Studies of distance geometry problems (DGP) have focused on cases where the vertices at the ends of all or most of the given distances are known or assigned, which we call assigned distance geometry problems (aDGPs). In this contribution we consider the unassigned distance geometry problem (uDGP) where the vertices associated with a given distance are unknown, so the graph structure has to be discovered. uDGPs arises when attempting to find the atomic structure of molecules and nanoparticles using X-ray or neutron diffraction data from non-crystalline materials. Rigidity theory provides a useful foundation for both aDGPs and uDGPs, though it is restricted to generic realizations of graphs, and key results are summarized. Conditions for unique realization are discussed for aDGP and uDGP cases, build-up algorithms for both cases are described and experimental results for uDGP are presented.

A genetic algorithm is proposed with real value variables, spatially based crossover operator, a small mutation, large scale mutation, vector sum local search and geometric only based objective function to generate candidate molecule conformations from atomic pair distance data. To better simulate experimental data only information from the pair distance data is used as constraints. Ideal Bucky ball with 60 atoms is used as the test case with both perfect pair distance data and Gaussian noise perturbed pair distance data. The GA generated result shows molecules close to ideal Bucky balls but with some defects. A description of the spatially based crossover operator is provided along with a local search based on vector summed error for each atom.