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Global Continuation For Distance Geometry Problems
 SIAM J. OPTIMIZATION
, 1995
"... Distance geometry problems arise in the interpretation of NMR data and in the determination of protein structure. We formulate the distance geometry problem as a global minimization problem with special structure, and show that global smoothing techniques and a continuation approach for global optim ..."
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Cited by 69 (7 self)
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Distance geometry problems arise in the interpretation of NMR data and in the determination of protein structure. We formulate the distance geometry problem as a global minimization problem with special structure, and show that global smoothing techniques and a continuation approach for global optimization can be used to determine solutions of distance geometry problems with a nearly 100% probability of success.
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 60 (9 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.
The Molecule Problem Exploiting Structure In Global Optimization
 SIAM Journal on Optimization
, 1995
"... . The molecule problem is that of determining the relative locations of a set of objects in Euclidean space relying only upon a sparse set of pairwise distance measurements. This NPhard problem has applications in the determination of molecular conformation. The molecule problem can be naturally e ..."
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Cited by 57 (0 self)
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. The molecule problem is that of determining the relative locations of a set of objects in Euclidean space relying only upon a sparse set of pairwise distance measurements. This NPhard problem has applications in the determination of molecular conformation. The molecule problem can be naturally expressed as a continuous, global optimization problem, but it also has a rich combinatorial structure. This paper investigates how that structure can be exploited to simplify the optimization problem. In particular, we present a novel divideandconquer algorithm in which a large global optimization problem is replaced by a sequence of smaller ones. Since the cost of the optimization can grow exponentially with problem size, this approach holds the promise of a substantial improvement in performance. Our algorithmic development relies upon some recently published results in graph theory. We describe an implementation of this algorithm and report some results of its performance on a sample ...
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.
A Subspace, Interior, and Conjugate Gradient Method for LargeScale BoundConstrained Minimization Problems
 SIAM Journal on Scientific Computing
, 1999
"... A subspace adaptation of the ColemanLi trust region and interior method is proposed for solving largescale boundconstrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergenc ..."
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Cited by 35 (1 self)
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A subspace adaptation of the ColemanLi trust region and interior method is proposed for solving largescale boundconstrained minimization problems. This method can be implemented with either sparse Cholesky factorization or conjugate gradient computation. Under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its fullspace version.
εOptimal Solutions To Distance Geometry Problems Via Global Continuation
, 1995
"... We show that a continuation approach to global optimization with global smoothing techniques can be used to obtain "optimal solutions to distance geometry problems. We show that determining an "optimal solution is still an NPhard problem when " is small. A discrete form of the Gaussian transform ..."
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Cited by 28 (6 self)
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We show that a continuation approach to global optimization with global smoothing techniques can be used to obtain "optimal solutions to distance geometry problems. We show that determining an "optimal solution is still an NPhard problem when " is small. A discrete form of the Gaussian transform is proposed based on the Hermite form of Gaussian quadrature. We show that the modified transform can be used whenever the transformed functions cannot be computed analytically. Our numerical results show that the discrete Gauss transform can be used to obtain "optimal solutions for general distance geometry problems, and in particular, to determine the threedimensional structure of protein fragments.
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.
Reconstructing a ThreeDimensional Model with Arbitrary Errors
 Journal of the ACM
, 1996
"... A number of current technologies allow for the determination of interatomic distance information in structures such as proteins and RNA. Thus, the reconstruction of a threedimensional set of points using information about its interpoint distances has become a task of basic importance in determini ..."
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Cited by 24 (0 self)
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A number of current technologies allow for the determination of interatomic distance information in structures such as proteins and RNA. Thus, the reconstruction of a threedimensional set of points using information about its interpoint distances has become a task of basic importance in determining molecular structure. The distance measurements one obtains from techniques such as NMR are typically sparse and errorprone, greatly complicating the reconstruction task. Many of these errors result in distance measurements that can be safely assumed to lie within certain fixed tolerances. But a number of sources of systematic error in these experiments lead to inaccuracies in the data that are very hard to quantify; in effect, one must treat certain entries of the measured distance matrix as being arbitrarily "corrupted." The existence of arbitrary errors leads to an interesting sort of errorcorrection problem  how many corrupted entries in a distance matrix can be efficiently corre...
Explicit Sensor Network Localization Using Semidefinite Representations and Clique Reductions
 Department of Combinatorics and Optimization, University of Waterloo
, 2009
"... AMS Subject Classification: The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solu ..."
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Cited by 18 (9 self)
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AMS Subject Classification: The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming, SDP,completion problem, by using the linear mapping between Euclidean distance matrices, EDM, and semidefinite matrices. The resulting SDP is solved using primaldual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances,
Pebble Game Algorithms and Sparse Graphs
, 2007
"... A multigraph G on n vertices is (k,ℓ)sparse if every subset of n ′ ≤ n vertices spans at most kn ′ − ℓ edges. G is tight if, in addition, it has exactly kn − ℓ edges. For integer values k and ℓ ∈ [0,2k), we characterize the (k,ℓ)sparse graphs via a family of simple, elegant and efficient algori ..."
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Cited by 17 (5 self)
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A multigraph G on n vertices is (k,ℓ)sparse if every subset of n ′ ≤ n vertices spans at most kn ′ − ℓ edges. G is tight if, in addition, it has exactly kn − ℓ edges. For integer values k and ℓ ∈ [0,2k), we characterize the (k,ℓ)sparse graphs via a family of simple, elegant and efficient algorithms called the (k,ℓ)pebble games.