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.

The molecular distance geometry problem can be formulated as the problem of finding an immersion in R 3 of a given undirected, nonnegatively weighted graph G. In this paper, we discuss a set of graphs G for which the problem may also be formulated as a combinatorial search in discrete space. This is theoretically interesting as an example of "combinatorialization" of a continuous nonlinear problem. It is also algorithmically interesting because the natural combinatorial solution algorithm performs much better than a global optimization approach on the continuous formulation. We present a Branch and Prune algorithm which can be used for obtaining a set of positions of the atoms of protein conformations when only some of the distances between the atoms are known.

The Molecular Distance Geometry Problem consists in finding the positions in R 3 of the atoms of a molecule, given some of the inter-atomic 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 Branch-and-Prune algorithm for the solution of this problem and report on very promising computational results.

Distance geometry problems arise from the need to position entities in the Euclidean K-space given some of their respective distances. Entities may be atoms (molecular distance geometry), wireless sensors (sensor network localization), or abstract vertices of a graph (graph drawing). In the context of molecular distance geometry, the distances are usually known because of chemical properties and Nuclear Magnetic Resonance experiments; sensor networks can estimate their relative distance by recording the power loss during a two-way exchange; finally, when drawing graphs in 2D or 3D, the graph to be drawn is given, and therefore distances between vertices can be computed. Distance geometry problems involve a search in a continuous Euclidean space, but sometimes the problem structure helps reduce the search to a discrete set of points. In this paper we survey some continuous and discrete methods for solving some problems of molecular distance geometry.

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