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.

Leishmune, the first prophylactic vaccine licensed against canine visceral leishmaniasis (CVL), has been used in Brazil since 2004, where seropositive dogs are sacrificed in order to control human visceral leishmaniasis (VL). We demonstrate here that vaccination with Leishmune does not interfere with the serological control campaign (110,000 dogs). Only 1.3% of positivity (76 among 5860) was detected among Leishmune uninfected vaccinees. We also analyzed the possible additive effect of Leishmune vaccination over dog culling, on the decrease of the incidence of CVL and VL in two Brazilian endemic areas, from 2004 to 2006. In Araçatuba, a 25% of decline was seen in CVL with a 61% decline in human cases, indicating the additive effect of Leishmune vaccination of 5.7% of the healthy dogs (1419 dogs), on regular dog culling. In Belo Horizonte (BH), rising curves of canine and human incidence were observed in the districts of Barreiro, Venda Nova and Noroeste, while the canine and human incidence of Centro Sul, Leste, Nordeste, Norte, Pampulha and Oeste, started to decrease or maintained a stabilized plateau after Leishmune vaccination. Among the districts showing a percent decrease of human incidence (-36.5%), Centro Sul and Pampulha showed the highest dog vaccination percents (63.27% and 27.27%, respectively) and the lowest dog incidence (-3.36% and 1.89%, respectively). They were followed by Oeste, that vaccinated 25.30% of the animals and experienced an increase of only 12.86% of dog incidence and by Leste and Nordeste, with lower proportions of vaccinees (11.72% and 10.76%, respectively) and probably because of that, slightly higher canine incidences (42.77% and 35.73%). The only exception was found in Norte district where the reduced human and canine incidence were not correlated to Leishmune vaccination. Much lower proportions of dogs were vaccinated in Venda Nova (4.35%), Noroeste (10.27%) and Barreiro (0.09%) districts, which according to that exhibited very increased canine incidences (24.48%, 21.85% and 328.57%, respectively), and pronounced increases in human incidence (14%, 4% and 17%, respectively). The decrease of canine (p=-0.008) and human incidences (p=-0.048) is directly correlated to the increase of the number of vaccinated dogs, confirming the additive control effect of Leishmune vaccination over dog culling, reducing the parasite reservoir, protecting dogs and, in this way, reducing the risk of transmission of VL to humans and becoming a new effective control tool.

The Distance Geometry Problem in three dimensions consists in finding an embedding in R 3 of a given nonnegatively weighted simple undirected graph such that edge weights are equal to the corresponding Euclidean distances in the embedding. This is a continuous search problem that can be discretized under some assumptions on the minimum degree of the vertices. In this paper we discuss the case where we consider the full-atom representation of the protein backbone and some of the edge weights are subject to uncertainty within a given nonnegative interval. We show that a discretization is still possible and propose the iBP algorithm to solve the problem. The approach is validated by some computational experiments on a set of artificially generated instances.