On the Edge-length Ratio of Outerplanar Graphs

International audienceWe show that any outerplanar graph admits a planar straight-line drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any ε > 0 there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than 2 −ε. We also show that this ratio cannot be bounded if the embeddings of the outerplanar graphs are given

Footprint Modeling and Connectivity Analysis for Wireless Sensor Networks

A wireless sensor network is a network consisting of spatially distributed, sometimeautonomous sensors, communicating wirelessly to cooperatively achieve some task. For example, a wireless sensor network may be used for habitat monitoring to ascertain the environment’s temperature, pressure, humidity, etc. In order for a wireless sensor network to provide such data, one needs to ensure there is connectivity between nodes. That is, nodes can communicate to exchange information. To analyze connectivity between sensors, the radio communication range of each sensor, also called the communication footprint, needs to be known. To date, the models used to analyze a sensor’s radio communication footprint have been overly simplistic (i.e., isotropic) and thus yield results not found in practice. Footprints are highly dependent on the deployment environments, which are typically heterogeneous and non-isotropic in structure. In this work, a ‘weak-monotonicity’ (W-M) model is leveraged to represent a footprint’s non-isotropic behavior. The work also considers the heterogeneity of the environment through the use of the log-normal shadowing model. In particular, the usable percentage of the W-M footprint (the area where the power exceeds the receiver threshold) in such environments is considered through analysis and simulation. We then develop an enhanced footprint model which overlays multiple W-M patterns and use this method to represent experimental propagation data. The work also considers the use of graph theory methods to analyze the connectivity of randomly deployed networks in nonhomogeneous, non-isotropic environments

Effective algorithms and protocols for wireless networking: a topological approach

Much research has been done on wireless sensor networks. However, most protocols and algorithms for such networks are based on the ideal model Unit Disk Graph (UDG) model or do not assume any model. Furthermore, many results assume the knowledge of location information of the network. In practice, sensor networks often deviate from the UDG model significantly. It is not uncommon to observe stable long links that are more than five times longer than unstable short links in real wireless networks. A more general network model, the quasi unit-disk graph (quasi-UDG) model, captures much better the characteristics of wireless networks. However, the understanding of the properties of general quasi-UDGs has been very limited, which is impeding the design of key network protocols and algorithms. In this dissertation we study the properties for general wireless sensor networks and develop new topological/geometrical techniques for wireless sensor networking. We assume neither the ideal UDG model nor the location information of the nodes. Instead we work on the more general quasi-UDG model and focus on figuring out the relationship between the geometrical properties and the topological properties of wireless sensor networks. Based on such relationships we develop algorithms that can compute useful substructures (planar subnetworks, boundaries, etc.). We also present direct applications of the properties and substructures we constructed including routing, data storage, topology discovery, etc. We prove that wireless networks based on quasi-UDG model exhibit nice properties like separabilities, existences of constant stretch backbones, etc. We develop efficient algorithms that can obtain relatively dense planar subnetworks for wireless sensor networks. We also present efficient routing protocols and balanced data storage scheme that supports ranged queries. We present algorithmic results that can also be applied to other fields (e.g., information management). Based on divide and conquer and improved color coding technique, we develop algorithms for path, matching and packing problem that significantly improve previous best algorithms. We prove that it is unlikely for certain problems in operation science and information management to have any relatively effective algorithm or approximation algorithm for them

Separability and Topology Control of Quasi Unit Disk Graphs

A deep understanding of the structural properties of wireless networks is critical for evaluating the performance of network protocols and improving their designs. Many protocols for wireless networks — routing, topology control, information storage/retrieval and numerous other applications — have been based on the idealized unit-disk graph (UDG) network model. The significant deviation of the UDG model from many real wireless networks is substantially limiting the applicability of such protocols. A more general network model, the quasi unitdisk graph (quasi-UDG) model, captures much better the characteristics of wireless networks. However, the understanding of the properties of general quasi-UDGs has been very limited, which is impeding the designs of key network protocols and algorithms. In this paper, we present results on two important properties of quasi-UDGs: separability and the existence of power efficient spanners. Network separability is a fundamental property leading to efficient network algorithms and fast parallel computation. We prove that every quasi-UDG has a corresponding grid graph with small balanced separators that captures its connectivity properties. We also study the problem of constructing an energyefficient backbone for a quasi-UDG. We present a distributed localized algorithm that, given a quasi-UDG, constructs a nearly planar backbone with a constant stretch factor and a bounded degree. We demonstrate the excellent performance of these auxiliary graphs through simulations and show their applications in efficient routing