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Universal Rigidity: Towards Accurate and Efficient Localization of Wireless Networks
, 2009
"... Abstract—A fundamental problem in wireless ad–hoc and sensor networks is that of determining the positions of nodes. Often, such a problem is complicated by the presence of nodes whose positions cannot be uniquely determined. Most existing work uses the notion of global rigidity from rigidity theory ..."
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Cited by 14 (4 self)
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Abstract—A fundamental problem in wireless ad–hoc and sensor networks is that of determining the positions of nodes. Often, such a problem is complicated by the presence of nodes whose positions cannot be uniquely determined. Most existing work uses the notion of global rigidity from rigidity theory to address the non–uniqueness issue. However, such a notion is not entirely satisfactory, as it has been shown that even if a network localization instance is known to be globally rigid, the problem of determining the node positions is still intractable in general. In this paper, we propose to use the notion of universal rigidity to bridge such disconnect. Although the notion of universal rigidity is more restrictive than that of global rigidity, it captures a large class of networks and is much more relevant to the efficient solvability of the network localization problem. Specifically, we show that both the problem of deciding whether a given network localization instance is universally rigid and the problem of determining the node positions of a universally rigid instance can be solved efficiently using semidefinite programming (SDP). Then, we give various constructions of universally rigid instances. In particular, we show that trilateration graphs are generically universally rigid, thus demonstrating not only the richness of the class of universally rigid instances, but also the fact that trilateration graphs possess much stronger geometric properties than previously known. Finally, we apply our results to design a novel edge sparsification heuristic that can reduce the size of the input network while provably preserving its original localization properties. One of the applications of such heuristic is to speed up existing convex optimization–based localization algorithms. Simulation results show that our speedup approach compares very favorably with existing ones, both in terms of accuracy and computation time.
EUCLIDEAN DISTANCE GEOMETRY AND APPLICATIONS
"... Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We surv ..."
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Cited by 6 (1 self)
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Abstract. Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the inputdataconsistsofanincompleteset of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of its most important applications, including molecular conformation, localization of sensor networks and statics. Key words. Matrix completion, barandjoint framework, graph rigidity, inverse problem, protein conformation, sensor network.
Lowdistortion inference of latent similarities from a multiplex social network
, 2012
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Mixed volume and distance geometry techniques for counting Euclidean embeddings of rigid graphs
"... A graph G is called generically minimally rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining tight bounds on the number of such embeddings, as a ..."
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Cited by 2 (1 self)
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A graph G is called generically minimally rigid in Rd if, for any choice of sufficiently generic edge lengths, it can be embedded in Rd in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining tight bounds on the number of such embeddings, as a function of the number of vertices. The study of rigid graphs is motivated by numerous applications, mostly in robotics, bioinformatics, sensor networks and architecture. We capture embeddability by polynomial systems with suitable structure, so that their mixed volume, which bounds the number of common roots, yields interesting upper bounds on the number of embeddings. We explore different polynomial formulations so as to reduce the corresponding mixed volume, namely by introducing new variables that remove certain spurious roots, and by applying the theory of distance geometry. We focus on R2 and R3, where Laman graphs and 1skeleta (or edge graphs) of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. Our implementation yields upper bounds for n ≤ 10 in R2 and R3, which reduce the existing gaps and lead to tight bounds for n ≤ 7 in both R2 and R3; in particular, we describe the recent settlement of the case of Laman graphs with 7 vertices. Our approach also yields a new upper bound for Laman graphs with 8 vertices, which is conjectured to be tight. We also establish the first lower bound in R3 of about 2.52n, where n denotes the number of vertices.
Comparing SOS and SDP relaxations of sensor network localization
, 2010
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On the number of realizations of certain Henneberg graphs arising in protein conformation
, 2012
"... Several application fields require finding Euclidean coordinates consistent with a set of distances. More precisely, given a simple undirected edgeweighted graph, we wish to find a realization in a Euclidean space so that adjacent vertices are placed at a distance which is equal to the correspondin ..."
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Cited by 1 (1 self)
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Several application fields require finding Euclidean coordinates consistent with a set of distances. More precisely, given a simple undirected edgeweighted graph, we wish to find a realization in a Euclidean space so that adjacent vertices are placed at a distance which is equal to the corresponding edge weight. Realizations of a graph can be either flexible or rigid. In certain cases, rigidity can be seen as a property of the graph rather than the realization. In the last decade, several advances have been made in graph rigidity, but most of these, for applicative reasons, focus on graphs having a unique realization. In this paper we consider a particular type of weighted Henneberg graphs that model protein backbones and show that almost all of them give rise to sets of incongruent realizations whose cardinality is a power of two.
IIITAllahabad
"... Localization in WSN involves the global discovery of node coordinates. In a network topology, a few nodes are deployed to known nodal locations and remaining node nodal location information are dynamically estimated using with algorithms. The nodes of sensor network are of random type and the topolo ..."
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Localization in WSN involves the global discovery of node coordinates. In a network topology, a few nodes are deployed to known nodal locations and remaining node nodal location information are dynamically estimated using with algorithms. The nodes of sensor network are of random type and the topology of network varied dynamically, as a result the most of the localization techniques fails in locating exact positions. The different solution’s that are already available succumbs to measure noise. We propose an algorithmic approach using the concept of graph rigidity, where a sensor graph is drawn to make use of network topology to form globally rigid subgraphs.
Selected Open Problems in Discrete Geometry and Optimization
, 2013
"... A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will ..."
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A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will contribute to further stimulate the interaction between geometers and optimizers.