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Given a simple weighted undirected graph G = 3 such that ||xu − xv|| = duv for each {u, v} ∈ E. We show that under a few assumptions usually satisfied in proteins, the MDGP can be formulated as a search in a discrete space. We call this MDGP subclass the Discretizable MDGP (DMDGP). We show that the DMDGP is NP-hard and we propose a solution algorithm called… (More)

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… (More)

We introduce the Discretizable Distance Geometry Problem in R 3 (DDGP 3), which consists in a subclass of instances of the Distance Geometry Problem for which an embedding in R 3 can be found by means of a discrete search. We show that the DDGP 3 is a generalization of the Discretizable Molecular Distance Geometry Problem (DMDGP), and we discuss the main… (More)

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… (More)

We consider the Discretizable Molecular Distance Geometry Problem (DMDGP), which consists in a subclass of instances of the distance geometry problem related to molecular conformations for which a combinatorial reformulation can be supplied. We investigate the performances of two different algorithms for solving the DMDGP. The first one is the Branch and… (More)

Given a weighted, undirected simple graph G = (V, E, d) (where d : E → R +), the Distance Geometry Problem (DGP) is to determine an embedding x : V → R K such that ∀{i, j} ∈ E x i − x j = d ij. Although, in general, the DGP is solved using continuous methods, under certain conditions the search is reduced to a discrete set of points. We give one such… (More)

—NMR experiments are able to provide some of the distances between pairs of hydrogen atoms in molecular conformations. The problem of finding the coordinates of such atoms is known as the molecular distance geometry problem. This problem can be reformulated as a combinatorial optimization problem and efficiently solved by an exact algorithm. To this… (More)

The Generalized Discretizable Molecular Distance Geometry Problem is a distance geometry problems that can be solved by a combinatorial algorithm called " Branch-and-Prune ". It was observed empirically that the number of solutions of YES instances is always a power of two. We give a proof that this event happens with probability one.