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Abstract

We consider the problems of computing $r$-approximate traveling salesman tours and $r$-approximate minimum spanning trees for a set of $n$ points in $\IR^d$, where $d \geq 1$ 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 $r = \Theta( n^{1-1/d} )$.