Bichromatic Geometric Spanners

Abstract

For an edge-weighted graph G = (V,E) and a stretch parameter t ≥ 1, a t-spanner is a subgraph H ⊆ G such that the shortest path distances in G and H satisfy δ_H(u,v) ≤ t δ_G(u,v) for all u,v ∈ V. In metric spanners, V is a finite metric space, and G is the complete graph with edge weights corresponding to the distances between the endpoints. When G is the complete graph on n points in the plane, O(n)-size t-spanners are possible for any t > 1: For every ε > 0, there is an (1+ε)-spanner with O(n/ε) edges (the stretch can be arbitrarily close to 1). 
When G = K(R,B) is the complete bipartite graph on n bichromatic points in the plane, in general, no spanner construction can achieve stretch t < 3 with o(n²) edges. Bose et al. (SICOMP 2009) constructed a (3+ε)-spanner with O(nlog n) edges for any constant ε > 0. Our main result is a new construction for a (3+ε)-spanner with O(√{1/ε} ⋅ n) edges. Eliminating the O(log n) factor resolves a problem left open for more than 17 years, and raises a new research problem about optimizing the dependence on ε. We also study spanners for G = K(R,B) on n bichromatic points on the real line: In this case, we show that the MST of K(R,B) is a 7-spanner, and we construct a 3-spanner with at most 2n-3 edges.