Ebook: Distance Geometry: Theory, Methods, and Applications

This volume is a collection of research surveys on the Distance Geometry Problem (DGP) and its applications. It will be divided into three parts: Theory, Methods and Applications. Each part will contain at least one survey and several research papers.

The first part, Theory, will deal with theoretical aspects of the DGP, including a new class of problems and the study of its complexities as well as the relation between DGP and other related topics, such as: distance matrix theory, Euclidean distance matrix completion problem, multispherical structure of distance matrices, distance geometry and geometric algebra, algebraic distance geometry theory, visualization of K-dimensional structures in the plane, graph rigidity, and theory of discretizable DGP: symmetry and complexity.

The second part, Methods, will discuss mathematical and computational properties of methods developed to the problems considered in the first chapter including continuous methods (based on Gaussian and hyperbolic smoothing, difference of convex functions, semidefinite programming, branch-and-bound), discrete methods (based on branch-and-prune, geometric build-up, graph rigidity), and also heuristics methods (based on simulated annealing, genetic algorithms, tabu search, variable neighborhood search).

Applications will comprise the third part and will consider applications of DGP to NMR structure calculation, rational drug design, molecular dynamics simulations, graph drawing and sensor network localization.

This volume will be the first edited book on distance geometry and applications. The editors are in correspondence with the major contributors to the field of distance geometry, including important research centers in molecular biology such as Institut Pasteur in Paris.

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Distance Geometry: Theory, Methods, and Applications is the first collection of research surveys dedicated to distance geometry and its applications. The first part of the book discusses theoretical aspects of the Distance Geometry Problem (DGP), where the relation between DGP and other related subjects are also presented. Covered topics include distance matrix theory, Euclidean distance matrix completion, multispherical structure of distance matrices, geometric algebra, algebraic distance geometry theory, visualization of K-dimensional structures in the plane, graph rigidity, and theory of discretizable DGP.

The second part of this volume presents mathematical and computational properties of methods developed to the problems discussed in the first portion, including continuous methods (based on Gaussian and hyperbolic smoothing, difference of convex functions, semidefinite programming, branch-and-bound), discrete methods (based on branch-and-prune, geometric build-up, graph rigidity), and also heuristics methods (based on simulated annealing, genetic algorithms, tabu search, variable neighborhood search).

Applications comprise the third part of the book, which is mainly devoted to the application of DGP to NMR structure calculation. This is an important and strongly multidisciplinary application in biology and biomedicine.

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Distance Geometry: Theory, Methods, and Applications is the first collection of research surveys dedicated to distance geometry and its applications. The first part of the book discusses theoretical aspects of the Distance Geometry Problem (DGP), where the relation between DGP and other related subjects are also presented. Covered topics include distance matrix theory, Euclidean distance matrix completion, multispherical structure of distance matrices, geometric algebra, algebraic distance geometry theory, visualization of K-dimensional structures in the plane, graph rigidity, and theory of discretizable DGP.

The second part of this volume presents mathematical and computational properties of methods developed to the problems discussed in the first portion, including continuous methods (based on Gaussian and hyperbolic smoothing, difference of convex functions, semidefinite programming, branch-and-bound), discrete methods (based on branch-and-prune, geometric build-up, graph rigidity), and also heuristics methods (based on simulated annealing, genetic algorithms, tabu search, variable neighborhood search).

Applications comprise the third part of the book, which is mainly devoted to the application of DGP to NMR structure calculation. This is an important and strongly multidisciplinary application in biology and biomedicine.

Content:
Front Matter....Pages i-xvi
Front Matter....Pages 1-1
Universal Rigidity of Bar Frameworks in General Position: A Euclidean Distance Matrix Approach....Pages 3-22
Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs....Pages 23-45
The Discretizable Molecular Distance Geometry Problem seems Easier on Proteins....Pages 47-60
Spheres Unions and Intersections and Some of their Applications in Molecular Modeling....Pages 61-83
Is the Distance Geometry Problem in NP?....Pages 85-93
Solving Spatial Constraints with Generalized Distance Geometry....Pages 95-120
A Topological Interpretation of the Walk Distances....Pages 121-135
Front Matter....Pages 137-137
Distance Geometry Methods for Protein Structure Determination....Pages 139-159
Solving the Discretizable Molecular Distance Geometry Problem by Multiple Realization Trees....Pages 161-176
ASAP: An Eigenvector Synchronization Algorithm for the Graph Realization Problem....Pages 177-195
Global Optimization for Atomic Cluster Distance Geometry Problems....Pages 197-212
Solving Molecular Distance Geometry Problems Using a Continuous Optimization Approach....Pages 213-224
DC Programming Approaches for Distance Geometry Problems....Pages 225-290
Stochastic Proximity Embedding: A Simple, Fast and Scalable Algorithm for Solving the Distance Geometry Problem....Pages 291-311
Front Matter....Pages 313-313
Distance Geometry for Realistic Molecular Conformations....Pages 315-328
Distance Geometry in Structural Biology: New Perspectives....Pages 329-350
Using a Distributed SDP Approach to Solve Simulated Protein Molecular Conformation Problems....Pages 351-376
An Overview on Protein Structure Determination by NMR: Historical and Future Perspectives of the use of Distance Geometry Methods....Pages 377-412
Back Matter....Pages 413-420

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Distance Geometry: Theory, Methods, and Applications is the first collection of research surveys dedicated to distance geometry and its applications. The first part of the book discusses theoretical aspects of the Distance Geometry Problem (DGP), where the relation between DGP and other related subjects are also presented. Covered topics include distance matrix theory, Euclidean distance matrix completion, multispherical structure of distance matrices, geometric algebra, algebraic distance geometry theory, visualization of K-dimensional structures in the plane, graph rigidity, and theory of discretizable DGP.

The second part of this volume presents mathematical and computational properties of methods developed to the problems discussed in the first portion, including continuous methods (based on Gaussian and hyperbolic smoothing, difference of convex functions, semidefinite programming, branch-and-bound), discrete methods (based on branch-and-prune, geometric build-up, graph rigidity), and also heuristics methods (based on simulated annealing, genetic algorithms, tabu search, variable neighborhood search).

Applications comprise the third part of the book, which is mainly devoted to the application of DGP to NMR structure calculation. This is an important and strongly multidisciplinary application in biology and biomedicine.

Content:
Front Matter....Pages i-xvi
Front Matter....Pages 1-1
Universal Rigidity of Bar Frameworks in General Position: A Euclidean Distance Matrix Approach....Pages 3-22
Mixed Volume and Distance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs....Pages 23-45
The Discretizable Molecular Distance Geometry Problem seems Easier on Proteins....Pages 47-60
Spheres Unions and Intersections and Some of their Applications in Molecular Modeling....Pages 61-83
Is the Distance Geometry Problem in NP?....Pages 85-93
Solving Spatial Constraints with Generalized Distance Geometry....Pages 95-120
A Topological Interpretation of the Walk Distances....Pages 121-135
Front Matter....Pages 137-137
Distance Geometry Methods for Protein Structure Determination....Pages 139-159
Solving the Discretizable Molecular Distance Geometry Problem by Multiple Realization Trees....Pages 161-176
ASAP: An Eigenvector Synchronization Algorithm for the Graph Realization Problem....Pages 177-195
Global Optimization for Atomic Cluster Distance Geometry Problems....Pages 197-212
Solving Molecular Distance Geometry Problems Using a Continuous Optimization Approach....Pages 213-224
DC Programming Approaches for Distance Geometry Problems....Pages 225-290
Stochastic Proximity Embedding: A Simple, Fast and Scalable Algorithm for Solving the Distance Geometry Problem....Pages 291-311
Front Matter....Pages 313-313
Distance Geometry for Realistic Molecular Conformations....Pages 315-328
Distance Geometry in Structural Biology: New Perspectives....Pages 329-350
Using a Distributed SDP Approach to Solve Simulated Protein Molecular Conformation Problems....Pages 351-376
An Overview on Protein Structure Determination by NMR: Historical and Future Perspectives of the use of Distance Geometry Methods....Pages 377-412
Back Matter....Pages 413-420
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