# Constructing degree-3 spanners with other sparseness properties

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## Abstract

Let *V* be any set of *n* points in *k*-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a *t-spanner* if for any *u* and *v* in *V*, the length of the shortest path from *u* to *v* in the spanner is at most *t* times *d(u, v)*. We show that for any *δ*>1, there exists a polynomial-time constructible *t*-spanner (where *t* is a constant that depends only on *δ* and *k*) with the following properties. Its maximum degree is 3, it has at most *n* · *δ* edges, and its total edge weight is comparable to the minimum spanning tree of *V* (for *k* ≤ 3 its weight is *O*(1) · *wt(MST)*, and for *k*>3 its weight is *O*(log *n*) · *wt(MST))*.

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## References

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© Springer-Verlag Berlin Heidelberg 1993