On Non-crossing (Projected) Spanning Trees of 3D point sets

We study the problem of optimizing the spanning tree of a set of points in IR3, whose projection onto a given plane contains no crossing edges. Denoted by NCMST (which stands for non-crossing minimum spanning trees), this problem is defined as follows. Given a set of points P in IR and a plane F , find a spanning tree whose projection onto F contains no crossing edges and its length is minimum among all such spanning trees. MCMST is motivated by areas such as shape modeling, surface reconstruction, and more. We prove that this problem is NP-complete and show that greedy algorithms (analogous to Prim and Kruskal), may perform arbitrarily bad. Nevertheless, we report that experimentally these algorithms perform well in practice.