Let V be a set of n points in 3-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 t > 1, a greedy algorithm produces a t-spanner with O(n) edges, and total edge weight O(1). tot(it4ST), where MST is a minimum spanning tree of V.
A major result is that for any constant-dimensional Euclidean metric and any ε > 0, there exists a (1 + ε)-spanner with lightness O(1) REF .
Optimally sparse spanners in 3-dimensional Euclidean space
SCG '93
Mathematics Computer Science