Fast Greedy Algorithms for Constructing Sparse Geometric Spanners

Given a set V of n points in ᵈ and a real constant t>1, we present the first O(nlog n)-time algorithm to compute a geometric t-spanner on V. A geometric t-spanner on V is a connected graph G = (V,E) with edge weights equal to the Euclidean distances between the endpoints, and with the property that, for all u,v∈ V, the distance between u and v in G is at most t times the Euclidean distance between u and v. The spanner output by the algorithm has O(n) edges and weight O(1)· wt(MST), and its degree is bounded by a constant.