Euclidean distance geometry and applications

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 consists of an incomplete set 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 the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Distance geometry in active structures

The final publication is available at link.springer.comDistance constraints are an emerging formulation that offers intuitive geometrical interpretation of otherwise complex problems. The formulation can be applied in problems such as position and singularity analysis and path planning of mechanisms and structures. This paper reviews the recent advances in distance geometry, providing a unified view of these apparently disparate problems. This survey reviews algebraic and numerical techniques, and is, to the best of our knowledge, the first attempt to summarize the different approaches relating to distance-based formulations.Peer ReviewedPostprint (author's final draft

Feasibility check for the distance geometry problem: an application to molecular conformations

The distance geometry problem (DGP) consists in finding an embedding in a metric space of a given weighted undirected graph such that for each edge in the graph, the corresponding distance in the embedding belongs to a given distance interval. We discuss the relationship between the existence of a graph embedding in a Euclidean space and the existence of a graph embedding in a lattice. Different approaches, including two integer programming (IP) models and a constraint programming (CP) approach, are presented to test the feasibility of the DGP. The two IP models are improved with the inclusion of valid inequalities, and the CP approach is improved using an algorithm to perform a domain reduction. The main motivation for this work is to derive new pruning devices within branch-and-prune algorithms for instances occurring in real applications related to determination of molecular conformations, which is a particular case of the DGP. A computational study based on a set of small-sized instances from molecular conformations is reported. This study compares the running times of the different approaches to check feasibility