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47
Rigidity, Computation, and Randomization in Network Localization
 In Proceedings of IEEE INFOCOM ’04, Hong Kong
, 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 82 (14 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 77 (5 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.
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 64 (6 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.
Connected Rigidity Matroids and Unique Realizations of Graphs
, 2004
"... A ddimensional framework is a straight line realization of a graph G in R d. We shall only consider generic frameworks, in which the coordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same le ..."
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Cited by 62 (10 self)
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A ddimensional framework is a straight line realization of a graph G in R d. We shall only consider generic frameworks, in which the coordinates of all the vertices of G are algebraically independent. Two frameworks for G are equivalent if corresponding edges in the two frameworks have the same length. A framework is a unique realization of G in R d if every equivalent framework can be obtained from it by an isometry of R d. Bruce Hendrickson proved that if G has a unique realization in R d then G is (d + 1)connected and redundantly rigid. He conjectured that every realization of a (d + 1)connected and redundantly rigid graph in R d is unique. This conjecture is true for d = 1 but was disproved by Robert Connelly for d ≥ 3. We resolve the remaining open case by showing that Hendrickson’s conjecture is true for d = 2. As a corollary we deduce that every realization of a 6connected graph as a 2dimensional generic framework is a unique realization. Our proof is based on a new inductive characterization of 3connected graphs whose rigidity matroid is connected.
Generic Global Rigidity
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2004
"... Suppose a finite configuration of labeled points p = (p1,...,pn) in Ed is given along with certain pairs of those points determined by a graph G such that the coordinates of the points of p are generic, i.e., algebraically independent over the integers. If another corresponding configuration q = (q ..."
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Cited by 48 (5 self)
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Suppose a finite configuration of labeled points p = (p1,...,pn) in Ed is given along with certain pairs of those points determined by a graph G such that the coordinates of the points of p are generic, i.e., algebraically independent over the integers. If another corresponding configuration q = (q1,...,qn) in Ed is given such that the corresponding edges of G for p and q have the same length, we provide a sufficient condition to ensure that p and q are congruent in Ed. This condition, together with recent results of Jackson and Jordán [JJ], give necessary and sufficient conditions for a graph being generically globally rigid in the plane.
Network localization in partially localizable networks
 in Proceedings of IEEE INFOCOM
, 2005
"... Abstract — Knowing the positions of the nodes in a network is essential to many next generation pervasive and sensor network functionalities. Although many network localization systems have recently been proposed and evaluated, there has been no systematic study of partially localizable networks, i. ..."
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Cited by 46 (10 self)
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Abstract — Knowing the positions of the nodes in a network is essential to many next generation pervasive and sensor network functionalities. Although many network localization systems have recently been proposed and evaluated, there has been no systematic study of partially localizable networks, i.e., networks in which there exist nodes whose positions cannot be uniquely determined. There is no existing study which correctly identifies precisely which nodes in a network are uniquely localizable and which are not. This absence of a sufficient uniqueness condition permits the computation of erroneous positions that may in turn lead applications to produce flawed results. In this paper, in addition to demonstrating the relevance of networks that may not be fully localizable, we design the first framework for two dimensional network localization with an efficient component to correctly determine which nodes are localizable and which are not. Implementing this system, we conduct comprehensive evaluations of network localizability, providing guidelines for both network design and deployment. Furthermore, we study an integration of traditional geographic routing with geographic routing over virtual coordinates in the partially localizable network setting. We show that this novel crosslayer integration yields good performance, and argue that such optimizations will be likely be necessary to ensure acceptable application performance in partially localizable networks. I.
Distributed graph layout for sensor networks
 In 12th Symposium on Graph Drawing (GD
, 2004
"... Sensor network applications frequently require that the sensors know their physical locations in some global coordinate system. This is usually achieved by equipping each sensor with a location measurement device, such as GPS. However, lowend systems or indoor systems, which cannot use GPS, must lo ..."
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Cited by 29 (2 self)
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Sensor network applications frequently require that the sensors know their physical locations in some global coordinate system. This is usually achieved by equipping each sensor with a location measurement device, such as GPS. However, lowend systems or indoor systems, which cannot use GPS, must locate themselves based only on crude information available locally, such as intersensor distances. We show how a collection of sensors, capable only of measuring distances to close neighbors, can compute their locations in a purely distributed manner, i.e. where each sensor communicates only with its neighbors. This can be viewed as a distributed graph drawing algorithm. We experimentally show that our algorithm consistently produces good results under a variety of simulated realworld conditions, and is relatively robust to the presence of noise in the distance measurements.
A BranchandPrune Algorithm for the Molecular Distance Geometry Problem
 International Transactions in Operational Research
, 2006
"... The Molecular Distance Geometry Problem consists in finding the positions in R 3 of the atoms of a molecule, given some of the interatomic distances. We show that under an additional requirement on the given distances (which is realistic from the chemical point of view) this can be transformed to a ..."
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Cited by 26 (23 self)
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The Molecular Distance Geometry Problem consists in finding the positions in R 3 of the atoms of a molecule, given some of the interatomic distances. We show that under an additional requirement on the given distances (which is realistic from the chemical point of view) this can be transformed to a combinatorial problem. We propose a BranchandPrune algorithm for the solution of this problem and report on very promising computational results.
Distance geometry optimization for protein structures
 Applied Mathematics Division, Argonne National Labs
, 1997
"... Abstract. We study the performance of the dgsol code for the solution of distance geometry problems with lower and upper bounds on distance constraints. The dgsol code uses only a sparse set of distance constraints, while other algorithms tend to work with a dense set of constraints either by imposi ..."
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Cited by 25 (3 self)
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Abstract. We study the performance of the dgsol code for the solution of distance geometry problems with lower and upper bounds on distance constraints. The dgsol code uses only a sparse set of distance constraints, while other algorithms tend to work with a dense set of constraints either by imposing additional bounds or by deducing bounds from the given bounds. Our computational results show that protein structures can be determined by solving a distance geometry problem with dgsol and that the approach based on dgsol is significantly more reliable and efficient than multistarts with an optimization code.
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 25 (2 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.