A divide-and-conquer strategy for embedding a distance-net point set into \(E^ n\) and its application (Q1344421)

This paper treats the problem of embedding a given finite and connected weighted graph into Euclidean \(n\)-space \(E^ n\) such that the weight of each edge equals the distance of its endpoints. The first theorem gives sufficient conditions for the embeddability of such a graph. The proof leads to a divide-and-conquer strategy for the actual construction of an embedding. The second theorem treats in a similar way the more general problem where lower and upper bounds of the weights are given, but only for \(n= 3\). It uses the so-called bound smoothing approach, which is based on the triangle and tetrangle inequalities. Finally, the importance of this problem for the investigation of the spatial structure of macromolecules is explained.