Molecular distance geometry methods: from continuous to discrete

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.