Euclidean Distance Geometry and Applications

  title={Euclidean Distance Geometry and Applications},
  author={Leo Liberti and Carlile Lavor and Nelson Maculan and Antonio Mucherino},
  journal={SIAM Rev.},
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in Euclidean space realizing those given distances. We survey the theory of Euclidean distance geometry and its most important applications, with special emphasis on molecular conformation problems. 
Open research areas in distance geometry
Distance geometry is based on the inverse problem that asks to find the positions of points, in a Euclidean space of given dimension, that are compatible with a given set of distances. We briefly
A Cycle-Based Formulation for the Distance Geometry Problem
This work proposes and test a new mathematical programming formulation based on the incidence between cycles and edges in the given graph, where the edges are realized as straight segments of length equal to the edge weight.
Distance Geometry on the Sphere
This paper generalizes a theorem of Godel about the case where the distances are spherical geodesics, and discusses a method for realizing cliques geodesically on a K-dimensional sphere.
Discretization orders and efficient computation of cartesian coordinates for distance geometry
New discretization assumptions that are weaker than previously proposed ones are discussed, and a greedy algorithm is presented for an automatic identification of discretized orders to motivate the development of a new method for computing point coordinates.
A constrained interval approach to the generalized distance geometry problem
This paper presents a new approach to the generalized distance geometry problem, based on a model that uses constraint interval arithmetic, and gives some computational experiments that illustrate the better performance of the proposed approach, compared to others from the literature.
Recent Advances on Oriented Conformal Geometric Algebra Applied to Molecular Distance Geometry
  • C. Lavor, R. Alves
  • Chemistry
    Systems, Patterns and Data Engineering with Geometric Calculi
  • 2021
Oriented Conformal Geometric Algebra was recently applied to Molecular Distance Geometry, where we want to determine 3D protein structures using distance information provided by Nuclear Magnetic
The Isomap Algorithm in Distance Geometry
This work starts from a simple observation, namely that Isomap can also be used to provide approximate realizations of weighted graphs very eciently, and then derive and benchmark six new heuristics.
Fisher information distance: a geometrical reading?


Computational Experience with the Molecular Distance Geometry Problem
This work applies three global optimization algorithms (spatial Branch-and-Bound, Variable Neighbourhood Search, Multi Level Single Linkage) to two sets of instances, one taken from the literature and the other new.
On the Discretization of Distance Geometry Problems
Two important aspects of the discretization are focused on: the identification of suitable vertex discretizing orderings and the analysis of the symmetries that can be found in BP trees.
An alternating projection algorithm for computing the nearest euclidean distance matrix
Recent extensions of von Neumann’s alternating projection algorithm permit an effective numerical approach to certain least squares problems subject to side conditions. This paper treats the problem
On the solution of molecular distance geometry problems with interval data
It is shown that a discretization is still possible and an algorithm to solve the Molecular Distance Geometry Problem is proposed and Computational experiments on a set of artificially generated instances are presented.
Convex Optimization & Euclidean Distance Geometry
This book is about convex optimization, convex geometry (with particular attention to distance geometry), geometric problems, and problems that can be transformed into geometrical problems.
New Foundation of Euclidean Geometry
My second paper on metrical geometry * contains a characterisation of the n-dimensional euclidean space among general semi-metrical spaces in terms of relations between the distances of its points.
Molecular distance geometry methods: from continuous to discrete
Some continuous and discrete methods for solving some problems of molecular distance geometry involve a search in a continuous Euclidean space but sometimes the problem structure helps reduce the search to a discrete set of points.
Solving the molecular distance geometry problem with inaccurate distance data
We present a new iterative algorithm for the molecular distance geometry problem with inaccurate and sparse data, which is based on the solution of linear systems, maximum cliques, and a minimization
Distance Geometry: Theory, Algorithms, and Chemical Applications
One of the best-known applications of distance geometry is the determination of molecular conformation from experimental data, most notably NMR spectroscopy and other important applications include enumerating the conformation spaces of small molecules, ligand docking and pharmacophore mapping in drug design, and the homology modeling of protein structure.