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21
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 (10 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.
Fast similarity search for learned metrics.
 IEEE Trans. Pattern Anal. Mach. Intell.,
, 2009
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Further relaxations of the SDP approach to sensor network localization
 SIAM J. Optim
"... Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating sensor’s locations, the speed of the SDP approach is not satisfactory for practical applications. In this paper we prop ..."
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Cited by 34 (0 self)
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Recently, a semidefinite programming (SDP) relaxation approach has been proposed to solve the sensor network localization problem. Although it achieves high accuracy in estimating sensor’s 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 decompose the single semidefinite matrix cone into a set of smallsize semidefinite matrix cones, which we call the smaller SDP (SSDP) approach. We present two such relaxations or decompositions; and they are, although weaker than SDP relaxation, tested to be both efficient and accurate in practical computations. The speed of the SSDP is much faster than that of the SDP approach as well as other approaches. We also prove several theoretical properties of the new SSDP relaxations.
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 32 (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.
GPSFree node localization in mobile wireless sensor networks
 PROCEEDINGS OF THE 5TH ACM INTERNATIONAL WORKSHOP ON DATA
, 2006
"... An important problem in mobile adhoc wireless sensor networks is the localization of individual nodes, i.e., each node’s awareness of its position relative to the network. In this paper, we introduce a variant of this problem (directional localization) where each node must be aware of both its posi ..."
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Cited by 26 (3 self)
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An important problem in mobile adhoc wireless sensor networks is the localization of individual nodes, i.e., each node’s awareness of its position relative to the network. In this paper, we introduce a variant of this problem (directional localization) where each node must be aware of both its position and orientation relative to the network. This variant is especially relevant for the applications in which mobile nodes in a sensor network are required to move in a collaborative manner. Using global positioning systems for localization in large scale sensor networks is not cost effective and may be impractical in enclosed spaces. On the other hand, a set of preexisting anchors with globally known positions may not always be available. To address these issues, in this work we propose an algorithm for directional node localization based on relative motion of neighboring nodes in an adhoc sensor network without an infrastructure of global positioning systems (GPS), anchor points, or even mobile seeds with known locations. Through simulation studies, we demonstrate that our algorithm scales well for large numbers of nodes and provides convergent localization over time, even with errors introduced by motion actuators and distance measurements. Furthermore, based on our localization algorithm, we introduce mechanisms to preserve network formation during directed mobility in mobile sensor networks. Our simulations confirm that, in a number of realistic scenarios, our algorithm provides for a mobile sensor network that is stable over time irrespective of speed, while using only constant storage per neighbor.
Lowdistortion embeddings of general metrics into the line
 In STOC’05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... A lowdistortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Lowdistortion embeddings have recently found numerous applications in computer science. Most of the known embedding results are ”absol ..."
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Cited by 25 (8 self)
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A lowdistortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Lowdistortion embeddings have recently found numerous applications in computer science. Most of the known embedding results are ”absolute”, that is, of the form: any metric Y from a given class of metrics C can be embedded into a metric X with low distortion c. This is beneficial if one can guarantee low distortion for all metrics Y in C. However, in many situations, the worstcase distortion is too large to be meaningful. For example, if X is a line metric, then even very simple metrics (an npoint star or an npoint cycle) are embeddable into X only with distortion linear in n. Nevertheless, embeddings into the line (or into lowdimensional spaces) are important for many applications. A solution to this issue is to consider ”relative ” (or ”approximation”) embedding problems, where the goal is to design an (aapproximation) algorithm which, given any metric X from C as an input, finds an embedding of X into Y which has distortion a ∗ cY (X), where cY (X) is the best possible distortion of an embedding of X into Y. In this paper we show algorithms and hardness results for relative embedding problems. In particular we give: • an algorithm that, given a general metric M, finds an embedding with distortion O( ∆ 4/5 poly(cline(M))), where ∆ is the spread of M • an algorithm that, given a weighted tree metric M, finds an embedding with distortion poly(cline(M)) • a hardness result, showing that computing minimum line distortion is hard to approximate up to a factor polynomial in n, even for weighted tree metrics with spread ∆ = n O(1). 1
Distributed localization using noisy distance and angle information
 PROCEEDINGS OF THE SEVENTH ACM INTERNATIONAL SYMPOSIUM ON MOBILE AD HOC NETWORKING AND COMPUTING
, 2006
"... Localization is an important and extensively studied problem in adhoc 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 th ..."
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Cited by 20 (5 self)
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Localization is an important and extensively studied problem in adhoc 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 NPhard. Localization with accurate distance information and relative angle information is also hard. These hardness results motivate our study of approximation schemes. We relax the nonconvex 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 LPbased 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.
Euclidean Distance Matrices and Applications
"... Over the past decade, Euclidean distance matrices, or EDMs, have been receiving increased attention for two main reasons. The first reason is that the many applications of EDMs, such as molecular conformation in bioinformatics, dimensionality reduction in machine learning and statistics, and especia ..."
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Cited by 14 (0 self)
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Over the past decade, Euclidean distance matrices, or EDMs, have been receiving increased attention for two main reasons. The first reason is that the many applications of EDMs, such as molecular conformation in bioinformatics, dimensionality reduction in machine learning and statistics, and especially
Ordinal embeddings of minimum relaxation: General properties, trees and ultrametrics
 Proceedings of the ACMSIAM Symposium on Discrete Algorithms
, 2005
"... We introduce a new notion of embedding, called minimumrelaxation ordinal embedding, parallel to the standard notion of minimumdistortion (metric) embedding. In an ordinal embedding, it is the relative order between pairs of distances, and not the distances themselves, that must be preserved as much ..."
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Cited by 12 (4 self)
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We introduce a new notion of embedding, called minimumrelaxation ordinal embedding, parallel to the standard notion of minimumdistortion (metric) embedding. In an ordinal embedding, it is the relative order between pairs of distances, and not the distances themselves, that must be preserved as much as possible. The (multiplicative) relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop several worstcase bounds and approximation algorithms on ordinal embedding. In particular, we establish that ordinal embedding has many qualitative differences from metric embedding, and capture the ordinal behavior of ultrametrics and shortestpath metrics of unweighted trees. 1