Finding a good embedding of a unit disk graph given by its connectivity information is a problem of practical importance in a variety of fields. In wireless ad hoc and sensor networks, such an embedding can be used to obtain virtual coordinates. In this paper, we prove a non-approximability result for the problem of embedding a given unit disk graph. Particularly, we show that if non-neighboring nodes are not allowed to be closer to each other than distance 1, then two neighbors can be as far apart as #, where # goes to 0 as n goes to infinity, unless P = NP . We further show that finding a realization of a d-quasi unit disk graph with 1/ # 2 is NP -hard.

Abstract — The localization problem is to determine an assignment of coordinates to nodes in a wireless ad-hoc or sensor network that is consistent with measured pairwise node distances. Most previously proposed solutions to this problem assume that the nodes can obtain pairwise distances to other nearby nodes using some ranging technology. However, for a variety of reasons that include obstructions and lack of reliable omnidirectional ranging, this distance information is hard to obtain in practice. Even when pairwise distances between nearby nodes are known, there may not be enough information to solve the problem uniquely. This paper describes MAL, a mobile-assisted localization method which employs a mobile user to assist in measuring distances between node pairs until these distance constraints form a “globally rigid ” structure that guarantees a unique localization. We derive the required constraints on the mobile’s movement and the minimum number of measurements it must collect; these constraints depend on the number of nodes visible to the mobile in a given region. We show how to guide the mobile’s movement to gather a sufficient number of distance samples for node localization. We use simulations and measurements from an indoor deployment using the Cricket location system to investigate the performance of MAL, finding in real-world experiments that MAL’s median pairwise distance error is less than 1.5 % of the true node distance. I.

Localization is an important and extensively studied problem in ad-hoc wireless sensor networks. Given the connectivity graph of the sensor nodes, along with additional local information (e.g. distances, angles, orientations etc.), the goal is to reconstruct the global geometry of the network. In this paper, we study the problem of localization with noisy distance and angle information. With no noise at all, the localization problem with both angle (with orientation) and distance information is trivial. However, in the presence of even a small amount of noise, we prove that the localization problem is NP-hard. Localization with accurate distance information and relative angle information is also hard. These hardness results motivate our study of approximation schemes. We relax the non-convex constraints to approximating convex constraints and propose linear programs (LP) for two formulations of the resulting localization problem, which we call the weak deployment and strong deployment problems. These two formulations give upper and lower bounds on the location uncertainty respectively: No sensor is located outside its weak deployment region, and each sensor can be anywhere in its strong deployment region without violating the approximate distance and angle constraints. Though LP-based algorithms are usually solved by centralized methods, we propose distributed, iterative methods, which are provably convergent to the centralized algorithm solutions. We give simulation results for the distributed algorithms, evaluating the convergence rate, dependence on measurement noises, and robustness to link dynamics.

rely on exact location information at the nodes, because when the greedy routing phase gets stuck at a local minimum, they require, as a fallback, a planar subgraph whose identification, in all existing methods, depends on exact node positions. In practice, however, location information at the network nodes is hardly precise; be it because the employed location hardware, such as GPS, exhibits an inherent measurement imprecision, or because the localization protocols which estimate positions of the network nodes cannot do so without errors. In this paper we propose a novel naming and routing scheme that can handle the uncertainty in location information. It is based on a macroscopic variant of geographic greedy routing, as well as a macroscopic planarization of the communication graph. If an upper bound on the deviation from true node locations is available, our routing protocol guarantees delivery of messages. Due to its macroscopic view, our routing scheme also produces shorter and more load-balanced paths than common geographic routing schemes, in particular in sparsely connected networks or in the presence of obstacles. I.

Abstract. While several important problems in the field of sensor networks have already been tackled, there is still a wide range of challenging, open problems that merit further attention. We present five theoretical problems that we believe to be essential to understanding sensor networks. The goal of this work is both to summarize the current state of research and, by calling attention to these fundamental problems, to spark interest in the networking community to attend to these and related problems in sensor networks.

Abstract — In this paper we present a simple and stateless

The quality of an embedding Φ: V ↦ → R 2 of a graph G = (V, E) into the Euclidean plane is the ratio of max{u,v}∈E ||Φ(u) − Φ(v)||2 to min{u,v}�∈E ||Φ(u) − Φ(v)||2. Given a graph G = (V, E), that is known to be a unit ball graph in fixed dimensional Euclidean space R d, we seek algorithms to compute an embedding Φ: V ↦ → R 2 of best (smallest) quality. Note that G comes with no associated geometric information and in this setting, related problems such as recognizing if G is a unit disk graph (UDG), are NP-hard. While any connected unit disk graph (UDG) has a 2-dimensional embedding with quality between 1/2 and 1, as far as we know, Vempala’s random projection approach (FOCS 1998) provides the best quality bound of O(log 3 n · √ log log n) for this problem. This paper presents a simple, combinatorial algorithm for computing a O(log 2.5 n)-quality 2-dimensional embedding of a given graph, that is known to be a UBG in fixed dimensional Euclidean space R d. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality-2 embedding in R O(d log d). The first step of our algorithm constructs a “growth-restricted approximation ” of the given UBG. While such a construction is trivial if the UBG comes with a geometric representation, we are not aware of any other algorithm that can perform this step without geometric information. Construction of a growth-restricted approximation permits us to bypass the standard and costly technique of solving a linear program with exponentially many “spreading constraints. ” As a side effect of our construction, we get a constant-factor approximation to the minimum clique cover problem for UBGs, described without geometry. The second step of our algorithm combines the probabilistic decomposition of growth-restricted graphs due to Lee and Krauthgamer (STOC 2003) with Rao’s embedding algorithm for planar graphs (SoCG 1999) to obtain a (k, O ( √ log n))-volume respecting embedding of growth-restricted graphs. Our problem is a version of the well known localization problem in wireless sensor networks, in which network nodes are required to compute virtual 2-dimensional Euclidean coordinates given little or (as in our case) no geometric information.

Localization is an essential service for many wireless sensor network applications. While several localization schemes rely on anchor nodes and range measurements to achieve fine-grained positioning, we propose a range-free, anchorfree solution that works using connectivity information only. The approach, suitable for deployments with strict cost constraints, is based on the neural network paradigm of Self-Organizing Maps (SOM). We present a lightweight SOMbased algorithm to compute virtual coordinates that are effective for location-aided routing. This algorithm can also exploit the location information, if available, of few anchor nodes to compute absolute positions. Results of extensive simulations show improvements over the popular Multi-Dimensional Scaling (MDS) scheme, especially for networks with low connectivity, which are intrinsically harder to localize, and in presence of irregular radio pattern or anisotropic deployment. We analytically demonstrate that the proposed scheme has low computation and communication overheads; hence, making it suitable for resource-constrained networks.

Abstract — Wireless routing based on an embedding of the connectivity graph is a very promising technique to overcome shortcomings of geographic routing and topology-based routing. This is of particular interest when either absolute coordinates for geographic routing are unavailable or when they poorly reflect the underlying connectivity in the network. We focus on dynamic networks induced by time-varying fading and mobility. This requires that the embedding is stable over time, whereas the focus of most existing embedding algorithms is on low distortion of single realizations of a graph. We develop a beaconbased distributed embedding algorithm that requires little control overhead, produces low distortion embeddings, and is stable. We also show that a low-dimensional embedding suffices, since at a sufficiently large scale, wireless connectivity graphs are dictated by geometry. The stability of the embedding allows us to combine georouting on the embedding with last encounter routing (LER) for node lookup, further reducing the control overhead. Our routing algorithm avoids dead ends through randomized greedy forwarding. We demonstrate through extensive simulations that our combined embedding and routing scheme outperforms existing algorithms. I.

Abstract—The one type of routing in ad hoc and sensor networks that currently appears to be most amenable to algorithmic analysis is geographic routing. This paper contains an introduction to the problem field of geographic routing, presents a specific routing algorithm based on a synthesis of the greedy forwarding and face routing approaches, and provides an algorithmic analysis of the presented algorithm from both a worst-case and an average-case perspective. Index Terms—Algorithmic analysis, routing, stretch, wireless networks.