Planar Geometric Spanners

Years and Authors of Summarized Original Work

• 2005; Bose, Smid, Gudmundsson

Problem Definition

Let S be a set of n points in the plane and let G be an undirected graph with vertex set S, in which each edge ${ (u,v) }$ has a weight, which is equal to the Euclidean distance |uv| between the points u and v. For any two points p and q in S, their shortest-path distance in G is denoted by ${ \delta_G(p,q) }$. If ${ t \geq 1 }$ is a real number, then G is a t-spanner for S if ${ \delta_G(p,q) \leq t|pq| }$ for any two points p and q in S. Thus, if t is close to 1, then the graph G contains close approximations to the ${ n \choose 2 }$ Euclidean distances determined by the pairs of points in S. If, additionally, G consists of O(n) edges, then this graph can be considered a sparse approximation to the complete graph on S. The smallest value of t for which G is a t-spanner is called the stretch factor (or dilation) of G. For a comprehensive overview of geometric spanners,...