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86
A Theory of Network Localization
, 2004
"... In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigid ..."
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Cited by 122 (12 self)
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In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.
Theory of semidefinite programming for sensor network localization
 IN SODA05
, 2005
"... We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior–point algorithm theories to prove that the SDP localizes any network or graph th ..."
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Cited by 120 (9 self)
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We analyze the semidefinite programming (SDP) based model and method for the position estimation problem in sensor network localization and other Euclidean distance geometry applications. We use SDP duality and interior–point algorithm theories to prove that the SDP localizes any network or graph that has unique sensor positions to fit given distance measures. Therefore, we show, for the first time, that these networks can be localized in polynomial time. We also give a simple and efficient criterion for checking whether a given instance of the localization problem has a unique realization in R 2 using graph rigidity theory. Finally, we introduce a notion called strong localizability and show that the SDP model will identify all strongly localizable sub–networks in the input network.
Localization in sparse networks using sweeps
 in Proceedings of ACM MobiCom
, 2006
"... Determining node positions is essential for many nextgeneration network functionalities. Previous localization algorithms lack correctness guarantees or require network density higher than required for unique localizability. In this paper, we describe a class of algorithms for finegrained localiza ..."
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Cited by 60 (6 self)
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Determining node positions is essential for many nextgeneration network functionalities. Previous localization algorithms lack correctness guarantees or require network density higher than required for unique localizability. In this paper, we describe a class of algorithms for finegrained localization called Sweeps. Sweeps correctly finitely localizes all nodes in bilateration networks. Sweeps also handles angle measurements and noisy measurements. We demonstrate the practicality of our algorithm through extensive simulations on a large number of networks, upon which it consistently localizes onethousandnode networks of average degree less than five in less than two minutes on a consumer PC.
Neighborhoodbased topology recognition in sensor networks
 In ALGOSENSORS04
, 2004
"... Abstract. We consider a crucial aspect of selforganization of a sensor network consisting of a large set of simple sensor nodes with no location hardware and only very limited communication range. After having been distributed randomly in a given twodimensional region, the nodes are required to de ..."
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Cited by 57 (1 self)
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Abstract. We consider a crucial aspect of selforganization of a sensor network consisting of a large set of simple sensor nodes with no location hardware and only very limited communication range. After having been distributed randomly in a given twodimensional region, the nodes are required to develop a sense for the environment, based on a limited amount of local communication. We describe algorithmic approaches for determining the structure of boundary nodes of the region, and the topology of the region. We also develop methods for determining the outside boundary, the distance to the closest boundary for each point, the Voronoi diagram of the different boundaries, and the geometric thickness of the network. Our methods rely on a number of natural assumptions that are present in densely distributed sets of nodes, and make use of a combination of stochastics, topology, and geometry. Evaluation requires only a limited number of simple local computations. ACM classification: C.2.1 Network architecture and design; F.2.2 Nonnumerical algorithms and problems; G.3 Probability and statistics
Further relaxation of the semidefinite programming approach to sensor network localization
 SIAM Journal on Optimization
, 2008
"... Abstract. Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating the sensor locations, the speed of the SDP approach is not satisfactory for practical applications. In this pa ..."
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Cited by 40 (3 self)
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Abstract. Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating the sensor locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we propose methods to further relax the SDP relaxation, more precisely, to relax the single semidefinite matrix cone into a set of smallsize semidefinite submatrix cones, which we call a subSDP (SSDP) approach. We present two such relaxations. Although they are weaker than the original SDP relaxation, they retain the key theoretical property, and numerical experiments show that they are both efficient and accurate. The speed of the SSDP is even faster than that of other approaches based on weaker relaxations. The SSDP approach may also pave a way to efficiently solving general SDP problems without sacrificing the solution quality.
Resilient localization for sensor networks in outdoor environments
 In International Conference on Distributed Computing Systems. IEEE Computer Society
, 2005
"... The process of determining the physical locations of nodes in a wireless sensor network is known as localization. Selflocalization is critical for largescale sensor networks, because manual or assisted localization is often impractical due to time requirements, economic constraints, or inherent li ..."
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Cited by 39 (1 self)
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The process of determining the physical locations of nodes in a wireless sensor network is known as localization. Selflocalization is critical for largescale sensor networks, because manual or assisted localization is often impractical due to time requirements, economic constraints, or inherent limitations of the deployment scenarios. We propose scalable solutions for reliably localizing wireless sensor networks in environments conducive to several types of ranging errors. We follow a hybrid hardwaresoftware approach for acoustic ranging or radio interferometry to acquire internode distance measurements, and a resilient selflocalization algorithm to compute the node location estimates. The acoustic ranging method improves on previous work, extending the practical measurement range up to 35m in grassy outdoor environments, achieving a distanceinvariant median measurement error of about 1 % (33cm). The localization algorithm is based on Least Squares Scaling with soft constraints. Empirical evaluation using ranging results obtained from sensor network field experiments and simulations confirms that our approach is more resilient than multidimensional scaling (MDS) algorithms against largemagnitude ranging errors and sparse range measurements: conditions that are common in largescale outdoor sensor
Approximation Accuracy, Gradient Methods, and Error Bound for Structured Convex Optimization
, 2009
"... Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this ..."
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Cited by 38 (1 self)
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Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this is an error bound for the linear convergence analysis of firstorder gradient methods for solving these problems. Example applications include compressed sensing, variable selection in regression, TVregularized image denoising, and sensor network localization.
Secondorder cone programming relaxation of sensor network localization
 SIAM J. Optimization
, 2007
"... Abstract. The sensor network localization problem has been much studied. Recently Biswas and Ye proposed a semidefinite programming (SDP) relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed. Here, we study a secondorder cone pro ..."
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Cited by 36 (2 self)
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Abstract. The sensor network localization problem has been much studied. Recently Biswas and Ye proposed a semidefinite programming (SDP) relaxation of this problem which has various nice properties and for which a number of solution methods have been proposed. Here, we study a secondorder cone programming (SOCP) relaxation of this problem, motivated by its simpler structure and its potential to be solved faster than SDP. We show that the SOCP relaxation, though weaker than the SDP relaxation, has nice properties that make it useful as a problem preprocessor. In particular, sensors that are uniquely positioned among interior solutions of the SOCP relaxation are accurate up to the square root of the distance error. Thus, these sensors, which are easily identified, are accurately positioned. In our numerical simulation, the interior solution found can accurately position up to 80–90 % of the sensors. We also propose a smoothing coordinate gradient descent method for finding an interior solution that is faster than an interiorpoint method. Key words. sensor network localization, semidefinite program, secondorder cone program, approximation algorithm, error bound
Connectivitybased Localization of Large Scale Sensor Networks with Complex Shape
"... Abstract—We study the problem of localizing a large sensor network having a complex shape, possibly with holes. A major challenge with respect to such networks is to figure out the correct network layout, i.e., avoid global flips where a part of the network folds on top of another. Our algorithm fir ..."
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Cited by 31 (4 self)
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Abstract—We study the problem of localizing a large sensor network having a complex shape, possibly with holes. A major challenge with respect to such networks is to figure out the correct network layout, i.e., avoid global flips where a part of the network folds on top of another. Our algorithm first selects landmarks on network boundaries with sufficient density, then constructs the landmark Voronoi diagram and its dual combinatorial Delaunay complex on these landmarks. The key insight is that the combinatorial Delaunay complex is provably globally rigid and has a unique realization in the plane. Thus an embedding of the landmarks by simply gluing the Delaunay triangles properly recovers the faithful network layout. With the landmarks nicely localized, the rest of the nodes can easily localize themselves by trilateration to nearby landmark nodes. This leads to a practical and accurate localization algorithm for large networks using only network connectivity. Simulations on various network topologies show surprisingly good results. In comparison, previous connectivitybased localization algorithms such as multidimensional scaling and rubberband representation generate globally flipped or distorted localization results. I.