On the number of minimal 1-Steiner trees (original) (raw)

We count the number of nonisomorphic geometric minimum spanning trees formed by adding a single point to an n-point set in d-dimensional space, by relating it to a family of convex decompositions of space. The O(n d log 2d2-d n) bound that we obtain significantly improves previously known bounds and is tight to within a polylogarithmic factor. The minimum Steiner tree of a point set is the tree of minimum total length, spanning the input points and possibly having some additional vertices. Georgakopulos and Papadimitriou [1] define the minimum 1-Steiner tree as the minimum spanning tree whose set of vertices consists of the input points and exactly one extra point. They describe an algorithm for constructing this tree in the plane by enumerating all combinatorially distinct minimal 1-Steiner trees, that is, trees formed by fixing the position of the extra vertex and considering the minimum spanning tree of the resulting set of points. They give an O(n 2) bound on the number of such trees and construct a matching lower bound. Independently, Monma and Suri [3] proved the same bounds in the plane and gave bounds of O(n 2a) and t'~(n ~) in any dimension d. We significantly improve Monma and Suri's upper bounds. We show that the maximum possible number of combinatorially distinct minimal 1-Steiner trees on n points in d-space is O(n d log 2d2-d n) for all * The research of D. Eppstein was performed in part while visiting Xerox PARC.