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

The problem of finding a minimum spaYlning tree connecting n points in a k-dimensional space is discussed under three comon distance metrics -Euclidean, rectilinear, and L . co By employing a subroutine that -=. solves the post office problem, we show that, for fixed k 2 3 , such a minimum spanning tree can be found in time O(n2’&Ck)(log n)l-a(k)) 9 where a(k) = 2 -(k+l) 1.8 . The bound can be improved to O((n log n) ) for points in the 3-dimensional Euclidean space. We also obtain o(n2) algorithms for finding a farthest pair in a set of n points and for other related problems. This research was supported in part by National Science Foundation grant MCS 72-03752 A03.