On the complexity of approximating Euclidean traveling salesman tours and minimum spanning trees

We consider the problems of computing r-approximate traveling salesman tours and r-approximate minimum spanning trees for a set of n points in R-d, where d greater than or equal to 1 is a constant. In the algebraic computation tree model, the complexities of both these problems are shown to be Theta(n log(n/r)), for all n and r such that r < n and r is larger than some constant. In the more powerful model of computation that additionally uses the floor function and random access, both problems can be solved in O(n) time if r = Theta(n(1-1/d)).