Convergence theorems for some layout measures on random lattice and random geometric graphs

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. For full square lattices, we give optimal layouts for the problems still open. Our convergence theorems can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidian TSP on random points in the $d$-dimensional cube. As the considered layout measures are non-subadditive, we use percolation theory to obtain our results on random lattices and random geometric graphs. In particular, we deal with the subcritical regimes on these class of graphs.Postprint (published version

L-Drawings of Directed Graphs

We introduce L-drawings, a novel paradigm for representing directed graphs aiming at combining the readability features of orthogonal drawings with the expressive power of matrix representations. In an L-drawing, vertices have exclusive $x$- and $y$-coordinates and edges consist of two segments, one exiting the source vertically and one entering the destination horizontally. We study the problem of computing L-drawings using minimum ink. We prove its NP-completeness and provide a heuristics based on a polynomial-time algorithm that adds a vertex to a drawing using the minimum additional ink. We performed an experimental analysis of the heuristics which confirms its effectiveness.Comment: 11 pages, 7 figure

The critical transmitting range for Connectivity in Mobile Packet Radio Networks

It is known that the critical transmitting range for connectivity in stationary packet radio networks is $r=c sqrt{frac{log n}{n}}$, for some constant $c!>!0$, under the assumption that $n$ nodes are uniformly distributed in $R=[0,1]^2$. In this note, we investigate how mobility affects this asymptotic result. We consider the well known random waypoint mobility model, whose asymptotic node spatial distribution has been recently derived. We prove that as long as the spatial distribution has a non-null uniform component, the mobile critical transmitting range differs from the stationary one at most by a constant factor. On the contrary, when the uniform component is null there is an asymptotic gap between the mobile and stationary case, i.e. $r!gg!c sqrt{frac{log n}{n}}$ for any constant $c!>!0$. Hence, the asymptotic behavior of the mobile critical transmitting range depends on the choice of the mobility parameters

The Critical Transmitting Range for Connectivity in Sparse Wireless Ad Hoc Networks

In this paper, we analyze the critical transmitting range for connectivity in wireless ad hoc networks. More specifically, we consider the following problem: assume $n$ nodes, each capable of communicating with nodes within a radius of $r$, are randomly and uniformly distributed in a $d$-dimensional region with a side of length $l$; how large must the transmitting range $r$ be to ensure that the resulting network is connected with high probability? First, we consider this problem for stationary networks, and we provide tight upper and lower bounds on the critical transmitting range for one-dimensional networks, and non-tight bounds for two and three-dimensional networks. Due to the presence of the geometric parameter $l$ in the model, our results can be applied to dense {em as well as sparse} ad hoc networks, contrary to existing theoretical results that apply only to dense networks. We also investigate several related questions through extensive simulations. First, we evaluate the relationship between the critical transmitting range and the minimum transmitting range that ensures formation of a connected component containing a large fraction (e.g. 90%) of the nodes. Then, we consider the mobile version of the problem, in which nodes are allowed to move during a time interval and the value of $r$ ensuring connectedness for a given fraction of the interval must be determined. These results yield insight into how mobility affects connectivit

Convergence theorems for some layout measures on random lattice and random geometric graphs

This work deals with convergence theorems and bounds on the cost of several layout measures for lattice graphs, random lattice graphs and sparse random geometric graphs. For full square lattices, we give optimal layouts for the problems still open. Our convergence theorems can be viewed as an analogue of the Beardwood, Halton and Hammersley theorem for the Euclidian TSP on random points in the $d$-dimensional cube. As the considered layout measures are non-subadditive, we use percolation theory to obtain our results on random lattices and random geometric graphs. In particular, we deal with the subcritical regimes on these class of graphs

THE EFFECT OF INTERACTIONS BETWEEN PROTOCOLS AND PHYSICAL TOPOLOGIES ON THE LIFETIME OF WIRELESS SENSOR NETWORKS

Wireless sensor networks enable monitoring and control applications such weather sensing, target tracking, medical monitoring, road monitoring, and airport lighting. Additionally, these applications require long term and robust sensing, and therefore require sensor networks to have long system lifetime. However, sensor devices are typically battery operated. The design of long lifetime networks requires efficient sensor node circuits, architectures, algorithms, and protocols. In this research, we observed that most protocols turn on sensor radios to listen or receive data then make a decision whether or not to relay it. To conserve energy, sensor nodes should consider not listening or receiving the data when not necessary by turning off the radio. We employ a cross layer scheme to target at the network layer issues. We propose a simple, scalable, and energy efficient forwarding scheme, which is called Gossip-based Sleep Protocol (GSP). Our proposed GSP protocol is designed for large low-cost wireless sensor networks with low complexity to reduce the energy cost for every node as much as possible. The analysis shows that allowing some nodes to remain in sleep mode improves energy efficiency and extends network lifetime without data loss in the topologies such as square grid, rectangular grid, random grid, lattice topology, and star topology. Additionally, GSP distributes energy consumption over the entire network because the nodes go to sleep in a fully random fashion and the traffic forwarding continuously via the same path can be avoided