Document Type

Article

Publication Date

4-1996

Publication Title

SIAM Journal on Computing

Abstract

Given n points in the plane, the degree-K spanning-tree problem asks for a spanning tree of minimum weight in which the degree of each vertex is at most K. This paper addresses the problem of computing low-weight degree-K spanning trees for $K > 2$. It is shown that for an arbitrary collection of n points in the plane, there exists a spanning tree of degree 3 whose weight is at most 1.5 times the weight of a minimum spanning tree. It is shown that there exists a spanning tree of degree 4 whose weight is at most 1.25 times the weight of a minimum spanning tree. These results solve open problems posed by Papadimitriou and Vazirani. Moreover, if a minimum spanning tree is given as part of the input, the trees can be computed in $O(n)$ time.

The results are generalized to points in higher dimensions. It is shown that for any $d \geqslant 3$, an arbitrary collection of points in $\Re ^d $ contains a spanning tree of degree 3 whose weight is at most ${5 / 3}$ times the weight of a minimum spanning tree. This is the first paper that achieves factors better than 2 for these problems.

DOI

10.1137/S0097539794264585

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