed (i.e., searchers can not jump but only move along the edges). The difficulty of the “connected” version and of the “monotone internal” version of the graph searching problem comes from the fact that none of these problems is minor closed for arbitrary graphs, as opposed to all known variants of graph searching. We prove that there is a unique chain of inequalities linking all the search numbers above. More precisely, for any graph G, s(G) = is(G) = ms(G) ≤ mis(G) ≤ cs(G) = ics(G) ≤ mcs(G) = mics(G). The first two inequalities can be strict. Motivated by the fact that connected graph searching and monotone internal graph searching are both minor closed in trees, we provide a complete characterization of the set of trees that can be cleared by a given number of searchers. In fact, we show that, in trees, there is exactly one obstruction for monotone internal search, as well as for connected search, and this obstruction is the same for the two problems. This allows us to prove that, for any tree T, mis(T)= cs(T) and cs(T) ≤ 2 s(T) − 2, using that ics(T)=mcs(T). This implies that there are only two different search numbers, and these search numbers differ by a factor of 2 at most.

Rendezvous and Election of Mobile Agents: Impact of Sense of Direction

Abstract Consider a collection of r identical asynchronous mobile agents dispersed on an arbitrary anonymous network of size n. The agents all execute the same protocol and move from node to neighboring node. At each node there is a whiteboard where the agents can write and read from. The topology of the network is unknown to the agents. We examine the problems of rendezvous (i.e., having the agents gather in the same node) and election (i.e., selecting a leader among those agents). These two problems are computationally equivalent in the context examined here. We study conditions for the existence of deterministic generic solutions, i.e., algorithms that solve the two problems regardless of the network topology and the initial placement of the agents. In particular, we study the impact of edge-labeling on the existence of such solutions. Rendezvous and election are unsolvable (i.e., there are no deterministic generic solutions) if gcd(r,n) > 1, regardless of whether or not the edge-labeling has sense of direction. On the other hand, if gcd(r,n) = 1 then the initial placement of the robots in the network creates topological asymmetries that could be exploited to solve the problems. We prove that these asymmetries can be exploited if the edge-labeling has sense of direction, but cannot if the edge-labeling is arbitrary. The possibility proof is constructive: we present a solution protocol and prove its correctness. The protocol, among other features, uses a dynamic naming mechanism based on sense of direction to overcome the complete anonymity of the system.

The generalized hierarchical product of graphs

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.

UNIFORM SCATTERING OF AUTONOMOUS MOBILE ROBOTS IN A GRID

We consider the uniform scattering problem for a set of autonomous mobile robots deployed in a grid network: starting from an arbitrary placement in the grid, using purely localized computations, the robots must move so to reach in finite time a state of static equilibrium in which they cover uniformly the grid. The theoretical quest is on determining the minimal capabilities needed by the robots to solve the problem. We prove that uniform scattering is indeed possible even for very weak robots. The proof is constructive. We present a provably correct protocol for uniform self-deployment in a grid. The protocol is fully localized, collision-free, and it makes minimal assumptions; in particular: (1) it does not require any direct or explicit communication between robots; (2) it makes no assumption on robots synchronization or timing, hence the robots can be fully asynchronous in all their actions; (3) it requires only a limited visibility range; (4) it uses at each robot only a constant size memory, hence computationally the robots can be simple Finite-State Machines; (5) it does not need a global localization system but only orientation in the grid (e.g., a compass); (6) it does not require identifiers, hence the robots can be anonymous and totally identical.

Fractality and the small-world effect in Sierpinski graphs

Although some real networks exhibit self-similarity, there is no standard definition of fractality in graphs. On the other hand, the small-world phenomenon is one of the most important common properties of real interconnection networks. In this paper we relate these two properties. In order to do so, we focus on the family of Sierpinski networks. For the Sierpinski gasket, the Sierpinski carpet and the Sierpinski tetra, we give the basic properties and we calculate the box-counting dimension as a measure of their fractality. We also define a deterministic family of graphs, which we call small-world Sierpinski graphs. We show that our construction preserves the structure of Sierpinski graphs, including its box-counting dimension, while the small-world phenomenon arises. Thus, in this family of graphs, fractality and small-world effect are simultaneously present.

Deterministic hierarchical networks

It has been shown that many networks associated with complex systems are small-world (they have both a large local clustering coefficient and a small diameter) and also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances the probabilistic and simulation techniques and, therefore, it provides a better understanding of the modeled systems. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems.It has been shown that many networks associated with complex systems are small-world (they have both a large local clustering coefficient and a small diameter) and also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances the probabilistic and simulation techniques and, therefore, it provides a better understanding of the modeled systems. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems.

Fault-tolerant routings in chordal ring networks

This paper studies routing vulnerability in networks modeled by chordal ring graphs. In a chordal ring graph, the vertices are labeled in ℤ2n and each even vertex i is adjacent to the vertices i + a, i + b, i, + c, where a, b, and c are different odd integers. Our study is based on a geometrical representation that associates to the graph a tile which periodically tessellates the plane. Using this approach, we present some previous results on triple-loop graphs, including an algorithm to calculate the coordinates of a given vertex in the tile. Then, an optimal consistent fault-tolerant routing of shortest paths is defined for a chordal ring graph with odd diameter and maximum order. This is accomplished by associating to the chordal ring graph a triple-loop one. When some faulty elements are present in the network, we give a method to obtain central vertices, which are vertices that can be used to reroute any communication affected by the faulty elements. This implies that the diameter of the corresponding surviving route graph is optimum.

On the Fiedler value of large planar graphs

Abstract The Fiedler value λ 2 , also known as algebraic connectivity, is the second smallest Laplacian eigenvalue of a graph. We study the maximum Fiedler value among all planar graphs G with n vertices, denoted by λ 2 max , and we show the bounds 2 + Θ ( 1 n 2 ) ⩽ λ 2 max ⩽ 2 + O ( 1 n ) . We also provide bounds on the maximum Fiedler value for the following classes of planar graphs: Bipartite planar graphs, bipartite planar graphs with minimum vertex-degree 3, and outerplanar graphs. Furthermore, we derive almost tight bounds on λ 2 max for two more classes of graphs, those of bounded genus and K h -minor-free graphs.