A fast approximation for minimum spanning trees in k-dimensional space

We study the problem of finding a minimum spanning tree on the complete graph on n points in E' , with the weight of an edge between any two points being the distance between the two points under some distance metric. A fast algorithm, which finds an approximate minimum spanning tree with wei h t a t most (1+c) times optimal in developed for the L,, q =2,3, ..., distance metrics. Moreover, if the n points are assumed to be independently and uniformly distributed in the box [0,lIk, then the probability tha t the approximate minimum spanning tree found is an exact minimum spanning tree is shown to be (1 o ( l / n ) ) . ~ ( n logn (( logn) 9 + log(€-')(logn)'.'a~('-')) time, is