On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems

The problem of finding a minimum spanning tree connecting n points in a k-dimensional space is discussed under three common distance metrics -- Euclidean, rectilinear, and $L_\infty$. By employing a subroutine that solves the post office problem, we show that, for fixed k $\geq$ 3, such a minimum spanning tree can be found in time O($n^{2-a(k)} {(log n)}^{1-a(k)}$), where a(k) = $2^{-(k+1)}$. The bound can be improved to O(${(n log n)}^{1.8}$) for points in the 3-dimensional Euclidean space. We also obtain o($n^2$) algorithms for finding a farthest pair in a set of n points and for other related problems.