On the Number of Minimal 1-Steiner Trees

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 log 2−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. Georgakopoulos 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(n2) 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(n2d) and Ω(nd) 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(nd log 2−d n) for all dimensions d. Our bounds require—as do the bounds in previous papers—either an assumption of general position or a tie-breaking rule, so that the minimum spanning tree is unique; otherwise even in the plane there can be exponentially many distinct topologies. We begin by stating a theorem of Kalai [2]: Theorem 1. In a family of n convex sets in d-space with the property that no c + 1 of the sets share a point, the number of k-tuples of sets that do have a point in common is bounded by ∑d i=0 (c−d k−i )(n+d−c i ) , whenever 0 ≤ k ≤ c. Here (c−d k−i ) = 0 if c− d < k − i. So in particular, putting k = c we have: Corollary 1. Given a family of convex decompositions of d-space with a total of n cells, the number of cells in the decomposition formed by overlaying them is less than (n d ) . A slightly weaker bound can be proved by an elementary argument that we now sketch. Assume that the convex decompositions are in general position with respect to each other; ∗Computer Science Dept., Polytechnic University, Six MetroTech Center, Brooklyn, NY 11201. †Xerox Palo Alto Research Center, 3333 Coyote Hill Rd., Palo Alto, CA 94304. ‡Dept. of Information and Computer Science, University of California, Irvine, CA 92717; research performed in part while visiting Xerox PARC.