#### 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

#### Recommended Citation

Khuller, Samir; Raghavachari, Balaji; and Young, Neal, "Low-Degree Spanning Trees of Small Weight" (1996). *Open Dartmouth: Faculty Open Access Articles*. 2067.

https://digitalcommons.dartmouth.edu/facoa/2067