Constructing degree-3 spanners with other sparseness properties

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 V1 the length of the shortest path from uto 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 isO(1) ωt(MST), and for k>3 its weight is O(log n) · ωt(MST)).